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Efficiency comparisons by Monte Carlo simulation

Overview

In this post, I show how to use Monte Carlo simulations to compare the efficiency of different estimators. I also illustrate what we mean by efficiency when discussing statistical estimators.

I wrote this post to continue a dialog with my friend who doubted the usefulness of the sample average as an estimator for the mean when the data-generating process (DGP) is a $$\chi^2$$ distribution with $$1$$ degree of freedom, denoted by a $$\chi^2(1)$$ distribution. The sample average is a fine estimator, even though it is not the most efficient estimator for the mean. (Some researchers prefer to estimate the median instead of the mean for DGPs that generate outliers. I will address the trade-offs between these parameters in a future post. For now, I want to stick to estimating the mean.)

In this post, I also want to illustrate that Monte Carlo simulations can help explain abstract statistical concepts. I show how to use a Monte Carlo simulation to illustrate the meaning of an abstract statistical concept. (If you are new to Monte Carlo simulations in Stata, you might want to see Monte Carlo simulations using Stata.)

Consistent estimator A is said to be more asymptotically efficient than consistent estimator B if A has a smaller asymptotic variance than B; see Wooldridge (2010, sec. 14.4.2) for an especially useful discussion. Theoretical comparisons can sometimes ascertain that A is more efficient than B, but the magnitude of the difference is rarely identified. Comparisons of Monte Carlo simulation estimates of the variances of estimators A and B give both sign and magnitude for specific DGPs and sample sizes.

The sample average versus maximum likelihood

Many books discuss the conditions under which the maximum likelihood (ML) estimator is the efficient estimator relative to other estimators; see Wooldridge (2010, sec. 14.4.2) for an accessible introduction to the modern approach. Here I compare the ML estimator with the sample average for the mean when the DGP is a $$\chi^2(1)$$ distribution.

Example 1 below contains the commands I used. For an introduction to Monte Carlo simulations see Monte Carlo simulations using Stata, and for an introduction to using mlexp to estimate the parameter of a $$\chi^2$$ distribution see Maximum likelihood estimation by mlexp: A chi-squared example. In short, the commands do the following $$5,000$$ times:

1. Draw a sample of 500 observations from a $$\chi^2(1)$$ distribution.
2. Estimate the mean of each sample by the sample average, and store this estimate in m_a in the dataset efcomp.dta.
3. Estimate the mean of each sample by ML, and store this estimate in m_ml in the dataset efcomp.dta.

Example 1: The distributions of the sample average and the ML estimators

. clear all
. set seed 12345
. postfile sim  mu_a mu_ml using efcomp, replace
. forvalues i = 1/5000 {
2.     quietly drop _all
3.     quietly set obs 500
4.     quietly generate double y = rchi2(1)
5.     quietly mean y
6.     local mu_a         =  _b[y]
7.     quietly mlexp (ln(chi2den({d=1},y)))
8.     local mu_ml   =  _b[d:_cons]
9.     post sim (mu_a') (mu_ml')
10. }
. postclose sim
. use efcomp, clear
. summarize

Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
mu_a |      5,000    .9989277    .0620524   .7792076   1.232033
mu_ml |      5,000    1.000988    .0401992   .8660786   1.161492


The mean of the $$5,000$$ sample average estimates and the mean of the $$5,000$$ ML estimates are each close to the true value of $$1.0$$. The standard deviation of the $$5,000$$ sample average estimates is $$0.062$$, and it approximates the standard deviation of the sampling distribution of the sample average for this DGP and sample size. Similarly, the standard deviation of the $$5,000$$ ML estimates is $$0.040$$, and it approximates the standard deviation of the sampling distribution of the ML estimator for this DGP and sample size.

We conclude that the ML estimator has a lower variance than the sample average for this DGP and this sample size, because $$0.040$$ is smaller than $$0.062$$.

To get a picture of this difference, we plot the density of the sample average and the density of the ML estimator. (Each of these densities is estimated from $$5,000$$ observations, but estimation error can be ignored because more data would not change the key results.)

Example 2: Plotting the densities of the estimators

. kdensity mu_a,   n(5000) generate(x_a   den_a)   nograph

. kdensity mu_ml,  n(5000) generate(x_ml  den_ml)  nograph

. twoway (line den_a x_a) (line den_ml x_ml)


Densities of the sample average and ML estimators

The plots show that the ML estimator is more tightly distributed around the true value than the sample average.

That the ML estimator is more tightly distributed around the true value than the sample average is what it means for one consistent estimator to be more efficient than another.

Done and undone

I used Monte Carlo simulation to illustrate what it means for one estimator to be more efficient than another. In particular, we saw that the ML estimator is more efficient than the sample average for the mean of a $$\chi^2(1)$$ distribution.

Many other estimators fall between these two estimators in an efficiency ranking. Generalized method of moments estimators and some quasi-maximum likelihood estimators come to mind and might be worth adding to these simulations.

Reference

Wooldridge, J. M. 2010. Econometric Analysis of Cross Section and Panel Data. 2nd ed. Cambridge, Massachusetts: MIT Press.

Categories: Statistics Tags:

Maximum likelihood estimation by mlexp: A chi-squared example

Overview

In this post, I show how to use mlexp to estimate the degree of freedom parameter of a chi-squared distribution by maximum likelihood (ML). One example is unconditional, and another example models the parameter as a function of covariates. I also show how to generate data from chi-squared distributions and I illustrate how to use simulation methods to understand an estimation technique.

The data

I want to show how to draw data from a $$\chi^2$$ distribution, and I want to illustrate that the ML estimator produces estimates close to the truth, so I use simulated data.

In the output below, I draw a $$2,000$$ observation random sample of data from a $$\chi^2$$ distribution with $$2$$ degrees of freedom, denoted by $$\chi^2(2)$$, and I summarize the results.

Example 1: Generating $$\chi^2(2)$$ data

. drop _all

. set obs 2000
number of observations (_N) was 0, now 2,000

. set seed 12345

. generate y = rchi2(2)

. summarize y

Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
y |      2,000    2.030865    1.990052   .0028283   13.88213


The mean and variance of the $$\chi^2(2)$$ distribution are $$2$$ and $$4$$, respectively. The sample mean of $$2.03$$ and the sample variance of $$3.96=1.99^2$$ are close to the true values. I set the random-number seed to $$12345$$ so that you can replicate my example; type help seed for details.

mlexp and the log-likelihood function

The log-likelihood function for the ML estimator for the degree of freedom parameter $$d$$ of a $$\chi^2(d)$$ distribution is

${\mathcal L}(d) = \sum_{i=1}^N \ln[f(y_i,d)]$

where $$f(y_i,d)$$ is the density function for the $$\chi^2(d)$$ distribution. See Trivedi, 2005 and Wooldridge, 2010 for instructions to ML.

The mlexp command estimates parameters by maximizing the specified log-likelihood function. You specify the contribution of an observation to the log-likelihood function inside parentheses, and you enclose parameters inside the curly braces $$\{$$ and $$\}$$. I use mlexp to estimate $$d$$ in example 2.

Example 2: Using mlexp to estimate $$d$$

. mlexp ( ln(chi2den({d},y)) )

initial:       log likelihood =     -  (could not be evaluated)
feasible:      log likelihood = -5168.1594
rescale:       log likelihood = -3417.1592
Iteration 0:   log likelihood = -3417.1592
Iteration 1:   log likelihood = -3416.7063
Iteration 2:   log likelihood = -3416.7063

Maximum likelihood estimation

Log likelihood = -3416.7063                     Number of obs     =      2,000

------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
/d |   2.033457   .0352936    57.62   0.000     1.964283    2.102631
------------------------------------------------------------------------------


The estimate of $$d$$ is very close to the true value of $$2.0$$, as expected.

Modeling the degree of freedom as a function of a covariate

When using ML in applied research, we almost always want to model the parameters of a distribution as a function of covariates. Below, I draw a covariate $$x$$ from Uniform(0,3) distribution, specify that $$d=1+x$$, and draw $$y$$ from a $$\chi^2(d)$$ distribution conditional on $$x$$. Having drawn data from the DGP, I estimate the parameters using mlexp.

Example 3: Using mlexp to estimate $$d=a+b x_i$$

. drop _all

. set obs 2000
number of observations (_N) was 0, now 2,000

. set seed 12345

. generate x = runiform(0,3)

. generate d = 1 + x

. generate y = rchi2(d)

. mlexp ( ln(chi2den({b}*x +{a},y)) )

initial:       log likelihood =     -  (could not be evaluated)
feasible:      log likelihood = -4260.0685
rescale:       log likelihood = -3597.6271
rescale eq:    log likelihood = -3597.6271
Iteration 0:   log likelihood = -3597.6271
Iteration 1:   log likelihood = -3596.5383
Iteration 2:   log likelihood =  -3596.538

Maximum likelihood estimation

Log likelihood =  -3596.538                     Number of obs     =      2,000

------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
/b |   1.061621   .0430846    24.64   0.000     .9771766    1.146065
/a |   .9524136   .0545551    17.46   0.000     .8454876     1.05934
------------------------------------------------------------------------------


The estimates of $$1.06$$ and $$0.95$$ are close to their true values.

mlexp makes this process easier by forming a linear combination of variables that you specify.

