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Resolving Inference Issues in Mixed Models

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Sam WeerahandiSam Weerahandi

April 2013

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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

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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

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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

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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

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

se

ise

kVV

MkVMV

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

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Introduction to Generalized Inference

Classical Pivotals for interval estimation are of the form Q=Q(X, )

Generalized Inference on a parameter , is a generalized pivotal of the form Q=Q(X, x, ) that is a function of Observable X, observed x, and nuisance parameters satisfying Q(x,x, , ) is free of having a distribution free of

Classical Extreme Regions are of the form Q(X, )<Q(x, ) cannot produce all extreme regions

Q( X,x, )< Q( x,x, 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, )= , then is estimated using E(Q), the expected value of Q, Median of Q, etc.

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Generalized Estimation (GE)

The case Q(x,x, )= is too restrictive except in location parameters

More generally, if Q(x,x, , ) = 0, then the solution of E{Q(X,x,)}=0 is said to be the Generalized Estimate of

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!

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Estimating Variance Components and BLUP

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

For simplicity consider a balanced Mixed Model The inference problems in canonical form reduces to:

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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”)

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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

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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