Multilevel models are regression models that incorporate group-specific effects. Groups may represent different levels of hierarchy such as hospitals, doctors nested within hospitals, and patients nested within doctors nested within hospitals. Group-specific effects are assumed to vary randomly across groups according to some a priori distribution, commonly a normal distribution. This assumption makes multilevel models natural candidates for Bayesian analysis. Bayesian multilevel models additionally assume that other model parameters such as regression coefficients and variance components—variances of group-specific effects—are also random.

Why use Bayesian multilevel models? In addition to standard reasons for Bayesian analysis, Bayesian multilevel modeling is often used when the number of groups is small or in the presence of many hierarchical levels. Bayesian information criteria such as deviance information criterion (DIC) are also popular for comparing multilevel models. When the comparison of groups is of main interest, Bayesian multilevel modeling can provide entire distributions of group-specific effects.

You can now fit Bayesian multilevel models in Stata and you can do this
easily—just prefix your multilevel command with **bayes:**

.bayes: mixed y x1 x2 || id:

Of course, when we say "easily", we refer to the model specification and not the model formulation. Just like any other modeling task, Bayesian multilevel modeling requires careful consideration.

Continuous, censored, binary, ordinal, count, GLM, and survival outcomes are
supported; see the
full list
of supported multilevel commands. All multilevel features such as multiple
levels of hierarchy, nested and crossed random effects, random intercepts and
coefficients, and random-effects covariance structures are available. All
Bayesian features as provided by the
[BAYES] **bayesmh** command
are supported when you use the **bayes** prefix with multilevel commands; read about general features of the **bayes prefix**.

- Outcomes: continuous, censored, binary, ordinal, count, GLM, survival
- Random intercepts and coefficients
- Nested and crossed effects
- Multiple levels of hierarchy
- Random-effects covariance structures
- Full Bayesian-features support

Now, let's look at several examples.

- Simple two-level models
- Random-effects covariance structures
- Three-level models
- Exponential survival model
- Log marginal likelihood
- Lognormal survival model
- Model comparison using posterior probabilities
- Crossed-effects logistic model

Consider data on math scores of pupils in the third and fifth years from different schools in Inner London (Mortimore et al. 1988). We want to investigate a school effect on math scores.

We can use Stata's **mixed** command to fit a two-level linear model of
five-year math scores (**math5**) on three-year math scores (**math3**)
with random intercepts for schools.

.mixed math5 math3 || school:Mixed-effects ML regression Number of obs = 887 Group variable: school Number of groups = 48 Obs per group: min = 5 avg = 18.5 max = 62 Wald chi2(1) = 347.92 Log likelihood = -2767.8923 Prob > chi2 = 0.0000

math5 | Coef. Std. Err. z P>|z| [95% Conf. Interval] | |

math3 | .6088066 .0326392 18.65 0.000 .5448349 .6727783 | |

_cons | 30.36495 .3491544 86.97 0.000 29.68062 31.04928 | |

Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] | |

school: Identity | ||

var(_cons) | 4.026853 1.189895 2.256545 7.186004 | |

var(Residual) | 28.12721 1.37289 25.5611 30.95094 | |

The likelihood-ratio test at the bottom and the estimate of the **school**
variance component suggest statistically significant variability between
schools in the **math5** scores after adjusting for the **math3**
scores.

To fit the corresponding Bayesian model, you can simply prefix the above
**mixed** command with **bayes:**.

.bayes: mixed math5 math3 || school:

But here, we will first use **bayes**'s **melabel** option to obtain
output similar to that of **mixed** for easier comparison of the results.

