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Numerical Example

To illustrate our response model with a delivery time, we use the response data collected by Huxley (1980) as a part of his dissertation research. He mailed out questionnaires to N=4,314 manufacturing firms, and he recorded the number of responses received by the end of each week during the 17-week period. Huxley’s response data has been extensively used as a benchmark in subsequent studies by Hill (1981), Parasuraman (1982), McGowan (1986), Bauer (1991), and others.

Using Huxley’s response data, we estimate the parameter values of our geometric response model. The results are given in Table 1.1. As a performance measure, we consider the sum of squared errors (SSE) of the cumulative number of responses si. The maximum value of the likelihood function in (1-11) is also considered as a performance measure.

Table 1.1 Various Delivery Time Models with Estimates of π, q, and d

Delivery Time Model

Response Rate, π

Delay Rate, q

Delivery Time, d

SSE

ML

No delivery time

0.5897

0.8836

428,948

–5959.4

Uniform distribution

0.5498

0.8365

u =2.000

129,894

–5628.6

Geometric distribution

0.5357

0.7414

r =0.742

135,515

–5700.5

Poisson distribution

0.5308

0.7746

s =2.163

91,876

–5578.3

Without the delivery time, the ultimate response rate is estimated as π=0.58972. The maximum likelihood estimate of the weekly delay rate is q=0.88355. The SSE of our geometric response model with an instant delivery time is 428,948, which is much better than the SSE=649,503 of Huxley’s (1980) classical regression model in (1-3). If we include a delivery time, the geometric response model performs even better, as shown in Table 1.1.

Among the three probability distributions of the delivery time, the Poisson distribution appears to be the best, followed by the uniform distribution. The Poisson delivery time model has the smallest SSE and the largest value of the likelihood function. The superior performance of the Poisson distribution is anticipated from Figure 1.3, where the Poisson delivery time looks more realistic than the uniform or geometric delivery distribution. By changing the parameter value of the Poisson distribution, we can represent a wide variety of delivery time distribution with different shapes and locations. In practice, we strongly suggest using the geometric response model with the Poisson delivery time.

Figure 1.4 illustrates Huxley’s (1980) original response data, along with the cumulative number of responses si predicted by our geometric response model with the Poisson delivery. The dotted curve in Figure 1.4 represents the predictions of Huxley’s (1980) classical response model. As contrasted in the figure, our S-shaped response curve with a delivery time is clearly a better choice than Huxley’s banana-shaped concave curve for the 17-week mail survey data.

Figure 1.4

Figure 1.4 Actual and fitted values of the cumulative number of responses at time k=17.

Figure 1.5 displays the cumulative number of responses si, up to k=25, predicted by Huxley’s model and by our geometric response model with the Poisson delivery. When the first k=10 week data is available, the Huxley’s growth curve has a negative value at k=0, as shown in Figure 1.5 (a), and it significantly overestimates the actual values from k=11 to 25. Furthermore, Huxley’s model approaches N=4,314 as k approaches infinity. On the other hand, our geometric response model with a Poisson delivery slightly underestimates the actual values from k=11 to 17, but it fits much better than Huxley’s response model.

Figure 1.5

Figure 1.5 Predictions of the cumulative number of responses.

The predicted values based on the first 15 weeks’ worth of data are shown in Figure 1.5(b). The S-shaped growth curve of our geometric response model predicts the cumulative number of responses by the end of the 25th week much better than Huxley’s banana-shaped concave curve.

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