[R-sig-ME] Crossing Interaction and lmer

[Johannes Schliesser]

Dear mixed model experts

I have troubles with a wired outcome with lmer(). 

I was testing effects on reaction times for two fixed factors, Identity 
(levels identical "id" and non-identical ("ni") and Bonus Size ("h"igh and 
"l"ow). Subjects "subj" and "items" were Random Effects. In fact, Bonus Size 
is nested within Items, as half of the Items were of high, the other half of 
low Bonus size.

The results indicate a strong crossing interaction:


       |    h   |    l   |   ALL  |
|id   |839.	 |861  |  850   |

|ni    |879    |829 |  854   |

|ALL| 859    |845  |852    |

Subjects tend to be about 40 ms faster for identical than for non-identical in 
the high bonus group, but about 30 ms slower for identical than for non-
identical in the low bonus group. Grand means for identical and non-identical 
items differ only for 4 ms.

So far so good.
Classic ANOVA rejects all main effects but confirmes the interaction.

The lmer analysis instead suggests a fixed effect for identity: 
----------
lmer.rtNA2 = lmer(rtNA~id*bonus+(1|subj)+(1|bonus/item), data = x4)
 lmer.rtNA
Linear mixed model fit by REML 
Formula: rtNA ~ id * bonus + (1 | subj) + (1 | item) 
   Data: x4 
   AIC   BIC logLik deviance REMLdev
 24346 24385 -12166    24360   24332
Random effects:
 Groups   Name        Variance Std.Dev.
 item     (Intercept)  1926.9   43.896 
 subj     (Intercept)  9648.6   98.227 
 Residual             19026.7  137.937 
Number of obs: 1905, groups: item, 80; subj, 24

Fixed effects:
            Estimate Std. Error t value
(Intercept)  839.804     22.132   37.95
idni          40.264      8.942    4.50
bonusl        22.841     13.278    1.72
idni:bonusl  -73.346     12.644   -5.80

Correlation of Fixed Effects:
            (Intr) idni   bonusl
idni        -0.202              
bonusl      -0.300  0.336       
idni:bonusl  0.143 -0.707 -0.476
-----------------------------------

The lmer model seems to test Identical vs. Non-Identical for the High-Bonus 
group only, but not overall (same as estimates for Bonus Size only in the "id" 
row). pvals.fnc() asssigns significance for Identity and the interaction 
Identity*Bonus Size.

I have the impression that this is missleading: we certainly cannot speak of  
slower reaction times for non-identical stimuli than for identical.

Leaving out the Bonus Size as a fixed factor "lmer(rt~id+(1|subj)+(1|item), 
data = x4)" gives a realistic t-value below 1 for Identity.

So, I ask: 
Is the missleading handling of crossing interactions a bug ? 
Or do I simply have to know that if there is a huge crossing interaction I 
should not generalize the Fixed Factor effects ?

And more relevant: How do I get an analysis that is appropriate to the data ?

Thanks for your help

Johannes