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Michael Friendly

# Part 7: Plots for logit models

### Contents

A contingency table gives the joint distribution of two or more discrete, categorical variables. In a two-way table, one typically uses the chi sup 2 test of association to determine if the row and column variables can be considered independent. Loglinear and logit models generalize this test of association to three- and higher-way tables; logit models are equivalent forms of a loglinear model when one variable is considered a response to the others.

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 lambda terms sum to zero, alpha = 2 lambda sub 1 sup C , beta sub i sup A = 2 lambda sub i1 sup AC , and beta sub j sup B = 2 lambda sub j1 sup BC .

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.

## Example

The table below shows data from the 1982 General Social Survey on votes in the 1980 US Presidential election for Reagan or for Carter or other in relation to race and political view (1=most liberal, 7=most conservative).
```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          4
```
Treating 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)

## Fitting the model with nominal main effects

The data are first read in to a data set vote in the frequency form that could be used as input to PROC LOGISTIC. PROC CATMOD, however, requires an explicit dependent variable and a separate observation for each value of the response. A second data step creates the response variable votefor.
```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
;
set vote;
```

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;
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 PREDICT contains observed ( _OBS_) and predicted ( _PRED_) values, and estimated standard errors. There are 3 observations for each race-view group: logit values have _TYPE_ = 'FUNCTION'; probabilities have _TYPE_ = 'PROB'.
```+-------------------------------------------------------------------+
|                                                                   |
|  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 _TYPE_ = 'FUNCTION' observations in a data step.
```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 _pred_ ± 1.645 _sepred_ , and a probability scale at the right as illustrated earlier.)
```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

## Other models

Again the predicted values in the output data set depend purely on the model specified in the PROC CATMOD step. The plotting steps remain the same.

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;
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;
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
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