[R-sig-ME] Fitting RT: underdispersion with gamma and identity link

[Baud-Bovy Gabriel]

Dear all,

I have updated a one-month old post on stakexchange about a GLMM model that I used to fit RT with gamma
distribution.

https://stats.stackexchange.com/questions/391076/how-to-interpret-significant-factors-in-a-glmm-gamma-model-that-appears-to-be-go

I mention this question in the mailing list because there are still things that I don't understand.  I have put some info at the end of this email it is probably best to answer me on stakeexchange, where it is also possible to see the plots
(otherwise, I'll cross post the answers).

- The DHARMa residual plot suggests underdispersion and I don't know how to deal with that.

- The results of the model don't make sense. For example, all fixed effect are statistically significant
   despite the fact that these factors explain little to nothing when looking at the plots. The random
   effects also appear to be  needed (statistically significant LRTs) and supported by the data  (rePCA)
   although I double that it is the case. Note that the model fits without warning.

-  I don't understand why the results are statistically significant. Intuitively, I would expect that
   estimates to be not statistically significant if there is a lot of unexplained variability.

-   Even if gamma distribution is not a perfect model of RT variability (because of underdispersion),
    it is a better model than Gaussian noise. I would expect therefore that the result of the model
     would be more trustworthy with gamma noise than with gaussian noise.

-  I also don't understand the value of residual SD, which seems to be on a different scale
   that I  would expect.

- As I have mentioned in a previous question to the list, my general goal is to be able to fit RT
  with identity link and welcome other suggestion but I would also like to understand what
  is happening in this case.

  If anybody is interested, I might share the
data privately. Thank you for any help.

Gabriel

>fitRespLat11 <- glmer(resp.lat ~ stake.i.c + dir.c + win.prob.c +
  (stake.i.c + dir.c + win.prob.c || su),
  data=tmp, family = Gamma(link = "identity"),
  control=glmerControl(optimizer =  "bobyqa"))

> summary(fitRespLat11)
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
 Family: Gamma  ( identity )
Formula: resp.lat ~ stake.i.c + dir.c + win.prob.c + (stake.i.c + dir.c + win.prob.c || su)
   Data: tmp
Control: glmerControl(optimizer = "bobyqa")

     AIC      BIC   logLik deviance df.resid
 58026.3  58083.4 -29004.2  58008.3     4160

Scaled residuals:
    Min      1Q  Median      3Q     Max
-1.7918 -0.6890 -0.2162  0.4502  4.8900

Random effects:
 Groups   Name        Variance  Std.Dev.
 su       (Intercept) 5.413e+03  73.576
 su.1     stake.i.c   2.008e+03  44.806
 su.2     dir.c       2.944e+03  54.260
 su.3     win.prob.c  7.514e+04 274.108
 Residual             2.172e-01   0.466
Number of obs: 4169, groups:  su, 120

Fixed effects:
            Estimate Std. Error t value Pr(>|z|)
(Intercept)  612.589      6.290  97.390  < 2e-16 ***
stake.i.c    -61.037      6.001 -10.171  < 2e-16 ***
dir.c        -33.530      5.481  -6.117 9.52e-10 ***
win.prob.c    34.071      8.262   4.124 3.72e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
           (Intr) stk..c dir.c
stake.i.c  -0.089
dir.c      -0.014  0.126
win.prob.c  0.191 -0.226 -0.108

> isSingular(fitRespLat11)
[1] FALSE

# Random structure checks

> rePCA(fitRespLat11)
$su
Standard deviations (1, .., p=4):
[1] 588.19493 157.88241 116.43287  96.14736

Rotation (n x k) = (4 x 4):
     [,1] [,2] [,3] [,4]
[1,]    0    1    0    0
[2,]    0    0    0    1
[3,]    0    0    1    0
[4,]    1    0    0    0

attr(,"class")
[1] "prcomplist"


> anova(fitRespLat11,fitRespLat12,fitRespLat13,fitRespLat14)
Data: tmp
Models:
fitRespLat14: resp.lat ~ stake.i.c + dir.c + win.prob.c + (1 | su)
fitRespLat13: resp.lat ~ stake.i.c + dir.c + win.prob.c + (stake.i.c || su)
fitRespLat12: resp.lat ~ stake.i.c + dir.c + win.prob.c + (stake.i.c + dir.c || su)
fitRespLat11: resp.lat ~ stake.i.c + dir.c + win.prob.c + (stake.i.c + dir.c + win.prob.c || su)
             Df   AIC   BIC logLik deviance  Chisq Chi Df Pr(>Chisq)
fitRespLat14  6 58203 58241 -29096    58191
fitRespLat13  7 58129 58173 -29057    58115 76.699      1  < 2.2e-16 ***
fitRespLat12  8 58056 58107 -29020    58040 74.820      1  < 2.2e-16 ***
fitRespLat11  9 58026 58083 -29004    58008 31.496      1  1.999e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

# See stackexchange link for plots and DHARMa residuals


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