.bayes, melabel: mixed math5 math3 || school:Bayesian multilevel regression MCMC iterations = 12,500 Metropolis-Hastings and Gibbs sampling Burn-in = 2,500 MCMC sample size = 10,000 Group variable: school Number of groups = 48 Obs per group: min = 5 avg = 18.5 max = 62 Number of obs = 887 Acceptance rate = .8091 Efficiency: min = .03366 avg = .3331 Log marginal likelihood max = .6298

Equal-tailed | ||

Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||

math5 | ||

math3 | .6087689 .0326552 .000436 .6087444 .5450837 .6729982 | |

_cons | 30.39202 .3597873 .01961 30.38687 29.67802 31.10252 | |

school | ||

var(_cons) | 4.272626 1.299061 .039697 4.122282 2.247659 7.220809 | |

var(Residual) | 28.23014 1.37812 .017365 28.18347 25.63394 31.04375 | |

Note: Default priors are used for some model parameters. |

The reported estimates of posterior means and posterior standard deviations
for model parameters are similar to the corresponding maximum-likelihood
estimates and standard errors reported by **mixed**.

Unlike **mixed**, which provided one estimate for each model parameter,
**bayes: mixed** provided, for each parameter, a sample of 10,000 Markov chain Monte Carlo (MCMC)
estimates from the simulated posterior distribution of the parameters. The
reported posterior means and posterior distributions are the corresponding
summaries of the marginal posterior distributions of the parameters.

Let's now see the output from **bayes: mixed** without the **melabel**
option. The output is lengthy, so we will describe it in pieces.

.bayes

Multilevel structure | ||

school | ||

{U0}: random intercepts | ||

Model summary | ||

Likelihood: | ||

math5 ~ normal(xb_math5,{e.math5:sigma2}) | ||

Priors: | ||

{math5:math3 _cons} ~ normal(0,10000) (1) | ||

{U0} ~ normal(0,{U0:sigma2}) (1) | ||

{e.math5:sigma2} ~ igamma(.01,.01) | ||

Hyperprior: | ||

{U0:sigma2} ~ igamma(.01,.01) | ||

The header now includes additional information about the fitted Bayesian
model. We can see, for example, that parameter **{U0}** represents random
intercepts in the model, that regression coefficients **{math5:math3}** and
**{math5:_cons}** are assigned default normal priors with zero means and
variances of 10,000, and that the variance component for schools,
**{U0:sigma2}**, is assigned the default inverse-gamma prior with 0.01 for
both the shape and scale parameters.

In the output table, the results are the same, but the parameter labels are different.

Equal-tailed | ||

Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||

math5 | ||

math3 | .6087689 .0326552 .000436 .6087444 .5450837 .6729982 | |

_cons | 30.39202 .3597873 .01961 30.38687 29.67802 31.10252 | |

school | ||

U0:sigma2 | 4.272626 1.299061 .039697 4.122282 2.247659 7.220809 | |

e.math5 | ||

sigma2 | 28.23014 1.37812 .017365 28.18347 25.63394 31.04375 | |

Note: Default priors are used for some model parameters. |

By default, **bayes: mixed** displays results using parameter names
as you would use when referring to these parameters in **bayes**'s
options or during postestimation. For example, you would use
**{U0:sigma2}** to refer to the variance component for schools
and **{e.math5:sigma2}** to refer to the error variance.

There is still one part of the output missing—the estimates of
random intercepts **{U0}**. In a Bayesian multilevel model,
random effects are model parameters just like regression coefficients
and variance components. **bayes** does not report them by default
because there are often too many of them. But you can display them
during or after estimation. Here we replay the estimation, adding
the **showreffects()** option to display the estimates of the
first 12 random intercepts.

.bayes, showreffects({U0[1/12]})Bayesian multilevel regression MCMC iterations = 12,500 Metropolis-Hastings and Gibbs sampling Burn-in = 2,500 MCMC sample size = 10,000 Group variable: school Number of groups = 48 Obs per group: min = 5 avg = 18.5 max = 62 Number of obs = 887 Acceptance rate = .8091 Efficiency: min = .03366 avg = .1749 Log marginal likelihood max = .6298

