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Lecture 5 “additional notes on crossed random effects models” Clustered versus non clustered random effects (Chap 11, new edition) • We have discussed higher-level hierarchical models where units are classified by some factors (for example schools) into top level clusters at level L. • The units in each top level cluster are then (sub)classified by a further factor (for example class) into clusters at level L-1. • The factors defining the classifications are nested in the same sense that a lower-level cluster can only belong to one higher level cluster (for example a class can only belong to one school) Non hierarchical models but random effects models • We now discuss non hierarchical models where units are cross-classified by two or more factors, with each unit potentially belonging to any combination of levels of the different factors Non Hierarchical Models • So far, we have treated occasions nested within individuals • However, if all individuals are affected similarly by some events or characteristics associated with the occasions, such as weather conditions, strikes, new legislation etc.. It seems reasonable to treat occasions as crossed with individuals, or to consider a “main effect” of time. Non hierarchical models • Factors are not always completely crossed. For example, the high schools and elementary schools attended by students are not clustered, but there are many combinations of high school and elementary school that do not occur in practice, perhaps because the schools are in different geographical regions. A psychological experiment with two potentially interacting factors (Gelman, sec 13.5) • Let y jk denotes the success rate of a pilots training on a flight simulator (j=1,2,3,4,5) in airport (k=1,….,8). • These 40 data points have two groupings - treatments and airports which are not nested Non nested random effects model Treatment random effects y jk Airport random effects j k jk j ~ N (0, ) 2 k ~ N (0, ) 2 jk ~ N (0, ) 2 j 1,2, 3, 4,5 k 1,..., 8 Estimates of the variance components ˆ 0.23 • ˆ 0.04 • 2 2 ˆ 0.32 2 The variance of the success rates is huge among airports - even larger than among the individual measurements. Whereas there is almost no differences across treatments How much do primary and secondary schools afflict attainment at age 16? • Data are cross-classified by 148 primary schools (elementary schools) and 19 secondary schools (middle/high schools) (fife.dta) • attain: attainment score at age 16 ( y ijk ) • pid: identifier for primary school (up to age 12) ( k ) • sid: identifier for secondary school (from age 12) ( j ) • vrq: verbal reasoning score from test taken in the last year of primary school • sex: gender (1:female; 0:male) Data characteristics • First, not every combination of primary and secondary school exists. • Second, many combinations of primary and secondary schools occur multiple times • For instance, students that attend elementary school 1 ended up in 3 secondary schools (1,9,18) • There are at most 6 secondary schools per primary schools, and for 90% of the primary schools there are at most 3 secondary schools per primary school • There are between 7 and 32 primary schools per secondary school, the median being between 13 and 14 An additive crossed random effects Attainment score at age 16 for model student i who went to secondary school j and primary school k Average score Random effects y ijk 1 1 j 2 k ijk 1 j ~ N (0, 1 ) 2 k ~ N (0, 2 ) Variance across secondary schools Variance across primary schools ijk ~ N (0, ) Residual variance Estimation using xtmixed Results for the additive model • The estimated standard deviation of the primary school random effect ( 2 ) is 1.06, which is considerably larger than the estimated standard deviation of the secondary school random effect, given by 0.59 ( 1 ) • Therefore elementary schools appear to be more variable in their effects than secondary schools. However neither of these estimates are precise • The standard deviation of the ( ijk) is estimated as 2.85( ). This number reflects any interactions between primary and secondary schools from the means implied by the additive effects and variability within groups of children belonging to the same combination of primary and secondary school Including a random interaction • For many combinations of primary and secondary school, we have several observations because more than one child attended that combination of schools y ijk 1 1 j 2 k 3 jk ijk 1 j ~ N (0, 1 ) 2 k ~ N (0, 2 ) 3 jk ~ N (0, 3 ) ijk ~ N (0, ) The random interaction term • The interaction term takes on a different value for each combination of secondary and primary school to allow the assumption of additive random effects to be relaxed. • For example, some secondary schools might be more beneficial to children who attended particular elementary schools, perhaps because of similar instructional practices • We could not include interaction terms in the pilot example, because there we have only one observation for each treatment, airport combination Intraclass correlations ( primary ) cor ( y ijk , y i' j ' k ) 2 IC among children for the same primary schools but different secondary schools 1 2 3 (sec ondary ) cor ( y ijk , y i' jk ' ) 1 IC among children for the same secondary schools but different primary schools 1 2 3 IC among children for the same primary and secondary schools ( primary ,sec ondary ) cor ( y ijk , y i' jk ) 1 2 3 1 2 3 ( primary | sec ondary ) cor ( y ijk , y i' jk | j ) Given the secondary school, this denotes the IC correlation among children that had the same primary school 2 3 2 3 Diagnostics • We can obtain the empirical Bayes estimates of both primary and secondary school random effects. If the model is correct, there EB estimates should have a normal distribution • We assess the normality of the EB estimates using a QQ plot