Multilevel Models: BasicsModel From Corn ExampleMultilevel ModelsBret LargetDepartments of Botany and of StatisticsUniversity of Wisconsin—MadisonApril 22, 20081 / 10Reminder about Corn ExampleWe consider a subset of a larger data set on corn grown on the islandAntigua.The response variable we consider is the harvest weight (harvwt) perplot (units unknown).There are eight sites with eight separate plots within each site wherethe corn is grown under the same treatment conditions.Multilevel Models: Basics Model From Corn Example 2 / 10Multilevel ModelIn a multilevel model, we may haveyi= αj[i]+ eiwhere i = 1, . . . , 64 indexes the observation and j[i] = 1, . . . , 8indicates which of the eight sites contains the ith observation.Iαj, ∼ N(µα, σ2α) are modeled;Iei∼ iid N(0, σ2);Iµα, σα, and σ2are unmodeled.Multilevel Models: Basics Model From Corn Example 3 / 10LikelihoodWe can express the likelihood of the data in terms of the unmodeledand modeled parameters.The normal density f (x | µ, σ) for mean µ and standard deviation σis large when the x is close to µ.The likeliho od has the formL =JYj=1f (αj| µα, σα) nYi=1f (yi| αj[i], σ)!The left factor is large when all of the αjare close to µα.The right factor is large when each αjis close to the yiin group j.The best estimate of µαwill be the overall mean ¯yall.The likeliho od is maximized at a balancing point when the αjissomewhere between the overall mean ¯yalland the sample means ¯yj.Multilevel Models: Basics Model From Corn Example 4 / 10PoolingComplete pooling is the estimate that assumes no difference betweengroups so thatα1= α2= · · · = αJ= ¯yallThis corresponds to an extreme multilevel model where σα= 0.No pooling estimates each αjusing only data from the jth sample.α1= ¯yj, . . . , αJ= ¯yJThis corresponds to an extreme multilevel model where σα= +∞.Multilevel models correspond to partial pooling where data from othersamples effects estimates for sample j, but the data within sample j ismost influential.The estimated coefficients are a weighted average between thecorresponding sample mean and the grad mean.ˆαj≈njσ2¯yj+1σ2α¯yallnjσ2+1σ2αMultilevel Models: Basics Model From Corn Example 5 / 10Weighted Averageˆαj≈njσ2¯yj+1σ2α¯yallnjσ2+1σ2α= njσ2njσ2+1σ2α!¯yj+ 1σ2αnjσ2+1σ2α!¯yallAn expression of the formw1A + w2Bwhere w1+ w2= 1 is a weighted average of A and B.The relative distance of the average to the endpoints is inverselyproportional to the weights.(If one weight is ten times the other, the distance of the average tothe end with the higher weight will be ten times smaller than to theother.)Multilevel Models: Basics Model From Corn Example 6 / 10Interpretationsˆαj≈njσ2¯yj+1σ2α¯yallnjσ2+1σ2αHere, when:Injgets bigger (more direct data in the group), or;Iσ2gets smaller (more likely for yito be close to group means)ˆαjmoves closer to ¯yj.WhenIσ2αgets smaller (more likely for all αjto be close to each other),ˆαjmoves closer to ¯yall.Multilevel Models: Basics Model From Corn Example 7 / 10Numerical Examplenj= 8 for j = 1, . . . , 8ˆµα= ¯yall= 4.29ˆσα= 1.55ˆσ = 0.87¯y1= 4.88ˆα1= 4.86Multilevel Models: Basics Model From Corn Example 8 / 10Site DBAN8(0.87)2= 10.571(1.55)2= 0.4210.5710.99(4.88) +0.4210.99(4.29) = 4.86The estimate is shrunk very little toward the grand mean.Multilevel Models: Basics Model From Corn Example 9 / 10Site NSAN¯y3= 2.09¯yall= 4.2910.5710.99(2.09) +0.4210.99(4.29) = 2.17The estimate is shrunk a little more toward the grand mean than forthe other site since the same proportion of a longer distance is larger.Multilevel Models: Basics Model From Corn Example 10 /
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