## How to simulate multilevel/longitudinal data

I was recently talking with my friend Rebecca about simulating multilevel data, and she asked me if I would show her some examples. It occurred to me that many of you might also like to see some examples, so I decided to post them to the Stata Blog.

## Introduction

We simulate data all the time at StataCorp and for a variety of reasons.

One reason is that real datasets that include the features we would like are often difficult to find. We prefer to use real datasets in the manual examples, but sometimes that isn’t feasible and so we create simulated datasets.

We also simulate data to check the coverage probabilities of new estimators in Stata. Sometimes the formulae published in books and papers contain typographical errors. Sometimes the asymptotic properties of estimators don’t hold under certain conditions. And every once in a while, we make coding mistakes. We run simulations during development to verify that a 95% confidence interval really is a 95% confidence interval.

Simulated data can also come in handy for presentations, teaching purposes, and calculating statistical power using simulations for complex study designs.

And, simulating data is just plain fun once you get the hang of it.

Some of you will recall Vince Wiggins’s blog entry from 2011 entitled “Multilevel random effects in xtmixed and sem — the long and wide of it” in which he simulated a three-level dataset. I’m going to elaborate on how Vince simulated multilevel data, and then I’ll show you some useful variations. Specifically, I’m going to talk about:

1. How to simulate single-level data
2. How to simulate two- and three-level data
3. How to simulate three-level data with covariates
4. How to simulate longitudinal data with random slopes
5. How to simulate longitudinal data with structured errors

## How to simulate single-level data

Let’s begin by simulating a trivially simple, single-level dataset that has the form

$y_i = 70 + e_i$

We will assume that e is normally distributed with mean zero and variance $$\sigma^2$$.

We’d want to simulate 500 observations, so let’s begin by clearing Stata’s memory and setting the number of observations to 500.

. clear
. set obs 500


Next, let’s create a variable named e that contains pseudorandom normally distributed data with mean zero and standard deviation 5:

. generate e = rnormal(0,5)


The variable e is our error term, so we can create an outcome variable y by typing

. generate y = 70 + e

. list y e in 1/5

+----------------------+
|        y           e |
|----------------------|
1. | 78.83927     8.83927 |
2. | 69.97774   -.0222647 |
3. | 69.80065   -.1993514 |
4. | 68.11398    -1.88602 |
5. | 63.08952   -6.910483 |
+----------------------+


We can fit a linear regression for the variable y to determine whether our parameter estimates are reasonably close to the parameters we specified when we simulated our dataset:

. regress y

Source |       SS       df       MS              Number of obs =     500
-------------+------------------------------           F(  0,   499) =    0.00
Model |           0     0           .           Prob > F      =       .
Residual |  12188.8118   499  24.4264766           R-squared     =  0.0000
Total |  12188.8118   499  24.4264766           Root MSE      =  4.9423

------------------------------------------------------------------------------
y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons |   69.89768    .221027   316.24   0.000     69.46342    70.33194
------------------------------------------------------------------------------


The estimate of _cons is 69.9, which is very close to 70, and the Root MSE of 4.9 is equally close to the error’s standard deviation of 5. The parameter estimates will not be exactly equal to the underlying parameters we specified when we created the data because we introduced randomness with the rnormal() function.

This simple example is just to get us started before we work with multilevel data. For familiarity, let’s fit the same model with the mixed command that we will be using later:

. mixed y, stddev

Mixed-effects ML regression                     Number of obs      =       500

Wald chi2(0)       =         .
Log likelihood = -1507.8857                     Prob > chi2        =         .

------------------------------------------------------------------------------
y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons |   69.89768   .2208059   316.56   0.000     69.46491    70.33045
------------------------------------------------------------------------------

------------------------------------------------------------------------------
Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
sd(Residual) |    4.93737   .1561334      4.640645    5.253068
------------------------------------------------------------------------------


The output is organized with the parameter estimates for the fixed part in the top table and the estimated standard deviations for the random effects in the bottom table. Just as previously, the estimate of _cons is 69.9, and the estimate of the standard deviation of the residuals is 4.9.

Okay. That really was trivial, wasn’t it? Simulating two- and three-level data is almost as easy.

## How to simulate two- and three-level data

I posted a blog entry last year titled “Multilevel linear models in Stata, part 1: Components of variance“. In that posting, I showed a diagram for a residual of a three-level model.

The equation for the variance-components model I fit had the form

$y_{ijk} = mu + u_i.. + u_{ij.} + e_{ijk}$

This model had three residuals, whereas the one-level model we just fit above had only one.

This time, let’s start with a two-level model. Let’s simulate a two-level dataset, a model for children nested within classrooms. We’ll index classrooms by i and children by j. The model is
$y_{ij} = mu + u_{i.} + e_{ij}$

For this toy model, let’s assume two classrooms with two students per classroom, meaning that we want to create a four-observation dataset, where the observations are students.

To create this four-observation dataset, we start by creating a two-observation dataset, where the observations are classrooms. Because there are two classrooms, we type

. clear
. set obs 2
. generate classroom = _n


From now on, we’ll refer to classroom as i. It’s easier to remember what variables mean if they have meaningful names.

Next, we’ll create a variable that contains each classroom’s random effect $$u_i$$, which we’ll assume follows an N(0,3) distribution.

. generate u_i = rnormal(0,3)

. list

+----------------------+
| classr~m         u_i |
|----------------------|
1. |        1    .7491351 |
2. |        2   -.0031386 |
+----------------------+


We can now expand our data to include two children per classroom by typing

. expand 2

. list

+----------------------+
| classr~m         u_i |
|----------------------|
1. |        1    .7491351 |
2. |        2   -.0031386 |
3. |        1    .7491351 |
4. |        2   -.0031386 |
+----------------------+


Now, we can think of our observations as being students. We can create a child ID (we’ll call it child rather than j), and we can create each child’s residual $$e_{ij}$$, which we will assume has an N(0,5) distribution:

. bysort classroom: generate child = _n

. generate e_ij = rnormal(0,5)

. list

+------------------------------------------+
| classr~m         u_i   child        e_ij |
|------------------------------------------|
1. |        1    .7491351       1    2.832674 |
2. |        1    .7491351       2    1.487452 |
3. |        2   -.0031386       1    6.598946 |
4. |        2   -.0031386       2   -.3605778 |
+------------------------------------------+


We now have nearly all the ingredients to calculate $$y_{ij}$$:

$$y_{ij} = mu + u_{i.} + e_{ij}$$

We’ll assume mu is 70. We type

. generate y = 70 + u_i + e_ij

. list y classroom u_i child e_ij, sepby(classroom)

+-----------------------------------------------------+
|        y   classr~m         u_i   child        e_ij |
|-----------------------------------------------------|
1. | 73.58181          1    .7491351       1    2.832674 |
2. | 72.23659          1    .7491351       2    1.487452 |
|-----------------------------------------------------|
3. | 76.59581          2   -.0031386       1    6.598946 |
4. | 69.63628          2   -.0031386       2   -.3605778 |
+-----------------------------------------------------+


Note that the random effect u_i is the same within each school, and each child has a different value for e_ij.

Our strategy was simple:

2. Create variables for the level ID and its random effect.
3. Expand the data by the number of observations within that level.
4. Repeat steps 2 and 3 until the bottom level is reached.

Let’s try this recipe for three-level data where children are nested within classrooms which are nested within schools. This time, I will index schools with i, classrooms with j, and children with k so that my model is

$y_{ijk} = mu + u_{i..} + u_{ij.} + e_{ijk}$

where

$$u_{i..}$$ ~ N(0,2)
$$u_{ij.}$$ ~ N(0,3)
$$u_{ijk}$$ ~ N(0,5)

Let’s create data for

(level 3, i)   2 schools

(level 2, j)   2 classrooms in each school

(level 1, k)  2 students in most classrooms; 3 students in i==2 & j==2

Begin by creating the level-three data for the two schools:

. clear
. set obs 2
. generate school = _n
. generate u_i = rnormal(0,2)
. list school u_i

+--------------------+
| school         u_i |
|--------------------|
1. |      1    3.677312 |
2. |      2   -3.193004 |
+--------------------+


Next, we expand the data so that we have the three classrooms nested within each of the schools, and we create its random effect:

. expand 2
. bysort school: generate classroom = _n
. generate u_ij = rnormal(0,3)
. list school u_i classroom u_ij, sepby(school)

+-------------------------------------------+
| school         u_i   classr~m        u_ij |
|-------------------------------------------|
1. |      1    3.677312          1    .9811059 |
2. |      1    3.677312          2   -3.482453 |
|-------------------------------------------|
3. |      2   -3.193004          1   -4.107915 |
4. |      2   -3.193004          2   -2.450383 |
+-------------------------------------------+


Finally, we expand the data so that we have three students in school 2′s classroom 2, and two students in all the other classrooms. Sorry for that complication, but I wanted to show you how to create unbalanced data.

In the previous examples, we’ve been typing things like expand 2, meaning double the observations. In this case, we need to do something different for school 2, classroom 2, namely,

. expand 3 if school==2 & classroom==2


and then we can just expand the rest:

. expand 2 if !(school==2 & clasroom==2)


Obviously, in a real simulation, you would probably want 16 to 25 students in each classroom. You could do something like that by typing

. expand 16+int((25-16+1)*runiform())


In any case, we will type

. expand 3 if school==2 & classroom==2

. expand 2 if !(school==2 & classroom==2)

. bysort school classroom: generate child = _n

. generate e_ijk = rnormal(0,5)

. generate y = 70 + u_i + u_ij + e_ijk

. list y school u_i classroom u_ij child e_ijk, sepby(classroom)

+------------------------------------------------------------------------+
|        y   school       u_i   classr~m        u_ij   child       e_ijk |
|------------------------------------------------------------------------|
1. | 76.72794        1  3.677312          1    .9811059       1    2.069526 |
2. | 69.81315        1  3.677312          1    .9811059       2   -4.845268 |
|------------------------------------------------------------------------|
3. | 74.09565        1  3.677312          2   -3.482453       1    3.900788 |
4. | 71.50263        1  3.677312          2   -3.482453       2    1.307775 |
|------------------------------------------------------------------------|
5. | 64.86206        2 -3.193004          1   -4.107915       1    2.162977 |
6. | 61.80236        2 -3.193004          1   -4.107915       2   -.8967164 |
|------------------------------------------------------------------------|
7. | 66.65285        2 -3.193004          2   -2.450383       1    2.296242 |
8. | 49.96139        2 -3.193004          2   -2.450383       2   -14.39522 |
9. | 64.41605        2 -3.193004          2   -2.450383       3    .0594433 |
+------------------------------------------------------------------------+


Regardless of how we generate the data, we must ensure that the school-level random effects u_i are the same within school and the classroom-level random effects u_ij are the same within classroom.

Concerning data construction, the example above we concocted to produce a dataset that would be easy to list. Let’s now create a dataset that is more reasonable:

$y_{ijk} = mu + u_{i..} + u_{ij.} + e_{ijk}$

where

$$u_{i..}$$ ~ N(0,2)
$$u_{ij.}$$ ~ N(0,3)
$$u_{ijk}$$ ~ N(0,5)

Let’s create data for

(level 3, i)   6 schools

(level 2, j)   10 classrooms in each school

(level 1, k)   16-25 students

. clear
. set obs 6
. generate school = _n
. generate u_i = rnormal(0,2)
. expand 10
. bysort school: generate classroom = _n
. generate u_ij = rnormal(0,3)
. expand 16+int((25-16+1)*runiform())
. bysort school classroom: generate child = _n
. generate e_ijk = rnormal(0,5)
. generate y = 70 + u_i + u_ij + e_ijk


We can use the mixed command to fit the model with our simulated data.

. mixed y || school: || classroom: , stddev

Mixed-effects ML regression                     Number of obs      =      1217

-----------------------------------------------------------
|   No. of       Observations per Group
Group Variable |   Groups    Minimum    Average    Maximum
----------------+------------------------------------------
school |        6        197      202.8        213
classroom |       60         16       20.3         25
-----------------------------------------------------------

Wald chi2(0)       =         .
Log likelihood = -3710.0673                     Prob > chi2        =         .

------------------------------------------------------------------------------
y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons |   70.25941   .9144719    76.83   0.000     68.46707    72.05174
------------------------------------------------------------------------------

------------------------------------------------------------------------------
Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
school: Identity             |
sd(_cons) |   2.027064   .7159027      1.014487    4.050309
-----------------------------+------------------------------------------------
classroom: Identity          |
sd(_cons) |   2.814152   .3107647       2.26647    3.494178
-----------------------------+------------------------------------------------
sd(Residual) |   4.828923   .1003814      4.636133     5.02973
------------------------------------------------------------------------------
LR test vs. linear regression:       chi2(2) =   379.37   Prob > chi2 = 0.0000


The parameter estimates from our simulated data match the parameters used to create the data pretty well: the estimate for _cons is 70.3, which is near 70; the estimated standard deviation for the school-level random effects is 2.02, which is near 2; the estimated standard deviation for the classroom-level random effects is 2.8, which is near 3; and the estimated standard deviation for the individual-level residuals is 4.8, which is near 5.

We’ve just done one reasonable simulation.

If we wanted to do a full simulation, we would need to do the above 100, 1,000, 10,000, or more times. We would put our code in a loop. And in that loop, we would keep track of whatever parameter interested us.

## How to simulate three-level data with covariates

Usually, we’re more interested in estimating the effects of the covariates than in estimating the variance of the random effects. Covariates are typically binary (such as male/female), categorical (such as race), ordinal (such as education level), or continuous (such as age).

Let’s add some covariates to our simulated data. Our model is

$y_{ijk} = mu + u_{i..} + u_{ij.} + e_{ijk}$

where

$$u_{i..}$$ ~ N(0,2)
$$u_{ij.}$$ ~ N(0,3)
$$u_{ijk}$$ ~ N(0,5)

We create data for

(level 3, i)   6 schools

(level 2, j)   10 classrooms in each school

(level 1, k)   16-25 students

(level 3, school i)       whether the school is in an urban environment

(level 2, classroom j)  teacher’s experience (years)

(level 1, student k)    student’s mother’s education level

We can create a binary covariate called urban at the school level that equals 1 if the school is located in an urban area and equals 0 otherwise.