Equal-tailed | ||

Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||

math5 | ||

math3 | .6087689 .0326552 .000436 .6087444 .5450837 .6729982 | |

_cons | 30.39202 .3597873 .01961 30.38687 29.67802 31.10252 | |

U0[school] | ||

1 | -2.685824 .9776969 .031227 -2.672364 -4.633162 -.7837494 | |

2 | .015465 1.290535 .03201 .0041493 -2.560203 2.556316 | |

3 | 1.049006 1.401383 .033731 1.021202 -1.534088 3.84523 | |

4 | -2.123055 .9921679 .028859 -2.144939 -4.069283 -.1507593 | |

5 | -.1504003 .9650027 .033881 -.1468966 -2.093015 1.721503 | |

6 | .5833945 1.192379 .032408 .5918357 -1.660335 3.049718 | |

7 | 1.490231 1.332917 .033846 1.481793 -1.095757 4.272903 | |

8 | .4198105 .9783772 .031891 .4579817 -1.496317 2.403908 | |

9 | -1.996105 1.02632 .035372 -2.001467 -4.037044 -.0296276 | |

10 | .6736806 1.249238 .031114 .660939 -1.70319 3.179273 | |

11 | -.5650109 .9926453 .031783 -.5839293 -2.646413 1.300388 | |

12 | -.3620733 1.090265 .033474 -.3203626 -2.550097 1.717532 | |

school | ||

U0:sigma2 | 4.272626 1.299061 .039697 4.122282 2.247659 7.220809 | |

e.math5 | ||

sigma2 | 28.23014 1.37812 .017365 28.18347 25.63394 31.04375 | |

Note: Default priors are used for some model parameters. |

We could have used **showreffects** to display all 48.

To compare schools, we can plot posterior distributions (our prior normal distributions updated based on the observed data) of random intercepts. We plot histograms for the same first 12 random intercepts.

.bayesgraph histogram {U0[1/12]}, byparm

We save the MCMC results and store the estimation results from our Bayesian random-intercept model for later model comparison.

.bayes, saving(ri_mcmc).estimates store ri

Our initial example used the default priors for model parameters. Just to show you how easy it is to specify custom priors, we specify a uniform on (-50,50) prior for the regression coefficients:

.bayes, prior({math5:}, uniform(-50,50)): mixed math5 math3 || school:

Model summary | ||

Likelihood: | ||

math5 ~ normal(xb_math5,{e.math5:sigma2}) | ||

Priors: | ||

{math5:math3 _cons} ~ uniform(-50,50) (1) | ||

{U0} ~ normal(0,{U0:sigma2}) (1) | ||

{e.math5:sigma2} ~ igamma(.01,.01) | ||

Hyperprior: | ||

{U0:sigma2} ~ igamma(.01,.01) | ||

Equal-tailed | ||

Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||

math5 | ||

math3 | .6094181 .0319517 .001432 .6085484 .5460873 .6732493 | |

_cons | 30.36818 .3290651 .022103 30.38259 29.73806 31.0131 | |

school | ||

U0:sigma2 | 4.261459 1.282453 .040219 4.084322 2.238583 7.218895 | |

e.math5 | ||

sigma2 | 28.24094 1.374732 .016577 28.20275 25.68069 31.01401 | |

Note: Default priors are used for some model parameters. |

**{math5:}** in the above is a shortcut for referring to all regression
coefficients associated with the dependent variable **math5**:
**{math5:math3}** and **{math5:_cons}**.

Similarly, we can use different priors for each regression coefficient, but we omit the output for brevity.

.bayes, prior({math5:_cons}, uniform(-50,50)) prior({math5:math3}, uniform(-5,5)): mixed math5 math3 || school:

Let's extend our simple random-intercept model to include a random coefficient.

Following **mixed**'s specification, we include **math3** in the
random-effects equation for the **school** level. We also store our
Bayesian estimation results for later comparison. This time, we save MCMC
results during estimation.

.bayes, saving(rc_mcmc): mixed math5 math3 || school: math3.estimates store rc

Here is abbreviated output from **bayes: mixed**, including a random
coefficient for **math3**.

