[R-sig-ME] interpreting random intercepts when no fixed intercept present

[Ben Bolker]

 <John.Morrongiello at ...> writes:

> 
> Hi list
 
> I was wondering how I interpret random effect intercepts in a model
> with no fixed intercept? Take for example the following model, based
> on those presented in Weisberg etal (2010):

> M1<-lmer(growth~0+Age+(1|ID)+(1|Year)
 
> Where growth is a repeatedly measured continuous variable, Age is a
> factor with 10 ordered levels (2:11) corresponding to each growth
> observation, 
  
  Hmm.  It would generally seem to make more sense to treat
age as numeric, since I would expect some sort of smooth, systematic
(linear/quadratic/spline) relationship between growth and age.
(Although I guess using an ordered factor does in some way allow
you to separate linear, quadratic, higher-order contributions
to the growth-age relationship)

>  ID and year are crossed random effects and represent
> individual animals (100) and the years in which they were
> sampled. This model provides a separate coefficient for each age;
> are the random effects deviations from just the Age2 (first)
> coefficient, or from the Age term in general? Each ID random effect
> has only one value, so they are obviously not unique deviations from
> each level of Age. Or are the random intercepts reflective of
> differences in average growth among individuals and years after the
> effect of age is 'accounted' for (i.e. not Age dependent)?

  Hmmm.  I think in order to answer this question I'd have to
figure out what model.matrix() is doing when we use
[ordered factor]+0 in a formula.  I thought I knew but now
I don't think I do ...

> d <- data.frame(f=ordered(rep(1:5,10)),y=runif(50))
> options(digits=3)
> coef(lm(y~f,data=d))
(Intercept)         f.L         f.Q         f.C         f^4 
      0.525      -0.064       0.154       0.144      -0.116 
> coef(lm(y~f+0,data=d))
   f1    f2    f3    f4    f5 
0.589 0.651 0.360 0.428 0.599 
> coef(lm(y~f,data=d,contrasts=list(f=contr.treatment)))
(Intercept)          f2          f3          f4          f5 
     0.5885      0.0625     -0.2286     -0.1602      0.0101 

 (It would probably be better to use an example with a clear
linear and quadratic term and nothing else, for clarity)
 
> Furthermore, if M1 was extended to include harvest (factor with
> three levels) to which the population was exposed (some to just one
> level, others all three):
 
> M2<-lmer(growth~0+Age+harvest+(1|ID)+(1|Year)
 
> Is the interpretation of random effects now different to that in M1
> in that they now include some harvest 'information'?

  I think the answer to this is going to have to involve more
searching into how model.matrix() parameterizes these models.
Basically, once you know how the fixed effects are parameterized,
you can interpret what it means to add a zero-mean random-effects
offset to it ...   
 
 
> Weisberg, S., Spangler, G., and Richmond, L.S. (2010). Mixed effects
> models for fish growth. Canadian Journal of Fisheries and Aquatic
> Sciences 67, 269-277.