## Example 29.11 Predicted Probabilities

Suppose you have collected marketing research data to examine the relationship between a prospect’s likelihood of buying your product and the person’s education and income. Specifically, the variables are as follows:

Variable

Levels

Interpretation

Education

high, low

Prospect’s education level

Income

high, low

Prospect’s income level

Purchase

yes, no

Did prospect purchase product?

The following statements first create a data set, loan, that contains the marketing research data. Then the CATMOD procedure fits a model, obtains the parameter estimates, and obtains the predicted probabilities of interest. These statements produce Output 29.11.1 and Output 29.11.2.

```data loan;
input Education \$ Income \$ Purchase \$ wt;
datalines;
high  high  yes    54
high  high  no     23
high  low   yes    41
high  low   no     12
low   high  yes    35
low   high  no     42
low   low   yes    19
low   low   no      8
;
```

```ods output PredictedValues=Predicted (keep=Education Income PredFunction);
proc catmod data=loan order=data;
weight wt;
response marginals;
model Purchase=Education Income / pred design;
run;
```
```proc sort data=Predicted;
by descending PredFunction;
run;
proc print data=Predicted;
run;
```

Notice that the preceding statements use the Output Delivery System (ODS) to output the parameter estimates instead of the OUT= option, though either can be used.

Output 29.11.1 Marketing Research Data: Obtaining Predicted Probabilities
 Complex Sample Survey Analysis

The CATMOD Procedure

Data Summary
Response Purchase Response Levels 2
Weight Variable wt Populations 4
Data Set LOAN Total Frequency 234
Frequency Missing 0 Observations 8

Population Profiles
Sample Education Income Sample Size
1 high high 77
2 high low 53
3 low high 77
4 low low 27

Response Profiles
Response Purchase
1 yes
2 no

Response Functions and Design Matrix
Sample Response
Function
Design Matrix
1 2 3
1 0.70130 1 1 1
2 0.77358 1 1 -1
3 0.45455 1 -1 1
4 0.70370 1 -1 -1

Analysis of Variance
Source DF Chi-Square Pr > ChiSq
Intercept 1 418.36 <.0001
Education 1 8.85 0.0029
Income 1 4.70 0.0302
Residual 1 1.84 0.1745

Analysis of Weighted Least Squares Estimates
Parameter   Estimate Standard
Error
Chi-
Square
Pr > ChiSq
Intercept   0.6481 0.0317 418.36 <.0001
Education high 0.0924 0.0311 8.85 0.0029
Income high -0.0675 0.0312 4.70 0.0302

Predicted Values for Response Functions
Education Income Function
Number
Observed Predicted Residual
Function Standard
Error
Function Standard
Error
high high 1 0.701299 0.052158 0.67294 0.047794 0.028359
high low 1 0.773585 0.057487 0.808034 0.051586 -0.03445
low high 1 0.454545 0.056744 0.48811 0.051077 -0.03356
low low 1 0.703704 0.087877 0.623204 0.064867 0.080499

Output 29.11.2 Predicted Probabilities Data Set
 Complex Sample Survey Analysis

Obs Education Income PredFunction
1 high low 0.808034
2 high high 0.67294
3 low low 0.623204
4 low high 0.48811

You can use the predicted values (values of PredFunction in Output 29.11.2) as scores representing the likelihood that a randomly chosen subject from one of these populations will purchase the product. Notice that the "Response Profiles" table in Output 29.11.1 shows you that the first sorted level of Purchase is 'yes', indicating that the predicted probabilities are for Pr(Purchase='yes'). For example, someone with high education and low income has an estimated probability of purchase of 0.808. Like any response function estimate given by PROC CATMOD, this estimate can be obtained by cross-multiplying the row from the design matrix corresponding to the sample (sample number 2 in this case) with the vector of parameter estimates: .

This ranking of scores can help in decision making (for example, with respect to allocation of advertising dollars, choice of advertising media, choice of print media, and so on).