. clear
. set obs 6
. generate school = _n
. generate u_i = rnormal(0,2)
. generate urban = runiform()<0.50


Here we assigned schools to one of the two groups with equal probability (runiform()<0.50), but we could have assigned 70% of the schools to be urban by typing

. generate urban = runiform()<0.70


At the classroom level, we could add a continuous covariate for the teacher's years of experience. We could generate this variable by using any of Stata's random-number functions (see help random_number_functions. In the example below, I've generated teacher's years of experience with a uniform distribution ranging from 5-20 years.

. expand 10
. bysort school: generate classroom = _n
. generate u_ij = rnormal(0,3)
. bysort school: generate teach_exp = 5+int((20-5+1)*runiform())


When we summarize our data, we see that teaching experience ranges from 6-20 years with an average of 13 years.

. summarize teach_exp

Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
teach_exp |        60    13.21667    4.075939          6         20


At the child level, we could add a categorical/ordinal covariate for mother's highest level of education completed. After we expand the data and create the child ID and error variables, we can generate a uniformly distributed random variable, temprand, on the interval [0,1].

. expand 16+int((25-16+1)*runiform())
. bysort school classroom: generate child = _n
. generate e_ijk = rnormal(0,5)
. generate temprand = runiform()


We can assign children to different groups by using the egen command with cutpoints. In the example below, children whose value of temprand is in the interval [0,0.5) will be assigned to mother_educ==0, children whose value of temprand is in the interval [0.5,0.9) will be assigned to mother_educ==1, and children whose value of temprand is in the interval [0.9,1) will be assigned to mother_educ==2.

. egen mother_educ = cut(temprand), at(0,0.5, 0.9, 1) icodes
. label define mother_educ 0 "HighSchool" 1 "College" 2 ">College"
. label values mother_educ mother_educ


The resulting frequencies of each category are very close to the frequencies we specified in our egen command.

. tabulate mother_educ, generate(meduc)

mother_educ |      Freq.     Percent        Cum.
------------+-----------------------------------
HighSchool |        602       50.17       50.17
College |        476       39.67       89.83
>College |        122       10.17      100.00
------------+-----------------------------------
Total |      1,200      100.00


We used the option generate(meduc) in the tabulate command above to create indicator variables for each category of mother_educ. This will allow us to specify an effect size for each category when we create our outcome variable.

. summarize meduc*

Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
meduc1 |      1200    .5016667    .5002057          0          1
meduc2 |      1200    .3966667    .4894097          0          1
meduc3 |      1200    .1016667    .3023355          0          1


Now, we can create an outcome variable called score by adding all our fixed and random effects together. We can specify an effect size (regression coefficient) for each fixed effect in our model.

. generate score = 70
+ (-2)*urban
+ 1.5*teach_exp
+ 0*meduc1
+ 2*meduc2
+ 5*meduc3
+ u_i + u_ij + e_ijk


I have specified that the grand mean is 70, urban schools will have scores 2 points lower than nonurban schools, and each year of teacher's experience will add 1.5 points to the students score.

Mothers whose highest level of education was high school (meduc1==1) will serve as the referent category for mother_educ(mother_educ==0). The scores of children whose mother completed college (meduc2==1 and mother_educ==1) will be 2 points higher than the children in the referent group. And the scores of children whose mother completed more than college (meduc3==1 and mother_educ==2) will be 5 points higher than the children in the referent group. Now, we can use the mixed command to fit a model to our simulated data. We used the indicator variables meduc1-meduc3 to create the data, but we will use the factor variable i.mother_educ to fit the model.

. mixed score urban teach_exp i.mother_educ  || school: ||
classroom: , stddev baselevel

Mixed-effects ML regression                     Number of obs      =      1259

-----------------------------------------------------------
|   No. of       Observations per Group
Group Variable |   Groups    Minimum    Average    Maximum
----------------+------------------------------------------
school |        6        200      209.8        217
classroom |       60         16       21.0         25
-----------------------------------------------------------

Wald chi2(4)       =    387.64
Log likelihood = -3870.5395                     Prob > chi2        =    0.0000

------------------------------------------------------------------------------
score |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
urban |  -2.606451    2.07896    -1.25   0.210    -6.681138    1.468237
teach_exp |   1.584759    .096492    16.42   0.000     1.395638     1.77388
|
mother_educ |
HighSchool  |          0  (base)
College  |   2.215281   .3007208     7.37   0.000     1.625879    2.804683
>College  |   5.065907   .5237817     9.67   0.000     4.039314      6.0925
|
_cons |   68.95018   2.060273    33.47   0.000     64.91212    72.98824
------------------------------------------------------------------------------

------------------------------------------------------------------------------
Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
school: Identity             |
sd(_cons) |   2.168154   .7713944      1.079559    4.354457
-----------------------------+------------------------------------------------
classroom: Identity          |
sd(_cons) |    3.06871   .3320171      2.482336    3.793596
-----------------------------+------------------------------------------------
sd(Residual) |   4.947779   .1010263      4.753681    5.149802
------------------------------------------------------------------------------
LR test vs. linear regression:       chi2(2) =   441.25   Prob > chi2 = 0.0000


“Close” is in the eye of the beholder, but to my eyes, the parameter estimates look remarkably close to the parameters that were used to simulate the data. The parameter estimates for the fixed part of the model are -2.6 for urban (parameter = -2), 1.6 for teach_exp (parameter = 1.5), 2.2 for the College category of mother_educ (parameter = 2), 5.1 for the >College category of mother_educ (parameter = 5), and 69.0 for the intercept (parameter = 70). The estimated standard deviations for the random effects are also very close to the simulation parameters. The estimated standard deviation is 2.2 (parameter = 2) at the school level, 3.1 (parameter = 3) at the classroom level, and 4.9 (parameter = 5) at the child level.

Some of you may disagree that the parameter estimates are close. My reply is that it doesn’t matter unless you’re simulating a single dataset for demonstration purposes. If you are, simply simulate more datasets until you get one that looks close enough for you. If you are simulating data to check coverage probabilities or to estimate statistical power, you will be averaging over thousands of simulated datasets and the results of any one of those datasets won’t matter.

## How to simulate longitudinal data with random slopes

Longitudinal data are often conceptualized as multilevel data where the repeated observations are nested within individuals. The main difference between ordinary multilevel models and multilevel models for longitudinal data is the inclusion of a random slope. If you are not familiar with random slopes, you can learn more about them in a blog entry I wrote last year (Multilevel linear models in Stata, part 2: Longitudinal data).

Simulating longitudinal data with a random slope is much like simulating two-level data, with a couple of modifications. First, the bottom level will be observations within person. Second, there will be an interaction between time (age) and a person-level random effect. So we will generate data for the following model:

$weight_{ij} = mu + age_{ij} + u_{0i.} + age*u_{1i.} + e_{ij}$

where

$$u_{0i.}$$ ~ N(0,3)   $$u_{1i.}$$ ~ N(0,1)   $$e_{ij}$$ ~ N(0,2)

Let’s begin by simulating longitudinal data for 300 people.

. clear
. set obs 300
. gen person = _n


For longitudinal data, we must create two person-level random effects: the variable u_0i is analogous to the random effect we created earlier, and the variable u_1i is the random effect for the slope over time.

. generate u_0i = rnormal(0,3)
. generate u_1i = rnormal(0,1)


Let’s expand the data so that there are five observations nested within each person. Rather than create an observation-level identification number, let’s create a variable for age that ranges from 12 to 16 years,

. expand 5
. bysort person: generate age = _n + 11


and create an observation-level error term from an N(0,2) distribution:

. generate e_ij = rnormal(0,2)

. list person u_0i u_1i age e_ij if person==1

+-------------------------------------------------+
| person       u_0i        u_1i   age        e_ij |
|-------------------------------------------------|
1. |      1   .9338312   -.3097848    12    1.172153 |
2. |      1   .9338312   -.3097848    13    2.935366 |
3. |      1   .9338312   -.3097848    14   -2.306981 |
4. |      1   .9338312   -.3097848    15   -2.148335 |
5. |      1   .9338312   -.3097848    16   -.4276625 |
+-------------------------------------------------+


The person-level random effects u_0i and u_1i are the same at all ages, and the observation-level random effects e_ij are different at each age. Now we’re ready to generate an outcome variable called weight, measured in kilograms, based on the following model:

$weight_{ij} = 3 + 3.6*age_{ij} + u_{0i} + age*u_{1i} + e_{ij}$

. generate weight = 3 + 3.6*age + u_0i + age*u_1i + e_ij


The random effect u_1i is multiplied by age, which is why it is called a random slope. We could rewrite the model as

$weight_{ij} = 3 + age_{ij}*(3.6 + u_{1i}) + u_{01} + e_{ij}$

Note that for each year of age, a person’s weight will increase by 3.6 kilograms plus some random amount specified by u_1j. In other words,the slope for age will be slightly different for each person.

We can use the mixed command to fit a model to our data:

. mixed weight age || person: age , stddev

Mixed-effects ML regression                     Number of obs      =      1500
Group variable: person                          Number of groups   =       300

Obs per group: min =         5
avg =       5.0
max =         5

Wald chi2(1)       =   3035.03
Log likelihood = -3966.3842                     Prob > chi2        =    0.0000

------------------------------------------------------------------------------
weight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
age |   3.708161   .0673096    55.09   0.000     3.576237    3.840085
_cons |   2.147311   .5272368     4.07   0.000     1.113946    3.180676
------------------------------------------------------------------------------

------------------------------------------------------------------------------
Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
person: Independent          |
sd(age) |   .9979648   .0444139      .9146037    1.088924
sd(_cons) |    3.38705   .8425298      2.080103    5.515161
-----------------------------+------------------------------------------------
sd(Residual) |   1.905885   .0422249      1.824897    1.990468
------------------------------------------------------------------------------
LR test vs. linear regression:       chi2(2) =  4366.32   Prob > chi2 = 0.0000


The estimate for the intercept _cons = 2.1 is not very close to the original parameter value of 3, but the estimate of 3.7 for age is very close (parameter = 3.6). The standard deviations of the random effects are also very close to the parameters used to simulate the data. The estimate for the person level _cons is 2.1 (parameter = 2), the person-level slope is 0.997 (parameter = 1), and the observation-level residual is 1.9 (parameter = 2).

## How to simulate longitudinal data with structured errors

Longitudinal data often have an autoregressive pattern to their errors because of the sequential collection of the observations. Measurements taken closer together in time will be more similar than measurements taken further apart in time. There are many patterns that can be used to descibe the correlation among the errors, including autoregressive, moving average, banded, exponential, Toeplitz, and others (see help mixed##rspec).

Let’s simulate a dataset where the errors have a Toeplitz structure, which I will define below.

We begin by creating a sample with 500 people with a person-level random effect having an N(0,2) distribution.

. clear
. set obs 500
. gen person = _n
. generate u_i = rnormal(0,2)


Next, we can use the drawnorm command to create error variables with a Toeplitz pattern.

A Toeplitz 1 correlation matrix has the following structure:

. matrix V = ( 1.0, 0.5, 0.0, 0.0, 0.0  \     ///
0.5, 1.0, 0.5, 0.0, 0.0  \     ///
0.0, 0.5, 1.0, 0.5, 0.0  \     ///
0.0, 0.0, 0.5, 1.0, 0.5  \     ///
0.0, 0.0, 0.0, 0.5, 1.0 )

. matrix list V

symmetric V[5,5]
c1  c2  c3  c4  c5
r1   1
r2  .5   1
r3   0  .5   1
r4   0   0  .5   1
r5   0   0   0  .5   1


The correlation matrix has 1s on the main diagonal, and each pair of contiguous observations will have a correlation of 0.5. Observations more than 1 unit of time away from each other are assumed to be uncorrelated.

We must also define a matrix of means to use the drawnorm command.

. matrix M = (0 \ 0 \ 0 \ 0 \ 0)

. matrix list M

M[5,1]
c1
r1   0
r2   0
r3   0
r4   0
r5   0


Now, we’re ready to use the drawnorm command to create five error variables that have a Toeplitz 1 structure.

. drawnorm e1 e2 e3 e4 e5, means(M) cov(V)

. list in 1/2

+---------------------------------------------------------------------------+
| person        u_i         e1         e2        e3          e4          e5 |
|---------------------------------------------------------------------------|
1. |      1   5.303562  -1.288265  -1.201399   .353249    .0495944   -1.472762 |
2. |      2  -.0133588   .6949759    2.82179  .7195075   -1.032395    .1995016 |
+---------------------------------------------------------------------------+


Let’s estimate the correlation matrix for our simulated data to verify that our simulation worked as we expected.

. correlate e1-e5
(obs=300)

|       e1       e2       e3       e4       e5
-------------+---------------------------------------------
e1 |   1.0000
e2 |   0.5542   1.0000
e3 |  -0.0149   0.4791   1.0000
e4 |  -0.0508  -0.0364   0.5107   1.0000
e5 |   0.0022  -0.0615   0.0248   0.4857   1.0000


The correlations are 1 along the main diagonal, near 0.5 for the contiguous observations, and near 0 otherwise.

Our data are currently in wide format, and we need them in long format to use the mixed command. We can use the reshape command to convert our data from wide to long format. If you are not familiar with the reshape command, you can learn more about it by typing help reshape.

. reshape long e, i(person) j(time)
(note: j = 1 2 3 4 5)

Data                               wide   ->   long
-----------------------------------------------------------------------------
Number of obs.                      300   ->    1500
Number of variables                   7   ->       4
j variable (5 values)                     ->   time
xij variables:
e1 e2 ... e5   ->   e
-----------------------------------------------------------------------------


Now, we are ready to create our age variable and the outcome variable weight.

