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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
------------------------------------------------------------------------------

file sim.dta not found; file saved

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.

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

. unicode erasebackups, badidea

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…

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
-------------+------------------------------           Adj 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 Read more…

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
headroom        float   %6.1f                 Headroom (in.)
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 dfbeta 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

dfbeta_mpg

Based on the impact on the Read more…

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 Read more…

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

{delta} = {{mu}_1 - {mu}_2} / {sigma}

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 Read more…

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.

GSP Data: http://www.stata-press.com/data/r12/productivity.dta

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.

Graph3

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.

Slide15

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

Slide16

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))

Graph4

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.

Slide19

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)

Slide21

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))

Graph6

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.

For more information

If you’d like to learn more about modeling multilevel and longitudinal data, check out

Multilevel and Longitudinal Modeling Using Stata, Third Edition
Volume I: Continuous Responses
Volume II: Categorical Responses, Counts, and Survival
by Sophia Rabe-Hesketh and Anders Skrondal

or sign up for our popular public training course Multilevel/Mixed Models Using Stata.

Multilevel linear models in Stata, part 1: Components of variance

In the last 15-20 years multilevel modeling has evolved from a specialty area of statistical research into a standard analytical tool used by many applied researchers.

Stata has a lot of multilevel modeling capababilities.

I want to show you how easy it is to fit multilevel models in Stata. Along the way, we’ll unavoidably introduce some of the jargon of multilevel modeling.

I’m going to focus on concepts and ignore many of the details that would be part of a formal data analysis. I’ll give you some suggestions for learning more at the end of the post.

    The videos

Stata has a 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 1: The xtmixed command

    Multilevel data

Multilevel data are characterized by a hierarchical structure. A classic example is children nested within classrooms and classrooms nested within schools. The test scores of students within the same classroom may be correlated due to exposure to the same teacher or textbook. Likewise, the average test scores of classes might be correlated within a school due to the similar socioeconomic level of the students.

You may have run across datasets with these kinds of structures in your own work. For our example, I would like to use a dataset that has both longitudinal and classical hierarchical features. You can access this dataset from within Stata by typing the following command:

use http://www.stata-press.com/data/r12/productivity.dta

We are going to build a model of gross state product for 48 states in the USA measured annually from 1970 to 1986. The states have been grouped into nine regions based on their economic similarity. For distributional reasons, we will be modeling the logarithm of annual Gross State Product (GSP) but in the interest of readability, I will simply refer to the dependent variable as GSP.

. describe gsp year state region

              storage  display     value
variable name   type   format      label      variable label
-----------------------------------------------------------------------------
gsp             float  %9.0g                  log(gross state product)
year            int    %9.0g                  years 1970-1986
state           byte   %9.0g                  states 1-48
region          byte   %9.0g                  regions 1-9

Let’s look at a graph of these data to see what we’re working with.

twoway (line gsp year, connect(ascending)), ///
        by(region, title("log(Gross State Product) by Region", size(medsmall)))

graph1

Each line represents the trajectory of a state’s (log) GSP over the years 1970 to 1986. The first thing I notice is that the groups of lines are different in each of the nine regions. Some groups of lines seem higher and some groups seem lower. The second thing that I notice is that the slopes of the lines are not the same. I’d like to incorporate those attributes of the data into my model.

    Components of variance

Let’s tackle the vertical differences in the groups of lines first. If we think about the hierarchical structure of these data, I have repeated observations nested within states which are in turn nested within regions. I used color to keep track of the data hierarchy.

slide2

We could compute the mean GSP within each state and note that the observations within in each state vary about their state mean.

slide3

Likewise, we could compute the mean GSP within each region and note that the state means vary about their regional mean.

slide4

We could also compute a grand mean and note that the regional means vary about the grand mean.

slide5

Next, let’s introduce some notation to help us keep track of our mutlilevel structure. In the jargon of multilevel modelling, the repeated measurements of GSP are described as “level 1″, the states are referred to as “level 2″ and the regions are “level 3″. I can add a three-part subscript to each observation to keep track of its place in the hierarchy.

slide7

Now let’s think about our model. The simplest regression model is the intercept-only model which is equivalent to the sample mean. The sample mean is the “fixed” part of the model and the difference between the observation and the mean is the residual or “random” part of the model. Econometricians often prefer the term “disturbance”. I’m going to use the symbol μ to denote the fixed part of the model. μ could represent something as simple as the sample mean or it could represent a collection of independent variables and their parameters.

slide8

Each observation can then be described in terms of its deviation from the fixed part of the model.

slide9

If we computed this deviation of each observation, we could estimate the variability of those deviations. Let’s try that for our data using Stata’s xtmixed command to fit the model:

. xtmixed gsp

Mixed-effects ML regression                     Number of obs      =       816

                                                Wald chi2(0)       =         .
Log likelihood = -1174.4175                     Prob > chi2        =         .