Multilevel structure | ||

school | ||

{U0}: random intercepts | ||

{U1}: random coefficients for math3 | ||

Model summary | ||

Likelihood: | ||

math5 ~ normal(xb_math5,{e.math5:sigma2}) | ||

Priors: | ||

{math5:math3 _cons} ~ normal(0,10000) (1) | ||

{U0} ~ normal(0,{U0:sigma2}) (1) | ||

{U1} ~ normal(0,{U1:sigma2}) (1) | ||

{e.math5:sigma2} ~ igamma(.01,.01) | ||

Hyperprior: | ||

{U0:sigma2} ~ igamma(.01,.01) | ||

{U1:sigma2} ~ igamma(.01,.01) | ||

Equal-tailed | ||

Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||

math5 | ||

math3 | .6143538 .0454835 .001655 .6137192 .5257402 .7036098 | |

_cons | 30.38813 .3577296 .019669 30.3826 29.71581 31.10304 | |

school | ||

U0:sigma2 | 4.551927 1.368582 .041578 4.361247 2.420075 7.722063 | |

U1:sigma2 | .0398006 .0194373 .001271 .0363514 .0131232 .0881936 | |

e.math5 | ||

sigma2 | 27.19758 1.354024 .021967 27.15869 24.71813 30.05862 | |

Note: Default priors are used for some model parameters. |

**mixed** assumes independence between random intercepts and coefficients.
**bayes: mixed** does too, to be consistent. We can relax this assumption by
specifying an unstructured variance–covariance as follows. We save the MCMC
results and store the estimation results from this model as well.

.bayes, saving(rcun_mcmc): mixed math5 math3 || school: math3, covariance(unstructured).estimates store rcun

Here is abbreviated output from **bayes: mixed** specifying the unstructured
covariance.

Multilevel structure | ||

school | ||

{U0}: random intercepts | ||

{U1}: random coefficients for math3 | ||

Model summary | ||

Likelihood: | ||

math5 ~ normal(xb_math5,{e.math5:sigma2}) | ||

Priors: | ||

{math5:math3 _cons} ~ normal(0,10000) (1) | ||

{U0}{U1} ~ mvnormal(2,{U:Sigma,m}) (1) | ||

{e.math5:sigma2} ~ igamma(.01,.01) | ||

Hyperprior: | ||

{U:Sigma,m} ~ iwishart(2,3,I(2)) | ||

Equal-tailed | ||

Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||

math5 | ||

math3 | .6234197 .0570746 .002699 .6228624 .5144913 .7365849 | |

_cons | 30.34691 .3658515 .021356 30.34399 29.62991 31.07312 | |

school | ||

U:Sigma_1_1 | 4.527905 1.363492 .046275 4.345457 2.391319 7.765521 | |

U:Sigma_2_1 | -.322247 .1510543 .004913 -.3055407 -.6683891 -.0679181 | |

U:Sigma_2_2 | .0983104 .0280508 .000728 .0941222 .0556011 .1649121 | |

e.math5 | ||

sigma2 | 26.8091 1.34032 .018382 26.76549 24.27881 29.53601 | |

Note: Default priors are used for some model parameters. |

We now use DIC to compare the three models: random-intercept model, random-coefficient model with independent covariance structure, and random-coefficient model with unstructured covariance structure.

.bayesstats ic ri rc rcun, diconlyDeviance information criterion

DIC | ||

ri | 5514.514 | |

rc | 5500.837 | |

rcun | 5494.349 | |

DIC is the smallest for the random-coefficient model with an unstructured random-effects covariance, so this model is preferable.

To demonstrate that you can specify custom priors for not only scalar
parameters but also matrices, we specify a custom inverse-Wishart prior for
the random-effects covariance matrix **{Sigma,m}**, which is short for
**{Sigma,matrix}**.

.matrix S = (5,-0.5\-0.5,1).bayes, iwishartprior(10 S): mixed math5 math3 || school: math3, covariance(unstructured)