. bysort person: generate age = _n + 11
. generate weight = 3 + 3.6*age + u_i + e

. list weight person u_i time age e if person==1

+-------------------------------------------------------+
|   weight   person        u_i   time   age           e |
|-------------------------------------------------------|
1. |  50.2153        1   5.303562      1    12   -1.288265 |
2. | 53.90216        1   5.303562      2    13   -1.201399 |
3. | 59.05681        1   5.303562      3    14     .353249 |
4. | 62.35316        1   5.303562      4    15    .0495944 |
5. |  64.4308        1   5.303562      5    16   -1.472762 |
+-------------------------------------------------------+


We can use the mixed command to fit a model to our simulated data.

. mixed weight age || person:, residual(toeplitz 1, t(time)) , stddev

Mixed-effects ML regression                     Number of obs      =      1500
Group variable: person                          Number of groups   =       300

Obs per group: min =         5
avg =       5.0
max =         5

Wald chi2(1)       =  33797.58
Log likelihood = -2323.9389                     Prob > chi2        =    0.0000

------------------------------------------------------------------------------
weight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
age |   3.576738   .0194556   183.84   0.000     3.538606     3.61487
_cons |   3.119974   .3244898     9.62   0.000     2.483985    3.755962
------------------------------------------------------------------------------

------------------------------------------------------------------------------
Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
person: Identity             |
sd(_cons) |   3.004718   .1268162      2.766166    3.263843
-----------------------------+------------------------------------------------
Residual: Toeplitz(1)        |
rho1 |   .4977523   .0078807      .4821492    .5130398
sd(e) |   .9531284   .0230028      .9090933    .9992964
------------------------------------------------------------------------------
LR test vs. linear regression:       chi2(2) =  3063.87   Prob > chi2 = 0.0000


Again, our parameter estimates match the parameters that were used to simulate the data very closely.

The parameter estimate is 3.6 for age (parameter = 3.6) and 3.1 for _cons (parameter = 3). The estimated standard deviations of the person-level random effect is 3.0 (parameter = 3). The estimated standard deviation for the errors is 0.95 (parameter = 1), and the estimated correlation for the Toeplitz structure is 0.5 (parameter = 0.5).

## Conclusion

I hope I’ve convinced you that simulating multilevel/longitudinal data is easy and useful. The next time you find yourself teaching a class or giving a talk that requires multilevel examples, try simulating the data. And if you need to calculate statistical power for a multilevel or longitudinal model, consider simulations.

Categories: Statistics Tags:

## Using resampling methods to detect influential points

As stated in the documentation for jackknife, an often forgotten utility for this command is the detection of overly influential observations.

Some commands, like logit or stcox, come with their own set of prediction tools to detect influential points. However, these kinds of predictions can be computed for virtually any regression command. In particular, we will see that the dfbeta statistics can be easily computed for any command that accepts the jackknife prefix. dfbeta statistics allow us to visualize how influential some observations are compared with the rest, concerning a specific parameter.

We will also compute Cook’s likelihood displacement, which is an overall measure of influence, and it can also be compared with a specific threshold.

### Using jackknife to compute dfbeta

The main task of jackknife is to fit the model while suppressing one observation at a time, which allows us to see how much results change when each observation is suppressed; in other words, it allows us to see how much each observation influences the results. A very intuitive measure of influence is dfbeta, which is the amount that a particular parameter changes when an observation is suppressed. There will be one dfbeta variable for each parameter. If $$\hat\beta$$ is the estimate for parameter $$\beta$$ obtained from the full data and $$\hat\beta_{(i)}$$ is the corresponding estimate obtained when the $$i$$th observation is suppressed, then the $$i$$th element of variable dfbeta is obtained as

$dfbeta = \hat\beta - \hat\beta_{(i)}$

Parameters $$\hat\beta$$ are saved by the estimation commands in matrix e(b) and also can be obtained using the _b notation, as we will show below. The leave-one-out values $$\hat\beta_{(i)}$$ can be saved in a new file by using the option saving() with jackknife. With these two elements, we can compute the dfbeta values for each variable.

Let’s see an example with the probit command.

. sysuse auto, clear
(1978 Automobile Data)

. *preserve original dataset
. preserve

. *generate a variable with the original observation number
. gen obs =_n

. probit foreign mpg weight

Iteration 0:   log likelihood =  -45.03321
Iteration 1:   log likelihood = -27.914626
Iteration 2:   log likelihood = -26.858074
Iteration 3:   log likelihood = -26.844197
Iteration 4:   log likelihood = -26.844189
Iteration 5:   log likelihood = -26.844189

Probit regression                                 Number of obs   =         74
LR chi2(2)      =      36.38
Prob > chi2     =     0.0000
Log likelihood = -26.844189                       Pseudo R2       =     0.4039

------------------------------------------------------------------------------
foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
mpg |  -.1039503   .0515689    -2.02   0.044    -.2050235   -.0028772
weight |  -.0023355   .0005661    -4.13   0.000     -.003445   -.0012261
_cons |   8.275464   2.554142     3.24   0.001     3.269437    13.28149
------------------------------------------------------------------------------

. *keep the estimation sample so each observation will be matched
. *with the corresponding replication
. keep if e(sample)
(0 observations deleted)

. *use jackknife to generate the replications, and save the values in
. *file b_replic
. jackknife, saving(b_replic, replace):  probit foreign mpg weight
(running probit on estimation sample)

Jackknife replications (74)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
..................................................    50
........................

Probit regression                               Number of obs      =        74
Replications       =        74
F(   2,     73)    =     10.36
Prob > F           =    0.0001
Log likelihood = -26.844189                     Pseudo R2          =    0.4039

------------------------------------------------------------------------------
|              Jackknife
foreign |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
mpg |  -.1039503   .0831194    -1.25   0.215     -.269607    .0617063
weight |  -.0023355   .0006619    -3.53   0.001    -.0036547   -.0010164
_cons |   8.275464   3.506085     2.36   0.021     1.287847    15.26308
------------------------------------------------------------------------------

. *verify that all the replications were successful
. assert e(N_misreps) ==0

. merge 1:1 _n using b_replic

Result                           # of obs.
-----------------------------------------
not matched                             0
matched                                74  (_merge==3)
-----------------------------------------

. *see how values from replications are stored
. describe, fullnames

Contains data from .../auto.dta
obs:            74                          1978 Automobile Data
vars:            17                          13 Apr 2013 17:45
size:         4,440                          (_dta has notes)
--------------------------------------------------------------------------------
storage   display    value
variable name   type    format     label      variable label
--------------------------------------------------------------------------------
make            str18   %-18s                 Make and Model
price           int     %8.0gc                Price
mpg             int     %8.0g                 Mileage (mpg)
rep78           int     %8.0g                 Repair Record 1978
trunk           int     %8.0g                 Trunk space (cu. ft.)
weight          int     %8.0gc                Weight (lbs.)
length          int     %8.0g                 Length (in.)
turn            int     %8.0g                 Turn Circle (ft.)
displacement    int     %8.0g                 Displacement (cu. in.)
gear_ratio      float   %6.2f                 Gear Ratio
foreign         byte    %8.0g      origin     Car type
obs             float   %9.0g
foreign_b_mpg   float   %9.0g                 [foreign]_b[mpg]
foreign_b_weight
float   %9.0g                 [foreign]_b[weight]
foreign_b_cons  float   %9.0g                 [foreign]_b[_cons]
_merge          byte    %23.0g     _merge
--------------------------------------------------------------------------------
Sorted by:
Note:  dataset has changed since last saved

. *compute the dfbetas for each covariate
. foreach var in mpg weight {
2.  gen dfbeta_var' = (_b[var'] -foreign_b_var')
3. }

. gen dfbeta_cons = (_b[_cons] - foreign_b_cons)

. label var obs "observation number"
. label var dfbeta_mpg "dfbeta for mpg"
. label var dfbeta_weight "dfbeta for weight"
. label var dfbeta_cons "dfbeta for the constant"

. *plot dfbeta values for variable mpg
. scatter dfbeta_mpg obs, mlabel(obs) title("dfbeta values for variable mpg")

. *restore original dataset
. restore


Based on the impact on the coefficient for variable mpg, observation 71 seems to be the most influential. We could create a similar plot for each parameter.

jackknife prints a dot for each successful replication and an ‘x’ for each replication that ends with an error. By looking at the output immediately following the jackknife command, we can see that all the replications were successful. However, we added an assert line in the code to avoid relying on visual inspection. If some replications failed, we would need to explore the reasons.

### A computational shortcut to obtain the dfbeta values

The command jackknife allows us to save the leave-one-out values in a different file. To use these, we would need to do some data management and merge the two files. On the other hand, the same command called with the option keep saves pseudovalues, which are defined as follows:

$\hat{\beta}_i^* = N\hat\beta - (N-1)\hat\beta_{(i)}$

where $$N$$ is the number of observations involved in the computation, returned as e(N). Therefore, using the pseudovalues, $$\beta_{(i)}$$ values can be computed as $\hat\beta_{(i)} = \frac{ N \hat\beta - \hat\beta^*_i}{N-1}$

Also, dfbeta values can be computed directly from the pseudovalues as $\hat\beta - \hat\beta_{(i)} = \frac{\hat\beta_{i}^* -\hat\beta} {N-1}$

Using the pseudovalues instead of the leave-one-out values simplifies our program because we don’t have to worry about matching each pseudovalue to the correct observation.

Let’s reproduce the previous example.

. sysuse auto, clear
(1978 Automobile Data)

. jackknife, keep: probit foreign  mpg weight
(running probit on estimation sample)

Jackknife replications (74)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
..................................................    50
........................

Probit regression                               Number of obs      =        74
Replications       =        74
F(   2,     73)    =     10.36
Prob > F           =    0.0001
Log likelihood = -26.844189                     Pseudo R2          =    0.4039

------------------------------------------------------------------------------
|              Jackknife
foreign |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
mpg |  -.1039503   .0831194    -1.25   0.215     -.269607    .0617063
weight |  -.0023355   .0006619    -3.53   0.001    -.0036547   -.0010164
_cons |   8.275464   3.506085     2.36   0.021     1.287847    15.26308
------------------------------------------------------------------------------

. *see how pseudovalues are stored
. describe, fullnames

> dta
obs:            74                          1978 Automobile Data
vars:            15                          13 Apr 2013 17:45
size:         4,070                          (_dta has notes)
--------------------------------------------------------------------------------
storage   display    value
variable name   type    format     label      variable label
--------------------------------------------------------------------------------
make            str18   %-18s                 Make and Model
price           int     %8.0gc                Price
mpg             int     %8.0g                 Mileage (mpg)
rep78           int     %8.0g                 Repair Record 1978
trunk           int     %8.0g                 Trunk space (cu. ft.)
weight          int     %8.0gc                Weight (lbs.)
length          int     %8.0g                 Length (in.)
turn            int     %8.0g                 Turn Circle (ft.)
displacement    int     %8.0g                 Displacement (cu. in.)
gear_ratio      float   %6.2f                 Gear Ratio
foreign         byte    %8.0g      origin     Car type
foreign_b_mpg   float   %9.0g                 pseudovalues: [foreign]_b[mpg]
foreign_b_weight
float   %9.0g                 pseudovalues: [foreign]_b[weight]
foreign_b_cons  float   %9.0g                 pseudovalues: [foreign]_b[_cons]
--------------------------------------------------------------------------------
Sorted by:  foreign
Note:  dataset has changed since last saved

. *verify that all the replications were successful
. assert e(N_misreps)==0

. *compute the dfbeta for each covariate
. local N = e(N)

. foreach var in  mpg weight {
2. gen dfbeta_var' = (foreign_b_var' - _b[var'])/(N'-1)
3. }

. gen dfbeta_cons' = (foreign_b_cons - _b[_cons])/(N'-1)

. *plot deff values for variable weight
. gen obs = _n

. label var obs "observation number"

. label var dfbeta_mpg "dfbeta for mpg"

. scatter dfbeta_mpg obs, mlabel(obs) title("dfbeta values for variable mpg")


### Dfbeta for grouped data

If you have panel data or a situation where each individual is represented by a group of observations (for example, conditional logit or survival models), you might be interested in influential groups. In this case, you would look at the changes on the parameters when each group is suppressed. Let’s see an example with xtlogit.

. webuse towerlondon, clear
. xtset family

. jackknife, cluster(family) idcluster(newclus) keep: xtlogit dtlm difficulty
. assert e(N_misreps)==0


The group-level pseudovalues will be saved on the first observations corresponding to each group, and there will be missing values on the rest. To compute the dfbeta value for the coefficient for difficulty, we type

. local N = e(N_clust)
. gen dfbeta_difficulty = (dtlm_b_difficulty - _b[difficulty])/(N'-1)


We can then plot those values:

. scatter dfbeta_difficulty newclus, mlabel(family) ///
title("dfbeta values for variable difficulty") xtitle("family")


Option idcluster() for jackknife generates a new variable that assigns consecutive integers to the clusters; using this variable produces a plot where families are equally spaced on the horizontal axis.

As before, we can see that some groups are more influential than others. It would require some research to find out whether this is a problem.

### Likelihood displacement

If we want a global measure of influence (that is, not tied to a particular parameter), we can compute the likelihood displacement values. We consider the likelihood displacement value as defined by Cook (1986):

$LD_i = 2[L(\hat\theta) - L(\hat\theta_{(i)})]$

where $$L$$ is the log-likelihood function (evaluated on the full dataset), $$\hat\theta$$ is the set of parameter estimates obtained from the full dataset, and $$\hat\theta_{(i)}$$ is the set of the parameter estimates obtained when leaving out the $$i$$th observation. Notice that what changes is the parameter vector. The log-likelihood function is always evaluated on the whole sample; provided that $$\hat\theta$$ is the set of parameters that maximizes the log likelihood, the log-likelihood displacement is always positive. Cook suggested, as a confidence region for this value, the interval $$[0, \chi^2_p(\alpha))$$, where $$\chi^2_p(\alpha)$$ is the ($$1-\alpha$$) quantile from a chi-squared distribution with $$p$$ degrees of freedom, and $$p$$ is the number of parameters in $$\theta$$.

