Case LR3: Preference Testing - Introduction to Predicting ...
Study Case: Preference Testing
Following exercises are from the study by Ries and Smith. (Chemical Engineering Progress, 59, 1963: 39-43).
An improved detergent, say X, has been formulated. It is considered a much better product than its old form, say M. The question is to test whether or not users prefer X over M in a blind test. For a random sample of 1008 consumers the following questions are asked.
1. Which of the products did you prefer?
2. What laundry detergent do you generally use?
3. What kind of water is used in your locality?
4. What is the temperature of the laundry water used?
TABLE 1
Cross-Classification of Sample of 1008 Consumers According to (1) the Softness of the Laundry Water Used, (2) the Previous Use of Detergent Brand M, (3) the Temperature of the Laundry Water Used, (4) the Preference for Detergent Brand X over Brand M in a Consumer Blind Trial
Previous User of M Previous Non-User of M
Water Brand High Low High Low
Softness Preference Temperature Temperature Temperature Temperature
Soft X 19 57 29 63
M 29 49 27 53
Medium X 23 47 33 66
M 47 55 23 50
Hard X 24 37 42 68
M 43 52 30 42
Weighted Least Squares Estimation Using SPSS
W = 1, 2, and 3 for soft to hard
Use = 1 if previous use of M, 0 otherwise
TH = 1 if temperature is high and 0 if low.
1. Full model
2. Reduced Model
Appendix: Logistic Regression Using Weighted Least Squares
1. Variance of Disturbances
Let:
R = Number of Successes in n independent trials, with probability of
success at each trial p.
R follows a binomial distribution (n, π).
E(R) = n π
σ(R) = [pic]
When n is large so that n π >5 and n(l-π) >5, R is approximately normal.
Let:
[pic]
Then:
(1) [pic] follows approx. N[pic]
(2) logit[pic]follows approx. N[pic]
For logistic regression using grouped data, estimate:
[pic]
2. Weighted Least Squares
Let the population regression model be defined as follows:
[pic] where [pic] follows random [pic]
Standard case
[pic]
B. Variances are different but known
The estimates of the regression coefficients are determined by the weighted least squares.
Minimize [pic] where [pic]
Define a diagonal matrix of weights, W as follows:
[pic]
Then [pic][pic]
.
[pic]
Note:
It is important to note that b follows normal distribution.
Here are key ANOVA results:
[pic]
[pic] where [pic] the weighted average of Y.
[pic]
For testing significance of the sample regression, we use the result:
Under [pic]
SSR follows a [pic] distribution with DF = K.
C. Variances are proportional to known values
[pic]
b is determined by the formula given in B. However,
[pic]
where [pic]
Note:
The weighted least squares routine in standard statistics packages assume this variance error model for estimation.
-----------------------
Use [pic] distribution DF=3 for computing the p-value.
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