Multilevel structure | ||

school | ||

{U0}: random intercepts | ||

{U1}: random coefficients for math3 | ||

Model summary | ||

Likelihood: | ||

math5 ~ normal(xb_math5,{e.math5:sigma2}) | ||

Priors: | ||

{math5:math3 _cons} ~ normal(0,10000) (1) | ||

{U0}{U1} ~ mvnormal(2,{U:Sigma,m}) (1) | ||

{e.math5:sigma2} ~ igamma(.01,.01) | ||

Hyperprior: | ||

{U:Sigma,m} ~ iwishart(2,10,S) | ||

Equal-tailed | ||

Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||

math5 | ||

math3 | .6130199 .0537473 .00282 .613916 .5058735 .7180286 | |

_cons | 30.3789 .3223274 .016546 30.3816 29.74903 31.02091 | |

school | ||

U:Sigma_1_1 | 3.482914 1.104742 .048864 3.344148 1.770735 6.0136 | |

U:Sigma_2_1 | -.2712029 .1169666 .004214 -.2596221 -.5337747 -.0745626 | |

U:Sigma_2_2 | .0775669 .0210763 .000651 .074876 .0443026 .1264642 | |

e.math5 | ||

sigma2 | 26.94206 1.342571 .022106 26.90405 24.4033 29.66083 | |

Note: Default priors are used for some model parameters. |

The **iwishartprior()** option overrides the parameters of the
default inverse-Wishart prior distributions used by **bayes:**. In
our example, this prior is used for the covariance matrix with the default
arguments of 3 degress of freedom and an identity scale matrix. In the above
example, we instead used 10 degrees of freedom and the scale matrix **S**.

Consider survival data that record durations (in months) of employment of
individuals from **tstart** to **tend**. Variable **failure**
contains 1 when the end date corresponds to the end of employment, and 0
otherwise. A two-level model would account for the variability between
individuals, who are identified by the **id** variable. To
demonstrate a three-level model, let's also account for the variability
between birth years of individuals, which may help explain some of
between-individual variability.

Here is how we would proceed with the standard multilevel analysis for these data.

. stset tend, origin(tstart) failure(failure). mestreg education njobs prestige i.female || birthyear: || id:, distribution(exponential)

We model the time to the end of employment as a function of the education level, the number of jobs held previously, the prestige of the current job, and gender.

In our Bayesian analysis, we will compare how well the two survival models, exponential and lognormal, fit these data.

First, we fit a Bayesian three-level exponential model by simply prefixing the
above **mestreg** command.

.bayes: mestreg education njobs prestige i.female || birthyear: || id:, distribution(exponential)

The output is lengthy, so we describe it in parts.

The multilevel summary provides the names of parameters, **{U0}** and
**{UU0}**, for random intercepts at the third and second levels of
hierarchy, respectively. (Observations compose the first level.)

Multilevel structure | ||

birthyear | ||

{U0}: random intercepts | ||

birthyear>id | ||

{UU0}: random intercepts | ||

The model summary describes the likelihood model and prior distributions used.
For example, both variance components, **{U0:sigma2}** and
**{UU0:sigma2}**, are assigned inverse-gamma prior distributions with scale
and shape parameters of 0.01.

Model summary | ||

Likelihood: | ||

_t ~ mestreg(xb__t) | ||

Priors: | ||

{_t:education njobs prestige 1.female _cons} ~ normal(0,10000) (1) | ||

{U0} ~ normal(0,{U0:sigma2}) (1) | ||

{UU0} ~ normal(0,{UU0:sigma2}) (1) | ||

Hyperpriors: | ||

{U0:sigma2} ~ igamma(.01,.01) | ||

{UU0:sigma2} ~ igamma(.01,.01) | ||

The header information now includes a group summary for each hierarchical level.

Bayesian multilevel exponential PH regression MCMC iterations = 12,500 Random-walk Metropolis-Hastings sampling Burn-in = 2,500 MCMC sample size = 10,000

No. of Observations per Group | ||

Group Variable | Groups Minimum Average Maximum | |

birthyear | 12 3 50.0 99 | |

id | 201 1 3.0 9 | |

Just like **mestreg**, **bayes: mestreg** by default reports hazard
ratios for the exponential survival model. You can use the **nohr** option
to instead obtain coefficients. You can specify this option with **bayes**
or **mestreg** during estimation or with **bayes** on replay.

