In Example 12.32, the probability of cancer metastasizing was given by the logistic regression model with β₀ = −2 and β₁ = 0.5.

Example 12.32

a.

b. Explain what happens to the odds when x is increased by 1. Your explanation  should involve the .5 that appears in the  formula for p(x).  

c. Support your answer to (b) algebraically,  starting from the formula for  p(x).
d. For what value of x are the odds 1? 5?  10?

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Example 12.32 It seems reasonable that the size of a cancerous tumor should be related to the likelihood that the cancer will spread (metastasize) to another site. The article "Molecular Detection of p16 Promoter Methylation in the Serum of Patients with Esophageal Squamous Cell Carcinoma" (Cancer Res. 2001: 3135-3138) investigated the spread of esophageal cancer to the lymph nodes. With x = size of a tumor (cm) and Y = 1 if the cancer does spread, consider the logistic regression model with P₁ = .5 and Bo=-2 (values suggested by data in the article). Then p(x) 1.0 p(x) = from which p(2) = 27 and p(8) = .88 (tumor sizes for patients in the study ranged from 1.7 to 9.0 cm). Because e 2 +.5(6.77) 4, the odds for a 6.77 cm tumor are 4, so that it is four times as likely as not that a tumor of this size will spread to the lymph nodes. Finally, for every 1-cm increase in tumor size, the odds of metastasis increase by a multiplicative factor of e5 1.65, or 65%. Be careful here: the probability of metastasis is not increasing by 65%, but rather the odds; under the logistic regression model, the probability of an outcome does not increase linearly with x (see Figure 12.36). .5 = e-2+.5x 1+e=2+.5x 10 20 30 40 50 60 70 80 X