Example 4: A linear combination in mlexp

. mlexp ( ln(chi2den({xb: x _cons},y)) )

initial:       log likelihood =     -  (could not be evaluated)
feasible:      log likelihood = -5916.7648
rescale:       log likelihood = -3916.6106
Iteration 0:   log likelihood = -3916.6106
Iteration 1:   log likelihood = -3621.2905
Iteration 2:   log likelihood = -3596.5845
Iteration 3:   log likelihood =  -3596.538
Iteration 4:   log likelihood =  -3596.538

Maximum likelihood estimation

Log likelihood =  -3596.538                     Number of obs     =      2,000

------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
x |   1.061621   .0430846    24.64   0.000     .9771766    1.146065
_cons |   .9524138   .0545551    17.46   0.000     .8454878     1.05934
------------------------------------------------------------------------------


The estimates are the same as in example 3, but the command was easier to write and the output is easier to read.

Done and undone

I have shown how to generate data from a $$\chi^2(d)$$ distribution when $$d$$ is a fixed number or a linear function of a covariate and how to estimate $$d$$ or the parameters of the model for $$d$$ by using mlexp.

The examples discussed above show how to use mlexp and illustrate an example of conditional maximum likelihood estimation.

mlexp can do much more than I have discussed here; see [R] mlexp for more details. Estimating the parameters of a conditional distribution is only the beginning of any research project. I will discuss interpreting these parameters in a future post.

References

Cameron, A. C., and P. K. Trivedi. 2005. Microeconometrics: Methods and applications. Cambridge: Cambridge University Press.

Wooldridge, J. M. 2010. Econometric Analysis of Cross Section and Panel Data. 2nd ed. Cambridge, Massachusetts: MIT Press.

Categories: Statistics Tags:

Introduction to treatment effects in Stata: Part 2

This post was written jointly with David Drukker, Director of Econometrics, StataCorp.

In our last post, we introduced the concept of treatment effects and demonstrated four of the treatment-effects estimators that were introduced in Stata 13.  Today, we will talk about two more treatment-effects estimators that use matching.

Introduction

Last time, we introduced four estimators for estimating the average treatment effect (ATE) from observational data.  Each of these estimators has a different way of solving the missing-data problem that arises because we observe only the potential outcome for the treatment level received.  Today, we introduce estimators for the ATE that solve the missing-data problem by matching.

Matching pairs the observed outcome of a person in one treatment group with the outcome of the “closest” person in the other treatment group. The outcome of the closest person is used as a prediction for the missing potential outcome. The average difference between the observed outcome and the predicted outcome estimates the ATE.

What we mean by “closest” depends on our data. Matching subjects based on a single binary variable, such as sex, is simple: males are paired with males and females are paired with females. Matching on two categorical variables, such as sex and race, isn’t much more difficult. Matching on continuous variables, such as age or weight, can be trickier because of the sparsity of the data. It is unlikely that there are two 45-year-old white males who weigh 193 pounds in a sample. It is even less likely that one of those men self-selected into the treated group and the other self-selected into the untreated group. So, in such cases, we match subjects who have approximately the same weight and approximately the same age.

This example illustrates two points. First, there is a cost to matching on continuous covariates; the inability to find good matches with more than one continuous covariate causes large-sample bias in our estimator because our matches become increasingly poor.

Second, we must specify a measure of similarity. When matching directly on the covariates, distance measures are used and the nearest neighbor selected. An alternative is to match on an estimated probability of treatment, known as the propensity score.

Before we discuss estimators for observational data, we note that matching is sometimes used in experimental data to define pairs, with the treatment subsequently randomly assigned within each pair. This use of matching is related but distinct.

Nearest-neighbor matching

Nearest-neighbor matching (NNM) uses distance between covariate patterns to define “closest”. There are many ways to define the distance between two covariate patterns. We could use squared differences as a distance measure, but this measure ignores problems with scale and covariance. Weighting the differences by the inverse of the sample covariance matrix handles these issues. Other measures are also used, but these details are less important than the costs and benefits of NNM dropping the functional-form assumptions (linear, logit, probit, etc.) used in the estimators discussed last time.

Dropping the functional-form assumptions makes the NNM estimator much more flexible; it estimates the ATE for a much wider class of models. The cost of this flexibility is that the NNM estimator requires much more data and the amount of data it needs grows with each additional continuous covariate.

In the previous blog entry, we used an example of mother’s smoking status on birthweight. Let’s reconsider that example.

. webuse cattaneo2.dta, clear


Now, we use teffects nnmatch to estimate the ATE by NNM.

. teffects nnmatch (bweight mmarried mage fage medu prenatal1) (mbsmoke)

Treatment-effects estimation                    Number of obs      =      4642
Estimator      : nearest-neighbor matching      Matches: requested =         1
Outcome model  : matching                                      min =         1
Distance metric: Mahalanobis                                   max =        16
------------------------------------------------------------------------------
|              AI Robust
bweight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ATE          |
mbsmoke |
(smoker  |
vs  |
nonsmoker)  |  -210.5435   29.32969    -7.18   0.000    -268.0286   -153.0584
------------------------------------------------------------------------------


The estimated ATE is -211, meaning that infants would weigh 211 grams less when all mothers smoked than when no mothers smoked.

The output also indicates that ties in distance caused at least one observation to be matched with 16 other observations, even though we requested only matching. NNM averages the outcomes of all the tied-in-distance observations, as it should. (They are all equally good and using all of them will reduce bias.)

NNM on discrete covariates does not guarantee exact matching. For example, some married women could be matched with single women. We probably prefer exact matching on discrete covariates, which we do now.

. teffects nnmatch (bweight mmarried mage fage medu prenatal1) (mbsmoke), ///
ematch(mmarried prenatal1)

Treatment-effects estimation                    Number of obs      =      4642
Estimator      : nearest-neighbor matching      Matches: requested =         1
Outcome model  : matching                                      min =         1
Distance metric: Mahalanobis                                   max =        16
------------------------------------------------------------------------------
|              AI Robust
bweight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ATE          |
mbsmoke |
(smoker  |
vs  |
nonsmoker)  |  -209.5726   29.32603    -7.15   0.000    -267.0506   -152.0946
------------------------------------------------------------------------------


Exact matching on mmarried and prenatal1 changed the results a little bit.

Using more than one continuous covariate introduces large-sample bias, and we have three. The option biasadj() uses a linear model to remove the large-sample bias, as suggested by Abadie and Imbens (2006, 2011).

. teffects nnmatch (bweight mmarried mage fage medu prenatal1) (mbsmoke), ///

Treatment-effects estimation                    Number of obs      =      4642
Estimator      : nearest-neighbor matching      Matches: requested =         1
Outcome model  : matching                                      min =         1
Distance metric: Mahalanobis                                   max =        16
------------------------------------------------------------------------------
|              AI Robust
bweight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ATE          |
mbsmoke |
(smoker  |
vs  |
nonsmoker)  |  -210.0558   29.32803    -7.16   0.000    -267.5377   -152.5739
------------------------------------------------------------------------------


In this case, the results changed by a small amount. In general, they can change a lot, and the amount increases with the number of continuous
covariates.

Propensity-score matching

NNM uses bias adjustment to remove the bias caused by matching on more than one continuous covariate. The generality of this approach makes it very appealing, but it can be difficult to think about issues of fit and model specification. Propensity-score matching (PSM) matches on an estimated probability of treatment known as the propensity score. There is no need for bias adjustment because we match on only one continuous covariate. PSM has the added benefit that we can use all the standard methods for checking the fit of binary regression models prior to matching.

We estimate the ATE by PSM using teffects psmatch.

. teffects psmatch (bweight) (mbsmoke mmarried mage fage medu prenatal1 )

Treatment-effects estimation                    Number of obs      =      4642
Estimator      : propensity-score matching      Matches: requested =         1
Outcome model  : matching                                      min =         1
Treatment model: logit                                         max =        16
------------------------------------------------------------------------------
|              AI Robust
bweight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ATE          |
mbsmoke |
(smoker  |
vs  |
nonsmoker)  |  -229.4492   25.88746    -8.86   0.000    -280.1877   -178.7107
------------------------------------------------------------------------------


The estimated ATE is now -229, larger in magnitude than the NNM estimates but not significantly so.

How to choose among the six estimatorsg

We now have six estimators:

2. IPW: Inverse probability weighting
3. IPWRA: Inverse probability weighting with regression adjustment
4. AIPW: Augmented inverse probability weighting
5. NNM: Nearest-neighbor matching
6. PSM: Propensity-score matching

The ATEs we estimated are

1. RA: -277.06
2. IPW: -275.56
3. IPWRA: -229.97
4. AIPW: -230.99
5. NNM: -210.06
6. PSM: -229.45

Which estimator should we use?