. bayes, nohr

Equal-tailed | ||

Haz. Ratio Std. Dev. MCSE Median [95% Cred. Interval] | ||

-t | ||

education | 1.10707 .040724 .004527 1.107038 1.024029 1.188931 | |

njobs | .7793024 .0354771 .002428 .7796324 .7114243 .8488112 | |

prestige | .9707042 .0064944 .000412 .9707677 .9580977 .9835198 | |

1.female | 1.664841 .2483748 .026186 1.642104 1.229063 2.192569 | |

_cons | .0137567 .0061351 .000673 .0126042 .0055763 .0287104 | |

birthyear | ||

U0:sigma2 | .1120806 .0969011 .006457 .0855433 .0092592 .382327 | |

birthyear>id | ||

UU0:sigma2 | .523587 .1229834 .010149 .5155754 .3040806 .8036657 | |

We save the MCMC results and store the estimation results from our Bayesian exponential
survival model for later model comparison.
We also specify the **remargl** option to compute the log
marginal likelihood, which we explain below.

.bayes, remargl saving(streg_exp_mcmc).estimates store streg_exp

Previously, we used **bayes: mestreg**
to fit a Bayesian exponential survival model to employment duration data.
If you look closely at the header output from **bayes: mestreg**,

Bayesian multilevel exponential PH regression MCMC iterations = 12,500 Random-walk Metropolis-Hastings sampling Burn-in = 2,500 MCMC sample size = 10,000 Number of obs = 600 Acceptance rate = .3045 Efficiency: min = .008093 avg = .01554 Log marginal likelihood max = .02482

you will notice that no value is reported for the log marginal likelihood
(LML). This is intentional. As we mentioned earlier, Bayesian multilevel
models treat random effects as parameters and thus may contain many model
parameters. For models with many parameters or high-dimensional models, the
computation of LML can be time consuming, and its accuracy may become
unacceptably low. For these reasons, the **bayes** prefix does not compute
the LML for multilevel models by default. You can specify the **remargl**
option during estimation or on replay to compute it.

Here is a subset of the relevant output after typing

.bayes, remarglBayesian multilevel exponential PH regression MCMC iterations = 12,500 Random-walk Metropolis-Hastings sampling Burn-in = 2,500 MCMC sample size = 10,000 Number of obs = 600 Acceptance rate = .3045 Efficiency: min = .008093 avg = .01554 Log marginal likelihood = -2530.4131 max = .02482

Notice that the LML value is now reported in the header output.

Why do we need the value of LML? Some of the Bayesian summaries used for model comparison such as Bayes factors and model posterior probabilities are based on LML. In this example, we want to demonstrate the use of model posterior probabilities to compare Bayesian models, and so we needed to compute LML.

We now fit a Bayesian three-level lognormal model by specifying the
corresponding distribution in the **distribution()** option. We also
specify the **remargl** and **saving()** options during estimation this
time.

.bayes, remargl saving(streg_lnorm_mcmc): mestreg education njobs prestige i.female || birthyear: || id:, distribution(lognormal)Bayesian multilevel lognormal AFT regression MCMC iterations = 12,500 Random-walk Metropolis-Hastings sampling Burn-in = 2,500 MCMC sample size = 10,000 Number of obs = 600 Acceptance rate = .3714 Efficiency: min = .004755 avg = .01522 Log marginal likelihood = -2496.4991 max = .04602

Equal-tailed | ||

Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||

-t | ||

education | -.0951555 .0314752 .002934 -.0947659 -.1563553 -.0348788 | |

njobs | .1733276 .0423128 .004265 .1745013 .0871177 .2545806 | |

prestige | .0272722 .0056387 .000263 .0272781 .0160048 .0384117 | |

1.female | -.3406851 .1323801 .018735 -.3383823 -.5965842 -.0912631 | |

_cons | 3.822971 .4130557 .052204 3.817862 3.027644 4.635152 | |

lnsigma | .0723268 .0419609 .002715 .0722478 -.0100963 .1569296 | |

birthyear | ||

U0:sigma2 | .1669752 .1566106 .013008 .120028 .0252243 .6019641 | |

birthyear>id | ||

UU0:sigma2 | .3061048 .1067159 .015475 .2919266 .1357847 .5485671 | |

Note: Default priors are used for some model parameters. |

We now compare models using model posterior probabilities.

.bayestest model streg_exp streg_lnormBayesian model tests

log(ML) P(M) P(M|y) | ||

streg_exp | -2.53e+03 0.5000 0.0000 | |

streg_lnorm | -2.50e+03 0.5000 1.0000 | |

The posterior probability for the lognormal model is essentially 1, so it is preferrable to the exponential model for these data.

