Introduction to Categorical Data Analysis

7.2 INFERENCE FOR LOGLINEAR MODELS 213
and indicate poorer fits. The models that lack any association term fit poorly, having
P -values below 0.001. The model (AC, AM, CM) that permits all pairwise associations
but assumes homogeneous association fits well (P = 0.54). Table 7.6 shows the
way PROC GENMOD in SAS reports the goodness-of-fit statistics for this model.
7.2.2 Loglinear Cell Residuals
Cell residuals show the quality of fit cell-by-cell. They show where a model fits
poorly. Sometimes they indicate that certain cells display lack of fit in an otherwise
good-fitting model. When a table has many cells, some residuals may be large purely
by chance.
Section 2.4.5 introduced standardized residuals for the independence model, and
Section 3.4.5 discussed them generally for GLMs. They divide differences between
observed and fitted counts by their standard errors. When the model holds, standardized
residuals have approximately a standard normal distribution. Lack of fit is
indicated by absolute values larger than about 2 when there are few cells or about 3
when there are many cells.
Table 7.8 shows standardized residuals for the model (AM, CM) ofAC conditional
independence with Table 7.3. This model has df = 2 for testing fit. The two
nonredundant residuals refer to checking AC independence at each level of M. The
large residuals reflect the overall poor fit. [In fact, X2 relates to the two nonredundant
residuals by X2 = (3.70) 2 + (12.80) 2 = 177.6.]. Extremely large residuals occur for
students who have not smoked marijuana. For them, the positive residuals occur when
A and C are both “yes” or both “no.” More of these students have used both or neither
of alcohol and cigarettes than one would expect if their usage were conditionally
independent. The same pattern persists for students who have smoked marijuana, but
the differences between observed and fitted counts are then not as striking.
Table 7.8 also shows standardized residuals for model (AC, AM, CM). Since df =
1 for this model, only one residual is nonredundant. Both G2 and X2 are small, so
Table 7.8. Standardized Residuals for Two Loglinear Models
Drug Use
Model (AM, CM) Model (AC, AM, CM)
Observed Fitted Standardized Fitted Standardized
A C M Count Count Residual Count Residual
Yes Yes Yes 911 909.2 3.70 910.4 0.63
No 538 438.8 12.80 538.6 −0.63
No Yes 44 45.8 −3.70 44.6 −0.63
No 456 555.2 −12.80 455.4 0.63
No Yes Yes 3 4.8 −3.70 3.6 −0.63
No 43 142.2 −12.80 42.4 0.63
No Yes 2 0.2 3.70 1.4 0.63
No 279 179.8 12.80 279.6 −0.63