Michael Friendly

A log-linear model expresses the relationship among all variables as a model for the log of the expected cell frequency. For example, for a three-way table, the hypothesis that of no three-way association can be expressed as the log-linear model,

(20)The log-linear model treats the variables symmetrically: none of the variables is distinguished as a response variable. However, the association parameters may be difficult to interpret, and the absence of a dependent variable makes it awkward to plot results in terms of the log-linear model. In this case, correspondence analysis and the mosaic display may provide a simpler way to display the patterns of association in a contingency table.

On the other hand, if one variable can be regarded as a response or dependent variable, and the others as independent variables, then the effects of the independent variables may be expressed as a logit model. For example, if variable C is a binary response, then model (20) can be expressed as an equivalent logit model,

(21)

where, because all

Both log-linear and logit models can be fit using PROC CATMOD in SAS. For logit models, the steps for fitting a model and plotting the results are similar to those used for logistic models with PROC LOGISTIC. The main differences are:

- For PROC CATMOD the independent variables can be categorical or quantitative, character or numeric. With PROC LOGISTIC the independent variables must be numeric. To use a categorical variables, we had to construct dummy variables.
- The input data set is arranged differently.
- The output data set from PROC CATMOD contains similar information, but the variable names are different and the information is arranged differently.

Political ---- White ---- --- Nonwhite --- View Reagan Carter Reagan Carter 1 1 12 0 6 2 13 57 0 16 3 44 71 2 23 4 155 146 1 31 5 92 61 0 8 6 100 41 2 7 7 18 8 0 4Treating the vote for Reagan vs. Carter or other as the response, a logit model with nominal main effects for race and political view is

(22)This model does not use the ordinal nature of political view. A model which uses the value of political view as a direct, quantitative independent variable can be expressed as

(23)

proc format; value race 0='NonWhite' 1='White'; data vote; input @10 race view reagan carter; format race race.; reagan= reagan + .5; *-- allow for 0 values; carter= carter + .5; total = reagan + carter; preagan = reagan / total; logit = log ( reagan / carter); cards; White 1 1 1 12 White 1 2 13 57 White 1 3 44 71 White 1 4 155 146 White 1 5 92 61 White 1 6 100 41 White 1 7 18 8 NonWhite 0 1 0 6 NonWhite 0 2 0 16 NonWhite 0 3 2 23 NonWhite 0 4 1 31 NonWhite 0 5 0 8 NonWhite 0 6 2 7 NonWhite 0 7 0 4 ; data votes; set vote; votes= reagan; votefor='REAGAN'; output; votes= carter; votefor='CARTER'; output;

Model (22) is fit using the statements below. The `
RESPONSE` statement is used to produce an output data set, `
PREDICT`, for plotting.

proc catmod data=votes order=data; weight votes; response / out=predict; model votefor = race view / noiter ; title2 f=duplex h=1.4 'Nominal Main Effects of Race and Political View (90% CI)';The results of the PROC CATMOD step include:

+-------------------------------------------------------------------+ | | | MAXIMUM-LIKELIHOOD ANALYSIS-OF-VARIANCE TABLE | | | | Source DF Chi-Square Prob | | -------------------------------------------------- | | INTERCEPT 1 43.75 0.0000 | | RACE 1 41.37 0.0000 | | VIEW 6 67.84 0.0000 | | | | LIKELIHOOD RATIO 6 3.45 0.7501 | | | +-------------------------------------------------------------------+

+-------------------------------------------------------------------+ | | | ANALYSIS OF MAXIMUM-LIKELIHOOD ESTIMATES | | | | Standard Chi- | | Effect Parameter Estimate Error Square Prob | | ----------------------------------------------------------- | | INTERCEPT 1 -1.4324 0.2166 43.75 0.0000 | | RACE 2 1.1960 0.1859 41.37 0.0000 | | VIEW 3 -1.6144 0.6551 6.07 0.0137 | | 4 -1.2000 0.2857 17.64 0.0000 | | 5 -0.1997 0.2083 0.92 0.3377 | | 6 0.2779 0.1672 2.76 0.0965 | | 7 0.6291 0.1941 10.51 0.0012 | | 8 1.1433 0.2052 31.03 0.0000 | | | +-------------------------------------------------------------------+The data set