We would never suggest searching the above table for the result that most closely fits your wishes and biases. The choice of estimator needs to be made beforehand.

So, how do we choose?

Here are some rules of thumb:

1. Under correct specification, all the estimators should produce similar results. (Similar estimates do not guarantee correct specification because all the specifications could be wrong.)
2. When you know the determinants of treatment status, IPW is a natural base-case estimator.
3. When you instead know the determinants of the outcome, RA is a natural base-case estimator.
4. The doubly robust estimators, AIPW and IPWRA, give us an extra shot at correct specification.
5. When you have lots of continuous covariates, NNM will crucially hinge on the bias adjustment, and the computation gets to be extremely difficult.
6. When you know the determinants of treatment status, PSM is another base-case estimator.
7. The IPW estimators are not reliable when the estimated treatment probabilities get too close to 0 or 1.

Final thoughts

Before we go, we reiterate the cautionary note from our last entry. Nothing about the mathematics of treatment-effects estimators magically extracts causal relationships from observational data. We cannot thoughtlessly analyze our data using Stata’s teffects commands and infer a causal relationship. The models must be supported by scientific theory.

If you would like to learn more about treatment effects in Stata, there is an entire manual devoted to the treatment-effects features in Stata 14; it includes a basic introduction, an advanced introduction, and many worked examples. In Stata, type help teffects:

.  help teffects


Title

[TE] teffects—Treatment-effects estimation for observational data

Syntax

… <output omitted> …

The title [TE] teffects will be in blue, which means it’s clickable. Click on it to go to the Treatment-Effects Reference Manual.

http://www.stata.com/manuals14/te/

References

Abadie, A., and Imbens, G. W. 2006. Large sample properties of matching estimators for average treatment effects. Econometrica 74: 235–267.

Abadie, A., and Imbens, G. W. 2011. Bias-corrected matching estimators for average treatment effects. Journal of Business and Economic Statistics 29: 1–11.

Cattaneo, M. D. 2010. Efficient semiparametric estimation of multi-valued treatment effects under ignorability. Journal of Econometrics 155: 138–154.

Categories: Statistics Tags:

Spotlight on irt

New to Stata 14 is a suite of commands to fit item response theory (IRT) models. IRT models are used to analyze the relationship between the latent trait of interest and the items intended to measure the trait. Stata’s irt commands provide easy access to some of the commonly used IRT models for binary and polytomous responses, and irtgraph commands can be used to plot item characteristic functions and information functions.

To learn more about Stata’s IRT features, I refer you to the [IRT] manual; here I want to go beyond the manual and show you a couple of examples of what you can do with a little bit of Stata code.

Example 1

To get started, I want to show you how simple IRT analysis is in Stata.

When I use the nine binary items q1q9, all I need to type to fit a 1PL model is

irt 1pl q*

Equivalently, I can use a dash notation or explicitly spell out the variable names:

irt 1pl q1-q9
irt 1pl q1 q2 q3 q4 q5 q6 q7 q8 q9

I can also use parenthetical notation:

irt (1pl q1-q9)

Parenthetical notation is not very useful for a simple IRT model, but comes in handy when you want to fit a single IRT model to combinations of binary, ordinal, and nominal items:

irt (1pl q1-q5) (1pl q6-q9) (pcm x1-x10) ...

IRT graphs are equally simple to create in Stata; for example, to plot item characteristic curves (ICCs) for all the items in a model, I type

irtgraph icc

Yes, that’s it!

Example 2

Sometimes, I want to fit the same IRT model on two different groups and see how the estimated parameters differ between the groups. The exercise can be part of investigating differential item functioning (DIF) or parameter invariance.

I split the data into two groups, fit two separate 2PL models, and create two scatterplots to see how close the parameter estimates for discrimination and difficulty are for the two groups. For simplicity, my group variable is 1 for odd-numbered observations and 0 for even-numbered observations.

We see that the estimated parameters for item q8 appear to differ between the two groups.

Here is the code used in this example.

webuse masc1, clear

gen odd = mod(_n,2)

irt 2pl q* if odd
mat b_odd = e(b)'

irt 2pl q* if !odd
mat b_even = e(b)'

svmat double b_odd, names(group1)
svmat double b_even, names(group2)
replace group11 = . in 19
replace group21 = . in 19

gen lab1 = ""
replace lab1 = "q8" in 15

gen lab2 = ""
replace lab2 = "q8" in 16

corr group11 group21 if mod(_n,2)
local c1 : display %4.2f r(rho)'

twoway (scatter group11 group21, mlabel(lab1) mlabsize(large) mlabpos(7)) ///
(function x, range(0 2)) if mod(_n,2), ///
name(discr,replace) title("Discrimination parameter; {&rho} = c1'") ///
xtitle("Group 1 observations") ytitle("Group 2 observations") ///
legend(off)

corr group11 group21 if !mod(_n,2)
local c2 : display %4.2f r(rho)'

twoway (scatter group11 group21, mlabel(lab2) mlabsize(large) mlabpos(7)) ///
(function x, range(-2 3)) if !mod(_n,2), ///
name(diff,replace) title("Difficulty parameter; {&rho} = c2'") ///
xtitle("Group 1 observations") ytitle("Group 2 observations") ///
legend(off)

graph combine discr diff, xsize(8)


Example 3

Continuing with the example above, I want to show you how to use a likelihood-ratio test to test for item parameter differences between groups.

Using item q8 as an example, I want to fit one model that constrains item q8 parameters to be the same between the two groups and fit another model that allows these parameters to vary.

The first model is easy. I can fit a 2PL model for the entire dataset, which implicitly constrains the parameters to be equal for both groups. I store the estimates under the name equal.

. webuse masc1, clear
(Data from De Boeck & Wilson (2004))

. generate odd = mod(_n,2)
. quietly irt 2pl q*
. estimates store equal


To estimate the second model, I need the following:

. irt (2pl q1-q7 q9) (2pl q8 if odd) (2pl q8 if !odd)


Unfortunately, this is illegal syntax. I can, however, split the item into two new variables where each variable is restricted to the required subsample:

. generate q8_1 = q8 if odd
(400 missing values generated)

. generate q8_2 = q8 if !odd
(400 missing values generated)


I estimate the second IRT model, this time with items q8_1 and q8_2 taking place of the original q8:

. quietly irt 2pl q1-q7 q8_1 q8_2 q9
. estat report q8_1 q8_2

Two-parameter logistic model                    Number of obs     =        800
Log likelihood = -4116.2064
------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
q8_1         |
Discrim |   1.095867   .2647727     4.14   0.000     .5769218    1.614812
Diff |  -1.886126   .3491548    -5.40   0.000    -2.570457   -1.201795
-------------+----------------------------------------------------------------
q8_2         |
Discrim |    1.93005   .4731355     4.08   0.000     1.002721    2.857378
Diff |  -1.544908   .2011934    -7.68   0.000     -1.93924   -1.150577
------------------------------------------------------------------------------


Now, I can perform the likelihood-ratio test:

. lrtest equal ., force

Likelihood-ratio test                                 LR chi2(2)  =      4.53
(Assumption: equal nested in .)                       Prob > chi2 =    0.1040


The test suggests the first model is preferable even though the two ICCs clearly differ:

. irtgraph icc q8_1 q8_2, ylabel(0(.25)1)


Summary

IRT models are used to analyze the relationship between the latent trait of interest and the items intended to measure the trait. Stata’s irt commands provide easy access to some of the commonly used IRT models, and irtgraph commands implement the most commonly used IRT plots. With just a few extra steps, you can easily create customized graphs, such as the ones demonstrated above, which incorporate information from separate IRT models.

Categories: Statistics Tags:

Introduction to treatment effects in Stata: Part 1

This post was written jointly with David Drukker, Director of Econometrics, StataCorp.

The topic for today is the treatment-effects features in Stata.

Treatment-effects estimators estimate the causal effect of a treatment on an outcome based on observational data.

In today’s posting, we will discuss four treatment-effects estimators:

2. IPW: Inverse probability weighting
3. IPWRA: Inverse probability weighting with regression adjustment
4. AIPW: Augmented inverse probability weighting

We’ll save the matching estimators for part 2.

We should note that nothing about treatment-effects estimators magically extracts causal relationships. As with any regression analysis of observational data, the causal interpretation must be based on a reasonable underlying scientific rationale.

Introduction

We are going to discuss treatments and outcomes.

A treatment could be a new drug and the outcome blood pressure or cholesterol levels. A treatment could be a surgical procedure and the outcome patient mobility. A treatment could be a job training program and the outcome employment or wages. A treatment could even be an ad campaign designed to increase the sales of a product.

Consider whether a mother’s smoking affects the weight of her baby at birth. Questions like this one can only be answered using observational data. Experiments would be unethical.

