Outline 1 Testing random effects ML and REML Testing random effects 2 Testing fixed effects Testing fixed effects with LRT Marmoset example Testing fixed effects with repeated measures Maximum Likelihood ML estimation Likelihood probability of the data given a specific model and parameter values Model for the corn example yi j i ei with j N 0 2 ei N 0 e2 where j i site i plot Marmoset example or more generally fixed effects and one or more level s of random effects yi X k j i j i ei with k N 0 2 j N 0 2 ei N 0 e2 Maximum Likelihood estimation We estimate fixed effects or and variances of random effects e etc by minimizing 2 log Likml n log 2 log V z n log e2 R t V 1 R z RSS e2 in standard regression where V V e is a matrix that depends on the variance components and R Y X is the vector of residuals when the predicted values are based on fixed effects only Maximum Likelihood estimation If we know the variance components with certainty then has a normal distribution if the model is true We use 1 var X t V 1 X to test hypotheses and get CI Corn example 4 29 with estimated SE of 0 56 Marmoset male 7 1 s with estimated SE of 22 7 s The estimates e2 2 etc are not normally distributed Biased estimates of e2 2 etc Most simple random sample yi ei With P case one 2 2 ML e yi y n which is too small on average We usually use Restricted Maximum Likelihood REML estimation Provides unbiased estimates of variance components if the model is correct Roughly restrict the data to n p modified observations which are independent of Then maximize the likelihood of these restricted modified observations Most simple case one random sample yi ei ei N 0 e2 We can restrict the data to the n 1 observations y2 y1 y3 y2 yn 1 yn Their distribution is independent of depends on e2 only The ML estimate of e2 based on these restricted data is X e2 yi y 2 n 1 Restricted Maximum Likelihood REML estimation P Standard linear regression we get 2 yi y 2 n p where p is the number of coefficients n p residual df We ve been using REML for a long time Mixed models we minimize the REML criterion 2 log Likreml n p log 2 log V R t V 1 R log X t V 1 X where the matrix V V 2 e2 is as before residuals R Y X as before X are the predictors with fixed effects ML versus REML REML based variances good for unbiased variance estimates same residual variance estimate as we have used before on fixed effect only models same variance estimates as ANOVA methods when the design is balanced more later Comparing ML and REML The extra term in REML last one depends on X The REML criterion can only be compared across models that have the same X to ensure we compare models based on the same restricted data ML the only method to perform likelihood ratio tests when testing fixed effects ML versus REML in R REML is the default lmer harvwt 1 site data corn Linear mixed model fit by REML AIC BIC logLik deviance REMLdev 194 9 201 4 94 47 189 6 188 9 Random effects Groups Name Variance Std Dev site Intercept 2 41652 1 55452 Residual 0 76477 0 87451 Number of obs 64 groups site 8 Fixed effects Estimate Std Error t value Intercept 4 2917 0 5603 7 659 ML versus REML in R Variances and SE are slightly smaller with ML lmer harvwt 1 site data corn REML FALSE Linear mixed model fit by maximum likelihood AIC BIC logLik deviance REMLdev 195 5 202 94 77 189 5 189 Random effects Groups Name Variance Std Dev site Intercept 2 10251 1 45000 Residual 0 76477 0 87451 Number of obs 64 groups site 8 Fixed effects Estimate Std Error t value Intercept 4 2917 0 5242 8 188 Testing random effects Corn example We might want to test the presence of site effects H0 0 versus the alternative HA 0 This question is about the whole population of sites not just the 8 sampled sites Warning variance estimates are usually not normally distributed For one thing they are always positive That s why lmer does not output any SE for variance components 2 2SE would a bad confidence intervals Testing random effects with a Likelihood ratio test Compare a simple null model with an alternative model which has k more variance parameters Test statistic X 2 2 log Lik null 2 log Lik alternative What is its null distribution Chi square based p value compare X 2 to 2df k But the chi square distribution is inappropriate when we test a borderline parameter 2 0 is borderline The resulting p value is too large the conclusion is conservative In simple cases the appropriate distribution is X 2 0 with 50 chance and X 2 21 with 50 chance If so the appropriate p value is half that obtained by comparing x 2 to 21 Parametric bootstrap based p value simulate data under the null model 2 0 to get the true distribution of X 2 under H0 More computer intensive but more accurate Flowering time Testing the inventory effect We can use the REML likelihood or the ML likelihood here anova uses ML on lmer models anova provides the chi square based p value fit lmer logdtf 1 subspecies 1 inventoryID data brassica2 fit noinventory update fit 1 inventoryID anova fit fit noinventory Data brassica2 Models fit noinventory logdtf 1 subspecies fit logdtf 1 subspecies 1 inventoryID fit noinventory fit Df AIC BIC logLik Chisq Chi Df Pr Chisq 3 23 33 14 66 14 66 4 117 97 106 40 62 98 96 64 1 2 2e 16 Since p 2 2 10 16 is conservative too high we can confidently say that there is strong evidence for an inventory 2 effect very strong evidence that inventory 0 Corn testing the site effect Need to use lm instead of lmer for the null model because it has no random effect make sure we use the same criterion in both LRT with p value based on the chi square distribution corn lmer lmer harvwt 1 site data corn corn lm lm harvwt 1 data corn x2 2 logLik corn lm REML T 2 logLik corn lmer REML T x2 1 61 368 pchisq x2 df 1 lower tail F 1 4 734755e 15 The appropriate p value would be even lower Strong evidence 2 0 that site Compare with the ANOVA F test on fixed effects anova lm harvwt site data corn Df Sum Sq Mean Sq F value Pr F site 7 140 679 20 0969 26 278 1 551e 15 Residuals 56 42 827 0 7648 Pygmy marmosets Testing group effects 5 populations 2 3 groups population 2 individuals group M F Response we …