[R-sig-ME] AIC in Cross-Classified Model

[Roberts, Kyle]

Dear All,

I was working an example for my class and came upon a problem. This example is the cross-classified model presented by Hox on p 127 of "Multilevel Analysis." The problem is that the AIC and BIC in the summary of the model are not the same AIC and BIC when the anova function is used. Any idea as to why? I think that the anova function gives the correct AIC, but I don't know why. I threw the dataset up on my website if anyone wants to replicate.

> sessionInfo()
R version 2.7.2 (2008-08-25) 
i386-pc-mingw32 

locale:
LC_COLLATE=English_United States.1252;LC_CTYPE=English_United States.1252;LC_MONETARY=English_United States.1252;LC_NUMERIC=C;LC_TIME=English_United States.1252

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] foreign_0.8-29     mlmRev_0.99875-1   lme4_0.999375-27   Matrix_0.999375-16
[5] lattice_0.17-15   

loaded via a namespace (and not attached):
[1] grid_2.7.2  tools_2.7.2

> pupils<-read.table("http://www.jkyleroberts.com/rfiles/mlm/HoxData/pupils.txt", header=T)
> 
> m0 <- lmer(ACHIEV ~ 1 + (1 | PSCHOOL) + (1 | SSCHOOL), pupils)
> summary(m0)
Linear mixed model fit by REML 
Formula: ACHIEV ~ 1 + (1 | PSCHOOL) + (1 | SSCHOOL) 
   Data: pupils 
  AIC  BIC logLik deviance REMLdev
 2329 2349  -1161     2318    2321
Random effects:
 Groups   Name        Variance Std.Dev.
 PSCHOOL  (Intercept) 0.171900 0.41461 
 SSCHOOL  (Intercept) 0.066652 0.25817 
 Residual             0.513128 0.71633 
Number of obs: 1000, groups: PSCHOOL, 50; SSCHOOL, 30

Fixed effects:
            Estimate Std. Error t value
(Intercept)  6.34861    0.07891   80.45
> 
> m1 <- lmer(ACHIEV ~ PUPSEX + PUPSES + (1 | PSCHOOL) + (1 | SSCHOOL), pupils)
> summary(m1)
Linear mixed model fit by REML 
Formula: ACHIEV ~ PUPSEX + PUPSES + (1 | PSCHOOL) + (1 | SSCHOOL) 
   Data: pupils 
  AIC  BIC logLik deviance REMLdev
 2270 2299  -1129     2243    2258
Random effects:
 Groups   Name        Variance Std.Dev.
 PSCHOOL  (Intercept) 0.171530 0.41416 
 SSCHOOL  (Intercept) 0.064809 0.25458 
 Residual             0.475236 0.68937 
Number of obs: 1000, groups: PSCHOOL, 50; SSCHOOL, 30

Fixed effects:
            Estimate Std. Error t value
(Intercept)  5.75548    0.10576   54.42
PUPSEXgirl   0.26132    0.04569    5.72
PUPSES       0.11407    0.01612    7.08

Correlation of Fixed Effects:
           (Intr) PUPSEX
PUPSEXgirl -0.254       
PUPSES     -0.641  0.075
> 
> m2 <- lmer(ACHIEV~PUPSEX+PUPSES+PDENOM+SDENOM+(1 | PSCHOOL) + (1 | SSCHOOL), pupils)
> summary(m2)
Linear mixed model fit by REML 
Formula: ACHIEV ~ PUPSEX + PUPSES + PDENOM + SDENOM + (1 | PSCHOOL) +      (1 | SSCHOOL) 
   Data: pupils 
  AIC  BIC logLik deviance REMLdev
 2273 2312  -1128     2238    2257
Random effects:
 Groups   Name        Variance Std.Dev.
 PSCHOOL  (Intercept) 0.165721 0.40709 
 SSCHOOL  (Intercept) 0.058446 0.24176 
 Residual             0.475207 0.68935 
Number of obs: 1000, groups: PSCHOOL, 50; SSCHOOL, 30

Fixed effects:
            Estimate Std. Error t value
(Intercept)  5.51888    0.14275   38.66
PUPSEXgirl   0.26309    0.04568    5.76
PUPSES       0.11352    0.01612    7.04
PDENOMyes    0.20400    0.12623    1.62
SDENOMyes    0.17572    0.09625    1.83

Correlation of Fixed Effects:
           (Intr) PUPSEX PUPSES PDENOM
PUPSEXgirl -0.200                     
PUPSES     -0.466  0.075              
PDENOMyes  -0.526  0.021  0.004       
SDENOMyes  -0.429  0.004 -0.025 -0.014
> 
> anova(m0, m1, m2)
Data: pupils
Models:
m0: ACHIEV ~ 1 + (1 | PSCHOOL) + (1 | SSCHOOL)
m1: ACHIEV ~ PUPSEX + PUPSES + (1 | PSCHOOL) + (1 | SSCHOOL)
m2: ACHIEV ~ PUPSEX + PUPSES + PDENOM + SDENOM + (1 | PSCHOOL) + 
m2:     (1 | SSCHOOL)
   Df     AIC     BIC  logLik   Chisq Chi Df Pr(>Chisq)    
m0  4  2325.9  2345.5 -1158.9                              
m1  6  2255.5  2284.9 -1121.7 74.3678      2    < 2e-16 ***
m2  8  2253.5  2292.8 -1118.8  5.9684      2    0.05058 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

*********************************************************
Dr. J. Kyle Roberts
Department of Teaching and Learning
Annette Caldwell Simmons School of Education 
   and Human Development
Southern Methodist University
P.O. Box 750381
Dallas, TX  75275
214-768-4494
http://www.hlm-online.com/