A study investigated characteristics associated with y = whether a cancer patient achieved remission (1 = yes, 0 = no). An important explanatory variable was a labeling index (LI = percentage of ‐labeled‐ cells) that measures proliferative activity of cells after a patient receives an injection of tritiated thymidine. Table 4.5 shows the data and R output for a logistic regression model. a. Show that P^(Y = 1) = 0.50 when LI = 26.0. b. When LI increases by 1, show that the estimated odds of remission multiply by 1.16. c. Summarize the LI effect by how P^(Y = 1) changes over the range or interquartile range of LI values. d. Show that the rate of change in P^(Y = 1) is 0.009 when LI = 8. e. Summarize the LI effect by the estimated average marginal effect.
Article by E.T. Lee, Computer Prog. Biomed. 4: 80–92 (1974).
LOGISTIC REGRESSION
Table 4.5 Software output for Exercise 4.1 on cancer remission.
> LI <- c(8,8,10,10,12,12,12,14,14,14,16,16,16,18,20,20,20,22,22,24,26,28,32,34,
+ 38,38,38)
> y <- c(0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,0,1,1,1,0)
> summary(glm(y ~ LI, family=binomial))
Estimate Std. Error z value Pr(>|z|)
(Intercept) -3.77714 1.37862 -2.740 0.00615
LI 0.14486 0.05934 2.441 0.01464
---
Null deviance: 34.372 on 26 degrees of freedom
Residual deviance: 26.073 on 25 degrees of freedom
> confint(glm(y ~ LI, family=binomial))
2.5 % 97.5 %
LI 0.04252 0.28467