Generalized Inference with Application to Small Sample Situations Sam Weerahandi

1. Generalized Inference with Application to Small Sample SituationsSam Weerahandi (Joint Work with Kawai, Yu et al., Mathew et al.)

2. Outline • Motivation: Why Generalize? • Problems with Classical & Bayesian Inferences • Introduction to Generalized inference • About Mixed Models • Mixed Models: An Overview • Issues with MLE based Inference • Application: BLUP in Mixed Models • Performance Comparison • An Application

3. Motivation: Why Generalize Classical InferenceSTAT 200 teach how to make ANY inference! Really? • Classical Approach to Inference (tests, confidence intervals, etc) works fine with mean μ, variance σ2 of Normal distribution • But it fails (MLE based inferences are asymptotics) with • most functions of the mean and variance, except for a few simple functions • advanced models such as Mixed Models and ANOVA with unequal variances • Classical Approach also fails to give small sample inferences with non-normal distributions: • Some functions of parameters of Uniform distribution, U(α,β), • Scale parameter of Gamma distribution, parameters of Weibull distribution, etc. • One can find various solutions in the literature, but approach vary from one to another • What is desirable is a systematic approach that works with greater class of functions of parameters

4. Further Issues With Classical Inferenceand Bayesian Inference • Classical Inference can provide only large sample inference for • ANOVA with unequal variances • Variance Components in Mixed Models • BLUPs in Mixed Models • Classical Inference could yield wrong signs in Small Sample Inference • In Multi-regional clinical trials, some regions could yield negative dose response due to chance • Estimated response to a TV Ad could become negative in some markets even if there is no reason why the Ad would alienate any demographic segment • Bayesian Inference can provide small sample inference, but • You need a prior • When non-informative prior with such algorithms as MCMC is used, it • takes days to estimate when model has large number of parameters • yields fairly different estimates somewhat different hyper parameters are used • yields fairly different estimates with different families of priors • Why not take the classical approach, but think like Bayesians?

5. Motivation (ctd.) • Multi-regional clinical trial example (ctd.) • If you run LSE you may not even get the right sign for some Regions • The problem could be alleviated using the same data in Mixed Model setting • Then you will get much more reasonable estimates (rather BLUPs), • In fact, LSE could yield the wrong sign even with two parameters: • Simulation from exact model, Y = 10 + .05 X +e when sample size 500 and e~N(0,1): • Mixed Models and BLUP (Best Linear Unbiased Predictor) are heavily used in high noise & small sample applications • 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

6. An 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 Inference: A Simple Example • Suppose, you have sample from X~N(μ, σ2 ) • How to make Inferences about ρ = μ/σ, the coefficient of variation based on Sample mean and the Sample Variance? • Despite simple distributional results and if you start out with the MLE, /S , it will lead to just asymptotic inferences • But note that is a Generalized Pivotal Quantity (GPQ), because (i) at the observed values R reduces to ρ, (ii) The distribution of R is free of unknown parameters • So any inference is possible. For example, Pr(R≤ ρ) yields an exact one-sided Generalized Confidence interval (GCI) • In fact above is a Classical CI, but MLE failed to produce it • Note: Exact CI always does not exist, but still you may be able to obtain an exact GCI. In such cases GCI tend to outperform more complicated approximations in terms of Repeated Sampling Properties

8. Generalized Inference (ctd.) • 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 • GE can produce inferences based on exact probabilities for Distributions such as Gamma, Weibull, Uniform • To do so you DO NOT need Prior or deal with hyper parameters • Read more about Generalized Inference at www.weerahandi.org and even read my second book FREE!

9. Application 1: Small Response Estimation when parameter sign is known • Problem with known sign of parameters of ten arise in practice: • Price Elasticity of demand • Response to promotional tactics • Difference between a Treatment and Placebo effects • Adverse effect of a treatment • Assume that a regression parameter, q is supposed to be positive; • Let be LSE of q. Then T= • Suppose q>a (e.g. a=0 if sign is known). • Kim (2008) showed that the Bayesian Estimate under appropriate non-informative prior is • The above estimate is always positive • The same estimate can be obtained by considering the Generalized Pivotal Q= - (T- q) with observed value q and taking the conditional expectation E(Q|Q>a)

10. Small Response Estimation (ctd) MSE Performance when s=1 Moreover, such classical estimators can be further improved by taking Stein (1961) like approach Consider the class of estimators of the form Find ks by Stein approach The resulting estimator is denoted as IGE As evident from the MSE (mean Squared Error) comparison IGE is uniformly better than LSE when the parameter is known to be positive MLE (truncated) can also be improved upon In Interval Estimation the approach provides shorter intervals

11. Applications in Mixed Models • 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 Field Rep Detailing • 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, the BLUP has replaced the LSE as the most widely used statistical technique

12. 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

13. 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

14. Problem in Mixed Model Inference • 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

15. Estimating Variance Components and BLUPs • 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

16. 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”)

17. Comparison of Variance Estimation Methods:Performance of ML/REML Vs. GE (ctd.) • Table below shows MSE performance of competing estimators of factor variance • Note that • Generalized estimate is better than any other estimate • REML is not as good as ML • For estimation of the BLUP, Yu, Zou, Carlson, and Weerahandi (2013) provides similar improvements over the ML and REML • GE based methods do not suffer from the zero variance drawback of ML and REML

18. 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

19. Application: Estimation of Response to TV Ads by Market • Data Preparation: • Obtain TV GRP and weekly/monthly Sales data by market • Ad-stock (e.g. http://en.wikipedia.org/wiki/Advertising_adstock) TV GRP • Obtain data for other variables that you want to control for • De-mean all variables including ad-stocked GRP • Approach to Modeling: • Model Sales or log sales as a linear function of all explanatory variables, including trend and seasonality in sales • Model the coefficients of ad-stocked GRP as random effects around the national average • Estimate the parameters of the Mixed Model by such methods as ML if there is no convergence problem, and by proposed generalized method otherwise • Use estimated responses to TV to write down the profit function • Demo