------------------------------------------------------------------------------
         gsp |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   10.50885   .0357249   294.16   0.000     10.43883    10.57887
------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
                sd(Residual) |   1.020506   .0252613      .9721766    1.071238
------------------------------------------------------------------------------

The top table in the output shows the fixed part of the model which looks like any other regression output from Stata, and the bottom table displays the random part of the model. Let’s look at a graph of our model along with the raw data and interpret our results.

predict GrandMean, xb
label var GrandMean "GrandMean"
twoway  (line GrandMean year, lcolor(black) lwidth(thick))              ///
        (scatter gsp year, mcolor(red) msize(tiny)),                    ///
        ytitle(log(Gross State Product), margin(medsmall))              ///
        legend(cols(4) size(small))                                     ///
        title("GSP for 1970-1986 by Region", size(medsmall))

graph1b

The thick black line in the center of the graph is the estimate of _cons, which is an estimate of the fixed part of model for GSP. In this simple model, _cons is the sample mean which is equal to 10.51. In “Random-effects Parameters” section of the output, sd(Residual) is the average vertical distance between each observation (the red dots) and fixed part of the model (the black line). In this model, sd(Residual) is the estimate of the sample standard deviation which equals 1.02.

At this point you may be thinking to yourself – “That’s not very interesting – I could have done that with Stata’s summarize command”. And you would be correct.

. summ gsp

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
         gsp |       816    10.50885    1.021132    8.37885   13.04882

But here’s where it does become interesting. Let’s make the random part of the model more complex to account for the hierarchical structure of the data. Consider a single observation, yijk and take another look at its residual.

slide11

The observation deviates from its state mean by an amount that we will denote eijk. The observation’s state mean deviates from the the regionals mean uij. and the observation’s regional mean deviates from the fixed part of the model, μ, by an amount that we will denote ui... We have partitioned the observation’s residual into three parts, aka “components”, that describe its magnitude relative to the state, region and grand means. If we calculated this set of residuals for each observation, wecould estimate the variability of those residuals and make distributional assumptions about them.

slide12

These kinds of models are often called “variance component” models because they estimate the variability accounted for by each level of the hierarchy. We can estimate a variance component model for GSP using Stata’s xtmixed command:

xtmixed gsp, || region: || state:

------------------------------------------------------------------------------
         gsp |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   10.65961   .2503806    42.57   0.000     10.16887    11.15035
------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
region: Identity             |   
                   sd(_cons) |   .6615227   .2038944       .361566    1.210325
-----------------------------+------------------------------------------------
state: Identity              |   
                   sd(_cons) |   .7797837   .0886614      .6240114    .9744415
-----------------------------+------------------------------------------------
                sd(Residual) |   .1570457   .0040071       .149385    .1650992
------------------------------------------------------------------------------

The fixed part of the model, _cons, is still the sample mean. But now there are three parameters estimates in the bottom table labeled “Random-effects Parameters”. Each quantifies the average deviation at each level of the hierarchy.

Let’s graph the predictions from our model and see how well they fit the data.

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(tiny)),            ///
        ytitle(log(Gross State Product), margin(medsmall))      ///
        legend(cols(4) size(small))                             ///
        by(region, title("Multilevel Model of GSP by Region", size(medsmall)))

graph2

Wow – that’s a nice graph if I do say so myself. It would be impressive for a report or publication, but it’s a little tough to read with all nine regions displayed at once. Let’s take a closer look at Region 7 instead.

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))

graph3

The red dots are the observations of GSP for each state within Region 7. The green lines are the estimated mean GSP within each State and the blue line is the estimated mean GSP within Region 7. The thick black line in the center is the overall grand mean for all nine regions. The model appears to fit the data fairly well but I can’t help noticing that the red dots seem to have an upward slant to them. Our model predicts that GSP is constant within each state and region from 1970 to 1986 when clearly the data show an upward trend.

So we’ve tackled the first feature of our data. We’ve succesfully incorporated the basic hierarchical structure into our model by fitting a variance componentis using Stata’s xtmixed command. But our graph tells us that we aren’t finished yet.

Next time we’ll tackle the second feature of our data — the longitudinal nature of the observations.

    For more information

If you’d like to learn more about modelling multilevel and longitudinal data, check out

Multilevel and Longitudinal Modeling Using Stata, Third Edition
Volume I: Continuous Responses
Volume II: Categorical Responses, Counts, and Survival
by Sophia Rabe-Hesketh and Anders Skrondal

or sign up for our popular public training course “Multilevel/Mixed Models Using Stata“.

There’s a course coming up in Washington, DC on February 7-8, 2013.