To perform our assessment based on the likelihood displacement, we will need to do the following:

1. Create an $$N\times p$$ matrix B, where the $$i$$th row contains the vector of parameter estimates obtained by leaving the $$i$$th observation out.
2. Create a new variable L1 such that its $$i$$th observation contains the log likelihood evaluated at the parameter estimates in the $$i$$th row of matrix B.
3. Use variable L1 to obtain the LD matrix, containing the likelihood displacement values.
4. Construct a plot for the values in LD, and add the $$\chi^2_p(\alpha)$$ as a reference.

Let's do it with our probit model.

#### Step 1.

We first create the macro cmdline containing the command line for the model we want to use. We fit the model and save the original log likelihood in macro ll0.

With a loop, the leave-one-out parameters are saved in consecutive rows of matrix B. It is useful to have those values in a matrix, because we will then extract each row to evaluate the log likelihood at those values.

**********Step 1
sysuse auto, clear
set more off
local cmdline probit foreign weight mpg
cmdline'
keep if e(sample)
local ll0 = e(ll)
mat b0 = e(b)
mat b = b0

local N = _N

forvalues i = 1(1)N'{
cmdline' if _n !=i'
mat b1 = e(b)
mat b = b \ b1
}

mat B = b[2...,1...]
mat list B


#### Step 2.

In each iteration of a loop, a row from B is stored as matrix b. To evaluate the log likelihood at these values, the trick is to use them as initial values and invoke the command with 0 iterations. This can be done for any command that is based on ml.

**********Step 2

gen L1 = .

forvalues i = 1(1)N'{
mat b = B[i',1...]
cmdline', from(b) iter(0)
local ll = e(ll)
replace L1 = ll' in i'
}


#### Step 3.

Using variable L1 and the macro with the original log likelihood, we compute Cook's likehood displacement.

**********Step 3

gen LD = 2*(ll0' - L1)


#### Step 4.

Create the plot, using as a reference the 90% quantile for the $$\chi^2$$ distribution. $$p$$ is the number of columns in matrix b0 (or equivalently, the number of columns in matrix B).

**********Step 4

local k = colsof(b0)
gen upper_bound = invchi2tail(k', .1)
gen n = _n

twoway scatter LD n, mlabel(n) || line upper_bound n, ///
title("Likelihood displacement")


We can see that observation 71 is the most influential, and its likelihood displacement value is within the range we would normally expect.

### Reference

Cook, D. 1986. Assessment of local influence. Journal of the Royal Statistical Society, Series B 48: 133–169.

Categories: Statistics Tags:

## How to create animated graphics using Stata

### Introduction

Today I want to show you how to create animated graphics using Stata. It’s easier than you might expect and you can use animated graphics to illustrate concepts that would be challenging to illustrate with static graphs. In addition to Stata, you will need a video editing program but don’t be concerned if you don’t have one. At the 2012 UK Stata User Group Meeting Robert Grant demonstrated how to create animated graphics from within Stata using a free software program called FFmpeg. I will show you how I create my animated graphs using Camtasia and how Robert creates his using FFmpeg.

I recently recorded a video for the Stata Youtube channel called “Power and sample size calculations in Stata: A conceptual introduction“. I wanted to illustrate two concepts: (1) that statistcal power increases as sample size increases, and (2) as effect size increases. Both of these concepts can be illustrated with a static graph along with the explanation “imagine that …”. Creating animated graphs allowed me to skip the explanation and just show what I meant.

### Creating the graphs

Videos are illusions. All videos — from Charles-Émile Reynaud’s 1877 praxinoscope to modern blu-ray movies — are created by displaying a series of ordered still images for a fraction of a second each. Our brains perceive this series of still images as motion.

To create the illusion of motion with graphs, we make an ordered series of slightly differing graphs. We can use loops to do this. If you are not familiar with loops in Stata, here’s one to count to five:

forvalues i = 1(1)5 {
disp "i = i'"
}

i = 1
i = 2
i = 3
i = 4
i = 5


We could place a graph command inside the loop. If, for each interation, the graph command created a slightly different graph, we would be on our way to creating our first video. The loop below creates a series of graphs of normal densities with means 0 through 1 in increments of 0.1.

forvalues mu = 0(0.1)1 {
twoway  function y=normalden(x,mu',1), range(-3 6) title("N(mu',1)")
}


You may have noticed the illusion of motion as Stata created each graph; the normal densities appeared to be moving to the right as each new graph appeared on the screen.

You may have also noticed that some of the values of the mean did not look as you would have wanted. For example, 1.0 was displayed as 0.999999999. That’s not a mistake, it’s because Stata stores numbers and performs calculations in base two and displays them in base ten; for a detailed explanation, see Precision (yet again), Part I.

We can fix that by reformating the means using the string() function.

forvalues mu = 0(0.1)1 {
local mu = string(mu', "%3.1f")
twoway  function y=normalden(x,mu',1), range(-3 6) title("N(mu',1)")
}


Next, we need to save our graphs. We can do this by adding graph export inside the loop.

forvalues mu = 0(0.1)1 {
local mu = string(mu', "%3.1f")
twoway  function y=normalden(x,mu',1), range(-3 6) title("N(mu',1)")
graph export graph_mu'.png, as(png) width(1280) height(720) replace
}


Note that the name of each graph file includes the value of mu so that we know the order of our files. We can view the contents of the directory to verify that Stata has created a file for each of our graphs.

. ls
<dir>   2/11/14 12:12  .
<dir>   2/11/14 12:12  ..
35.6k   2/11/14 12:11  graph_0.0.png
35.6k   2/11/14 12:11  graph_0.1.png
35.7k   2/11/14 12:11  graph_0.2.png
35.7k   2/11/14 12:11  graph_0.3.png
35.7k   2/11/14 12:11  graph_0.4.png
35.8k   2/11/14 12:11  graph_0.5.png
35.9k   2/11/14 12:12  graph_0.6.png
35.7k   2/11/14 12:12  graph_0.7.png
35.8k   2/11/14 12:12  graph_0.8.png
35.9k   2/11/14 12:12  graph_0.9.png
35.6k   2/11/14 12:12  graph_1.0.png


Now that we have created our graphs, we need to combine them into a video.

There are many commercial, freeware, and free software programs available that we could use. I will outline the basic steps using two of them, one a commerical GUI based product (not free) called Camtasia, and the other a free command-based program called FFmpeg.

### Creating videos with Camtasia

Most commercial video editing programs have similar interfaces. The user imports image, sound and video files, organizes them in tracks on a timeline and then previews the resulting video. Camtasia is a commercial video program that I use to record videos for the Stata Youtube channel and its interface looks like this.

We begin by importing the graph files into Camtasia:

Next we drag the images onto the timeline:

And then we make the display time for each image very short…in this case 0.1 seconds or 10 frames per second.

After previewing the video, we can export it to any of Camtasia’s supported formats. I’ve exported to a “.gif” file because it is easy to view in a web browser.

We just created our first animated graph! All we have to do to make it look as professional as the power-and-sample size examples I showed you earlier is go back into our Stata program and modify the graph command to add the additional elements we want to display!

### Creating videos with FFmpeg

Stata user and medical statistician Robert Grant gave a presentation at the 2012 UK Stata User Group Meeting in London entitled “Producing animated graphs from Stata without having to learn any specialized software“. You can read more about Robert by visiting his blog and clicking on About.

In his presentation, Robert demonstrated how to combine graph images into a video using a free software program called FFmpeg. Robert followed the same basic strategy I demonstrated above, but Robert’s choice of software has two appealing features. First, the software is readily available and free. Second, FFmpeg can be called from within the Stata environment using the winexec command. This means that we can create our graphs and combine them into a video using Stata do files. Combining dozens or hundreds of graphs into a single video with a program is faster and easier than using a drag-and-drop interface.

Let’s return to our previous example and combine the files using FFmpeg. Recall that we inserted the mean into the name of each file (e.g. “graph_0.4.png”) so that we could keep track of the order of the files. In my experience, it can be difficult to combine files with decimals in their names using FFmpeg. To avoid the problem, I have added a line of code between the twoway command and the graph export command that names the files with sequential integers which are padded with zeros.

forvalues mu = 0(0.1)1 {
local mu = string(mu', "%3.1f")
twoway function y=normalden(x,mu',1), range(-3 6) title("N(mu',1)")
local mu = string(mu'*10+1, "%03.0f")
graph export graph_mu'.png, as(png) width(1280) height(720) replace
}

. ls
<dir>   2/12/14 12:21  .
<dir>   2/12/14 12:21  ..
35.6k   2/12/14 12:21  graph_001.png
35.6k   2/12/14 12:21  graph_002.png
35.7k   2/12/14 12:21  graph_003.png
35.7k   2/12/14 12:21  graph_004.png
35.7k   2/12/14 12:21  graph_005.png
35.8k   2/12/14 12:21  graph_006.png
35.9k   2/12/14 12:21  graph_007.png
35.7k   2/12/14 12:21  graph_008.png
35.8k   2/12/14 12:21  graph_009.png
35.9k   2/12/14 12:21  graph_010.png
35.6k   2/12/14 12:21  graph_011.png


We can then combine these files into a video with FFmpeg using the following commands

local GraphPath "C:\Users\jch\AnimatedGraphics\example\"
winexec "C:\Program Files\FFmpeg\bin\ffmpeg.exe" -i GraphPath'graph_%03d.png
-b:v 512k GraphPath'graph.mpg


The local macro GraphPath contains the path for the directory where my graphics files are stored.

The Stata command winexec whatever executes whatever. In our case, whatever is ffmpeg.exe, preceeded by ffmpeg.exe‘s path, and followed by the arguments FFmpeg needs. We specify two options, -i and -b.

The -i option is followed by a path and filename template. In our case, the path is obtained from the Stata local macro GraphPath and the filename template is “graph_%03d.png”. This template tells FFmpeg to look for a three digit sequence of numbers between “graph_” and “.png” in the filenames. The zero that precedes the three in the template tells FFmpeg that the three digit sequence of numbers is padded with zeros.

The -b option specifies the path and filename of the video to be created along with some attributes of the video.

Once we have created our video, we can use FFmpeg to convert our video to other video formats. For example, we could convert “graph.mpg” to “graph.gif” using the following command:

winexec "C:\Program Files\FFmpeg\bin\ffmpeg.exe" -r 10 -i GraphPath'graph.mpg
-t 10 -r 10 GraphPath'graph.gif


which creates this graph:

FFmpeg is a very flexible program and there are far too many options to discuss in this blog entry. If you would like to learn more about FFmpeg you can visit their website at www.ffmpeg.org.

### More Examples

I made the preceding examples as simple as possible so that we could focus on the mechanics of creating videos. We now know that, if we want to make professional looking videos, all the complication comes on the Stata side. We leave our loop alone but change the graph command inside it to be more complicated.

So here’s how I created the two animated-graphics videos that I used to create the overall video “Power and sample size calculations in Stata: A conceptual introduction” on our YouTube channel.

The first demonstrated that increasing the effect size (the difference between the means) results in increased statistical power.

local GraphCounter = 100
local mu_null = 0
local sd = 1
local z_crit = round(-1*invnormal(0.05), 0.01)
local z_crit_label = z_crit' + 0.75

forvalues mu_alt = 1(0.01)3 {
twoway  ///
function y=normalden(x,mu_null',sd'),                    ///
range(-3 z_crit') color(red) dropline(0)   ||    ///
function y=normalden(x,mu_alt',sd'),                     ///
range(-3 5) color(green) dropline(mu_alt') ||    ///
function y=normalden(x,mu_alt',sd'),                     ///
range(z_crit' 6) recast(area) color(green) ||    ///
function y=normalden(x,mu_null',sd'),                    ///
range(z_crit' 6) recast(area) color(red)         ///
title("Power for {&mu}={&mu}{subscript:0} versus {&mu}={&mu}{subscript:A}") ///
xtitle("{it: z}") xlabel(-3 -2 -1 0 1 2 3 4 5 6)           ///
legend(off)                                                ///
ytitle("Density") yscale(range(0 0.6))                     ///
ylabel(0(0.1)0.6, angle(horizontal) nogrid)                ///
text(0.45 0 "{&mu}{subscript:0}", color(red))              ///
text(0.45 mu_alt' "{&mu}{subscript:A}", color(green))

graph export mu_alt_GraphCounter'.png, as(png) width(1280) height(720) replace

local ++GraphCounter
}


The above Stata code created the *.png files that I then combined using Camtasia to produce this gif:

The second video demonstrated that power increases as the sample size increases.

local GraphCounter = 301
local mu_label = 0.45
local power_label = 2.10
local mu_null = 0
local mu_alt = 2

forvalues sd = 1(-0.01)0.5 {
local z_crit = round(-1*invnormal(0.05)*sd', 0.01)
local z_crit_label = z_crit' + 0.75

twoway                                                                        ///
function y=normalden(x,mu_null',sd'),                                     ///
range(-3 z_crit') color(red) dropline(0)  ||                      ///
function y=normalden(x,mu_alt',sd'),                                      ///
range(-3 5) color(green)  dropline(mu_alt')      ||               ///
function y=normalden(x,mu_alt',sd'),                                      ///
range(z_crit' 6) recast(area) color(green)       ||               ///
function y=normalden(x,mu_null',sd'),                                     ///
range(z_crit' 6) recast(area) color(red)                          ///
title("Power for {&mu}={&mu}{subscript:0} versus {&mu}={&mu}{subscript:A}") ///
xtitle("{it: z}") xlabel(-3 -2 -1 0 1 2 3 4 5 6)                            ///
legend(off)                                                                 ///
ytitle("Density") yscale(range(0 0.6))                                      ///
ylabel(0(0.1)0.6, angle(horizontal) nogrid)                                 ///
text(mu_label' 0 "{&mu}{subscript:0}", color(red))                         ///
text(mu_label' mu_alt' "{&mu}{subscript:A}", color(green))
graph export mu_alt_GraphCounter'.png, as(png) width(1280) height(720) replace

local ++GraphCounter
local mu_label = mu_label' + 0.005
local power_label = power_label' + 0.03
}


Just as previously, the above Stata code creates the *.png files that I then combine using Camtasia to produce a gif:

Let me show you some more examples.

