multilevelmodels2

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[95% Conf. Interval] + meanage | .0268506 .0239453 .0737825 withinage | .0008156 male | .0981351 .0229623 .0531299 .1431403 dmar | .003459 .0252057 .0528612 demp | .02528 educ | .0856712 .0061483 .0736207 .0977216 inerel | .008957 .0059298 .0205792 ses | .131454 .0134228 .1051458 .1577622 _cons | .9703564 Between amp。 within effects are opposite. Older countries are MORE environmental, but older people are LESS. Omitted variables? Wealthy European countries with strong green parties have older populations! • Example: Proenvironmental attitudes Generalizing: Random Coefficients • Linear random intercept model allows random variation in intercept (mean) for groups • But, the same idea can be applied to other coefficients • That is, slope coefficients can ALSO be random! ijijjijjij XXY   2211Random Coefficient Model     ijijjjij XY   2211Which can be written as: • Where zeta1 is a random intercept ponent • Zeta2 is a random slope ponent. Linear Random Coefficient Model RabeHesketh amp。 Skrondal 2020, p. 63 Both intercepts and slopes vary randomly across j groups Random Coefficients Summary • Some things to remember: • Dummy variables allow fixed estimates of intercepts across groups • Interactions allow fixed estimates of slopes across groups – Random coefficients allow intercepts and/or slopes to have random variability • The model does not directly estimate those effects – Just as we don‟t estimate coefficients of “e” for each case… • BUT, random ponents can be predicted after you run a model – Just as you can pute residuals – random error – This allows you to examine some assumptions (normality). STATA Notes: xtreg, xtmixed • xtreg – allows estimation of between, within (fixed), and random intercept models • xtreg y x1 x2 x3, i(groupid) fe fixed (within) model • xtreg y x1 x2 x3, i(groupid) be between model • xtreg y x1 x2 x3, i(groupid) re random intercept (GLS) • xtreg y x1 x2 x3, i(groupid) mle random intercept (MLE) • xtmixed – allows random intercepts amp。 slopes • “Mixed” models refer to models that have both fixed and random ponents • xtmixed [depvar] [fixed equation] || [random eq], options • Ex: xtmixed y x1 x2 x3 || groupid: x2 – Random intercept is assumed. Random coef for X2 specified. STATA Notes: xtreg, xtmixed • Random intercepts • xtreg y x1 x2 x3, i(groupid) mle – Is equivalent to • xtmixed y x1 x2 x3 || groupid: , mle • xtmixed assumes random intercept – even if no other random effects are specified after “groupid” – But, we can add random coefficients for all Xs: • xtmixed y x1 x2 x3 || groupid: x1 x2 x3 , mle cov(unstr) –Useful to add: “cov(unstructured)” • Stata default treats random terms (intercept, slope) as totally uncorrelated… not always reasonable • “cov(unstr) relaxes constraints regarding covariance among random effects (See RabeHesketh amp。 Skrondal). STATA Notes: GLLAMM • Note: xtmixed can do a lot… but GLLAMM can do even more! • “General linear amp。 latent mixed models” • Must be downloaded into stata. Type “search gllamm” and follow instructions to install… – GLLAMM can do a wide range of mixed amp。 latentvariable models • Multilevel models。 Some kinds of latent class models。 Confirmatory factor analysis。 Some kinds of Structural Equation Models with latent variables… and others… • Documentation available via Stata help – And, in the RabeHesketh amp。 Skrondal text. Random intercepts: xtmixed . xtmixed supportenv age male dmar demp educ inerel ses || country: , mle Mixedeffects ML regression Number of obs = 27807 Group variable: country Number of groups = 26 Obs per group: min = 511 avg = max = 2154 Wald chi2(7) = Log likelihood = Prob chi2 = supportenv | Coef. Std. Err. z P|z| [95% Conf. Interval] + age | .0008151 male | .0978558 .0229613 .0528524 .1428592 dmar | .0031799 .0252041 .0525791 demp | .0252797 educ | .0857707 .0061482 .0737204 .097821 inerel | .0090639 .0059295 .0206856 ses | .1314591 .0134228 .1051509 .1577674 _cons | .118294 [remainder of output cut off] Note: xtmixed yields identical results to xtreg , mle • Example: Proenvironmental attitudes Random intercepts: xtmixed supportenv | Coef. Std. Err. z P|z| [95% Conf. Interval] + age | .0008151 male | .0978558 .0229613 .0528524 .1428592 dmar | .0031799 .0252041 .0525791 demp | .0252797 educ | .0857707 .0061482 .0737204 .097821 inerel | .0090639 .0059295。