【文章內(nèi)容簡介】 [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