[R] lmer, gamma family, log link: interpreting random effects

[Andrew Robinson]

Dear all,

I'm curious about how to interpret the results of the following code.
The first model is directly from the help page of lmer; the second is
the same model but using the Gamma family with log link.  The fixed
effects make sense, because

y = 251.40510 + 10.46729 * Days

is about the same as 

log(y) = 5.53613298  + 0.03502057 * Days

but the random effects seem quite disproportionate. Am I missing
something obvious, or is this a bug?

Cheers

Andrew

> require(lme4)
> fm1 <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy)
> fixef(fm1)
(Intercept)        Days 
  251.40510    10.46729 
> ranef(fm1)
An object of class "ranef.lmer"
[[1]]
    (Intercept)        Days
308   2.2713451   9.1966961
309 -40.3825921  -8.6223099
310 -38.9403378  -5.4521894
330  23.6656443  -4.8100611
331  22.2391295  -3.0662861
332   9.0324969  -0.2709760
333  16.8277808  -0.2214775
334  -7.2256113   1.0733822
335  -0.3503176 -10.7492136
337  34.8784025   8.6302064
349 -25.1898574   1.1699674
350 -13.0500361   6.6107496
351   4.5697621  -3.0138529
352  20.8539011   3.5376139
369   3.2744289   0.8723798
370 -25.5865985   4.8179542
371   0.8049003  -0.9877807
372  12.3075594   1.2851975

> fm1.g.log <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy,  family=Gamma(link="log"))
> fixef(fm1.g.log)
(Intercept)        Days 
 5.53613298  0.03502057 
> ranef(fm1.g.log)
An object of class "ranef.lmer"
[[1]]
      (Intercept)          Days
308  6.817799e-10  4.462209e-09
309 -1.367226e-09 -6.921681e-09
310 -1.120295e-09 -5.303576e-09
330  1.173973e-10 -6.049948e-10
331  2.104704e-10  1.531865e-10
332  1.501432e-10  6.721697e-10
333  3.048322e-10  1.103687e-09
334 -6.193655e-11 -7.056224e-11
335 -7.539673e-10 -4.999876e-09
337  1.263556e-09  6.502735e-09
349 -3.974501e-10 -1.294252e-09
350  2.112016e-10  2.125564e-09
351 -1.220139e-10 -1.095394e-09
352  6.412044e-10  3.215876e-09
369  1.231353e-10  6.292502e-10
370 -1.520576e-10  4.003904e-10
371 -5.220764e-11 -4.587212e-10
372  3.234287e-10  1.484018e-09


-- 
Andrew Robinson  
Department of Mathematics and Statistics            Tel: +61-3-8344-9763
University of Melbourne, VIC 3010 Australia         Fax: +61-3-8344-4599
http://www.ms.unimelb.edu.au/~andrewpr
http://blogs.mbs.edu/fishing-in-the-bay/