Multilevel Models 4 Sociology 8811 Class 26 Copyright

Multilevel Models 4 Sociology 8811, Class 26 Copyright © 2007 by Evan Schofer Do not copy or distribute without permission

Announcements • Paper #2 due April 26 – 1 week! • Any questions? • Class topic: More multilevel models • Next week: Guest speaker, start structural equation models…

Review • 1. Separating between vs. within effects • Create variables reflecting: – Level 2 means – Level 1 deviations (group-mean centering) • Include both in the model • 2. Hausman specification test • To help you choose between FEM & REM.

Linear Random Intercept Model • A linear random intercept model: Linear Random Intercept Model • Zeta (z) is a random effect for each group – Allowing each of j groups to have its own intercept – Assumed to be independent & normally distributed – Note: Other texts refer to random intercepts as uj or nj.

Extensions of Random Intercept Model • Linear random intercept model has been extended to address non-linear outcomes… • Dichotomous: Logit, probit, cloglog – Stata: xtlogit, xtprobit, xtcloglog • Count: Poisson / NBREG – xtpoisson, xtnbreg • EHA: Cox & parametric models with shared frailty – Stcox … , shared(groupid); streg … , shared(groupid)

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! Random Coefficient Model Which can be written as: • Where zeta-1 is a random intercept component • Zeta-2 is a random slope component.

Linear Random Coefficient Model Both intercepts and slopes vary randomly across j groups Rabe-Hesketh & Skrondal 2004, p. 63

Random Coefficients Summary • Some things to remember: • Dummy variables allow fixed estimates of intercepts across groups • Dummy interactions allow fixed estimates of slopes across groups • Random coefficients allow intercepts and/or slopes to vary across groups randomly! – The model does not directly estimate those effects, just as a model does not estimate coefficients for each case residual – BUT, random components can be predicted after the fact (just as you can compute residuals – random error).

STATA Notes: xtreg, xtmixed • xtreg – allows estimation of between, within (fixed), and random intercept models • • xtreg y x 1 x 2 x 3, i(groupid) fe - fixed (within) model xtreg y x 1 x 2 x 3, i(groupid) be - between model xtreg y x 1 x 2 x 3, i(groupid) re - random intercept (GLS) xtreg y x 1 x 2 x 3, i(groupid) mle - random intercept (MLE) • xtmixed – allows random slopes & intercepts • “Mixed” models refer to models that have both fixed and random components • xtmixed [depvar] [fixed equation] || [random eq], options • Ex: xtmixed y x 1 x 2 x 3 || groupid: x 2 – Random intercept is assumed. Random coef for X 2 specified.

STATA Notes: xtreg, xtmixed • Random intercepts • xtreg y x 1 x 2 x 3, i(groupid) mle – Is equivalent to: • xtmixed y x 1 x 2 x 3 || 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 x 1 x 2 x 3 || groupid: x 1 x 2 x 3 , mle – Note: xtmixed can do a lot… but GLLAMM can do even more! • “General linear & latent mixed models” • Must be downloaded into stata. Type “search gllamm” and follow instructions to install…

Random intercepts: xtmixed • Example: Pro-environmental attitudes. xtmixed supportenv age male dmar demp educ incomerel ses || country: , mle Mixed-effects ML regression Group variable: country Wald chi 2(7) = 625. 75 Log likelihood = -56919. 098 Number of obs Number of groups = = 27807 26 Obs per group: min = avg = max = 511 1069. 5 2154 Prob > chi 2 0. 0000 = ---------------------------------------supportenv | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------+--------------------------------age | -. 0038662. 0008151 -4. 74 0. 000 -. 0054638 -. 0022687 male |. 0978558. 0229613 4. 26 0. 000. 0528524. 1428592 dmar |. 0031799. 0252041 0. 13 0. 900 -. 0462193. 0525791 demp | -. 0738261. 0252797 -2. 92 0. 003 -. 1233734 -. 0242788 educ |. 0857707. 0061482 13. 95 0. 000. 0737204. 097821 incomerel |. 0090639. 0059295 1. 53 0. 126 -. 0025578. 0206856 ses |. 1314591. 0134228 9. 79 0. 000. 1051509. 1577674 _cons | 5. 924237. 118294 50. 08 0. 000 5. 692385 6. 156089 ---------------------------------------[remainder of output cut off] Note: xtmixed yields identical results to xtreg , mle