+-------------------------------------------------------------------+ | | | RACE VIEW _TYPE_ _NUMBER_ _OBS_ _PRED_ _SEPRED_ | | | | White 1 FUNCTION 1 -2.120 -1.851 0.758 | | White 1 PROB 1 0.107 0.136 0.089 | | White 1 PROB 2 0.893 0.864 0.089 | | White 2 FUNCTION 1 -1.449 -1.437 0.297 | | White 2 PROB 1 0.190 0.192 0.046 | | White 2 PROB 2 0.810 0.808 0.046 | | ... | | | +-------------------------------------------------------------------+To plot the fitted logits, select the

data predict; set predict; if _type_ = 'FUNCTION';A simple plot of predicted logits can then be obtained with the following PROC GPLOT step. (The plots displayed use the Annotate facility to add 90% confidence limits, calculated as

proc gplot data=predict; plot _pred_ * view = race / haxis=axis1 hminor=0 vaxis=axis2; symbol1 i=none v=+ h=1.5 c=black; symbol2 i=none v=square h=1.5 c=red ; axis1 label=(h=1.4 'Conservativism') offset=(2); axis2 order=(-5 to 2) offset=(0,3) label=(h=1.4 a=90 'LOGIT(Reagan / Carter)');

Figure 34: Observed log odds

Figure 35: Fitted logits for main effects model

For example, to test and plot results under the assumption that
political view has a linear effect on the logit scale, as in model
(23), we use the same ` MODEL` statement, but specify`
VIEW` as a direct (quantitative) predictor.

proc catmod data=votes order=data; direct view; weight votes; response / out=predict; model votefor = race view / noiter ; title2 'Linear Effect for Political View (90% CI)'; run;The results indicate that this model fits nearly as well as the nominal main effects model, and is preferred since it is more parsimonious.

+-------------------------------------------------------------------+ | | | MAXIMUM-LIKELIHOOD ANALYSIS-OF-VARIANCE TABLE | | | | Source DF Chi-Square Prob | | -------------------------------------------------- | | INTERCEPT 1 101.39 0.0000 | | RACE 1 42.86 0.0000 | | VIEW 1 67.13 0.0000 | | | | LIKELIHOOD RATIO 11 9.58 0.5688 | | | | | | ANALYSIS OF MAXIMUM-LIKELIHOOD ESTIMATES | | | | Standard Chi- | | Effect Parameter Estimate Error Square Prob | | ----------------------------------------------------------------| | INTERCEPT 1 -3.1645 0.3143 101.39 0.0000| | RACE 2 1.2213 0.1865 42.86 0.0000| | VIEW 3 0.4719 0.0576 67.13 0.0000| | | +-------------------------------------------------------------------+

To test whether a single slope for political view, * beta sup
VIEW * is adequate for both races, fit a model which allows
separate slopes (an interaction between race and view):

proc catmod data=votes order=data; direct view; weight votes; response / out=predict; model votefor = race view race*view; title2 'Separate slopes for Political View (90% CI)';The results show that this model does not offer a significant improvement in goodness of fit. The plot nevertheless indicates a slightly steeper slope for white voters than for non-white voters.

+-------------------------------------------------------------------+ | | | MAXIMUM-LIKELIHOOD ANALYSIS-OF-VARIANCE TABLE | | | | Source DF Chi-Square Prob | | -------------------------------------------------- | | INTERCEPT 1 26.55 0.0000 | | RACE 1 2.02 0.1556 | | VIEW 1 9.91 0.0016 | | VIEW*RACE 1 0.78 0.3781 | | | | LIKELIHOOD RATIO 10 8.81 0.5504 | | | | | | ANALYSIS OF MAXIMUM-LIKELIHOOD ESTIMATES | | | | Standard Chi- | | Effect Parameter Estimate Error Square Prob | | ----------------------------------------------------------------| | INTERCEPT 1 -2.7573 0.5351 26.55 0.0000| | RACE 2 0.7599 0.5351 2.02 0.1556| | VIEW 3 0.3787 0.1203 9.91 0.0016| | VIEW*RACE 4 0.1060 0.1203 0.78 0.3781| | | +-------------------------------------------------------------------+

Figure 36: Fitted logits for linear effect of Conservativism

Figure 37: Fitting separate slopes

[Previous] [Next] [Up] [Top]