In example 5 of [ME] melogit, we fit a
crossed-effects model to the data from a study measuring students' attainment
scores in primary and secondary schools from Fife, Scotland. The binary
outcome of interest is whether the score is greater than 6. The model includes
**sex** as the covariate and the effects of primary and secondary schools
as crossed random effects.

.melogit attain_gt_6 sex || _all:R.sid || pid:

We fit the corresponding Bayesian crossed-effects model by simply prefixing
the above **melogit** command with **bayes:**

.bayes: melogit attain_gt_6 sex || _all: R.sid || pid:

The output is lengthy, so as before, we describe it in parts.

Multilevel structure | ||

sid | ||

{U0}: random intercepts | ||

pid | ||

{V0}: random intercepts | ||

We have two sets of random intercepts, **{U0}** and **{V0}**, at the
secondary **sid** and primary **pid** levels, respectively.

Model summary | ||

Likelihood: | ||

attain_gt_6 ~ melogit(xb_attain_gt_6) | ||

Priors: | ||

{attain_gt_6:sex _cons} ~ normal(0,10000) (1) | ||

{U0} ~ normal(0,{U0:sigma2}) (1) | ||

{V0} ~ normal(0,{V0:sigma2}) (1) | ||

Hyperpriors: | ||

{U0:sigma2} ~ igamma(.01,.01) | ||

{V0:sigma2} ~ igamma(.01,.01) | ||

The likelihood model is a multilevel logistic model. The prior distributions are normal for regression coefficients and random intercepts and are inverse-gamma for the variance components.

No. of Observations per Group | ||

Group Variable | Groups Minimum Average Maximum | |

_all | 1 3,435 3,435.0 3,435 | |

pid | 148 1 23.2 72 | |

The header information includes the MCMC simulation summary as well as other details about the fitted Bayesian model.

Equal-tailed | ||

Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||

attain_gt_6 | ||

sex | .2898934 .0731912 .004749 .2879629 .1568701 .4429235 | |

_cons | -.6502969 .109456 .010693 -.6497501 -.8509903 -.4300285 | |

sid | ||

U0:sigma2 | .1484609 .0826556 .005049 .1336719 .0241995 .3532284 | |

pid | ||

V0:sigma2 | .4777177 .0954019 .004961 .4701 .3228973 .6864048 | |

Note: Default priors are used for some model parameters. |

The results suggest that both primary and secondary schools contribute to the variation in whether the attainment score is greater than 6 after adjusting for sex in the model.

Instead of the estimates of coefficients, we can obtain the estimates of odds ratios.

.bayes, or

Equal-tailed | ||

Odds Ratio Std. Dev. MCSE Median [95% Cred. Interval] | ||

attain_gt_6 | ||

sex | 1.339886 .0988964 .00647 1.333708 1.169844 1.557253 | |

_cons | .5250545 .0584524 .005902 .5221763 .4269919 .6504906 | |

sid | ||

U0:sigma2 | .1484609 .0826556 .005049 .1336719 .0241995 .3532284 | |

pid | ||

V0:sigma2 | .4777177 .0954019 .004961 .4701 .3228973 .6864048 | |

Note: Estimates are transformed only in the first equation. | ||

Note: _cons estimates baseline odds (conditional on zero random effects). | ||

Note: Default priors are used for model parameters. |

We could have also specified the **or** option during estimation either
with the **bayes** prefix

.bayes, or: melogit attain_gt_6 sex || _all: R.sid || pid:

or with **melogit**

.bayes: melogit attain_gt_6 sex || _all: R.sid || pid:, or

Mortimore, P., P. Sammons, L. Stoll, D. Lewis, and R. Ecob. 1988.
*School Matters: The Junior Years*. Wells, Somerset, UK:
Open Books.

Learn mode about the general features of the bayes prefix.

Learn more about Stata's Bayesian analysis features.

Read more about the **bayes** prefix and Bayesian analysis in the *Stata Bayesian Analysis Reference Manual*.