The problem with observational data is that the subjects choose whether to get the treatment. For example, a mother decides to smoke or not to smoke. The subjects are said to have self-selected into the treated and untreated groups.

In an ideal world, we would design an experiment to test cause-and-effect and treatment-and-outcome relationships. We would randomly assign subjects to the treated or untreated groups. Randomly assigning the treatment guarantees that the treatment is independent of the outcome, which greatly simplifies the analysis.

Causal inference requires the estimation of the unconditional means of the outcomes for each treatment level. We only observe the outcome of each subject conditional on the received treatment regardless of whether the data are observational or experimental. For experimental data, random assignment of the treatment guarantees that the treatment is independent of the outcome; so averages of the outcomes conditional on observed treatment estimate the unconditional means of interest. For observational data, we model the treatment assignment process. If our model is correct, the treatment assignment process is considered as good as random conditional on the covariates in our model.

Let’s consider an example. Figure 1 is a scatterplot of observational data similar to those used by Cattaneo (2010). The treatment variable is the mother’s smoking status during pregnancy, and the outcome is the birthweight of her baby.

The red points represent the mothers who smoked during pregnancy, while the green points represent the mothers who did not. The mothers themselves chose whether to smoke, and that complicates the analysis.

We cannot estimate the effect of smoking on birthweight by comparing the mean birthweights of babies of mothers who did and did not smoke. Why not? Look again at our graph. Older mothers tend to have heavier babies regardless of whether they smoked while pregnant. In these data, older mothers were also more likely to be smokers. Thus, mother’s age is related to both treatment status and outcome. So how should we proceed?

RA estimators model the outcome to account for the nonrandom treatment assignment.

We might ask, “How would the outcomes have changed had the mothers who smoked chosen not to smoke?” or “How would the outcomes have changed had the mothers who didn’t smoke chosen to smoke?”. If we knew the answers to these counterfactual questions, analysis would be easy: we would just subtract the observed outcomes from the counterfactual outcomes.

The counterfactual outcomes are called unobserved potential outcomes in the treatment-effects literature. Sometimes the word unobserved is dropped.

We can construct measurements of these unobserved potential outcomes, and our data might look like this:

In figure 2, the observed data are shown using solid points and the unobserved potential outcomes are shown using hollow points. The hollow red points represent the potential outcomes for the smokers had they not smoked. The hollow green points represent the potential outcomes for the nonsmokers had they smoked.

We can estimate the unobserved potential outcomes then by fitting separate linear regression models with the observed data (solid points) to the two treatment groups.

In figure 3, we have one regression line for nonsmokers (the green line) and a separate regression line for smokers (the red line).

Let’s understand what the two lines mean:

The green point on the left in figure 4, labeled Observed, is an observation for a mother who did not smoke. The point labeled E(y0) on the green regression line is the expected birthweight of the baby given the mother’s age and that she didn’t smoke. The point labeled E(y1) on the red regression line is the expected birthweight of the baby for the same mother had she smoked.

The difference between these expectations estimates the covariate-specific treatment effect for those who did not get the treatment.

Now, let’s look at the other counterfactual question.

The red point on the right in figure 4, labeled Observed in red, is an observation for a mother who smoked during pregnancy. The points on the green and red regression lines again represent the expected birthweights — the potential outcomes — of the mother’s baby under the two treatment conditions.

The difference between these expectations estimates the covariate-specific treatment effect for those who got the treatment.

Note that we estimate an average treatment effect (ATE), conditional on covariate values, for each subject. Furthermore, we estimate this effect for each subject, regardless of which treatment was actually received. Averages of these effects over all the subjects in the data estimate the ATE.

We could also use figure 4 to motivate a prediction of the outcome that each subject would obtain for each treatment level, regardless of the treatment recieved. The story is analogous to the one above. Averages of these predictions over all the subjects in the data estimate the potential-outcome means (POMs) for each treatment level.

It is reassuring that differences in the estimated POMs is the same estimate of the ATE discussed above.

The ATE on the treated (ATET) is like the ATE, but it uses only the subjects who were observed in the treatment group. This approach to calculating treatment effects is called regression adjustment (RA).

Let’s open a dataset and try this using Stata.

. webuse cattaneo2.dta, clear
(Excerpt from Cattaneo (2010) Journal of Econometrics 155: 138-154)

To estimate the POMs in the two treatment groups, we type

. teffects ra (bweight mage) (mbsmoke), pomeans


We specify the outcome model in the first set of parentheses with the outcome variable followed by its covariates. In this example, the outcome variable is bweight and the only covariate is mage.

We specify the treatment model — simply the treatment variable — in the second set of parentheses. In this example, we specify only the treatment variable mbsmoke. We’ll talk about covariates in the next section.

The result of typing the command is

. teffects ra (bweight mage) (mbsmoke), pomeans

Iteration 0:   EE criterion =  7.878e-24
Iteration 1:   EE criterion =  8.468e-26

Treatment-effects estimation                    Number of obs      =      4642
Outcome model  : linear
Treatment model: none
------------------------------------------------------------------------------
|               Robust
bweight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
POmeans      |
mbsmoke |
nonsmoker  |   3409.435   9.294101   366.84   0.000     3391.219    3427.651
smoker  |   3132.374   20.61936   151.91   0.000     3091.961    3172.787
------------------------------------------------------------------------------


The output reports that the average birthweight would be 3,132 grams if all mothers smoked and 3,409 grams if no mother smoked.

We can estimate the ATE of smoking on birthweight by subtracting the POMs: 3132.374 – 3409.435 = -277.061. Or we can reissue our teffects ra command with the ate option and get standard errors and confidence intervals:

. teffects ra (bweight mage) (mbsmoke), ate

Iteration 0:   EE criterion =  7.878e-24
Iteration 1:   EE criterion =  5.185e-26

Treatment-effects estimation                    Number of obs      =      4642
Outcome model  : linear
Treatment model: none
-------------------------------------------------------------------------------
|               Robust
bweight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
ATE           |
mbsmoke |
(smoker vs    |
nonsmoker)  |  -277.0611   22.62844   -12.24   0.000    -321.4121   -232.7102
--------------+----------------------------------------------------------------
POmean        |
mbsmoke |
nonsmoker  |   3409.435   9.294101   366.84   0.000     3391.219    3427.651
-------------------------------------------------------------------------------


The output reports the same ATE we calculated by hand: -277.061. The ATE is the average of the differences between the birthweights when each mother smokes and the birthweights when no mother smokes.

We can also estimate the ATET by using the teffects ra command with option atet, but we will not do so here.

IPW: The inverse probability weighting estimator

RA estimators model the outcome to account for the nonrandom treatment assignment. Some researchers prefer to model the treatment assignment process and not specify a model for the outcome.

We know that smokers tend to be older than nonsmokers in our data. We also hypothesize that mother’s age directly affects birthweight. We observed this in figure 1, which we show again below.

This figure shows that treatment assignment depends on mother’s age. We would like to have a method of adjusting for this dependence. In particular, we wish we had more upper-age green points and lower-age red points. If we did, the mean birthweight for each group would change. We don’t know how that would affect the difference in means, but we do know it would be a better estimate of the difference.

To achieve a similar result, we are going to weight smokers in the lower-age range and nonsmokers in the upper-age range more heavily, and weight smokers in the upper-age range and nonsmokers in the lower-age range less heavily.

We will fit a probit or logit model of the form

Pr(woman smokes) = F(a + b*age)

teffects uses logit by default, but we will specify the probit option for illustration.

Once we have fit that model, we can obtain the prediction Pr(woman smokes) for each observation in the data; we’ll call this pi. Then, in making our POMs calculations — which is just a mean calculation — we will use those probabilities to weight the observations. We will weight observations on smokers by 1/pi so that weights will be large when the probability of being a smoker is small. We will weight observations on nonsmokers by 1/(1-pi) so that weights will be large when the probability of being a nonsmoker is small.

That results in the following graph replacing figure 1:

In figure 5, larger circles indicate larger weights.

To estimate the POMs with this IPW estimator, we can type

. teffects ipw (bweight) (mbsmoke mage, probit), pomeans


The first set of parentheses specifies the outcome model, which is simply the outcome variable in this case; there are no covariates. The second set of parentheses specifies the treatment model, which includes the outcome variable (mbsmoke) followed by covariates (in this case, just mage) and the kind of model (probit).

The result is

. teffects ipw (bweight) (mbsmoke mage, probit), pomeans

Iteration 0:   EE criterion =  3.615e-15
Iteration 1:   EE criterion =  4.381e-25

Treatment-effects estimation                    Number of obs      =      4642
Estimator      : inverse-probability weights
Outcome model  : weighted mean
Treatment model: probit
------------------------------------------------------------------------------
|               Robust
bweight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
POmeans      |
mbsmoke |
nonsmoker  |   3408.979   9.307838   366.25   0.000     3390.736    3427.222
smoker  |   3133.479   20.66762   151.61   0.000     3092.971    3173.986
------------------------------------------------------------------------------


Our output reports that the average birthweight would be 3,133 grams if all the mothers smoked and 3,409 grams if none of the mothers smoked.