The next example demonstrates the basic idea of lowess smoothing.

sysuse auto
local WindowWidth = 500
forvalues WindowUpper = 2200(25)5000 {
local WindowLower = WindowUpper' - WindowWidth'
twoway (scatter mpg weight)                                             ///
(lowess mpg weight if weight < (WindowUpper'-250), lcolor(green))    ///
(lfit mpg weight if weight>WindowLower' & weight<WindowUpper',      ///
lwidth(medium) lcolor(red))                                      ///
, xline(WindowLower' WindowUpper', lwidth(medium) lcolor(black))    ///
legend(on order(1 2 3) cols(3))
graph export lowess_WindowUpper'.png, as(png) width(1280) height(720) replace
}


The result is,

The animated graph I created is not yet a perfect analogy to what lowess actually does, but it comes close. It has two problems. The lowess curve changes outside of the sliding window, which it should not and the animation does not illustrate the weighting of the points within the window, say by using differently sized markers for the points in the sliding window. Even so, the graph does a far better job than the usual explanaton that one should imagine sliding a window across the scatterplot.

As yet another example, we can use animated graphs to demonstrate the concept of convergence. There is a FAQ on the Stata website written by Bill Gould that explains the relationship between the chi-squared and F distributions. The animated graph below shows that F(d1, d2) converges to d1*χ^2 as d2 goes to infinity:

forvalues df = 1(1)100 {
twoway function  y=chi2(2,2*x), range(0 6) color(red) ||                                       ///
function y=F(2,df',x), range(0 6) color(green)                                              ///
title("Cumulative distributions for {&chi}{sup:2}{sub:df} and {it:F}{subscript:df,df2}")     ///
xtitle("{it: denominator df}") xlabel(0 1 2 3 4 5 6) legend(off)                             ///
text(0.45 4 "df2 = df'",  size(huge) color(black))                                          ///
legend(on order(1 "{&chi}{sup:2}{sub:df}" 2 "{it:F}{subscript:df,df2}") cols(2) position(5) ring(0))

local df = string(df', "%03.0f")
graph export converge2_df'.png, as(png) width(1280) height(720) replace
}


The t distribution has a similar relationship with the normal distribution.

forvalues df = 1(1)100 {
twoway  function y=normal(x), range(-3 3) color(red)   ||                     ///
function y=t(df',x), range(-3 3) color(green)                              ///
title("Cumulative distributions for Normal(0,1) and {it:t}{subscript:df}")  ///
xtitle("{it: t/z}") xlabel(-3 -2 -1 0 1 2 3) legend(off)                    ///
text(0.45 -2 "df = df'",  size(huge) color(black))                         ///
legend(on order(1 "N(0,1)" 2 "{it:t}{subscript:df}") cols(2) position(5) ring(0))

local df = string(df', "%03.0f")
graph export converge_df'.png, as(png) width(1280) height(720) replace
}


The result is

### Final thoughts

I have learned through trial and error two things that improve the quality of my animated graphs. First, note that the axes of the graphs in most of the examples above are explicitly defined in the graph commands. This is often necessary to keep the axes stable from graph to graph. Second, videos have a smoother, higher quality appearance when there are many graphs with very small changes from graph to graph.

I hope I have convinced you that creating animated graphics with Stata is easier than you imagined. If the old saying that “a picture is worth a thousand words” is true, imagine how many words you can save using animated graphs.

### Other resources

FFmpeg

Camtasia

Relationship between chi-squared and F distributions

Robert Grant’s blog and examples

Hans Rosling’s 200 Countries recreated using only Stata

Categories: Graphics Tags:

## Retaining an Excel cell’s format when using putexcel

In a previous blog entry, I talked about the new Stata 13 command putexcel and how we could use putexcel with a Stata command’s stored results to create tables in an Excel file.

After the entry was posted, a few users pointed out two features they wanted added to putexcel:

1. Retain a cell’s format after writing numeric data to it.
2. Allow putexcel to format a cell.

In Stata 13.1, we added the new option keepcellformat to putexcel. This option retains a cell’s format after writing numeric data to it. keepcellformat is useful for people who want to automate the updating of a report or paper.

To review, the basic syntax of putexcel is as follows:

putexcel excel_cell=(expression) … using filename[, options]


If you are working with matrices, the syntax is

putexcel excel_cell=matrix(expression) … using filename[, options]


In the previous blog post, we exported a simple table created by the correlate command by using the commands below.

. sysuse auto
(1978 Automobile Data)

. correlate foreign mpg
(obs=74)

|  foreign      mpg
-------------+------------------
foreign |   1.0000
mpg |   0.3934   1.0000

. putexcel A1=matrix(r(C), names) using corr


These commands created the file corr.xlsx, which contained the table below in the first worksheet.

As you can see, this table is not formatted. So, I formatted the table by hand in Excel so that the correlations were rounded to two digits and the column and row headers were bold with a blue background.

putexcel‘s default behavior is to remove the formatting of cells. Thus, if we want to change the correlated variables in our command from foreign and mpg to foreign and weight using the below commands, the new correlations shown in Excel will revert to the default format:

. sysuse auto, clear
(1978 Automobile Data)

. correlate foreign weight
(obs=74)

|  foreign   weight
-------------+------------------
foreign |   1.0000
weight |  -0.5928   1.0000

. putexcel A1=matrix(r(C), names) using corr, modify


As of Stata 13.1, you can now use the keepcellformat option to preserve a numeric cell’s format when writing to it. For example, the command

. putexcel A1=matrix(r(C), names) using corr, modify keepcellformat


will produce

Let’s look at a real-world problem and really see how the keepcellformat option can help us. Suppose we need to export the following tabulate table to a report we wrote in Word.

. webuse auto2, clear
(1978 Automobile Data)

. label variable rep78 "Repair Record"

. tabulate rep78

Repair |
Record |      Freq.     Percent        Cum.
------------+-----------------------------------
Poor |          2        2.90        2.90
Fair |          8       11.59       14.49
Average |         30       43.48       57.97
Good |         18       26.09       84.06
Excellent |         11       15.94      100.00
------------+-----------------------------------
Total |         69      100.00


In the previous putexcel blog post, I mentioned my user-written command tab2xl, which exports a one-way tabulation to an Excel file. I have since updated the command so that it uses the new keepcellformat option to preserve cell formatting. You can download the updated tab2xl command by typing the following:

. net install http://www.stata.com/users/kcrow/tab2xl, replace


Using this command, I can now export my tabulate table to Excel by typing

. tab2xl rep78 using tables, row(1) col(1)


Once the table is in Excel, I format it by hand so that it looks like this:

I then link this Excel table to a Word document. When you link an Excel table to a Word document, it

1. preserves the formatting of the table and
2. automatically updates the Word document when you update the Excel table.

It is fairly easy to link an Excel table to a Word document or PowerPoint presentation. In Excel/Word 2010, you would do as follows:

1. Highlight the table/data in Excel.
2. On the Home tab, click on the Copy button.
3. Open the Word document and scroll to where you want the table pasted.
4. On the Home tab of Word, click on the Paste button.
5. Select Link & Keep Source Formatting, , from the Paste icon menu.

My report now looks like this:

With the Excel table linked into Word, any time we update our Excel table using putexcel, we also update our table in Word.

Suppose that after a few weeks, we get more repair record data. We now need to update our report, and our new tabulate table looks like this:

. tabulate rep78

Repair |
Record |      Freq.     Percent        Cum.
------------+-----------------------------------
Poor |          4        2.90        2.90
Fair |          8        5.80        8.70
Average |         60       43.48       52.17
Good |         44       31.88       84.06
Excellent |         22       15.94      100.00
------------+-----------------------------------
Total |        138      100.00


To update the report, we simply need to reissue the putexcel command after tabulate.

. tabulate rep78
. tab2xl rep78 using tables, row(1) col(1)


The linked Word report will automatically reflect the changes:

Categories: Programming Tags:

## Fitting ordered probit models with endogenous covariates with Stata’s gsem command

The new command gsem allows us to fit a wide variety of models; among the many possibilities, we can account for endogeneity on different models. As an example, I will fit an ordinal model with endogenous covariates.

### Parameterizations for an ordinal probit model

The ordinal probit model is used to model ordinal dependent variables. In the usual parameterization, we assume that there is an underlying linear regression, which relates an unobserved continuous variable $$y^*$$ to the covariates $$x$$.

$y^*_{i} = x_{i}\gamma + u_i$

The observed dependent variable $$y$$ relates to $$y^*$$ through a series of cut-points $$-\infty =\kappa_0<\kappa_1<\dots< \kappa_m=+\infty$$ , as follows:

$y_{i} = j {\mbox{ if }} \kappa_{j-1} < y^*_{i} \leq \kappa_j$

Provided that the variance of $$u_i$$ can’t be identified from the observed data, it is assumed to be equal to one. However, we can consider a re-scaled parameterization for the same model; a straightforward way of seeing this, is by noting that, for any positive number $$M$$:

$\kappa_{j-1} < y^*_{i} \leq \kappa_j \iff M\kappa_{j-1} < M y^*_{i} \leq M\kappa_j$

that is,

$\kappa_{j-1} < x_i\gamma + u_i \leq \kappa_j \iff M\kappa_{j-1}< x_i(M\gamma) + Mu_i \leq M\kappa_j$

In other words, if the model is identified, it can be represented by multiplying the unobserved variable $$y$$ by a positive number, and this will mean that the standard error of the residual component, the coefficients, and the cut-points will be multiplied by this number.

Let me show you an example; I will first fit a standard ordinal probit model, both with oprobit and with gsem. Then, I will use gsem to fit an ordinal probit model where the residual term for the underlying linear regression has a standard deviation equal to 2. I will do this by introducing a latent variable $$L$$, with variance 1, and coefficient $$\sqrt 3$$. This will be added to the underlying latent residual, with variance 1; then, the ‘new’ residual term will have variance equal to $$1+((\sqrt 3)^2\times Var(L))= 4$$, so the standard deviation will be 2. We will see that as a result, the coefficients, as well as the cut-points, will be multiplied by 2.

. sysuse auto, clear
(1978 Automobile Data)

. oprobit rep mpg disp , nolog

Ordered probit regression                         Number of obs   =         69
LR chi2(2)      =      14.68
Prob > chi2     =     0.0006
Log likelihood = -86.352646                       Pseudo R2       =     0.0783

------------------------------------------------------------------------------
rep78 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
mpg |   .0497185   .0355452     1.40   0.162    -.0199487    .1193858
displacement |  -.0029884   .0021498    -1.39   0.165     -.007202    .0012252
-------------+----------------------------------------------------------------
/cut1 |  -1.570496   1.146391                      -3.81738    .6763888
/cut2 |  -.7295982   1.122361                     -2.929386     1.47019
/cut3 |   .6580529   1.107838                     -1.513269    2.829375
/cut4 |    1.60884   1.117905                     -.5822132    3.799892
------------------------------------------------------------------------------

. gsem (rep <- mpg disp, oprobit), nolog

Generalized structural equation model             Number of obs   =         69
Log likelihood = -86.352646

--------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
rep78 <-       |
mpg |   .0497185   .0355452     1.40   0.162    -.0199487    .1193858
displacement |  -.0029884   .0021498    -1.39   0.165     -.007202    .0012252
---------------+----------------------------------------------------------------
rep78          |
/cut1 |  -1.570496   1.146391    -1.37   0.171     -3.81738    .6763888
/cut2 |  -.7295982   1.122361    -0.65   0.516    -2.929386     1.47019
/cut3 |   .6580529   1.107838     0.59   0.553    -1.513269    2.829375
/cut4 |    1.60884   1.117905     1.44   0.150    -.5822132    3.799892
--------------------------------------------------------------------------------

. local a = sqrt(3)

. gsem (rep <- mpg disp L@a'), oprobit var(L@1) nolog

Generalized structural equation model             Number of obs   =         69
Log likelihood = -86.353008

( 1)  [rep78]L = 1.732051
( 2)  [var(L)]_cons = 1
--------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
rep78 <-       |
mpg |    .099532     .07113     1.40   0.162    -.0398802    .2389442
displacement |  -.0059739   .0043002    -1.39   0.165    -.0144022    .0024544
L |   1.732051  (constrained)
---------------+----------------------------------------------------------------
rep78          |
/cut1 |  -3.138491   2.293613    -1.37   0.171     -7.63389    1.356907
/cut2 |  -1.456712   2.245565    -0.65   0.517    -5.857938    2.944513
/cut3 |   1.318568    2.21653     0.59   0.552     -3.02575    5.662887
/cut4 |   3.220004   2.236599     1.44   0.150     -1.16365    7.603657
---------------+----------------------------------------------------------------
var(L)|          1  (constrained)
--------------------------------------------------------------------------------

### Ordinal probit model with endogenous covariates

This model is defined analogously to the model fitted by -ivprobit- for probit models with endogenous covariates; we assume an underlying model with two equations,

$\begin{eqnarray} y^*_{1i} =& y_{2i} \beta + x_{1i} \gamma + u_i & \\ y_{2i} =& x_{1i} \pi_1 + x_{2i} \pi_2 + v_i & \,\,\,\,\,\, (1) \end{eqnarray}$

where $$u_i \sim N(0, 1)$$, $$v_i\sim N(0,s^2)$$, and $$corr(u_i, v_i) = \rho$$.

We don’t observe $$y^*_{1i}$$; instead, we observe a discrete variable $$y_{1i}$$, such as, for a set of cut-points (to be estimated) $$\kappa_0 = -\infty < \kappa_1 < \kappa_2 \dots < \kappa_m = +\infty$$,

$y_{1i} = j {\mbox{ if }} \kappa_{j-1} < y^*_{1i} \leq \kappa_j$

### The parameterization we will use

I will re-scale the first equation, preserving the correlation. That is, I will consider the following system:

$\begin{eqnarray} z^*_{1i} =& y_{2i}b +x_{1i}c + t_i + \alpha L_i &\\ y_{2i} = &x_{1i}\pi_1 + x_{2i}\pi_2 + w_i + \alpha L_i & \,\,\,\,\,\, (2) \end{eqnarray}$

where $$t_i, w_i, L_i$$ are independent, $$t_i \sim N(0, 1)$$ , $$w_i \sim N(0,\sigma^2)$$, $$L_i \sim N(0, 1)$$

$y_{1i} = j {\mbox{ if }} \lambda_{j-1} < z^*_{1i} \leq \lambda_j$

By introducing a latent variable in both equations, I am modeling a correlation between the error terms. The fist equation is a re-scaled version of the original equation, that is, $$z^*_1 = My^*_1$$,

$y_{2i}b +x_{1i}c + t_i + \alpha_i L_i = M(y_{2i}\beta) +M x_{1i}\gamma + M u_i$

This implies that
$M u_i = t_i + \alpha_i L_i,$
where $$Var(u_i) = 1$$ and $$Var(t_i + \alpha L_i) = 1 + \alpha^2$$, so the scale is $$M = \sqrt{1+\alpha^2}$$.

