Multilevel Models Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison April 22 2008 1 10 Reminder about Corn Example We consider a subset of a larger data set on corn grown on the island Antigua The response variable we consider is the harvest weight harvwt per plot units unknown There are eight sites with eight separate plots within each site where the corn is grown under the same treatment conditions Multilevel Models Basics Model From Corn Example 2 10 Multilevel Model In a multilevel model we may have yi j i ei where i 1 64 indexes the observation and j i 1 8 indicates which of the eight sites contains the ith observation I I I j N 2 are modeled ei iid N 0 2 and 2 are unmodeled Multilevel Models Basics Model From Corn Example 3 10 Likelihood We can express the likelihood of the data in terms of the unmodeled and modeled parameters The normal density f x for mean and standard deviation is large when the x is close to The likelihood has the form J n Y Y L f j f yi j i j 1 i 1 The left factor is large when all of the j are close to The right factor is large when each j is close to the yi in group j The best estimate of will be the overall mean y all The likelihood is maximized at a balancing point when the j is somewhere between the overall mean y all and the sample means y j Multilevel Models Basics Model From Corn Example 4 10 Pooling Complete pooling is the estimate that assumes no difference between groups so that 1 2 J y all This corresponds to an extreme multilevel model where 0 No pooling estimates each j using only data from the jth sample 1 y j J y J This corresponds to an extreme multilevel model where Multilevel models correspond to partial pooling where data from other samples effects estimates for sample j but the data within sample j is most influential The estimated coefficients are a weighted average between the corresponding sample mean and the grad mean nj 1 y y all j 2 2 j nj 12 2 Multilevel Models Basics Model From Corn Example 5 10 Weighted Average j nj y j 12 2 nj 1 2 2 y all nj 2 nj 2 1 2 y j nj 2 1 2 1 2 y all An expression of the form w1 A w2 B where w1 w2 1 is a weighted average of A and B The relative distance of the average to the endpoints is inversely proportional to the weights If one weight is ten times the other the distance of the average to the end with the higher weight will be ten times smaller than to the other Multilevel Models Basics Model From Corn Example 6 10 Interpretations j nj y j 12 2 nj 1 2 2 y all Here when I I nj gets bigger more direct data in the group or 2 gets smaller more likely for yi to be close to group means j moves closer to y j When I 2 gets smaller more likely for all j to be close to each other j moves closer to y all Multilevel Models Basics Model From Corn Example 7 10 Numerical Example nj 8 for j 1 8 y all 4 29 1 55 0 87 y 1 4 88 1 4 86 Multilevel Models Basics Model From Corn Example 8 10 Site DBAN 8 0 87 2 1 1 55 2 10 57 0 42 10 57 0 42 4 88 4 29 4 86 10 99 10 99 The estimate is shrunk very little toward the grand mean Multilevel Models Basics Model From Corn Example 9 10 Site NSAN y 3 2 09 y all 4 29 10 57 0 42 2 09 4 29 2 17 10 99 10 99 The estimate is shrunk a little more toward the grand mean than for the other site since the same proportion of a longer distance is larger Multilevel Models Basics Model From Corn Example 10 10
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