Random Effects... and more about pigs G G G G G G G G G G G

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1 et s examine the random effects model in terms of the pig weight example. This had eight litters, and in the first analysis we were willing to think of as fixed effects. This means that we might want to breed the same animals again. et s express the model as Y ij = μ + i + ε ij, where we now think of i as the litter effect. We ll say that Var( i ) = σ. Reminder: When we did this as a fixed effects story, we got this output from Stat ANOVA One Way : One-way ANOVA: Weight versus itter Source DF SS MS F P itter Error Total S = R-Sq = 30.38% R-Sq(adj) = 0.% We also got confidence intervals separately for each litter, but we need not show that here. Now, with random effects, Var(Y ij ) = σ + σ, where σ is the variability contribution from the litter (mother) and σ is the variability within litter. We can get this analysis through Stat ANOVA General inear Model. Set up the panel as follows: 1

2 The output is the following: Random Effects... and more about pigs General inear Model: Weight versus itter Factor Type evels Values itter random 8 1,, 3, 4, 5, 6, 7, 8 Analysis of Variance for Weight, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P itter Error Total S = R-Sq = 30.38% R-Sq(adj) = 0.% Unusual Observations for Weight Obs Weight Fit SE Fit Residual St Resid R R R denotes an observation with a large standardized residual. * WARNING * No multiple comparisons were calculated for the following terms which contain or interact with random factors. itter Observe that the analysis of variance table is (here) exactly the same as for the fixed effect version. In this case, however, the F is a test of H 0 : σ = 0. In complicated unbalanced designs, constructing tests of random effects can get very tricky, and it will go way beyond this course. But there was more to this experiment. The litters were sired by two different boars, whom we named Herman and Mortimer. We would probably want to regard the boars as fixed effects. We would then note that the litters are nested within this classification. The specific model is Y ijk = weight from (boar = i, litter = j, piglet k within litter) = μ + β i + ij + ε ijk The letters have been chose to reflect the identities of the effect (as β for boar and for litter ). We have Var( ij ) = σ and Var(ε ijk ) = σ. Yes, we can get an analysis for this too. Use Stat ANOVA General inear Model and set up the panel as indicated.

3 Note that * This names litter as nested within boar. That s the c1(c3) notation, which you can think of as saying that the boar factor contains the random litter factor. * The litter is named as a random factor. This is the output: General inear Model: Weight versus Boar, itter Factor Type evels Values Boar fixed Herman, Mortimer itter(boar) random 8 1, 3, 4,, 5, 6, 7, 8 Analysis of Variance for Weight, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Boar x itter(boar) Error Total x Not an exact F-test. S = R-Sq = 30.38% R-Sq(adj) = 0.% Unusual Observations for Weight Obs Weight Fit SE Fit Residual St Resid R R R denotes an observation with a large standardized residual. * WARNING * No multiple comparisons were calculated for the following terms which contain or interact with random factors. (Boar)itter 3

4 There is the interesting note that the F test for Boar is not exact. Since F = MS Treatment, we see MSError that the corresponding MS Error must have been , a number which does not 7.50 appear in the analysis of variance. Here s what s going on. The MS Treatment is estimating something like ( coefficient ) ( coefficient ) ( boar effect) σ + σ + 1 but there is no MS line in the table that estimates ( coefficient ) σ + σ to be used as 1 a denominator. The software constructs such a statistic from SS itter(boar) and SS Error, but it gets for the denominator a quantity which is not independent of the numerator. Thus the resulting F test is only approximate. In addition, there is no obvious way to set up multiple comparisons between the boars. Can we get a test directly from first principles? Get back to the model Y ijk = weight from (boar = i, litter = j, piglet k within litter) Average over litter: = μ + β i + ij + ε ijk Y ij = μ + β i + ij + ε ij Note that Var(Y ij ) = σ σ +. The imbalance, meaning unequal n ij values, makes it n ij impossible to construct a standard test. 4

5 Suppose that you would also want to declare the boars to be random. This changes the model to Y ijk = weight from (boar = i, litter = j, piglet k within litter) = μ + b i + ij + ε ijk Now the use of b i (rather than β i ) reminds us that the boar effect is random. We have Var(b i ) = σ, Var( ij ) = σ and Var(ε ijk ) = σ. B Here is the Stat ANOVA General inear Model panel: 5

6 Here s the output: General inear Model: Weight versus Boar, itter Factor Type evels Values Boar random Herman, Mortimer itter(boar) random 8 1, 3, 4,, 5, 6, 7, 8 Analysis of Variance for Weight, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Boar x itter(boar) Error Total x Not an exact F-test. S = R-Sq = 30.38% R-Sq(adj) = 0.% Unusual Observations for Weight Obs Weight Fit SE Fit Residual St Resid R R R denotes an observation with a large standardized residual. * WARNING * No multiple comparisons were calculated for the following terms which contain or interact with random factors. (Boar)itter The numbers come out the same, but the interpretations have changed. In the Boar line.. with boar as a random effect, the test is about H 0 : σ B = 0 with boar as a fixed effect (as earlier), the test is about H 0 : β 1 = β. 6