I have a question that I think will interest many 
on this list. It is not intrinsically related to 
mixed models, although they may generate some 
additional complications, as I discuss at the end.

Here's a toy example that illustrates the issue. 
Suppose one has a data frame containing successes 
associated with a binomial r.v., e.g.,

Sex	Factor	Weight
10	1	20                   e.g., 10 
boys out of 20 children in study 1
15	2	20

Call this the "AGGREGATED" form of the data. One 
could proceed to analyze this as follows:

short
Call:  glm(formula = Sex ~ 1, family = binomial, data = test2, weights = Total)
Coefficients:
(Intercept)
      0.5108
Degrees of Freedom: 1 Total (i.e. Null);  1 Residual
Null Deviance:	    2.706
Residual Deviance: 2.706	AIC: 11.37

One could look for an effect of factor

short_factor
Call:  glm(formula = Sex ~ Factor, family = 
binomial, data = test2,      weights = Total)
Coefficients:
(Intercept)      Factor2
  -3.753e-16    1.099e+00
Degrees of Freedom: 1 Total (i.e. Null);  0 Residual
Null Deviance:	    2.706
Residual Deviance: 2.22e-15	AIC: 10.67

Of course, 10 boys out of 20 children means that 
there were 20 Bernoulli trials, so the 
"DISAGGREGATED" form of the data frame above is

Sex Factor
1      1
1      1
1      1
1      1
1      1
1      1
1      1
1      1
1      1
1      1
0      1
0      1
0      1
0      1
0      1
0      1
0      1
0      1
0      1
0      1
1      2
1      2
1      2
1      2
1      2
1      2
1      2
1      2
1      2
1      2
1      2
1      2
1      2
1      2
1      2
0      2
0      2
0      2
0      2
0      2

Now, we have

long
Call:  glm(formula = Sex ~ 1, family = binomial, data = test)
Coefficients:
(Intercept)
      0.5108
Degrees of Freedom: 39 Total (i.e. Null);  39 Residual
Null Deviance:	    52.93
Residual Deviance: 52.93	AIC: 54.93

And

long_factor
Call:  glm(formula = Sex ~ Factor, family = binomial, data = test)
Coefficients:
(Intercept)      Factor2
   7.506e-16    1.099e+00
Degrees of Freedom: 39 Total (i.e. Null);  38 Residual
Null Deviance:	    52.93
Residual Deviance: 50.22	AIC: 54.22

There is nothing eyecatching here in that we get 
identical estimates of the overall proportion of 
boys (in the only intercept model) and identical 
estimates of the factor-specific intercepts (in 
the factor model) (ignoring error). In addition, 
the AIC values are such that one would select the 
same model regardless of whether the data were 
aggregated or not.

However, it is not obvious to me that one 
generally gets the "same" results regardless of 
the aggregation. Therefore, my question is: is 
there a preferred or canonical input form from a 
statistical point of view? aggregated or 
disaggregated or ? I am interested to hear about 
the statistical ins and outs of this. I have 
often obtained disaggregated data (as defined 
above) and not thought twice about the 
consequences of aggregating them for analysis (or 
vice-versa).

Of course, I think the answer to my question will 
be that "it depends"Š.on what one is trying to 
estimate. To this extent, there may be extra 
wrinkles to this issue in the context of 
analyzing mixed models. Consider the following:


AGGREGATED
shortlmer
Generalized linear mixed model fit by the Laplace approximation
Formula: Sex ~ 1 + (1 | Factor)
    Data: test2
    AIC   BIC logLik deviance
  6.612 3.998 -1.306    2.612
Random effects:
  Groups Name        Variance Std.Dev.
  Factor (Intercept) 0.074601 0.27313
Number of obs: 2, groups: Factor, 2
Fixed effects:
             Estimate Std. Error z value Pr(>|z|)
(Intercept)   0.5202     0.3803   1.368    0.171

shortlmer_factor
Generalized linear mixed model fit by the Laplace approximation
Formula: Sex ~ Factor + (1 | Factor)
    Data: test2
  AIC   BIC     logLik  deviance
    6 2.079 -1.524e-12 3.048e-12
Random effects:
  Groups Name        Variance   Std.Dev.
  Factor (Intercept) 5.6766e-17 7.5343e-09
Number of obs: 2, groups: Factor, 2
Fixed effects:
              Estimate Std. Error  z value Pr(>|z|)
(Intercept) 4.903e-07  4.472e-01 1.10e-06    1.000
Factor2     1.099e+00  6.831e-01    1.608    0.108
Correlation of Fixed Effects:
         (Intr)
Factor2 -0.655

In this case, note that AIC of the constant model 
is greater than the AIC of the factor model AND 
the BIC of the constant model is greater than the 
BIC of the factor model. So, the two criteria 
tend in the same direction in terms of model 
selection.

DISAGGREGATED
longlmer
Generalized linear mixed model fit by the Laplace approximation
Formula: Sex ~ 1 + (1 | Factor)
    Data: test
    AIC   BIC logLik deviance
  56.83 60.21 -26.42    52.83
Random effects:
  Groups Name        Variance Std.Dev.
  Factor (Intercept) 0.074601 0.27313
Number of obs: 40, groups: Factor, 2
Fixed effects:
             Estimate Std. Error z value Pr(>|z|)
(Intercept)   0.5201     0.3803   1.368    0.171

  longlmer_factor
Generalized linear mixed model fit by the Laplace approximation
Formula: Sex ~ Factor + (1 | Factor)
    Data: test
    AIC   BIC logLik deviance
  56.22 61.29 -25.11    50.22
Random effects:
  Groups Name        Variance Std.Dev.
  Factor (Intercept)  0        0
Number of obs: 40, groups: Factor, 2
Fixed effects:
              Estimate Std. Error  z value Pr(>|z|)
(Intercept) 4.856e-07  4.472e-01 1.09e-06    1.000
Factor2     1.099e+00  6.831e-01    1.608    0.108
Correlation of Fixed Effects:
         (Intr)
Factor2 -0.655

We get identical fixed-effect estimates 
regardless of aggregation/disaggregation.

Note that AIC of the constant model is greater 
than the AIC of the factor model, but BIC of the 
constant model is LESS than the BIC of the factor 
model. Of course, the differences are not great 
(magnitude „ 2) but the two criteria do NOT tend 
in the same direction in terms of model selection.

This is one context in which the data 
aggregation/disaggregation could influence the 
results of analysis even when the analysis is at 
a level "above" the level of the aggregation.

Thoughts about all of this as well as pointers to 
any relevant discussions in the literature would 
be much appreciated.

--
Steven Orzack

The Fresh Pond Research Institute
173 Harvey Street
Cambridge, MA. 02140
617 864-4307

www.freshpond.org
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