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The CATMOD Procedure

Example 22.1: Linear Response Function, r=2 Responses

In an example from Ries and Smith (1963), the choice of detergent brand (Brand= M or X) is related to three other categorical variables: the softness of the laundry water (Softness= soft, medium, or hard), the temperature of the water (Temperature= high or low), and whether the subject was a previous user of brand M (Previous= yes or no). The linear response function, which could also be specified as RESPONSE MARGINALS, yields one probability, Pr(brand preference=M), as the response function to be analyzed. Two models are fit in this example: the first model is a saturated one, containing all of the main effects and interactions, while the second is a reduced model containing only the main effects. The following statements produce Output 22.1.1 through Output 22.1.4: 

   title 'Detergent Preference Study';
   data detergent;
      input Softness $ Brand $ Previous $ Temperature $ Count @@;
      datalines;
   soft X yes high 19   soft X yes low 57
   soft X no  high 29   soft X no  low 63 
   soft M yes high 29   soft M yes low 49
   soft M no  high 27   soft M no  low 53
   med  X yes high 23   med  X yes low 47
   med  X no  high 33   med  X no  low 66
   med  M yes high 47   med  M yes low 55
   med  M no  high 23   med  M no  low 50
   hard X yes high 24   hard X yes low 37
   hard X no  high 42   hard X no  low 68
   hard M yes high 43   hard M yes low 52
   hard M no  high 30   hard M no  low 42
   ;

   proc catmod data=detergent;
      response 1 0;
      weight Count;
      model Brand=Softness|Previous|Temperature 
            / freq prob nodesign;
      title2 'Saturated Model';
   run;

Output 22.1.1: Detergent Preference Study: Linear Model Analysis
 
Detergent Preference Study
Saturated Model

The CATMOD Procedure

Response Brand Response Levels 2
Weight Variable Count Populations 12
Data Set DETERGENT Total Frequency 1008
Frequency Missing 0 Observations 24

The "Data Summary" table (Output 22.1.1) indicates that you have two response levels and twelve populations.

Output 22.1.2: Population Profiles
 
Detergent Preference Study
Saturated Model

The CATMOD Procedure

Population Profiles
Sample Softness Previous Temperature Sample Size
1 hard no high 72
2 hard no low 110
3 hard yes high 67
4 hard yes low 89
5 med no high 56
6 med no low 116
7 med yes high 70
8 med yes low 102
9 soft no high 56
10 soft no low 116
11 soft yes high 48
12 soft yes low 106

The "Population Profiles" table in Output 22.1.2 displays the ordering of independent variable levels as used in the table of parameter estimates.

Output 22.1.3: Response Profiles, Frequencies, and Probabilities
 
Detergent Preference Study
Saturated Model

The CATMOD Procedure

Response Profiles
Response Brand
1 M
2 X
 
Response Frequencies
Sample Response Number
1 2
1 30 42
2 42 68
3 43 24
4 52 37
5 23 33
6 50 66
7 47 23
8 55 47
9 27 29
10 53 63
11 29 19
12 49 57
 
Response Probabilities
Sample Response Number
1 2
1 0.41667 0.58333
2 0.38182 0.61818
3 0.64179 0.35821
4 0.58427 0.41573
5 0.41071 0.58929
6 0.43103 0.56897
7 0.67143 0.32857
8 0.53922 0.46078
9 0.48214 0.51786
10 0.45690 0.54310
11 0.60417 0.39583
12 0.46226 0.53774

Since Brand M is the first level in the "Response Profiles" table (Output 22.1.3), the RESPONSE statement causes Pr(Brand=M) to be the single response function modeled.

Output 22.1.4: Analysis of Variance and WLS Estimates
 
Detergent Preference Study
Saturated Model

The CATMOD Procedure

Analysis of Variance
Source DF Chi-Square Pr > ChiSq
Intercept 1 983.13 <.0001
Softness 2 0.09 0.9575
Previous 1 22.68 <.0001
Softness*Previous 2 3.85 0.1457
Temperature 1 3.67 0.0555
Softness*Temperature 2 0.23 0.8914
Previous*Temperature 1 2.26 0.1324
Softnes*Previou*Temperat 2 0.76 0.6850
Residual 0 . .
 
