Statistical Methods for Psychology

Examining the Saturated Model

In considering two-way tables, we defined a saturated model as one that includes all possi-

ble effects. The same holds for three-way and higher-order tables. Consider the model that

can be designated as FMVor written as

This is the saturated model for our data. It includes all possible effects and exhausts the

degrees of freedom available in the data. (One degree of freedom goes to estimating l, one

each to estimating F, V, and FV, and two each to estimating M, FM, MV, and FMV; Mhas

three levels and thus two degrees of freedom for it and its interactions.) These sum to 12,

and because we have 12 cells there isn’t anything left over. If we knew the values of the

various lambdas, and eventually we will, the resultant expected frequencies would exactly

equal the observed frequencies, leaving nothing else to be explained. For this reason we

know without even looking at the data that the likelihood ratio for this model will be

exactly 0.00. We should not be any happier with this perfect fit than we are when we draw

a straight line to fit perfectly any two points, and for the same reason—the model exhausts

the degrees of freedom.

We do not fit a saturated model to data just because we hope that it will fit—we know

that before we start. We usually fit it hoping that it will help us identify simpler models by

revealing nonsignificant effects. If we could show, for example, that we could do about as

well by eliminating the three-way interaction and two of the two-way interactions, we

would be well on our way to representing the data by a relatively simple model.

In Exhibit 17.1 you will see part of the printout from the SPSS GENLOG analysis of

the saturated model.^6 You can either run GENLOG from syntax or from drop-down

menus. The syntax is given first, and the only line that will change in further analyses is

the /Design statement. You can see from the output that chi-square is precisely 0.000, as it

should be, and that the expected frequencies exactly match the obtained frequencies. What

we would like to do is to find a model that fits nearly as well but has fewer components.

x^2

ln(Fijk)=l1lF1lM1lV1lMV1lFM1lFV1lFMV

Section 17.6 Three-Way Tables 647

GENLOG

Verdict Fault Moral

/PRINT = FREQ RESID ADJRESID ZRESID DEV ESTIM CORR COV

/CRITERIA = CIN(95) ITERATE(20) CONVERGE(.001) DELTA(.5)

/DESIGN Fault Moral Verdict Fault*Moral Fault*Verdict Moral*Verdict Fault*Moral*Verdict

.

aModel: Poisson

bDesign: Constant 1 Fault 1 Moral 1 Verdict 1 Fault*

Moral 1 Verdict*Fault 1 Verdict*Moral 1 Verdict*

Fault*Moral

Goodness-of-Fit Testsa,b

Likelihood Ratio

Pearson Chi-Square

Value

.000

.000

df

0

0

Sig.

.

.

(^6) If you generate the model from drop-down menus, be sure to specify that you want it to print out estimates in
the options menu. By default the options menu adds 0.5 to every cell if you request a saturated model. This is to
prevent trying to calculate ln(0), which is undefined.
Exhibit 17.1 Saturated model applied to Pugh’s data on three variables
(continues)