The second equation remains the same, we just express $$v_i$$ as $$w_i + \alpha L_i$$. Now, after estimating the system (2), we can recover the parameters in (1) as follows:

$\beta = \frac{1}{\sqrt{1+ \alpha^2}} b$
$\gamma = \frac{1}{\sqrt{1+ \alpha^2}} c$
$\kappa_j = \frac{1}{\sqrt{1+ \alpha^2}} \lambda_j$

$V(v_i) = V(w_i + \alpha L_i) =V(w_i) + \alpha^2$.

$\rho = Cov(t_i + \alpha L_i, w_i + \alpha L_i) = \frac{\alpha^2}{(\sqrt{1+\alpha^2}\sqrt{V(w_i)+\alpha^2)}}$

Note: This parameterization assumes that the correlation is positive; for negative values of the correlation, $$L$$ should be included in the second equation with a negative sign (that is, L@(-a) instead of L@a). When trying to perform the estimation with the wrong sign, the model most likely won’t achieve convergence. Otherwise, you will see a coefficient for L that is virtually zero. In Stata 13.1 we have included features that allow you to fit the model without this restriction. However, this time we will use the older parameterization, which will allow you to visualize the different components more easily.

### Simulating data, and performing the estimation

clear
set seed 1357
set obs 10000
forvalues i = 1(1)5 {
gen xi' =2* rnormal() + _n/1000
}

mat C = [1,.5 \ .5, 1]
drawnorm z1 z2, cov(C)

gen y2 = 0
forvalues i = 1(1)5 {
replace y2 = y2 + xi'
}
replace y2 = y2 + z2

gen y1star = y2 + x1 + x2 + z1
gen xb1 = y2 + x1 + x2

gen y1 = 4
replace y1 = 3 if xb1 + z1 <=.8
replace y1 = 2 if xb1 + z1 <=.3
replace y1 = 1 if xb1 + z1 <=-.3
replace y1 = 0 if xb1 + z1 <=-.8

gsem (y1 <- y2 x1 x2 L@a, oprobit) (y2 <- x1 x2 x3 x4 x5 L@a), var(L@1)

local y1 y1
local y2 y2

local xaux  x1 x2 x3 x4 x5
local xmain  y2 x1 x2

local s2 sqrt(1+_b[y1':L]^2)
foreach v in xmain'{
local trans trans' (y1'_v': _b[y1':v']/s2')
}

foreach v in xaux' _cons {
local trans trans' (y2'_v': _b[y2':v'])
}

qui tab y1' if e(sample)
local ncuts = r(r)-1
forvalues i = 1(1) ncuts'{
local trans trans' (cut_i': _b[y1'_cuti':_cons]/s2')
}

local s1 sqrt(  _b[var(e.y2'):_cons]  +_b[y1':L]^2)

local trans trans' (sig_2: s1')
local trans trans' (rho_12: _b[y1':L]^2/(s1'*s2'))
nlcom trans'


### Results

This is the output from gsem:

Generalized structural equation model             Number of obs   =      10000
Log likelihood = -14451.117

( 1)  [y1]L - [y2]L = 0
( 2)  [var(L)]_cons = 1
------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
y1 <-        |
y2 |   1.379511   .0775028    17.80   0.000     1.227608    1.531414
x1 |   1.355687   .0851558    15.92   0.000     1.188785    1.522589
x2 |   1.346323   .0833242    16.16   0.000      1.18301    1.509635
L |   .7786594   .0479403    16.24   0.000     .6846982    .8726206
-------------+----------------------------------------------------------------
y2 <-        |
x1 |   .9901353   .0044941   220.32   0.000      .981327    .9989435
x2 |   1.006836   .0044795   224.76   0.000      .998056    1.015615
x3 |   1.004249   .0044657   224.88   0.000     .9954963    1.013002
x4 |   .9976541   .0044783   222.77   0.000     .9888767    1.006431
x5 |   .9987587   .0044736   223.26   0.000     .9899907    1.007527
L |   .7786594   .0479403    16.24   0.000     .6846982    .8726206
_cons |   .0002758   .0192417     0.01   0.989    -.0374372    .0379887
-------------+----------------------------------------------------------------
y1           |
/cut1 |  -1.131155   .1157771    -9.77   0.000    -1.358074   -.9042358
/cut2 |  -.5330973   .1079414    -4.94   0.000    -.7446585    -.321536
/cut3 |   .2722794   .1061315     2.57   0.010     .0642654    .4802933
/cut4 |     .89394   .1123013     7.96   0.000     .6738334    1.114047
-------------+----------------------------------------------------------------
var(L)|          1  (constrained)
-------------+----------------------------------------------------------------
var(e.y2)|   .3823751    .074215                      .2613848    .5593696
------------------------------------------------------------------------------


These are the results we obtain when we transform the values reported by gsem to the original parameterization:

------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
y1_y2 |   1.088455   .0608501    17.89   0.000     .9691909    1.207719
y1_x1 |   1.069657   .0642069    16.66   0.000      .943814    1.195501
y1_x2 |   1.062269   .0619939    17.14   0.000      .940763    1.183774
y2_x1 |   .9901353   .0044941   220.32   0.000      .981327    .9989435
y2_x2 |   1.006836   .0044795   224.76   0.000      .998056    1.015615
y2_x3 |   1.004249   .0044657   224.88   0.000     .9954963    1.013002
y2_x4 |   .9976541   .0044783   222.77   0.000     .9888767    1.006431
y2_x5 |   .9987587   .0044736   223.26   0.000     .9899907    1.007527
y2__cons |   .0002758   .0192417     0.01   0.989    -.0374372    .0379887
cut_1 |   -.892498   .0895971    -9.96   0.000    -1.068105   -.7168909
cut_2 |  -.4206217   .0841852    -5.00   0.000    -.5856218   -.2556217
cut_3 |   .2148325   .0843737     2.55   0.011     .0494632    .3802018
cut_4 |    .705332   .0905974     7.79   0.000     .5277644    .8828997
sig_2 |   .9943267    .007031   141.42   0.000     .9805462    1.008107
rho_12 |   .4811176   .0477552    10.07   0.000     .3875191     .574716
------------------------------------------------------------------------------

The estimates are quite close to the values used for the simulation. If you try to perform the estimation with the wrong sign for the coefficient for L, you will get a number that is virtually zero (if you get convergence at all). In this case, the evaluator is telling us that the best value it can find, provided the restrictions we have imposed, is zero. If you see such results, you may want to try the opposite sign. If both give a zero coefficient, it means that this is the solution, and there is not endogeneity at all. If one of them is not zero, it means that the non-zero value is the solution. As stated before, in Stata 13.1, the model can be fitted without this restriction.

Categories: Statistics Tags:

## Export tables to Excel

There is a new command in Stata 13, putexcel, that allows you to easily export matrices, expressions, and stored results to an Excel file. Combining putexcel with a Stata command’s stored results allows you to create the table displayed in your Stata Results window in an Excel file.

A stored result is simply a scalar, macro, or matrix stored in memory after you run a Stata command. The two main types of stored results are e-class (for estimation commands) and r-class (for general commands). You can list a command’s stored results after it has been run by typing ereturn list (for estimation commands) and return list (for general commands). Let’s try a simple example by loading the auto dataset and running correlate on the variables foreign and mpg

. sysuse auto
(1978 Automobile Data)

. correlate foreign mpg
(obs=74)

|  foreign      mpg
-------------+------------------
foreign |   1.0000
mpg |   0.3934   1.0000


Because correlate is not an estimation command, use the return list command to see its stored results.

. return list

scalars:
r(N) =  74
r(rho) =  .3933974152205484

matrices:
r(C) :  2 x 2


Now we can use putexcel to export these results to Excel. The basic syntax of putexcel is

putexcel excel_cell=(expression) … using filename [, options]

If you are working with matrices, the syntax is

putexcel excel_cell=matrix(expression) … using filename [, options]

It is easy to build the above syntax in the putexcel dialog. There is a helpful video on Youtube about the dialog here. Let’s list the matrix r(C) to see what it contains.

. matrix list r(C)

symmetric r(C)[2,2]
foreign        mpg
foreign          1
mpg  .39339742          1


To re-create the table in Excel, we need to export the matrix r(C) with the matrix row and column names. The command to type in your Stata Command window is

putexcel A1=matrix(r(C), names) using corr


Note that to export the matrix row and column names, we used the names option after we specifed the matrix r(C). When I open the file corr.xlsx in Excel, the table below is displayed.

Next let’s try a more involved example. Load the auto dataset, and run a tabulation on the variable foreign. Because tabulate is not an estimation command, use the return list command to see its stored results.

. sysuse auto
(1978 Automobile Data)

. tabulate foreign

Car type |      Freq.     Percent        Cum.
------------+-----------------------------------
Domestic |         52       70.27       70.27
Foreign |         22       29.73      100.00
------------+-----------------------------------
Total |         74      100.00

. return list

scalars:
r(N) =  74
r(r) =  2


tabulate is different from most commands in Stata in that it does not automatically save all the results we need into the stored results (we will use scalar r(N)). We need to use the matcell() and matrow() options of tabulate to save the results produced by the command into two Stata matrices.

. tabulate foreign, matcell(freq) matrow(names)

Car type |      Freq.     Percent        Cum.
------------+-----------------------------------
Domestic |         52       70.27       70.27
Foreign |         22       29.73      100.00
------------+-----------------------------------
Total |         74      100.00

. matrix list freq

freq[2,1]
c1
r1  52
r2  22

. matrix list names

names[2,1]
c1
r1   0
r2   1


The putexcel commands used to create a basic tabulation table in Excel column 1 row 1 are

putexcel A1=("Car type") B1=("Freq.") C1=("Percent") using results, replace
putexcel A2=matrix(names) B2=matrix(freq) C2=matrix(freq/r(N)) using results,
modify


Below is the table produced in Excel by these commands.

Again this is a basic tabulation table. You probably noticed that we did not have the Cum. column or the Total row in the export table. Also our Car type column contains the numeric values (0,1), not the value lables (Domestic, Foreign) of the variable foreign, and our Percent column is not formatted correctly. To get the exact table displayed in the Results window into an Excel file takes a little programming. With a few functions and a forvalues loop, we can easily export any table produced by running the tabulate command on a numeric variable.

There are two extended macro functions, label and display, that can help us. The label function can extract the value labels for each variable, and the display function can correctly format numbers for our numeric columns. Last, we use forvalues to loop over the rows of the returned matrices to produce our final tables. Our do-file to produce the tabulate table in Excel looks like

sysuse auto
tabulate foreign, matcell(freq) matrow(names)

putexcel A1=("Car type") B1=("Freq.") C1=("Percent") D1=("Cum.") using results, replace

local rows = rowsof(names)
local row = 2
local cum_percent = 0

forvalues i = 1/rows' {

local val = names[i',1]
local val_lab : label (foreign) val'

local freq_val = freq[i',1]

local percent_val = freq_val'/r(N)'*100
local percent_val : display %9.2f percent_val'

local cum_percent : display %9.2f (cum_percent' + percent_val')

putexcel Arow'=("val_lab'") Brow'=(freq_val') Crow'=(percent_val') ///
Drow'=(cum_percent') using results, modify
local row = row' + 1
}

putexcel Arow'=("Total") Brow'=(r(N)) Crow'=(100.00) using results, modify


The above commands produce this table in Excel:

The solution above works well for this one table, but what if we need to export the tabulation table for 100 variables to the same Excel spreadsheet? It would be very tedious to run the same do-file 100 times, each time changing the cell and row numbers. Now we could easily change our do-file into the Stata command (ado-file) called tab2xl. The syntax for our new command could be

tab2xl varname using filename, row(rownumber) col(colnumber) [replace sheet(name)]


The pseudocode of our program (file tab2xl.ado) looks like

program tab2xl
/* parse command syntax */

/* tabulate varname */

/* get column letters based on starting column number passed in */

/* write header row to filename in starting row number passed in */

/* loop over rows of returned matrix and calculate/write values to filename */

/* write total row to filename */
end


If you would like to download a working version of our tab2xl command, type

net install http://www.stata.com/users/kcrow/tab2xl


in Stata.

Categories: Programming Tags:

## Measures of effect size in Stata 13

Today I want to talk about effect sizes such as Cohen’s d, Hedges’s g, Glass’s Δ, η2, and ω2. Effects sizes concern rescaling parameter estimates to make them easier to interpret, especially in terms of practical significance.

Many researchers in psychology and education advocate reporting of effect sizes, professional organizations such as the American Psychological Association (APA) and the American Educational Research Association (AERA) strongly recommend their reporting, and professional journals such as the Journal of Experimental Psychology: Applied and Educational and Psychological Measurement require that they be reported.

Anyway, today I want to show you

1. What effect sizes are.
2. How to calculate effect sizes and their confidence intervals in Stata.
3. How to calculate bootstrap confidence intervals for those effect sizes.
4. How to use Stata’s effect-size calculator.

## 1. What are effect sizes?

The importance of research results is often assessed by statistical significance, usually that the p-value is less than 0.05. P-values and statistical significance, however, don’t tell us anything about practical significance.

What if I told you that I had developed a new weight-loss pill and that the difference between the average weight loss for people who took the pill and the those who took a placebo was statistically significant? Would you buy my new pill? If you were overweight, you might reply, “Of course! I’ll take two bottles and a large order of french fries to go!”. Now let me add that the average difference in weight loss was only one pound over the year. Still interested? My results may be statistically significant but they are not practically significant.