Analysis of Weighted Least Squares Estimates
Effect Parameter Estimate Standard
Error		 Chi-
Square		 Pr > ChiSq
Intercept 1 0.5069 0.0162 983.13 <.0001
Softness 2 -0.00073 0.0225 0.00 0.9740
  3 0.00623 0.0226 0.08 0.7830
Previous 4 -0.0770 0.0162 22.68 <.0001
Softness*Previous 5 -0.0299 0.0225 1.77 0.1831
  6 -0.0152 0.0226 0.45 0.5007
Temperature 7 0.0310 0.0162 3.67 0.0555
Softness*Temperature 8 -0.00786 0.0225 0.12 0.7265
  9 -0.00298 0.0226 0.02 0.8953
Previous*Temperature 10 -0.0243 0.0162 2.26 0.1324
Softnes*Previou*Temperat 11 0.0187 0.0225 0.69 0.4064
  12 -0.0138 0.0226 0.37 0.5415

The "Analysis of Variance" table in Output 22.1.4 shows that all of the interactions are nonsignificant. Therefore, a main-effects model is fit with the following statements: 

      model Brand=Softness Previous Temperature / noprofile;
      title2 'Main-Effects Model';
   run;
   quit;

The PROC CATMOD statement is not required due to the interactive capability of the CATMOD procedure. The NOPROFILE option suppresses the redisplay of the "Response Profiles" table. Output 22.1.5 through Output 22.1.7 are produced.

Output 22.1.5: Main-Effects Design Matrix
 
Detergent Preference Study
Main-Effects Model

The CATMOD Procedure

Response Brand Response Levels 2
Weight Variable Count Populations 12
Data Set DETERGENT Total Frequency 1008
Frequency Missing 0 Observations 24
 
Sample Response
Function		 Design Matrix
1 2 3 4 5
1 0.41667 1 1 0 1 1
2 0.38182 1 1 0 1 -1
3 0.64179 1 1 0 -1 1
4 0.58427 1 1 0 -1 -1
5 0.41071 1 0 1 1 1
6 0.43103 1 0 1 1 -1
7 0.67143 1 0 1 -1 1
8 0.53922 1 0 1 -1 -1
9 0.48214 1 -1 -1 1 1
10 0.45690 1 -1 -1 1 -1
11 0.60417 1 -1 -1 -1 1
12 0.46226 1 -1 -1 -1 -1

The design matrix in Output 22.1.5 displays the results of the factor effects modeling used in PROC CATMOD.

Output 22.1.6: ANOVA Table for the Main-Effects Model
 
Detergent Preference Study
Main-Effects Model

The CATMOD Procedure

Analysis of Variance
Source DF Chi-Square Pr > ChiSq
Intercept 1 1004.93 <.0001
Softness 2 0.24 0.8859
Previous 1 20.96 <.0001
Temperature 1 3.95 0.0468
Residual 7 8.26 0.3100

The analysis of variance table in Output 22.1.6 shows that previous use of Brand M, together with the temperature of the laundry water, are significant factors in preferring Brand M laundry detergent. The table also shows that the additive model fits since the goodness-of-fit statistic (the Residual Chi-Square) is nonsignificant.

Output 22.1.7: WLS Estimates for the Main-Effects Model
 
Detergent Preference Study
Main-Effects Model

The CATMOD Procedure

Analysis of Weighted Least Squares Estimates
Effect Parameter Estimate Standard
Error		 Chi-
Square		 Pr > ChiSq
Intercept 1 0.5080 0.0160 1004.93 <.0001
Softness 2 -0.00256 0.0218 0.01 0.9066
  3 0.0104 0.0218 0.23 0.6342
Previous 4 -0.0711 0.0155 20.96 <.0001
Temperature 5 0.0319 0.0161 3.95 0.0468

The negative coefficient for Previous (-0.0711) in Output 22.1.7 indicates that the first level of Previous (which, from the table of population profiles, is `no') is associated with a smaller probability of preferring Brand M than the second level of Previous (with coefficient constrained to be 0.0711 since the parameter estimates for a given effect must sum to zero). In other words, previous users of Brand M are much more likely to prefer it than those who have never used it before.

Similarly, the positive coefficient for Temperature indicates that the first level of Temperature (which, from the "Population Profiles" table, is `high') has a larger probability of preferring Brand M than the second level of Temperature. In other words, those who do their laundry in hot water are more likely to prefer Brand M than those who do their laundry in cold water.

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