Resolving Inference Issues in Mixed Models by Sam Weerahandi April 2013 Draft – Work in Progress– Confidential – Do Not Distribute 1 Outline Why Mixed Models are Important Mixed Models: An Overview Issues with MLE based Inference Introduction to Generalized Inference Application: BLUP in Mixed Models Performance Comparison 2 Why Mixed Models Are Important! Mixed Models are especially useful in applications involving large samples with noisy data small samples with low noise In Clinical Research & Public Health Studies, Mixed Model can yield results of greater accuracy in estimating effects by treatment levels Patient groups In Sales & Marketing Mixed Models are heavily used to estimate Response due to promotional tactics: – Advertisements (TV, Magazine, Web) by Market – Doctors Response to Detailing/Starters. In fact, if you don’t use Mixed Models in this type of applications you may get unreliable or junk estimates, tests, and intervals So, BLUP (and hence SAS PROC MIXED) has replaced LSE as the most widely used statistical technique by Management Science groups of Pharmaceutical companies, in particular 3 An Example Suppose you are asked to estimate effect of a TV/Magazine Ad by every Market/District using a model of longitudinal sales data on ad-stocked exposure If you run LSE you may not even get the right sign of estimates for 40% of Markets If you formulate in a Mixed Model setting you will get much more reliable estimates So, use Mixed Models and BLUP instead of LSE Mixed Models and the BLUP (Best Linear Unbiased Predictor) are heavily used in high noise & small sample applications In analysis of promotions, SAS Proc Mixed or R/S+ Lme is used more than any other procedure But REML/ML frequently yield zero/negative variance components BLUPs fail or all become equal REML/ML could be inaccurate when factor variance is relatively small 4 Overview of Mixed Models Suppose certain groups/segments distributed around their parent Assumption in Mixed Models: Random effects are Normally distributed around the mean, the parent estimate, say M Suppose Regression By Groups yield estimate Mi for Segment i Let Vs be the between segment variance and Ve be the error variance, which are known as Variance Components It can be shown that the Best Unbiased Predictor (BLUP) of Segment i effect is Ve M kVs M i Ve kVs a weighted average of the two estimates, and k is a known constant that depends on sample size and group data The above is a shrinkage estimate that move extreme estimates towards the parent estimate 5 Problem BLUP in Mixed model is a function of Variance Components Classical estimates of Factor variance can become negative when noise (error variance) is large and/or sample size is small Then, ML and REML fails: PROC Mixed will complaint about non-convergence or will yield equal BLUPs for all segments I tried the Bayesian approach with MCMC, but when I did a sanity check (i) by changing the hyper parameters OR (ii) by using Gamma type prior in place of log-normal, I got very different estimates After both the Classical & Bayesian Approaches failed me, I wrote a paper about “Generalized Point Estimation”, which can Assure estimates fall into the parameter space Can take advantage of known signs of parameters without any prior Can improve MSE of estimates by taking such classical methods as Stein method 6 Introduction to Generalized Inference Classical Pivotals for interval estimation are of the form Q=Q(X, q) Generalized Inference on a parameter q, is a generalized pivotal of the form Q=Q(X, x, q,z) that is a function of Observable X, observed x, and nuisance parameters satisfying Q(x,x, q, z) is free of z having a distribution free of z Classical Extreme Regions are of the form Q(X, q0)<Q(x, q0) cannot produce all extreme regions Q( X,x, q0, z)< Q( x,x, q0, z) greater class of extreme regions Generalized Test and Intervals are based on exact probability statements on Q Generalized Estimators are based on transformed Generalized Pivotals If Q or a transformation satisfy Q(x,x, z)= q, then q is estimated using E(Q), the expected value of Q, Median of Q, etc. 7 Generalized Estimation (GE) The case Q(x,x, z)= q is too restrictive except in location parameters More generally, if Q(x,x, q, z) = 0, then the solution of E{Q(X,x,q,z)}=0 is said to be the Generalized Estimate of q Note: As in classical estimation, one will have a choice of estimates and need to find one satisfying such desirable conditions as minimum MSE Major advantage of GE is that, as in Bayesian Inference, it can assure, via conditional expectation, any known signs of parameters Variance components are positive Variance ratio in BLUP is between 0 and 1 Can produce inferences based on exact probabilities for Distributions such as Gamma, Weibull, Uniform To do so you DO NOT need Prior or specify values of hyper parameters Read more about Generalized Inference at www.weerahandi.org and even read my second book FREE! 8 Estimating Variance Components and BLUP For simplicity consider a balanced Mixed Model The inference problems in canonical form reduces to: Generalized approach can produce the above estimate or better estimates Generalized pivotal quantity is a Generalized Estimator and E(Q)=0 yields the classical estimate But the drawback of the classical estimate is that MLE/UE frequently yields negative estimates The conditional E(Q|C)=0 with known knowledge C yields BLUPs are then obtained as weighted average Least Squares Estimates of Parent and Child 9 Comparison of Variance Estimation Methods (based on 10,000 simulated samples): Performance of MLE Vs. GE Assume One-Way Random Effects model with k segments n data from each segment Degrees of freedom a=k-1 and e=n(k-1) The variance component is estimated by the MLE and GE Note that with small sample sizes MLE/UE yield negative estimates for Variance Component In such situations SAS does not provide estimates or BLUP (just say “did not converge”) 10 Comparison of Variance Estimation Methods: Performance of ML/REML Vs. GE (ctd.) Table below shows MSE performance of competing estimates Note that Generalized estimate is better than any other estimate REML is not as good as ML Only GE can yield unequal BLUPs with any sample 11 Further Issues with BLUP ML and REML Prediction Intervals for BLUP are highly conservative: Actual coverage of 95% intended intervals area as large as 100% This implies serious lack of power in Testing of Hypotheses The drawback prevails unless number of groups tend to infinity Generalized Intervals proposed by Mathew, Gamage, and Weerahandi (2012) can rectify the drawback Table below shows Performance of competing estimates 12