Or what if I told you that the difference in weight loss was not statistically significant — the p-value was “only” 0.06 — but the average difference over the year was 20 pounds? You might very well be interested in that pill.

The size of the effect tells us about the practical significance. P-values do not assess practical significance.

All of which is to say, one should report parameter estimates along with statistical significance.

In my examples above, you knew that 1 pound over the year is small and 20 pounds is large because you are familiar with human weights.

In another context, 1 pound might be large, and in yet another, 20 pounds small.

Formal measures of effects sizes are thus usually presented in unit-free but easy-to-interpret form, such as standardized differences and proportions of variability explained.

### The “d” family

Effect sizes that measure the scaled difference between means belong to the “d” family. The generic formula is

The estimators differ in terms of how sigma is calculated.

Cohen’s d, for instance, uses the pooled sample standard deviation.

Hedges’s g incorporates an adjustment which removes the bias of Cohen’s d.

Glass’s Δ was originally developed in the context of experiments and uses the “control group” standard deviation in the denominator. It has subsequently been generalized to nonexperimental studies. Because there is no control group in observational studies, Kline (2013) recommends reporting Glass’s Δ using the standard deviation for each group. Glass’s Delta_1 uses one group’s standard deviation and Delta_2 uses the other group’s.

Although I have given definitions to Cohen’s d, Hedges’s g, and Glass’s Δ, different authors swap the definitions around! As a result, many authors refer to all of the above as just Delta.

Be careful when using software to know which Delta you are getting. I have used Stata terminology, of course.

Anyway, the use of a standardized scale allows us to assess of practical significance. Delta = 1.5 indicates that the mean of one group is 1.5 standard deviations higher than that of the other. A difference of 1.5 standard deviations is obviously large, and a difference of 0.1 standard deviations is obviously small.

### The “r” family

The r family quantifies the ratio of the variance attributable to an effect to the total variance and is often interpreted as the “proportion of variance explained”. The generic estimator is known as eta-squared,

η2 is equivalent to the R-squared statistic from linear regression.

ω2 is a less biased variation of η2 that is equivalent to the adjusted R-squared.

Both of these measures concern the entire model.

Partial η2 and partial ω2 are like partial R-squareds and concern individual terms in the model. A term might be a variable or a variable and its interaction with another variable.

Both the d and r families allow us to make an apples-to-apples comparison of variables measured on different scales. For example, an intervention could affect both systolic blood pressure and total cholesterol. Comparing the relative effect of the intervention on the two outcomes would be difficult on their original scales.

How does one compare mm/Hg and mg/dL? It is straightforward in terms of Cohen’s d or ω2 because then we are comparing standard deviation changes or proportion of variance explained.

## 2. How to calculate effect sizes and their confidence intervals in Stata

Consider a study where 30 school children are randomly assigned to classrooms that incorporated web-based instruction (treatment) or standard classroom environments (control). At the end of the school year, the children were given tests to measure reading and mathematics skills. The reading test is scored on a 0-15 point scale and, the mathematics test, on a 0-100 point scale.

Let’s download a dataset for our fictitious example from the Stata website by typing:

. use http://www.stata.com/videos13/data/webclass.dta

Contains data from http://www.stata.com/videos13/data/webclass.dta
obs:            30                          Fictitious web-based learning
experiment data
vars:             5                          5 Sep 2013 11:28
size:           330                          (_dta has notes)
-------------------------------------------------------------------------------
storage   display    value
variable name   type    format     label      variable label
-------------------------------------------------------------------------------
id              byte    %9.0g                 ID Number
treated         byte    %9.0g      treated    Treatment Group
agegroup        byte    %9.0g      agegroup   Age Group
math            float   %9.0g                 Math Score
-------------------------------------------------------------------------------

. notes

_dta:
1.  Variable treated records 0=control, 1=treated.
2.  Variable agegroup records 1=7 years old, 2=8 years old, 3=9 years old.


We can compute a t-statistic to test the null hypothesis that the average math scores are the same in the treatment and control groups.

. ttest math, by(treated)

Two-sample t test with equal variances
------------------------------------------------------------------------------
Group |     Obs        Mean    Std. Err.   Std. Dev.   [95% Conf. Interval]
---------+--------------------------------------------------------------------
Control |      15    69.98866    3.232864    12.52083    63.05485    76.92246
Treated |      15    79.54943    1.812756    7.020772    75.66146     83.4374
---------+--------------------------------------------------------------------
combined |      30    74.76904    2.025821    11.09588    70.62577    78.91231
---------+--------------------------------------------------------------------
diff |           -9.560774    3.706412               -17.15301   -1.968533
------------------------------------------------------------------------------
diff = mean(Control) - mean(Treated)                          t =  -2.5795
Ho: diff = 0                                     degrees of freedom =       28

Ha: diff < 0                 Ha: diff != 0                 Ha: diff > 0
Pr(T < t) = 0.0077         Pr(|T| > |t|) = 0.0154          Pr(T > t) = 0.9923


The treated students have a larger mean, yet the difference of -9.56 is reported as negative because -ttest- calculated Control minus Treated. So just remember, negative differences mean Treated > Control in this case.

The t-statistic equals -2.58 and its two-sided p-value of 0.0154 indicates that the difference between the math scores in the two groups is statistically significant.

Next, let’s calculate effect sizes from the d family:

. esize twosample math, by(treated) cohensd hedgesg glassdelta

Effect size based on mean comparison

Obs per group:
Control =     15
Treated =     15
---------------------------------------------------------
Effect Size |   Estimate     [95% Conf. Interval]
--------------------+------------------------------------
Cohen's d |  -.9419085    -1.691029   -.1777553
Hedges's g |   -.916413    -1.645256   -.1729438
Glass's Delta 1 |  -.7635896     -1.52044    .0167094
Glass's Delta 2 |  -1.361784    -2.218342   -.4727376
---------------------------------------------------------


Cohen’s d and Hedges’s g both indicate that the average reading scores differ by approximately -0.93 standard deviations with 95% confidence intervals of (-1.69, -0.18) and (-1.65, -0.17) respectively.

Since this is an experiment, we are interested in Glass’s Delta 1 because it is calculated using the control group standard deviation. Average reading scores differ by -0.76 and the confidence interval is (-1.52, 0.02).

The confidence intervals for Cohen’s d and Hedges’s g do not include the null value of zero but the confidence interval for Glass’s Delta 1 does. Thus we cannot completely rule out the possibility that the treatment had no effect on math scores.

Next we could incorporate the age group of the children into our analysis by using a two-way ANOVA to test the null hypothesis that the mean math scores are equal for all groups.

. anova math treated##agegroup

Number of obs =      30     R-squared     =  0.2671
Root MSE      = 10.4418     Adj R-squared =  0.1144

Source |  Partial SS    df       MS           F     Prob > F
-----------------+----------------------------------------------------
Model |  953.697551     5   190.73951       1.75     0.1617
|
treated |  685.562956     1  685.562956       6.29     0.0193
agegroup |  47.7059268     2  23.8529634       0.22     0.8051
treated#agegroup |  220.428668     2  110.214334       1.01     0.3789
|
Residual |  2616.73825    24  109.030761
-----------------+----------------------------------------------------
Total |   3570.4358    29  123.118476


The F-statistic for the entire model is not statistically significant (F=1.75, ndf=5, ddf=24, p=0.1617) but the F-statistic for the main effect of treatment is statistically significant (F=6.29, ndf=1, ddf=24, p=0.0193).

We can compute the η2 and partial η2 estimates for this model using the estat esize command immediately after our anova command (note that estat esize works after the regress command too).

. estat esize

Effect sizes for linear models

---------------------------------------------------------------------
Source |   Eta-Squared     df     [95% Conf. Interval]
----------------------+----------------------------------------------
Model |   .2671096         5            0    .4067062
|
treated |   .2076016         1     .0039512    .4451877
agegroup |   .0179046         2            0    .1458161
treated#agegroup |   .0776932         2            0     .271507
---------------------------------------------------------------------


The overall η2 indicates that our model accounts for approximately 26.7% of the variablity in math scores though the 95% confidence interval includes the null value of zero (0.00%, 40.7%). The partial η2 for treatment is 0.21 (21% of the variability explained) and its 95% confidence interval excludes zero (0.3%, 20%).

We could calculate the alternative r-family member ω2 rather than η2 by typing

. estat esize, omega

Effect sizes for linear models

---------------------------------------------------------------------
Source | Omega-Squared     df     [95% Conf. Interval]
----------------------+----------------------------------------------
Model |   .1144241         5            0    .2831033
|
treated |    .174585         1            0    .4220705
agegroup |          0         2            0    .0746342
treated#agegroup |   .0008343         2            0    .2107992
---------------------------------------------------------------------


The overall ω2 indicates that our model accounts for approximately 11.4% of the variability in math scores and treatment accounts for 17.5%. This perplexing result stems from the way that ω2 and partial ω2 are calculated. See Pierce, Block, & Aguinis (2004) for a thorough explanation.

Except for the η2 for treatment, the confidence intervals include 0 so we cannot rule out the possibility that there is no effect. Whether results are practically significant is generically a matter context and opinion. In some situations, accounting for 5% of the variability in an outcome could be very important and in other situations accounting for 30% may not be.

We could repeat the same analyses for the reading scores using the following commands:

. ttest reading, by(treated)
. esize twosample reading, by(treated) cohensd hedgesg glassdelta
. estat esize
. estat esize, omega


None of the t- or F-statistics for reading scores were statistically significant at the 0.05 level.

Even though the reading and math scores were measured on two different scales, we can directly compare the relative effect of the treatment using effect sizes:

        Effect Size   |     Reading Score          Math Score
------------------------------------------------------------
Cohen's d     |   -0.23 (-0.95 - 0.49)  -0.94 (-1.69 - -0.18)
Hedges's g    |   -0.22 (-0.92 - 0.48)  -0.92 (-1.65 - -0.17)
Glass's Delta |   -0.21 (-0.93 - 0.51)  -0.76 (-1.52 -  0.02)
Eta-squared   |    0.02 ( 0.00 - 0.20)   0.21 ( 0.00 -  0.44)
Omega-squared |    0.00 ( 0.00 - 0.17)   0.17 ( 0.00 -  0.42)


The results show that the average reading scores in the treated and control groups differ by approximately 0.22 standard deviations while the average math scores differ by approximately 0.92 standard deviations. Similarly, treatment status accounted for almost none of the variability in reading scores while it accounted for roughly 17% of the variability in math scores. The intervention clearly had a larger effect on math scores than reading scores. We also know that we cannot completely rule out an effect size of zero (no effect) for both reading and math scores because several confidence intervals included zero. Whether or not the effects are practically significant is a matter of interpretation but the effect sizes provide a standardized metric for evaluation.

## 3. How to calculate bootstrap confidence intervals

Simulation studies have shown that bootstrap confidence intervals for the d family may be preferable to confidence intervals based on the noncentral t distribution when the variable of interest does not have a normal distribution (Kelley 2005; Algina, Keselman, and Penfield 2006). We can calculate bootstrap confidence intervals for Cohen’s d and Hedges’s g using Stata’s bootstrap prefix:

. bootstrap r(d) r(g), reps(500) nowarn:  esize twosample reading, by(treated)
(running esize on estimation sample)

Bootstrap replications (500)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
..................................................    50
..................................................   100
..................................................   150
..................................................   200
..................................................   250
..................................................   300
..................................................   350
..................................................   400
..................................................   450
..................................................   500

Bootstrap results                               Number of obs      =        30
Replications       =       500

_bs_1:  r(d)
_bs_2:  r(g)

------------------------------------------------------------------------------
|   Observed   Bootstrap                         Normal-based
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
_bs_1 |   -.228966   .3905644    -0.59   0.558    -.9944582    .5365262
_bs_2 |  -.2227684   .3799927    -0.59   0.558    -.9675403    .5220036
------------------------------------------------------------------------------


The bootstrap estimate of the 95% confidence interval for Cohen’s d is -0.99 to 0.54 which is slightly wider than the earlier estimate based on the non-central t distribution (see [R] esize for details). The bootstrap estimate is slightly wider for Hedges’s g as well.

## 4. How to use Stata’s effect-size calculator

You can use Stata’s effect size calculators to estimate them using summary statistics. If we know that the mean, standard deviation and sample size for one group is 70, 12.5 and 15 respectively and 80, 7 and 15 for another group, we can use esizei to estimate effect sizes from the d family:

. esizei 15 70 12.5 15 80 7, cohensd hedgesg glassdelta

Effect size based on mean comparison

Obs per group:
Group 1 =     15
Group 2 =     15
---------------------------------------------------------
Effect Size |   Estimate     [95% Conf. Interval]
--------------------+------------------------------------
Cohen's d |  -.9871279    -1.739873   -.2187839
Hedges's g |  -.9604084    -1.692779   -.2128619
Glass's Delta 1 |        -.8    -1.561417   -.0143276
Glass's Delta 2 |  -1.428571    -2.299112   -.5250285
---------------------------------------------------------


We can estimate effect sizes from the r family using esizei with slightly different syntax. For example, if we know the numerator and denominator degrees of freedom along with the F statistic, we can calculate η2 and ω2 using the following command:

. esizei 1 28 6.65

Effect sizes for linear models

---------------------------------------------------------
Effect Size |   Estimate     [95% Conf. Interval]
--------------------+------------------------------------
Eta-Squared |   .1919192     .0065357    .4167874
Omega-Squared |   .1630592            0    .3959584
---------------------------------------------------------


## Video demonstration

Stata has dialog boxes that can assist you in calculating effect sizes. If you would like a brief introduction using the GUI, you can watch a demonstration on Stata’s YouTube Channel:

Tour of effect sizes in Stata

## Final thoughts and further reading

Most older papers and many current papers do not report effect sizes. Nowadays, the general consensus among behavioral scientists, their professional organizations, and their journals is that effect sizes should always be reported in addition to tests of statistical significance. Stata 13 now makes it easy to compute most popular effects sizes.

