Hello,
I'm fitting success or failure data using binomial-normal models. The data
are clustered by Reg, and I'm interesting in evaluating the effect of Ox
(0, 1), Ab (0, 1), and Mn (0, 1).

I'm interested in these three models:

fm1<-glmer(cbind(succ, fail)~1+Ox+Ab+Mn+(1|Reg), family="binomial", mydf)
fm2<-glmer(cbind(succ,
fail)~1+Ox+Ab+Mn+(1|Reg)+(0+Ox|Reg)+(0+Ab|Reg)+(0+Mn|Reg),
family="binomial", mydf)
fm3<-glmer(cbind(succ, fail)~1+Ox+Ab+Mn+(1+Ox+Ab+Mn|Reg),
family="binomial", mydf)

The residual deviance is constantly decreasing when I move from fm1, to
fm2, to fm3 (see summaries reported at the end of this message).
The summary of fm3 gives the following estimates for the random effects:

Random effects:
 Groups Name        Variance Std.Dev. Corr
 Reg    (Intercept) 1.385    1.177
        Ox          0.184    0.429    -1.000
        Ab          1.501    1.225    -0.991  0.991
        Mn          0.335    0.579    -0.841  0.841  0.834

My questions is:
1) Why is the correlation associated with Ox equal to -1 ?
2) Can someone explain to me why I have pairs of correlations with opposite
signs for Ab and Mn. How to interpret this?

Thanks a lot for your help,
Best regards


#####################################################

> sessionInfo()
R version 2.13.2 (2011-09-30)
Platform: i386-pc-mingw32/i386 (32-bit)

locale:
[1] LC_COLLATE=English_United States.1252  LC_CTYPE=English_United
States.1252
[3] LC_MONETARY=English_United States.1252 LC_NUMERIC=C

[5] LC_TIME=English_United States.1252

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

other attached packages:
 [1] arm_1.5-02         survival_2.36-9    mvtnorm_0.9-9992   abind_1.4-0
     R2WinBUGS_2.1-18
 [6] coda_0.14-6        MASS_7.3-14        lme4_0.999375-42
Matrix_0.9996875-3 lattice_0.19-33
[11] reshape2_1.2.1     ggplot2_0.8.9      proto_0.3-9.2      reshape_0.8.4
     plyr_1.7.1

loaded via a namespace (and not attached):
[1] cluster_1.14.0 digest_0.5.1   Hmisc_3.9-1    nlme_3.1-102
stats4_2.13.2  stringr_0.5    tools_2.13.2


Summaries:
> fm1<-glmer(cbind(succ, fail)~1+Ox+Ab+Mn+(1|Reg), family="binomial",
mydf); summary(fm1)
Generalized linear mixed model fit by the Laplace approximation
Formula: cbind(succ, fail) ~ 1 + Ox + Ab + Mn + (1 | Reg)
   Data: mydf
 AIC BIC logLik deviance
 293 305   -141      283
Random effects:
 Groups Name        Variance Std.Dev.
 Reg    (Intercept) 0.713    0.844
Number of obs: 81, groups: Reg, 37

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)
(Intercept)   -3.173      0.164   -19.4   <2e-16 ***
Ox             1.577      0.110    14.4   <2e-16 ***
Ab             1.321      0.130    10.2   <2e-16 ***
Mn             1.302      0.143     9.1   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
   (Intr) Ox     Ab
Ox -0.304
Ab -0.260  0.693
Mn -0.361  0.110  0.094

> fm2<-glmer(cbind(succ,
fail)~1+Ox+Ab+Mn+(1|Reg)+(0+Ox|Reg)+(0+Ab|Reg)+(0+Mn|Reg),
family="binomial", mydf); summary(fm2)
Generalized linear mixed model fit by the Laplace approximation
Formula: cbind(succ, fail) ~ 1 + Ox + Ab + Mn + (1 | Reg) + (0 + Ox |
 Reg) + (0 + Ab | Reg) + (0 + Mn | Reg)
   Data: mydf
 AIC BIC logLik deviance
 286 305   -135      270
Random effects:
 Groups Name        Variance Std.Dev.
 Reg    (Intercept) 0.8066   0.898
 Reg    Ox          0.1609   0.401
 Reg    Ab          0.0128   0.113
 Reg    Mn          0.0414   0.204
Number of obs: 81, groups: Reg, 37

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)
(Intercept)   -3.230      0.176  -18.37   <2e-16 ***
Ox             1.666      0.165   10.09   <2e-16 ***
Ab             1.365      0.166    8.22   <2e-16 ***
Mn             1.329      0.159    8.38   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
   (Intr) Ox     Ab
Ox -0.270
Ab -0.230  0.426
Mn -0.344  0.093  0.079

> fm3<-glmer(cbind(succ, fail)~1+Ox+Ab+Mn+(1+Ox+Ab+Mn|Reg),
family="binomial", mydf); summary(fm3)
Generalized linear mixed model fit by the Laplace approximation
Formula: cbind(succ, fail) ~ 1 + Ox + Ab + Mn + (1 + Ox + Ab + Mn | Reg)
   Data: mydf
 AIC BIC logLik deviance
 271 305   -122      243
Random effects:
 Groups Name        Variance Std.Dev. Corr
 Reg    (Intercept) 1.385    1.177
        Ox          0.184    0.429    -1.000
        Ab          1.501    1.225    -0.991  0.991
        Mn          0.335    0.579    -0.841  0.841  0.834
Number of obs: 81, groups: Reg, 37

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)
(Intercept)   -3.450      0.221  -15.58  < 2e-16 ***
Ox             2.030      0.129   15.75  < 2e-16 ***
Ab             1.386      0.245    5.67  1.5e-08 ***
Mn             1.453      0.188    7.72  1.2e-14 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
   (Intr) Ox     Ab
Ox -0.774
Ab -0.930  0.712
Mn -0.688  0.532  0.639
>

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