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【正文】 on math test scores ? Is it different in coed vs. singlesex schools? – Can you think of others? Crosslevel interactions ? Idea: specify a level2 variable that affects a level1 slope ju 111 ?? ??Intercept equation ijijijXY ??? ??? 21Level 1 equation jj uZ 2322 ??? ???Slope equation with interaction Crosslevel interaction: Level2 variable Z affects slope (B2) of a level1 X variable Coefficient ?3 reflects size of interaction (effect on B2 per unit change in Z) Crosslevel Interactions ? Crosslevel interaction in singleequation form: ijijjijjij XXY ?????? ??????? jij32211 ZXRandom Coefficient Model with crosslevel interaction – Stata strategy: manually pute crosslevel interaction variables ? Ex: Poverty*WelfareState, Gender*SingleSexSchool ? Then, put interaction variable in the “fixed” model – Interpretation: B3 coefficient indicates the impact of each unit change in Z on slope B2 ? If B3 is positive, increase in Z results in larger B2 slope. Crosslevel Interactions . xtmixed supportenv age male dmar demp educ ine_dev inc_meanXeduc ses || country: ine_mean , mle cov(unstr) Mixedeffects ML regression Number of obs = 27807 Group variable: country Number of groups = 26 supportenv | Coef. Std. Err. z P|z| [95% Conf. Interval] + age | .0008148 male | .1006206 .0229617 .0556165 .1456246 dmar | .0041417 .025195 .0535229 demp | .0252727 educ | .0297683 .0233227 ine_dev | .0081591 .005936 .0197934 inc_meanXeduc| .0265714 .0064013 .0140251 .0391177 ses | .1307931 .0134189 .1044926 .1570936 _cons | .107474 ? Proenvironmental attitudes Interaction: inc_meanXeduc has a positive effect… The education slope is bigger in wealthy countries Note: main effects change. “educ” indicates slope when inc_mean = 0 Interaction between country mean ine and individuallevel education Crosslevel Interactions . xtmixed supportenv age male dmar demp educ ine_dev inc_meanXeduc ses || country: ine_mean , mle cov(unstr) Randomeffects Parameters | Estimate Std. Err. [95% Conf. Interval] + country: Unstructured | sd(ine~n) | .5419256 .2095339 .253995 sd(_cons) | .8679172 corr(ine~n,_cons) | .0143006 + sd(Residual) | .0079307 LR test vs. linear regression: chi2(3) = Prob chi2 = ? Random part of output (cont?d from last slide) Random ponents: Ine_mean slope allowed to have random variation Interceps (“cons”) allowed to have random variation “cov(unstr)” allows for the possibility of correlation between random slopes amp。 intercepts… generally a good idea. Beyond 2level models ? Sometimes data has 3 levels or more ? Ex: School, classroom, individual ? Ex: Family, individual, time (repeated measures) ? Can be dealt with in xtmixed, GLLAMM, HLM ? Note: stata manual doesn?t count lowest level – What we call 3level is described as “2level” in stata manuals –xtmixed syntax: specify “fixed” equation and then random effects starting with “top” level ? xtmixed var1 var2 var3 || schoolid: var2 || classid:var3 – Again, specify unstructured covariance: cov(unstr) Crossed Effects / 2way models ? Sometimes data are not really nested… but crossed ? Example: Longitudinal data: Individuals nested within countries and years ? Strategies: – 1. Use a bination of fixed/random effects amp。 manual dummies ? Ex: Random effects for country, but dummies for years –2. Estimate “twoway” variance ponent model ? Random effects for country amp。 year ? In stata you have to do this manually (3level model) – See RabeHesketh amp。 Skrondal for an example. Advice about building models ? Raudenbush amp。 Bryk 2020 – Start building the level 1 model first – Then build level 2 model ? Keeping a close eye on level 2 N. Beyond Linear Models ? Stata can specify multilevel models for dichotomous amp。 count variables – Random intercept models ? xtlogit – logistic regression – dichotomous ? xtpois – poisson regression – counts ? xtnbreg – negative binomial – counts ? xtgee – any family, link… w/random intercept – Random intercept amp。 coefficient models – Plus, allows more than 2 levels… ? xtmelogit – mixed logit model ? xtmepoisson – mixed poisson model Shared Frailty Models: EHA ? Shared frailty model = random intercept in an event history model ? Stata: stcox var1 var2 var3, shared(clusterID) ? Cluster ID variable could be country id, school id, etc… ? Formula: Cox model with shared frailty )e x p()()( 0 iij uXthth ?? ?? Where ui is a random variable for i groups ? Parametric shared frailty models are similar… Shared Frailty Models: EHA ? Shared frailty (random effects) are useful for: – 1. Clustered data ? Just like prior examples – 2. Models with repeated events ? Repeated events is a kind of clustering within caseid ? Again, dummy variables (FEM) is a reasonable option – In stata, you?d have to enter the dummies manually ? Stata: specify