This time, the ATE is -275.5, and if we typed

. teffects ipw (bweight) (mbsmoke mage, probit), ate
(Output omitted)


we would learn that the standard error is 22.68 and the 95% confidence interval is [-319.9,231.0].

Just as with teffects ra, if we wanted ATET, we could specify the teffects ipw command with the atet option.

IPWRA: The IPW with regression adjustment estimator

RA estimators model the outcome to account for the nonrandom treatment assignment. IPW estimators model the treatment to account for the nonrandom treatment assignment. IPWRA estimators model both the outcome and the treatment to account for the nonrandom treatment assignment.

IPWRA uses IPW weights to estimate corrected regression coefficients that are subsequently used to perform regression adjustment.

The covariates in the outcome model and the treatment model do not have to be the same, and they often are not because the variables that influence a subject’s selection of treatment group are often different from the variables associated with the outcome. The IPWRA estimator has the double-robust property, which means that the estimates of the effects will be consistent if either the treatment model or the outcome model — but not both — are misspecified.

Let’s consider a situation with more complex outcome and treatment models but still using our low-birthweight data.

The outcome model will include

1. mage: the mother’s age
2. prenatal1: an indicator for prenatal visit during the first trimester
3. mmarried: an indicator for marital status of the mother
4. fbaby: an indicator for being first born

The treatment model will include

1. all the covariates of the outcome model
2. mage^2
3. medu: years of maternal education

We will also specify the aequations option to report the coefficients of the outcome and treatment models.

. teffects ipwra (bweight mage prenatal1 mmarried fbaby)                ///
(mbsmoke mmarried c.mage##c.mage fbaby medu, probit)   ///
, pomeans aequations

Iteration 0:   EE criterion =  1.001e-20
Iteration 1:   EE criterion =  1.134e-25

Treatment-effects estimation                    Number of obs      =      4642
Outcome model  : linear
Treatment model: probit
-------------------------------------------------------------------------------
|               Robust
bweight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
POmeans       |
mbsmoke |
nonsmoker  |   3403.336    9.57126   355.58   0.000     3384.576    3422.095
smoker  |   3173.369   24.86997   127.60   0.000     3124.624    3222.113
--------------+----------------------------------------------------------------
OME0          |
mage |   2.893051   2.134788     1.36   0.175    -1.291056    7.077158
prenatal1 |   67.98549   28.78428     2.36   0.018     11.56933    124.4017
mmarried |   155.5893   26.46903     5.88   0.000      103.711    207.4677
fbaby |   -71.9215   20.39317    -3.53   0.000    -111.8914   -31.95162
_cons |   3194.808   55.04911    58.04   0.000     3086.913    3302.702
--------------+----------------------------------------------------------------
OME1          |
mage |  -5.068833   5.954425    -0.85   0.395    -16.73929    6.601626
prenatal1 |   34.76923   43.18534     0.81   0.421    -49.87248    119.4109
mmarried |   124.0941   40.29775     3.08   0.002     45.11193    203.0762
fbaby |   39.89692   56.82072     0.70   0.483    -71.46966    151.2635
_cons |   3175.551   153.8312    20.64   0.000     2874.047    3477.054
--------------+----------------------------------------------------------------
TME1          |
mmarried |  -.6484821   .0554173   -11.70   0.000     -.757098   -.5398663
mage |   .1744327   .0363718     4.80   0.000     .1031452    .2457202
|
c.mage#c.mage |  -.0032559   .0006678    -4.88   0.000    -.0045647   -.0019471
|
fbaby |  -.2175962   .0495604    -4.39   0.000    -.3147328   -.1204595
medu |  -.0863631   .0100148    -8.62   0.000    -.1059917   -.0667345
_cons |  -1.558255   .4639691    -3.36   0.001    -2.467618   -.6488926
-------------------------------------------------------------------------------


The POmeans section of the output displays the POMs for the two treatment groups. The ATE is now calculated to be 3173.369 – 3403.336 = -229.967.

The OME0 and OME1 sections display the RA coefficients for the untreated and treated groups, respectively.

The TME1 section of the output displays the coefficients for the probit treatment model.

Just as in the two previous cases, if we wanted the ATE with standard errors, etc., we would specify the ate option. If we wanted ATET, we would specify the atet option.

AIPW: The augmented IPW estimator

IPWRA estimators model both the outcome and the treatment to account for the nonrandom treatment assignment. So do AIPW estimators.

The AIPW estimator adds a bias-correction term to the IPW estimator. If the treatment model is correctly specified, the bias-correction term is 0 and the model is reduced to the IPW estimator. If the treatment model is misspecified but the outcome model is correctly specified, the bias-correction term corrects the estimator. Thus, the bias-correction term gives the AIPW estimator the same double-robust property as the IPWRA estimator.

The syntax and output for the AIPW estimator is almost identical to that for the IPWRA estimator.

. teffects aipw (bweight mage prenatal1 mmarried fbaby)                 ///
(mbsmoke mmarried c.mage##c.mage fbaby medu, probit)    ///
, pomeans aequations

Iteration 0:   EE criterion =  4.632e-21
Iteration 1:   EE criterion =  5.810e-26

Treatment-effects estimation                    Number of obs      =      4642
Estimator      : augmented IPW
Outcome model  : linear by ML
Treatment model: probit
-------------------------------------------------------------------------------
|               Robust
bweight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
POmeans       |
mbsmoke |
nonsmoker  |   3403.355   9.568472   355.68   0.000     3384.601    3422.109
smoker  |   3172.366   24.42456   129.88   0.000     3124.495    3220.237
--------------+----------------------------------------------------------------
OME0          |
mage |   2.546828   2.084324     1.22   0.222    -1.538373    6.632028
prenatal1 |   64.40859   27.52699     2.34   0.019     10.45669    118.3605
mmarried |   160.9513    26.6162     6.05   0.000     108.7845    213.1181
fbaby |   -71.3286   19.64701    -3.63   0.000     -109.836   -32.82117
_cons |   3202.746   54.01082    59.30   0.000     3096.886    3308.605
--------------+----------------------------------------------------------------
OME1          |
mage |  -7.370881    4.21817    -1.75   0.081    -15.63834    .8965804
prenatal1 |   25.11133   40.37541     0.62   0.534    -54.02302    104.2457
mmarried |   133.6617   40.86443     3.27   0.001      53.5689    213.7545
fbaby |   41.43991   39.70712     1.04   0.297    -36.38461    119.2644
_cons |   3227.169   104.4059    30.91   0.000     3022.537    3431.801
--------------+----------------------------------------------------------------
TME1          |
mmarried |  -.6484821   .0554173   -11.70   0.000     -.757098   -.5398663
mage |   .1744327   .0363718     4.80   0.000     .1031452    .2457202
|
c.mage#c.mage |  -.0032559   .0006678    -4.88   0.000    -.0045647   -.0019471
|
fbaby |  -.2175962   .0495604    -4.39   0.000    -.3147328   -.1204595
medu |  -.0863631   .0100148    -8.62   0.000    -.1059917   -.0667345
_cons |  -1.558255   .4639691    -3.36   0.001    -2.467618   -.6488926
-------------------------------------------------------------------------------


The ATE is 3172.366 – 3403.355 = -230.989.

Final thoughts

The example above used a continuous outcome: birthweight. teffects can also be used with binary, count, and nonnegative continuous outcomes.

The estimators also allow multiple treatment categories.

An entire manual is devoted to the treatment-effects features in Stata 13, and it includes a basic introduction, advanced discussion, and worked examples. If you would like to learn more, you can download the [TE] Treatment-effects Reference Manual from the Stata website.

More to come

Next time, in part 2, we will cover the matching estimators.

Reference

Cattaneo, M. D. 2010. Efficient semiparametric estimation of multi-valued treatment effects under ignorability. Journal of Econometrics 155: 138–154.

Bayesian modeling: Beyond Stata’s built-in models

This post was written jointly with Nikolay Balov, Senior Statistician and Software Developer, StataCorp.

A question on Statalist motivated us to write this blog entry.

A user asked if the churdle command (http://www.stata.com/stata14/hurdle-models/) for fitting hurdle models, new in Stata 14, can be combined with the bayesmh command (http://www.stata.com/stata14/bayesian-analysis/) for fitting Bayesian models, also new in Stata 14:

http://www.statalist.org/forums/forum/general-stata-discussion/general/1290426-comibining-bayesmh-and-churdle

Our initial reaction to this question was ‘No’ or, more precisely, ‘Not easily’ — hurdle models are not among the likelihood models supported by bayesmh. One can write a program to compute the log likelihood of the double hurdle model and use this program with bayesmh (in the spirit of http://www.stata.com/stata14/bayesian-evaluators/), but this may seem like a daunting task if you are not familiar with Stata programming.