Some methodologists believe that effect sizes with confidence intervals should always be reported and that statistical hypothesis tests should be abandoned altogether; see Cumming (2012) and Kline (2013). While this may sound like a radical notion, other fields such as epidemiology have been moving in this direction since the 1990s. Cumming and Kline offer compelling arguments for this paradigm shift as well as excellent introductions to effect sizes.

American Psychological Association (2009). Publication Manual of the American Psychological Association, 6th Ed. Washington, DC: American Psychological Association.

Algina, J., H. J. Keselman, and R. D. Penfield. (2006). Confidence interval coverage for Cohen’s effect size statistic. Educational and Psychological Measurement, 66(6): 945–960.

Cumming, G. (2012). Understanding the New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis. New York: Taylor & Francis.

Kelley, K. (2005). The effects of nonnormal distributions on confidence intervals around the standardized mean difference: Bootstrap and parametric confidence intervals. Educational and Psychological Measurement 65: 51–69.

Kirk, R. (1996). Practical significance: A concept whose time has come. Educational and Psychological Measurement, 56, 746-759.

Kline, R. B. (2013). Beyond Significance Testing: Statistics Reform in the Behavioral Sciences. 2nd ed. Washington, DC: American Psychological Association.

Pierce, C.A., Block, R. A., and Aguinis, H. (2004). Cautionary note on reporting eta-squared values from multifactor ANOVA designs. Educational and Psychological Measurement, 64(6) 916-924

Thompson, B. (1996) AERA Editorial Policies regarding Statistical Significance Testing: Three Suggested Reforms. Educational Researcher, 25(2) 26-30

Wilkinson, L., & APA Task Force on Statistical Inference. (1999). Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54, 594-604

Categories: Statistics Tags:

## Stata 13 ships June 24

There’s a new release of Stata. You can order it now, it starts shipping on June 24, and you can find out about it at www.stata.com/stata13/.

Well, we sure haven’t made that sound exciting when, in fact, Stata 13 is a big — we mean really BIG — release, and we really do want to tell you about it.

Rather than summarizing, however, we’ll send you to the website, which in addition to the standard marketing materials, has technical sheets, demonstrations, and even videos of the new features.

And all 11,000 pages of the manuals are now online.

## Update on the Stata YouTube Channel

What is it about round numbers that compels us to pause and reflect? We celebrate 20-year school reunions, 25-year wedding anniversaries, 50th birthdays and other similar milestones. I don’t know the answer but the Stata YouTube Channel recently passed several milestones – more than 1500 subscribers, over 50,000 video views and it was launched six months ago. We felt the need for a small celebration to mark the occasion, and I thought that I would give you a brief update.

I could tell you about re-recording the original 24 videos with a larger font to make them easier to read. I could tell you about the hardware and software that we use to record them including our experiments with various condenser and dynamic microphones. I could share quotes from some of the nice messages we’ve received. But I think it would be more fun to talk about….you!

YouTube collects data about the number of views each video receives as well as summary data about who, what, when, where, and how you are watching them. There is no need to be concerned about your privacy; there are no personal identifiers of any kind associated with these data. But the summary data are interesting, and I thought it might be fun to share some of the data with you.

## Who’s watching?

Figure 1 shows the age distribution of Stata YouTube Channel viewers. If you have ever attended a Stata Conference, you will not be surprised by this graph…until you notice the age group at the bottom. I would not have guessed that 13-17 year olds are watching our videos. Perhaps they saw Stata in the movie “Moneyball” with Brad Pitt and wanted to learn more. Or maybe they were influenced by the latest fashion craze sweeping the youth of the world.

## What are you watching?

We have posted more than 50 videos over a wide range of topics. Figure 2 shows the total number of views for the ten most popular videos. The more popular of the ten are about broad topics. These broader videos are mostly older and have thus had time to accumulate more views.

Even so, these videos receive more views per day currently than do the special topic videos that have been posted more recently. This supports my belief that Stata YouTube Channel viewers tend to be relatively new Stata users who want to learn about general topics, and that means more generic videos in the future. So you and your two post-docs will just have to read the manual if you want to learn how to fit asymmetric power ARCH models with outer-product gradient standard errors.

## When are you watching?

We usually post new videos on Tuesday mornings which might lead you to believe that the peak viewing day would also be Tuesday. Figure 3, however, shows us that the average number of views per day (vpd) is higher on Wednesdays at 420 vpd and in fact peaks on Thursdays at 430 vpd before declining Friday through Sunday.

Figure 4 also shows us that late September may have been not the best time to launch the Stata YouTube Channel. Our early momentum in September and October slowed during the November and December holiday seasons. We were, however, pleased to see that 49 of you spent New Years Eve watching our videos. Perhaps next year we’ll prepare something more festive just for you!

## Where are you watching?

What do the Czech Republic, Pakistan, Uganda, Madagascar, the United Kingdom, the Bahamas, the United States, Montenegro, and Italy have in common? Correct! They are all countries in which you are watching our videos. They are also locations depicted in one of my favorite action films but I’ll leave that to the trivia buffs. I think the most exciting information that we found in our data is that the Stata YouTube Channel is being viewed in 164 countries!

You might not be surprised to learn that roughly half of the people watching the videos live in the United States, the United Kingdom, or Canada. The results may be unexpected when we consider the “view rate” defined as the number of views per 100,000 residents. Figure 5 shows the top 20 countries ranked by view rate for countries with at least four million residents. Denmark had the highest view rate which was nearly twice the rate of Norway which had the second highest view rate. The view rate in Denmark was more than three times the rate in the US and the UK.

## How are you watching?

You might think that I would have anything to report about “how” you are watching the videos, but it turns out that 5.2% of you are watching on mobile devices. Perhaps this explains the 13-17 year old demographic or the 49 people watching on New Year’s Eve. Or maybe we are helping you pass the time in the dentist office waiting room.

## Final thoughts

Six months isn’t much of a milestone. We Stata folk will use any excuse to break out the cake and ice cream. Even so, the Stata YouTube Channel began as an experiment and often experiments do not work out as we would like. This experiment has exceeded our expectations and, as a result, we have started taking requests for videos on our Facebook page and we’ll be adding more videos every week. So thanks for watching and stay tuned!

Now if you will excuse me, I’m going to get some cake and ice cream.

Categories: Resources Tags:

## Multilevel linear models in Stata, part 2: Longitudinal data

In my last posting, I introduced you to the concepts of hierarchical or “multilevel” data. In today’s post, I’d like to show you how to use multilevel modeling techniques to analyse longitudinal data with Stata’s xtmixed command.

Last time, we noticed that our data had two features. First, we noticed that the means within each level of the hierarchy were different from each other and we incorporated that into our data analysis by fitting a “variance component” model using Stata’s xtmixed command.

The second feature that we noticed is that repeated measurement of GSP showed an upward trend. We’ll pick up where we left off last time and stick to the concepts again and you can refer to the references at the end to learn more about the details.

## The videos

Stata has a very friendly dialog box that can assist you in building multilevel models. If you would like a brief introduction using the GUI, you can watch a demonstration on Stata’s YouTube Channel:

Introduction to multilevel linear models in Stata, part 2: Longitudinal data

## Longitudinal data

I’m often asked by beginning data analysts – “What’s the difference between longitudinal data and time-series data? Aren’t they the same thing?”.

The confusion is understandable — both types of data involve some measurement of time. But the answer is no, they are not the same thing.

Univariate time series data typically arise from the collection of many data points over time from a single source, such as from a person, country, financial instrument, etc.

Longitudinal data typically arise from collecting a few observations over time from many sources, such as a few blood pressure measurements from many people.

There are some multivariate time series that blur this distinction but a rule of thumb for distinguishing between the two is that time series have more repeated observations than subjects while longitudinal data have more subjects than repeated observations.

Because our GSP data from last time involve 17 measurements from 48 states (more sources than measurements), we will treat them as longitudinal data.

## Random intercept models

As I mentioned last time, repeated observations on a group of individuals can be conceptualized as multilevel data and modeled just as any other multilevel data. We left off last time with a variance component model for GSP (Gross State Product, logged) and noted that our model assumed a constant GSP over time while the data showed a clear upward trend.

If we consider a single observation and think about our model, nothing in the fixed or random part of the models is a function of time.

Let’s begin by adding the variable year to the fixed part of our model.

As we expected, our grand mean has become a linear regression which more accurately reflects the change over time in GSP. What might be unexpected is that each state’s and region’s mean has changed as well and now has the same slope as the regression line. This is because none of the random components of our model are a function of time. Let’s fit this model with the xtmixed command:

. xtmixed gsp year, || region: || state:

------------------------------------------------------------------------------
gsp |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
year |   .0274903   .0005247    52.39   0.000     .0264618    .0285188
_cons |  -43.71617   1.067718   -40.94   0.000    -45.80886   -41.62348
------------------------------------------------------------------------------

------------------------------------------------------------------------------
Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
region: Identity             |
sd(_cons) |   .6615238   .2038949      .3615664    1.210327
-----------------------------+------------------------------------------------
state: Identity              |
sd(_cons) |   .7805107   .0885788      .6248525    .9749452
-----------------------------+------------------------------------------------
sd(Residual) |   .0734343   .0018737      .0698522    .0772001
------------------------------------------------------------------------------

The fixed part of our model now displays an estimate of the intercept (_cons = -43.7) and the slope (year = 0.027). Let’s graph the model for Region 7 and see if it fits the data better than the variance component model.

predict GrandMean, xb
label var GrandMean "GrandMean"
predict RegionEffect, reffects level(region)
predict StateEffect, reffects level(state)
gen RegionMean = GrandMean + RegionEffect
gen StateMean = GrandMean + RegionEffect + StateEffect

twoway  (line GrandMean year, lcolor(black) lwidth(thick))      ///
(line RegionMean year, lcolor(blue) lwidth(medthick))   ///
(line StateMean year, lcolor(green) connect(ascending)) ///
(scatter gsp year, mcolor(red) msize(medsmall))         ///
if region ==7,                                          ///
ytitle(log(Gross State Product), margin(medsmall))      ///
legend(cols(4) size(small))                             ///
title("Multilevel Model of GSP for Region 7", size(medsmall))

That looks like a much better fit than our variance-components model from last time. Perhaps I should leave well enough alone, but I can’t help noticing that the slopes of the green lines for each state don’t fit as well as they could. The top green line fits nicely but the second from the top looks like it slopes upward more than is necessary. That’s the best fit we can achieve if the regression lines are forced to be parallel to each other. But what if the lines were not forced to be parallel? What if we could fit a “mini-regression model” for each state within the context of my overall multilevel model. Well, good news — we can!

## Random slope models

By introducing the variable year to the fixed part of the model, we turned our grand mean into a regression line. Next I’d like to incorporate the variable year into the random part of the model. By introducing a fourth random component that is a function of time, I am effectively estimating a separate regression line within each state.

Notice that the size of the new, brown deviation u1ij. is a function of time. If the observation were one year to the left, u1ij. would be smaller and if the observation were one year to the right, u1ij.would be larger.

It is common to “center” the time variable before fitting these kinds of models. Explaining why is for another day. The quick answer is that, at some point during the fitting of the model, Stata will have to compute the equivalent of the inverse of the square of year. For the year 1986 this turns out to be 2.535e-07. That’s a fairly small number and if we multiply it by another small number…well, you get the idea. By centering age (e.g. cyear = year – 1978), we get a more reasonable number for 1986 (0.01). (Hint: If you have problems with your model converging and you have large values for time, try centering them. It won’t always help, but it might).

So let’s center our year variable by subtracting 1978 and fit a model that includes a random slope.

gen cyear = year - 1978
xtmixed gsp cyear, || region: || state: cyear, cov(indep)

I’ve color-coded the output so that we can match each part of the output back to the model and the graph. The fixed part of the model appears in the top table and it looks like any other simple linear regression model. The random part of the model is definitely more complicated. If you get lost, look back at the graphic of the deviations and remind yourself that we have simply partitioned the deviation of each observation into four components. If we did this for every observation, the standard deviations in our output are simply the average of those deviations.

Let’s look at a graph of our new “random slope” model for Region 7 and see how well it fits our data.

predict GrandMean, xb
label var GrandMean "GrandMean"
predict RegionEffect, reffects level(region)
predict StateEffect_year StateEffect_cons, reffects level(state)

gen RegionMean = GrandMean + RegionEffect
gen StateMean_cons = GrandMean + RegionEffect + StateEffect_cons
gen StateMean_year = GrandMean + RegionEffect + StateEffect_cons + ///
(cyear*StateEffect_year)

twoway  (line GrandMean cyear, lcolor(black) lwidth(thick))             ///
(line RegionMean cyear, lcolor(blue) lwidth(medthick))          ///
(line StateMean_cons cyear, lcolor(green) connect(ascending))   ///
(line StateMean_year cyear, lcolor(brown) connect(ascending))   ///
(scatter gsp cyear, mcolor(red) msize(medsmall))                ///
if region ==7,                                                  ///
ytitle(log(Gross State Product), margin(medsmall))              ///
legend(cols(3) size(small))                                     ///
title("Multilevel Model of GSP for Region 7", size(medsmall))`

The top brown line fits the data slightly better, but the brown line below it (second from the top) is a much better fit. Mission accomplished!

## Where do we go from here?

I hope I have been able to convince you that multilevel modeling is easy using Stata’s xtmixed command and that this is a tool that you will want to add to your kit. I would love to say something like “And that’s all there is to it. Go forth and build models!”, but I would be remiss if I didn’t point out that I have glossed over many critical topics.

In our GSP example, we would still like to consider the impact of other independent variables. I haven’t mentioned choice of estimation methods (ML or REML in the case of xtmixed). I’ve assessed the fit of our models by looking at graphs, an approach important but incomplete. We haven’t thought about hypothesis testing. Oh — and, all the usual residual diagnostics for linear regression such as checking for outliers, influential observations, heteroskedasticity and normality still apply….times four! But now that you understand the concepts and some of the mechanics, it shouldn’t be difficult to fill in the details. If you’d like to learn more, check out the links below.

I hope this was helpful…thanks for stopping by.