A study used logistic regression to determine characteristics associated with Y = whether a cancer patient achieved remission (1 = yes). The most important explanatory variable was a labeling index (LI) that measures proliferative activity of cells after a patient receives an injection of tritiated thymidine. It represents the percentage of cells that are “labeled.” Table 1 shows the grouped data. Software reports Table 2 for a logistic regression model using LI to predict π = P(Y = 1). Show how software obtained = 0.068 when LI = 8. Show that = 0.50 when LI = 26.0. Show that the rate of change in is 0.009 when LI = 8 and is 0.036 when LI = 26. The lower quartile and upper quartile for LI are 14 and 28. Show that increases by 0.42, from 0.15 to 0.57, between those values. When LI increases by 1, show the estimated odds of remission multiply by 1.16. Using information from Table 2, conduct a Wald test for the LI effect. Interpret. Using information from Table 2, construct a Wald confidence interval for the odds ratio corresponding to a 1-unit increase in LI. Interpret. Using information from Table 2, conduct a likelihood-ratio test for the LI effect. Interpret. Using information from Table 2, construct the likelihood-ratio confidence interval for the odds ratio. Interpret.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 21E

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Question

A study used logistic regression to determine characteristics associated with Y = whether a cancer patient achieved remission (1 = yes). The most important explanatory variable was a labeling index (LI) that measures proliferative activity of cells after a patient receives an injection of tritiated thymidine. It represents the percentage of cells that are “labeled.” Table 1 shows the grouped data. Software reports Table 2 for a logistic regression model using LI to predict π = P(Y = 1).

Show how software obtained = 0.068 when LI = 8.
Show that = 0.50 when LI = 26.0.
Show that the rate of change in is 0.009 when LI = 8 and is 0.036 when LI = 26.
The lower quartile and upper quartile for LI are 14 and 28. Show that increases by 0.42, from 0.15 to 0.57, between those values.
When LI increases by 1, show the estimated odds of remission multiply by 1.16.
Using information from Table 2, conduct a Wald test for the LI effect. Interpret.
Using information from Table 2, construct a Wald confidence interval for the odds ratio corresponding to a 1-unit increase in LI. Interpret.
Using information from Table 2, conduct a likelihood-ratio test for the LI effect. Interpret.
Using information from Table 2, construct the likelihood-ratio confidence interval for the odds ratio. Interpret.

LI
8
10
12
Number of Number of
Cases
2233 m
2
Remissions LI
0
0
0
18
62228
0
0
20
Number of Number of
Cases
Remissions
24
1
3
1
2
1
LI
28
32
34
38
0
1
Number of Number of
Cases Remissions
2
14
1
16
3
1
Source: Reprinted with permission from E. T. Lee, Computer Prog. Biomed., 4: 80-92, 1974.
1
1
1
3
1
0
1
2

Parameter Estimate
Intercept
-3.7771
0.1449
li
Table 2. Computer Output
Obs
1
2
li
8
10
Source
li
Standard
Error
remiss
0
0
1.3786
0.0593
DF
1
n
LR Statistic
Chi-Square
8.30
22
2
Likelihood Ratio
95% Conf. Limits
2
-6.9946 -1.4097
0.0425
0.2846
pi_hat
0.06797
0.08879
Pr > ChiSq
0.0040
lower
0.01121
0.01809
Chi-Square
7.51
5.96
upper
0.31925
0.34010

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