And then we realized, why not simply call churdle from the evaluator to compute the log likelihood? All we need is for churdle to evaluate the log likelihood at specific values of model parameters without performing iterations. This can be achieved by specifying churdle‘s options from() and iterate(0).

Let’s look at an example. We consider a simple hurdle model using a subset of the fitness dataset from [R] churdle:

. webuse fitness
. set seed 17653
. sample 10
. churdle linear hours age, select(commute) ll(0)

Iteration 0:   log likelihood = -2783.3352
Iteration 1:   log likelihood =  -2759.029
Iteration 2:   log likelihood = -2758.9992
Iteration 3:   log likelihood = -2758.9992

Cragg hurdle regression                         Number of obs     =      1,983
LR chi2(1)        =       3.42
Prob > chi2       =     0.0645
Log likelihood = -2758.9992                     Pseudo R2         =     0.0006

------------------------------------------------------------------------------
hours |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
hours        |
age |   .0051263   .0028423     1.80   0.071    -.0004446    .0106971
_cons |   1.170932   .1238682     9.45   0.000     .9281548    1.413709
-------------+----------------------------------------------------------------
selection_ll |
commute |  -.0655171   .1561046    -0.42   0.675    -.3714765    .2404423
_cons |   .1421166   .0882658     1.61   0.107    -.0308813    .3151144
-------------+----------------------------------------------------------------
lnsigma      |
_cons |   .1280215     .03453     3.71   0.000      .060344     .195699
-------------+----------------------------------------------------------------
/sigma |   1.136577    .039246                      1.062202    1.216161
------------------------------------------------------------------------------


Let’s assume for a moment that we already have an evaluator, mychurdle1, that returns the corresponding log-likelihood value. We can fit a Bayesian hurdle model using bayesmh as follows:

. gen byte hours0 = (hours==0) //dependent variable for the selection equation
. set seed 123
. bayesmh (hours age) (hours0 commute),
llevaluator(mychurdle1, parameters({lnsig}))
prior({hours:} {hours0:} {lnsig}, flat)
saving(sim, replace) dots

(output omitted)


We use a two-equation specification to fit this model. The main regression is specified first, and the selection regression is specified next. The additional parameter, log of the standard deviation associated with the main regression, is specified in llevaluator()‘s suboption parameters(). All parameters are assigned flat priors to obtain results similar to churdle. MCMC results are saved in sim.dta.

Here is the actual output from bayesmh:

. bayesmh (hours age) (hours0 commute), llevaluator(mychurdle1, parameters({lns
> ig})) prior({hours:} {hours0:} {lnsig}, flat) saving(sim, replace) dots

Burn-in 2500 aaaaaaaaa1000aaaaaa...2000..... done
Simulation 10000 .........1000.........2000.........3000.........4000.........5
> 000.........6000.........7000.........8000.........9000.........10000 done

Model summary
------------------------------------------------------------------------------
Likelihood:
hours hours0 ~ mychurdle1(xb_hours,xb_hours0,{lnsig})

Priors:
{hours:age _cons} ~ 1 (flat)                                        (1)
{hours0:commute _cons} ~ 1 (flat)                                        (2)
{lnsig} ~ 1 (flat)
------------------------------------------------------------------------------
(1) Parameters are elements of the linear form xb_hours.
(2) Parameters are elements of the linear form xb_hours0.

Bayesian regression                              MCMC iterations  =     12,500
Random-walk Metropolis-Hastings sampling         Burn-in          =      2,500
MCMC sample size =     10,000
Number of obs    =      1,983
Acceptance rate  =      .2889
Efficiency:  min =     .05538
avg =     .06266
Log marginal likelihood = -2772.3953                          max =     .06945

------------------------------------------------------------------------------
|                                                Equal-tailed
|      Mean   Std. Dev.     MCSE     Median  [95% Cred. Interval]
-------------+----------------------------------------------------------------
hours        |
age |  .0050916   .0027972   .000106   .0049923  -.0000372   .0107231
_cons |  1.167265    .124755   .004889   1.175293   .9125105   1.392021
-------------+----------------------------------------------------------------
hours0       |
commute | -.0621282   .1549908   .006585  -.0613511  -.3623891   .2379805
_cons |  .1425693   .0863626   .003313   .1430396  -.0254507   .3097677
-------------+----------------------------------------------------------------
lnsig |  .1321532   .0346446   .001472   .1326704   .0663646   .2015249
------------------------------------------------------------------------------

file sim.dta saved


The results are similar to those produced by churdle, as one would expect with noninformative priors.

If desired, we can use bayesstats summary to obtain the estimate of the standard deviation:

. bayesstats summary (sigma: exp({lnsig}))

Posterior summary statistics                      MCMC sample size =    10,000

sigma : exp({lnsig})

------------------------------------------------------------------------------
|                                                Equal-tailed
|      Mean   Std. Dev.     MCSE     Median  [95% Cred. Interval]
-------------+----------------------------------------------------------------
sigma |  1.141969   .0396264   .001685   1.141874   1.068616   1.223267
------------------------------------------------------------------------------


Let’s now talk in more detail about a log-likelihood evaluator. We will consider two evaluators: one using churdle and one directly implementing the log likelihood of the considered hurdle model.

Log-likelihood evaluator using churdle

Here we demonstrate how to write a log-likelihood evaluator that calls an existing Stata estimation command, churdle in our example, to compute the log likelihood.

program mychurdle1
version 14.0
args llf
tempname b
mat b' = ($MH_b,$MH_p)
capture churdle linear $MH_y1$MH_y1x1 if $MH_touse, /// select($MH_y2x1) ll(0) from(b') iterate(0)
if _rc {
if (_rc==1) { // handle break key
exit _rc
}
scalar llf' = .
}
else {
scalar llf' = e(ll)
}
end


The mychurdle1 program returns the log-likelihood value computed by churdle at the current values of model parameters. This program accepts one argument — a temporary scalar to contain the log-likelihood value llf. We stored current values of model parameters (regression coefficients from two equations stored in vector MH_b and the extra parameter, log standard-deviation, stored in vector MH_p) in a temporary matrix b. We specified churdle‘s options from() and iterate(0) to evaluate the log likelihood at the current parameter values. Finally, we stored the resulting log-likelihood value in llf (or missing value if the command failed to evaluate the log likelihood).

Log-likelihood evaluator directly computing log likelihood

Here we demonstrate how to write a log-likelihood evaluator that computes the likelihood of the fitted hurdle model directly rather than calling churdle.

program mychurdle2
version 14.0
args lnf xb xg lnsig
tempname sig
scalar sig' = exp(lnsig')
tempvar lnfj
qui gen double lnfj' = normal(xg')  if $MH_touse qui replace lnfj' = log(1 - lnfj') if$MH_y1 <= 0 & $MH_touse qui replace lnfj' = log(lnfj') - log(normal(xb'/sig')) /// + log(normalden($MH_y1,xb',sig'))       ///
if $MH_y1 > 0 &$MH_touse
summarize lnfj' if $MH_touse, meanonly if r(N) <$MH_n {
scalar lnf' = .
exit
}
scalar lnf' = r(sum)
end


The mychurdle2 program accepts four arguments: a temporary scalar to contain the log-likelihood value llf, temporary variables xb and xg containing linear predictors from the corresponding main and selection equations evaluated at the current values of model parameters, and temporary scalar lnsig containing the current value of the log standard-deviation parameter. We compute and store the observation-level log likelihood in a temporary variable lnfj. Global MH_y1 contains the name of the dependent variable from the first (main) equation, and global MH_touse marks the estimation sample. If all observation-specific log likelihood contributions are nonmissing, we store the overall log-likelihood value in lnf or we otherwise store missing.

We fit our model using the same syntax as earlier, except we use mychurdle2 as the program evaluator.

. set seed 123
. bayesmh (hours age) (hours0 commute), llevaluator(mychurdle2, parameters({lns
> ig})) prior({hours:} {hours0:} {lnsig}, flat) saving(sim, replace) dots

Burn-in 2500 aaaaaaaaa1000aaaaaa...2000..... done
Simulation 10000 .........1000.........2000.........3000.........4000.........5
> 000.........6000.........7000.........8000.........9000.........10000 done

Model summary
------------------------------------------------------------------------------
Likelihood:
hours hours0 ~ mychurdle2(xb_hours,xb_hours0,{lnsig})

Priors:
{hours:age _cons} ~ 1 (flat)                                        (1)
{hours0:commute _cons} ~ 1 (flat)                                        (2)
{lnsig} ~ 1 (flat)
------------------------------------------------------------------------------
(1) Parameters are elements of the linear form xb_hours.
(2) Parameters are elements of the linear form xb_hours0.