Random intercepts: xtmixed • Ex: Pro-environmental attitudes (cont’d) supportenv | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------+--------------------------------age | -. 0038662. 0008151 -4. 74 0. 000 -. 0054638 -. 0022687 male |. 0978558. 0229613 4. 26 0. 000. 0528524. 1428592 dmar |. 0031799. 0252041 0. 13 0. 900 -. 0462193. 0525791 demp | -. 0738261. 0252797 -2. 92 0. 003 -. 1233734 -. 0242788 educ |. 0857707. 0061482 13. 95 0. 000. 0737204. 097821 incomerel |. 0090639. 0059295 1. 53 0. 126 -. 0025578. 0206856 ses |. 1314591. 0134228 9. 79 0. 000. 1051509. 1577674 _cons | 5. 924237. 118294 50. 08 0. 000 5. 692385 6. 156089 -----------------------------------------------------------------------------Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] ---------------+------------------------country: Identity | sd(_cons) |. 5397758. 0758083. 4098899. 7108199 ---------------+------------------------sd(Residual) | 1. 869954. 0079331 1. 85447 1. 885568 ---------------------------------------LR test vs. linear regression: chibar 2(01) = 2128. 07 Prob >= chibar 2 = 0. 0000 xtmixed output puts all random effects below main coefficients. Here, they are “cons” (constant) for groups defined by “country”, plus residual (e) Non-zero SD indicates that intercepts vary

Random Coefficients: xtmixed • Ex: Pro-environmental attitudes (cont’d). xtmixed supportenv age male dmar demp educ incomerel ses || country: educ, mle [output omitted] supportenv | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------+--------------------------------age | -. 0035122. 0008185 -4. 29 0. 000 -. 0051164 -. 001908 male |. 1003692. 0229663 4. 37 0. 000. 0553561. 1453824 dmar |. 0001061. 0252275 0. 00 0. 997 -. 0493388. 049551 demp | -. 0722059. 0253888 -2. 84 0. 004 -. 121967 -. 0224447 educ |. 081586. 0115479 7. 07 0. 000. 0589526. 1042194 incomerel |. 008965. 0060119 1. 49 0. 136 -. 0028181. 0207481 ses |. 1311944. 0134708 9. 74 0. 000. 1047922. 1575966 _cons | 5. 931294. 132838 44. 65 0. 000 5. 670936 6. 191652 ---------------------------------------Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] ---------------+------------------------country: Independent | sd(educ) |. 0484399. 0087254. 0340312. 0689492 sd(_cons) |. 6179026. 0898918. 4646097. 821773 ---------------+------------------------sd(Residual) | 1. 86651. 0079227 1. 851046 1. 882102 ---------------------------------------LR test vs. linear regression: chi 2(2) = 2187. 33 Prob > chi 2 = 0. 0000 Here, we have allowed the slope of educ to vary randomly across countries Educ (slope) varies!

Random Coefficients: xtmixed • What are random coefficients doing? • Let’s look at results from a simplified model – Only random slope & intercept for education Model fits a different slope & intercept for each group!

Random Coefficients • Why bother with random coefficients? • 1. A solution for clustering (non-independence) – Usually people just use random intercepts, but slopes may be an issue also • 2. You can create a better-fitting model – If slopes & intercepts vary, a random coefficient model may fit better – Assuming distributional assumptions are met – Model fit compared to OLS can be tested…. • 3. Better predictions – Attention to group-specific random effects can yield better predictions (e. g. , slopes) for each group » Rather than just looking at “average” slope for all groups • 4. Helps us think about multilevel data » Level 2 predictors & cross-level interactions

Multilevel Model Notation • So far, we have expressed random effects in a single equation: Random Coefficient Model • However, it is common to separate the fixed and random parts into multiple equations: Just a basic OLS model… Intercept equation Slope Equation But, intercept & slope are each specified separately as having a random component

Multilevel Model Notation • Substituting equations results in similar form: Intercept equation Slope Equation • Which is equivalent to: Random Coefficient Model

Multilevel Model Notation • The “separate equation” formulation is no different from what we did before… • But it is a vivid & clear way to present your models • All random components are obvious because they are stated in separate equations • NOTE: Some software (e. g. , HLM) requires this format – Rules: • 1. Specify an OLS model, just like normal • 2. Specify an additional equation for each coefficient – i. e. , for the intercept and any X variable (slope) • 3. Include a random term in the level-2 equation – Note: You don’t have to include random term if you don’t want

Multilevel Model Notation • Every level-1 b justifies a level-2 equation • Level 2 equations include random term… Equation for intercept Equation for SES Equation for AGE Note: If you don’t wish to include a random term for any level-2 equation, you don’t have to!

Cross-Level Interactions • Finally, we can specify predictors of slope coefficients • That is, look at effect of level-2 variables on slope of level-1 coefficients • Strategy: Include variables in level-2 equations… Intercept equation Slope equation with predictor W is a variable that predicts b 1 (slope) g 11 coefficient indicates effect of each unit change of W on slope b 1

Cross-Level Interactions • Implementation in Stata • 1. Compute an interaction term in STATA manually – Ex: Interaction of SCHOOLSIZE * SES • 2. Include interaction in model via xtreg or xtmixed • 3. Interpret results like any other interaction term.