Hi,

I have the following type of data: 86 subjects in three independent groups (high power vs low power vs control). Each subject solves 8 reasoning problems of two kinds: conflict problems and noconflict problems. I measure accuracy in solving the reasoning problems. To summarize: binary response, 1 within subject var (TYPE), 1 between subject var (POWER).

I wanted to fit the following model: for problem i, person j:
logodds ( Y_ij ) = b_0j + b_1j TYPE_ij
with b_0j = b_00 + b_01 POWER_j + u_0j
and b_1j = b_10 + b_11 POWER_j

I think it makes sense, but I'm not sure.
Here are the observed cell means:
                conflict                 noconflict
control 0.6896552            0.9568966
high      0.6935484            0.9677419
low         0.8846154            0.9903846

GLMER gives me:
summary(glmer(accuracy~f_power*f_type + (1|subject), family=binomial,data=syllogisms))
Generalized linear mixed model fit by the Laplace approximation
Formula: accuracy ~ f_power * f_type + (1 | subject)
   Data: syllogisms
 AIC   BIC logLik deviance
406 437.7   -196      392
Random effects:
Groups  Name        Variance Std.Dev.
subject (Intercept) 4.9968   2.2353
Number of obs: 688, groups: subject, 86

Fixed effects:
                            Estimate Std. Error z value Pr(>|z|)
(Intercept)                  1.50745    0.50507   2.985  0.00284 **
f_powerhp                    0.13083    0.70719   0.185  0.85323
f_powerlow                   2.04121    0.85308   2.393  0.01672 *
f_typenoconflict             3.28715    0.64673   5.083 3.72e-07 ***
f_powerhp:f_typenoconflict   0.21680    0.93165   0.233  0.81599
f_powerlow:f_typenoconflict -0.01199    1.45807  -0.008  0.99344
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
            (Intr) f_pwrh f_pwrl f_typn f_pwrh:_
f_powerhp   -0.714
f_powerlow  -0.592  0.423
f_typncnflc -0.185  0.132  0.109
f_pwrhp:f_t  0.128 -0.170 -0.076 -0.694
f_pwrlw:f_t  0.082 -0.059 -0.144 -0.444  0.308

glmmPQL gives me:
summary(glmmPQL(fixed=accuracy~f_power*f_type, random=~1|subject, family=binomial, data=syllogisms))
iteration 1
iteration 2
iteration 3
iteration 4
iteration 5
iteration 6
Linear mixed-effects model fit by maximum likelihood
Data: syllogisms
  AIC BIC logLik
   NA  NA     NA

Random effects:
Formula: ~1 | subject
        (Intercept)  Residual
StdDev:    1.817202 0.8045027

Variance function:
Structure: fixed weights
Formula: ~invwt
Fixed effects: accuracy ~ f_power * f_type
                                Value Std.Error  DF  t-value p-value
(Intercept)                 1.1403334 0.4064642 599 2.805496  0.0052
f_powerhp                   0.0996481 0.5683296  83 0.175335  0.8612
f_powerlow                  1.5358270 0.6486150  83 2.367856  0.0202
f_typenoconflict            3.0096016 0.4769761 599 6.309754  0.0000
f_powerhp:f_typenoconflict  0.1856061 0.6790046 599 0.273350  0.7847
f_powerlow:f_typenoconflict 0.0968204 1.0318659 599 0.093830  0.9253
Correlation:
                            (Intr) f_pwrh f_pwrl f_typn f_pwrh:_
f_powerhp                   -0.715
f_powerlow                  -0.627  0.448
f_typenoconflict            -0.194  0.138  0.121
f_powerhp:f_typenoconflict   0.136 -0.182 -0.085 -0.702
f_powerlow:f_typenoconflict  0.089 -0.064 -0.153 -0.462  0.325

Standardized Within-Group Residuals:
         Min           Q1          Med           Q3          Max
-12.43735991   0.06243699   0.22966010   0.33106978   2.23942234

Number of Observations: 688
Number of Groups: 86


Strange thing is that when you convert the estimates to probabilities, they are quite far off. For control, no conflict (intercept), the estimation from glmer is 1.5 -> 81% and for glmmPQL is 1.14 -> 75%, whereas the observed is: 68%.

Am I doing something wrong?

Any help is very much appreciated.
Sam.

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