Bayesian regression                              MCMC iterations  =     12,500
Random-walk Metropolis-Hastings sampling         Burn-in          =      2,500
MCMC sample size =     10,000
Number of obs    =      1,983
Acceptance rate  =      .2889
Efficiency:  min =     .05538
avg =     .06266
Log marginal likelihood = -2772.3953                          max =     .06945

------------------------------------------------------------------------------
|                                                Equal-tailed
|      Mean   Std. Dev.     MCSE     Median  [95% Cred. Interval]
-------------+----------------------------------------------------------------
hours        |
age |  .0050916   .0027972   .000106   .0049923  -.0000372   .0107231
_cons |  1.167265    .124755   .004889   1.175293   .9125105   1.392021
-------------+----------------------------------------------------------------
hours0       |
commute | -.0621282   .1549908   .006585  -.0613511  -.3623891   .2379805
_cons |  .1425693   .0863626   .003313   .1430396  -.0254507   .3097677
-------------+----------------------------------------------------------------
lnsig |  .1321532   .0346446   .001472   .1326704   .0663646   .2015249
------------------------------------------------------------------------------



We obtain the same results as those obtained using approach 1, and we obtain them much faster.

Final remarks

Approach 1 is very straightforward. It can be applied to any Stata command that returns the log likelihood and allows you to specify parameter values at which this log likelihood must be evaluated. Without too much programming effort, you can use almost any existing Stata maximum likelihood estimation command with bayesmh. A disadvantage of approach 1 is slower execution compared with programming the likelihood directly, as in approach 2. For example, the command using the mychurdle1 evaluator from approach 1 took about 25 minutes to run, whereas the command using the mychurdle2 evaluator from approach 2 took only 20 seconds.

Categories: Statistics Tags:

Stata 14 announced, ships

We’ve just announced the release of Stata 14. Stata 14 ships and downloads starting now.

I just posted on Statalist about it. Here’s a copy of what I wrote.

Stata 14 is now available. You heard it here first.

There’s a long tradition that Statalisters hear about Stata’s new releases first. The new forum is celebrating its first birthday, but it is a continuation of the old Statalist, so the tradition continues, but updated for the modern world, where everything happens more quickly. You are hearing about Stata 14 roughly a microsecond before the rest of the world. Traditions are important.

Here’s yet another example of everything happening faster in the modern world. Rather than the announcement preceding shipping by a few weeks as in previous releases, Stata 14 ships and downloads starting now. Or rather, a microsecond from now.

Some things from the past are worth preserving, however, and one is that I get to write about the new release in my own idiosyncratic way. So let me get the marketing stuff out of the way and then I can tell you about a few things that especially interest me and might interest you.

MARKETING BEGINS.

Here’s a partial list of what’s new, a.k.a. the highlights:

• Unicode
• More than 2 billion observations (Stata/MP)
• Bayesian analysis
• IRT (Item Response Theory)
• Panel-data survival models
• Treatment effects
• Treatment effects for survival models
• Endogenous treatments
• Probability weights
• Balance analysis
• Multilevel mixed-effects survival models
• Small-sample inference for multilevel models
• SEM (structural equation modeling)
• Survival models
• Satorra-Bentler scaled chi-squared test
• Survey data
• Multilevel weights
• Power and sample size
• Survival models
• Contingency (epidemiological) tables
• Markov-switching regression models
• Tests for structural breaks in time-series
• Fractional outcome regression models
• Hurdle models
• Censored Poisson regression
• Survey support & multilevel weights for multilevel models
• New random-number generators
• Estimated marginal means and marginal effects
• Tables for multiple outcomes and levels
• Integration over unobserved and latent variables
• ICD-10
• Stata in Spanish and in Japanese

The above list is not complete; it lists about 30% of what’s new.

For all the details about Stata 14, including purchase and update information, and links to distributors outside of the US, visit stata.com/stata14.

If you are outside of the US, you can order from your authorized Stata distributor. They will supply codes so that you can access and download from stata.com.

MARKETING ENDS.

I want to write about three of the new features ‒ Unicode, more than 2-billion observations, and Bayesian analysis.

Unicode is the modern way that computers encode characters such as the letters in what you are now reading. Unicode encodes all the world’s characters, meaning I can write Hello, Здравствуйте, こんにちは, and lots more besides. Well, the forum software is modern and I always could write those words here. Now I can write them in Stata, too.

For those who care, Stata uses Unicode’s UTF-8 encoding.

Anyway, you can use Unicode characters in your data, of course; in your variable labels, of course; and in your value labels, of course. What you might not expect is that you can use Unicode in your variable names, macro names, and everywhere else Stata wants a name or identifier.

Here’s the auto data in Japanese:

Your use of Unicode may not be as extreme as the above. It might be enough just to make tables and graphs labeled in languages other than English. If so, just set the variable labels and value labels. It doesn’t matter whether the variables are named übersetzung and kofferraum or gear_ratio and trunkspace or 変速比 and トランク.

I want to remind English speakers that Unicode includes mathematical symbols. You can use them in titles, axis labels, and the like.

Few good things come without cost. If you have been using Extended ASCII to circumvent Stata’s plain ASCII limitations, those files need to be translated to Unicode if the strings in them are to display correctly in Stata 14. This includes .dta files, do-files, ado-files, help files, and the like. It’s easier to do than you might expect. A new unicode analyze command will tell you whether you have files that need fixing and, if so, the new unicode translate command will fix them for you. It’s almost as easy as typing

. unicode translate *

This command translates your files and that has got to concern you. What if it mistranslates them? What if the power fails? Relax. unicode translate makes backups of the originals, and it keeps the backups until you delete them, which you have to do by typing

Yes, the option really is named badidea and it is not optional. Another unicode command can restore the backups.

The difficult part of translating your existing files is not performing the translation, it’s determining which Extended ASCII encoding your files used so that the translation can be performed. We have advice on that in the help files but, even so, some of you will only be able to narrow down the encoding to a few choices. The good news is that it is easy to try each one. You just type

. unicode retranslate *

It won’t take long to figure out which encoding works best.

Stata/MP now allows you to process datasets containing more than 2.1-billion observations. This sounds exciting, but I suspect it will interest only a few of you. How many of us have datasets with more than 2.1-billion observations? And even if you do, you will need a computer with lots of memory. This feature is useful if you have access to a 512-gigabyte, 1-terabyte, or 1.5-terabyte computer. With smaller computers, you are unlikely to have room for 2.1 billion observations. It’s exciting that such computers are available.

We increased the limit on only Stata/MP because, to exploit the higher limit, you need multiple processors. It’s easy to misjudge how much larger a 2-billion observation dataset is than a 2-million observation one. On my everyday 16 gigabyte computer ‒ which is nothing special ‒ I just fit a linear regression with six RHS variables on 2-million observations. It ran in 1.2 seconds. I used Stata/SE, and the 1.2 seconds felt fast. So, if my computer had more memory, how long would it take to fit a model on 2-billion observations? 1,200 seconds, which is to say, 20 minutes! You need Stata/MP. Stata/MP4 will reduce that to 5 minutes. Stata/MP32 will reduce that to 37.5 seconds.

By the way, if you intend to use more than 2-billion observations, be sure to click on help obs_advice that appears in the start-up notes after Stata launches. You will get better performance if you set min_memory and segmentsize to larger values. We tell you what values to set.

There’s quite a good discussion about dealing with more than 2-billion observations at stata.com/stata14/huge-datasets.

After that, it’s statistics, statistics, statistics.

Which new statistics will interest you obviously depends on your field. We’ve gone deeper into a number of fields. Treatment effects for survival models is just one example. Multilevel survival models is another. Markov-switching models is yet another. Well, you can read the list above.

Two of the new statistical features are worth mentioning, however, because they simply weren’t there previously. They are Bayesian analysis and IRT models, which are admittedly two very different things.

IRT is a highlight of the release and for some of it you will be the highlight, so I mention it, and I’ll just tell you to see stata.com/stata14/irt for more information.

Bayesian analysis is the other highlight as far as I’m concerned, and it will interest a lot of you because it cuts across fields. Many of you are already knowledgeable about this and I can just hear you asking, “Does Stata include …?” So here’s the high-speed summary:

Stata fits continuous-, binary-, ordinal-, and count-outcome models. And linear and nonlinear models. And generalized nonlinear models. Univariate, multivariate, and multiple-equation. It provides 10 likelihood models and 18 prior distributions. It also allows for user-defined likelihoods combined with built-in priors, built-in likelihoods combined with user-defined priors, and a roll-your-own programming approach to calculate the posterior density directly. MCMC methods are provided, including Adaptive Metropolis-Hastings (MH), Adaptive MH with Gibbs updates, and full Gibbs sampling for certain likelihoods and priors.

It’s also easy to use and that’s saying something.

