145 - 146). The chi square difference test (Δχ^2 ) can be used to indicate the point
at which model trimming has reached an optimal stage. This is done simply by
taking the difference in the model chi-square value for the two models (simple
model / larger chi-square minus complex model / smaller chi-square value) and
then using the difference in degrees of freedom between the two models to
calculate a chi-square probability value. Where the difference is found to be
significant (p ≤ 0.05), it can be concluded that the more complex model is
preferred over the simplified model. Garson writes:
...the goal is to find the most parsimonious model which is not
significantly different from the saturated model, which fully but trivially
explains the data. After dropping a path, a significant chi-square
difference indicates the fit of the simpler model is significantly worse
than for the more complex model and the complex model may be
retained. ...as paths are trimmed, chi-square tends to increase,
indicating a worse model fit and also increasing chi-square difference.
(Garson 2011b)Put another way, because more complex models (for the same data) are expected
to have a lower χ^2 value, and less complex models (with fewer parameters to be
estimated) are expected to have a higher χ^2 value, if a model is trimmed (a
parameter is removed) it is expected that χ^2 will rise. Therefore, when we remove
a path from a model, then we expect fit to worsen and we hope to observe a non-
significant worsening (indicating that the more parsimonious model fits the data
equally well in comparison with the more complex model). It is the χ^2 difference
test and its p value that is used to indicate whether or not the worsening of fit is
statistically significant.
The WV→EPA path was removed from model SEM 5:1 and the constrained model
(SEM 5:1 1 ) was re-estimated. The results of the chi square difference test are
described in Table 7 - 27.
Constrained model
(SEM 5:1 1 with
WV→EPA fixed to
zero)
Initial model
(SEM 5:1 with
WV→EPA free)Chi square difference (Δχ^2 )chi sq d.f. chi sq d.f. chi sq diff d.f. diff sig
161.745 144 160.472 143 1.273 1 0.259