Mixed Models II - Behind the Scenes Report Revised June 11, 2002 by G. Monette

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1 Mixed Models II - Behind the Scenes Report Revised June 11, 2002 by G. Monette What is a mixed model "really" estimating? Paradox lost - paradox regained - paradox lost again. "Simple example": 4 patients each observed at 3 dosages of a drug for depression: Q: Is the drug effective? Problem: Data "observational" not "experimental". Dosage might be affected by severity of illness. 1

2 OLS approaches: 1) Pooled (ignore subjects) Drug bad Model: Y X 2) Aggregate + regress Model Ȳ X By Subject Drug even worse 2

3 3) Within Subject Y X + Sub Drug is good Note: 2 ecological correlation 2 vs 3: Robinson s Paradox e.g. Life Expectancy vs. Smoking aggregated by country 1 vs 3: Simpson s Paradox Call ŝlope in 1. ˆγ Pooled 2. ˆγ Between 3. ˆγ Within Let W B be weight (precision) of ˆγ B W W be weight (precision) of ˆγ W Then ˆγ P =(W B + W W ) 1 (W Bˆγ B + W W ˆγ W ) 3

4 So, ˆγ P is an optimal combination of ˆγ B and ˆγ W Note: "optimal" in common model is connect. In our example ˆγ W is probably a "good" estimator, ˆγ B is NOT. So, ˆγ P mixes the good and the bad. What does a Mixed Model do? Does it give us ˆγ W? Or ˆγ P? Answer: Something in between. With a random intercept model: Y X / 1 Sub ˆγ MM is also an "optimal" combination of ˆγ W and ˆγ B If Y ij = γ 00 + γ 10 X ij + u 0 j + ε ij Var: τ 00 σ 2 then, σ 2 /n Weight on ˆγ B = σ 2 OLS weight /n + τ 00 Weight on ˆγ W = OLS weight I.E. [ ˆγ MM =( ) 1 σ 2 ] /n W Bˆγ σ 2 /n + τ B + W W ˆγ W 00 Note: Here n is # of obs/subject. So, [ ˆγ MM =( ) 1 σ 2 ] /n W Bˆγ σ 2 /n + τ B + W W ˆγ W 00 If σ 2 /n very small / τ 00 ( 0) = ˆγ W 4

5 If σ 2 /n big / τ 00 ( 1) = ˆγ P So ˆγ MM is between ˆγ P and ˆγ W depending on σ 2 / (nτ 00 ) If the between subject effect is not the same as the within subject effect (often the case with observational data) then MixedModelcangivebiasedresults. In our example: 5

6 To the Rescue: Contextual variable and within subject effects. Idea: decompose X ij : X ij = X j +(X ij X j ) }{{} mean within S deviation from within subject mean Use X j as a level-2 variable and X ij X j as a level-1 variable Consider OLS: E(Y )=γ 0 + γ 1 X j + γ 2 ( X ij X j ) What is γ 1? E(Y )/ X j when X ij X j =0 i.e. between S effect when X ij = X j What is γ 2? Answer: ˆγ 2 =ˆγ W! 6

7 So a possible approach for Mixed Models: 1. Transform inner variables into outer means and inner deviations. Handy trick for small datasets: PROC GLM; CLASS SUB; OUTPUT OUT = dsname P = X_B R = X_W; 2. Model with both 3. Test equality of parameters. If equal can revert to raw inner variable. 4. If not equal, rich possibilities for interpretation. Note: Much discussion on "centering" Should we use X ij X j or X ij X Often seems inconclusive. [See centering on X min in next section] Problem: Centering X ij defines γ for X j has little effect on γ for X ij - "center"!! Essentially 1. Controlling for X j defines γ for X ij whatever 2. Controlling for X ij whatever defines γ for X j Compositional vs contextual effects We have three variable to consider (note that we can start with GMC (grand mean centering) of X ij. This only affect the meaning of the intercept (and effects of other interacting variables) but does affect the effects below. 3variables: 7

8 1. X ij (raw variables) 2. X ij X j (CWC: centered within contexts) 3. X j (contextual mean) Note that we can t use all 3 variables (why not?) 3effects: 1. γ W within context 2. γ B between contexts = compositional effect 3. γ C contextual effect Diagram: (R&B p. 140) How to get what: γ B = γ W + γ C Raw CWC Mean Variables: X ij X ij X j X j Contextual model: γ W γ C Compositional model: γ W γ B 8

9 Hausmanspecificationtest: (S&Bp. 87) Random vs Fixed: Is γ X in random intercept model an unbiased estimator of γ W? Idea: equivalent to testing γ C =0. If so, then γ B = γ W and random intercept model is ok. If not, then use fixed effects model. Thirdchoice:use contextual or compositional model. Contextual or compositional? Fixed part is equivalent (same except for labelling of effects) since X C = X B A for some non-singular matrix A. Equivalently we note that span(x C )=span(x B ) But the random part is different. Two random models are equivalent if there is a single A such Z Cj = Z Bj A Consider using X ij X j and 1 in the random model instead of X ij and 1: Two solutions: 9

10 1. Choose according to desired random model. (S&B prefer contextual, p. 80ff.) Then estimate desired parameters with ESTIMATE statements. 2. Choose variables for fixed part to get estimates you want. Choose variables for random part according to desired pattern of variance. Variables don t have to be the same! As long as the fixed model is equivalent to a model with the random effects, then the models are equivalent. 10

11 Multivariate EBLUPs eg. β 0j = γ 00 u 0j β 1j = γ 10 u 1j with: Let estimation of be OLS 11

12 Combine OLS with Empirical Prior Amount and direction of shrinkage depends on shape of T. Note what happens if we drop a random effect: Note: Shrinkage of picture at works the same way. Just centre Note: If T is "small" in some direciton, collapse into T. 12

13 Appendix : Creating contextual variables in large data sets: Suppose we want to create a contextual variable and a within-cluster deviation for a variable X DATA MYDATA; INPUT GROUP X; CARDS; ;; PROC MEANS; BY GROUP; VAR X; OUPUT OUT=NEW MEAN = M; RUN; DATA NEW; SET NEW; KEEP M GROUP; DATA TOGETHER; MERGE MYDATA NEW; BY GROUP; RUN; PROC PRINT DATA TOGETHER; RUN; 13