There’s a great example of the new Bayes features in The Stata News. I mention this because including the example there is nearly a proof of ease of use. The example looks at the number of disasters in the British coal mining industry. There was a fairly abrupt decrease in the rate sometime between 1887 and 1895, which you see if you eyeballed a graph. In the example, we model the number of disasters before the change point as one Poisson process; the number after, as another Poisson process; and then we fit a model of the two Poisson parameters and the date of change. For the change point it uses a uniform prior on [1851, 1962] ‒ the range of the data ‒ and obtains a posterior mean estimate of 1890.4 and a 95% credible interval of [1886, 1896], which agrees with our visual assessment.

I hope something I’ve written above interests you. Visit stata.com/stata14 for more information.

‒ Bill
wgould@stata.com

Using gmm to solve two-step estimation problems

Two-step estimation problems can be solved using the gmm command.

When a two-step estimator produces consistent point estimates but inconsistent standard errors, it is known as the two-step-estimation problem. For instance, inverse-probability weighted (IPW) estimators are a weighted average in which the weights are estimated in the first step. Two-step estimators use first-step estimates to estimate the parameters of interest in a second step. The two-step-estimation problem arises because the second step ignores the estimation error in the first step.

One solution is to convert the two-step estimator into a one-step estimator. My favorite way to do this conversion is to stack the equations solved by each of the two estimators and solve them jointly. This one-step approach produces consistent point estimates and consistent standard errors. There is no two-step problem because all the computations are performed jointly. Newey (1984) derives and justifies this approach.

I’m going to illustrate this approach with the IPW example, but it can be used with any two-step problem as long as each step is continuous.

IPW estimators are frequently used to estimate the mean that would be observed if everyone in a population received a specified treatment, a quantity known as a potential-outcome mean (POM). A difference of POMs is called the average treatment effect (ATE). Aside from all that, it is the mechanics of the two-step IPW estimator that interest me here. IPW estimators are weighted averages of the outcome, and the weights are estimated in a first step. The weights used in the second step are the inverse of the estimated probability of treatment.

Let’s imagine we are analyzing an extract of the birthweight data used by Cattaneo (2010). In this dataset, bweight is the baby’s weight at birth, mbsmoke is 1 if the mother smoked while pregnant (and 0 otherwise), mmarried is 1 if the mother is married, and prenatal1 is 1 if the mother had a prenatal visit in the first trimester.

Let’s imagine we want to estimate the mean when all pregnant women smoked, which is to say, the POM for smoking. If we were doing substantive research, we would also estimate the POM when no pregnant women smoked. The difference between these estimated POMs would then estimate the ATE of smoking.

In the IPW estimator, we begin by estimating the probability weights for smoking. We fit a probit model of mbsmoke as a function of mmarried and prenatal1.

. use cattaneo2
(Excerpt from Cattaneo (2010) Journal of Econometrics 155: 138-154)

. probit mbsmoke mmarried prenatal1, vce(robust)

Iteration 0:   log pseudolikelihood = -2230.7484
Iteration 1:   log pseudolikelihood = -2102.6994
Iteration 2:   log pseudolikelihood = -2102.1437
Iteration 3:   log pseudolikelihood = -2102.1436

Probit regression                                 Number of obs   =       4642
Wald chi2(2)    =     259.42
Prob > chi2     =     0.0000
Log pseudolikelihood = -2102.1436                 Pseudo R2       =     0.0577

------------------------------------------------------------------------------
|               Robust
mbsmoke |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
mmarried |  -.6365472   .0478037   -13.32   0.000    -.7302407   -.5428537
prenatal1 |  -.2144569   .0547583    -3.92   0.000    -.3217811   -.1071327
_cons |  -.3226297   .0471906    -6.84   0.000    -.4151215   -.2301379
------------------------------------------------------------------------------


The results indicate that both mmarried and prenatal1 significantly predict whether the mother smoked while pregnant.

We want to calculate the inverse probabilities. We begin by getting the probabilities:

. predict double pr, pr


Now, we can obtain the inverse probabilities by typing

. generate double ipw = (mbsmoke==1)/pr


We can now perform the second step: calculate the mean for smokers by using the IPWs.

. mean bweight [pw=ipw]

Mean estimation                     Number of obs    =     864

--------------------------------------------------------------
|       Mean   Std. Err.     [95% Conf. Interval]
-------------+------------------------------------------------
bweight |   3162.868   21.71397      3120.249    3205.486
--------------------------------------------------------------
. mean bweight [pw=ipw] if mbsmoke


The point estimate reported by mean is consistent; the reported standard error Read more…

Categories: Statistics Tags:

Using gsem to combine estimation results

gsem is a very flexible command that allows us to fit very sophisticated models. However, it is also useful in situations that involve simple models.

For example, when we want to compare parameters among two or more models, we usually use suest, which combines the estimation results under one parameter vector and creates a simultaneous covariance matrix of the robust type. This covariance estimate is described in the Methods and formulas of [R] suest as the robust variance from a “stacked model”. Actually, gsem can estimate these kinds of “stacked models”, even if the estimation samples are not the same and eventually overlap. By using the option vce(robust), we can replicate the results from suest if the models are available for gsem. In addition, gsem allows us to combine results from some estimation commands that are not supported by suest, like models including random effects.

Example: Comparing parameters from two models

Let’s consider the childweight dataset, described in [ME] mixed. Consider the following models, where weights of boys and girls are modeled using the age and the age-squared:

. webuse childweight, clear
(Weight data on Asian children)

. regress  weight age c.age#c.age if girl == 0, noheader
------------------------------------------------------------------------------
weight |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
age |   7.985022   .6343855    12.59   0.000     6.725942    9.244101
|
c.age#c.age |   -1.74346   .2374504    -7.34   0.000    -2.214733   -1.272187
|
_cons |   3.684363   .3217223    11.45   0.000     3.045833    4.322893
------------------------------------------------------------------------------

. regress  weight age c.age#c.age if girl == 1, noheader
------------------------------------------------------------------------------
weight |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
age |   7.008066   .5164687    13.57   0.000     5.982746    8.033386
|
c.age#c.age |  -1.450582   .1930318    -7.51   0.000    -1.833798   -1.067365
|
_cons |   3.480933   .2616616    13.30   0.000     2.961469    4.000397
------------------------------------------------------------------------------


To test whether birthweights are the same for the two groups, we need to test whether the intercepts in the two regressions are the same. Using suest, we would proceed as follows:

. quietly regress weight age c.age#c.age if girl == 0, noheader

. estimates store boys

. quietly regress weight age c.age#c.age if girl == 1, noheader

. estimates store girls

. suest boys girls

Simultaneous results for boys, girls

Number of obs   =        198

------------------------------------------------------------------------------
|               Robust
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
boys_mean    |
age |   7.985022   .4678417    17.07   0.000     7.068069    8.901975
|
c.age#c.age |   -1.74346   .2034352    -8.57   0.000    -2.142186   -1.344734
|
_cons |   3.684363   .1719028    21.43   0.000      3.34744    4.021286
-------------+----------------------------------------------------------------
boys_lnvar   |
_cons |   .4770289   .1870822     2.55   0.011     .1103546    .8437032
-------------+----------------------------------------------------------------
girls_mean   |
age |   7.008066   .4166916    16.82   0.000     6.191365    7.824766
|
c.age#c.age |  -1.450582   .1695722    -8.55   0.000    -1.782937   -1.118226
|
_cons |   3.480933   .1556014    22.37   0.000      3.17596    3.785906
-------------+----------------------------------------------------------------
girls_lnvar  |
_cons |   .0097127   .1351769     0.07   0.943    -.2552292    .2746545
------------------------------------------------------------------------------


Invoking an estimation command with the option coeflegend will give us a legend we can use to refer to the parameters when we use postestimation commands like test.

. suest, coeflegend

Simultaneous results for boys, girls

Number of obs   =        198

------------------------------------------------------------------------------
|      Coef.  Legend
-------------+----------------------------------------------------------------
boys_mean    |
age |   7.985022  _b[boys_mean:age]
|
c.age#c.age |   -1.74346  _b[boys_mean:c.age#c.age]
|
_cons |   3.684363  _b[boys_mean:_cons]
-------------+----------------------------------------------------------------
boys_lnvar   |
_cons |   .4770289  _b[boys_lnvar:_cons]
-------------+----------------------------------------------------------------
girls_mean   |
age |   7.008066  _b[girls_mean:age]
|
c.age#c.age |  -1.450582  _b[girls_mean:c.age#c.age]
|
_cons |   3.480933  _b[girls_mean:_cons]
-------------+----------------------------------------------------------------
girls_lnvar  |
_cons |   .0097127  _b[girls_lnvar:_cons]
------------------------------------------------------------------------------

. test  _b[boys_mean:_cons] = _b[girls_mean:_cons]

( 1)  [boys_mean]_cons - [girls_mean]_cons = 0

chi2(  1) =    0.77
Prob > chi2 =    0.3803


We find no evidence that the intercepts are different.

Now, let’s replicate those results Read more…

Categories: Statistics Tags:

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.