Biology Reference
In-Depth Information
FIGURE 12.16
The saturated model.
This model fits the data perfectly because
all the edges are free parameters, esti-
mated from the data. Its deviance is there-
fore zero and it has zero degrees of
freedom. This is the null model; the objec-
tive of the analysis is to reproduce the
observed covariance with as few edges as
possible.
the two models. The saturated model has zero degrees of freedom, so the degrees of free-
dom for the chi-square are the number of fixed parameters in the model.
Models that contain few fixed parameters are likely to fit well, so we are looking not
only for a model that fits well but one that fits well using as few edges as possible. When
models are nested, meaning that one model is included within the other, we can compare
the two by the chi-square difference test, subtracting the chi-square of the more complex
model from that of the simpler model, and also subtracting the degrees of freedom of the
more complex model from those of the simpler model. The resultant
chi-square is dis-
tributed as a chi-square with the degrees of freedom given by the difference in degrees of
freedom of the two models. If statistically significant, the more complex model improves
upon the simpler one. When models are not nested, we need an alternative approach for
judging the relative fit of two models. One is to use the Akaike Information Criterion
(AIC), ranking models by their AIC. The AIC was introduced in Chapter 11, but we sum-
marize it here for purposes of convenience. AIC is a function of the log-likelihood of the
parameters given the data and the number of parameters in the model (
Akaike, 1974
), cal-
culated as:
Δ
2
k
ð
Þ
AIC
5
2
2ln
likelihood
(12.10)
where
k
is the number of parameters in the model. To compare models, we can compute
the difference in their AIC (
AIC).
We can also use exploratory methods to find the model that reproduces the observed
correlation matrix using as few edges as possible. Alternatively, we can search for a model
that reproduces the inverse correlation matrix, which gives the pairwise correlations hold-
ing all other variables constant. When the analysis is done using the inverse correlation
matrix, the models are usually termed “Gaussian graphical models” and the search for the
best-fitting model is called “concentration model selection”. When the models are instead
Δ
