Comments about Ch. 3 of Applied Logistic Regression, 2nd Ed.

  1. (p. 50) I think the relative risk is a more natural value/function of interest than is the odds ratio. In a lot of cases the odds ratio is described as if it was the relative risk, since it often approximates the relative risk. But if one group has 75% fatalities, and another group has 25% fatalities, then the odds ratio is (0.75/0.25)/(0.25/0.75) = 9, while the ralative risk is 3. In this case the odds ratio is not a good approximation of the relative risk (and it's a bit hard to grasp what the odds ratio represents). An odds ratio of 9 can also correspond to a relative risk of about 9, or a relative risk of 1.8, 27/11, or some other value.
  2. (pp. 51-52) The difference between the odds ratio estimates of 8.1 and 8.11 is just due to the number of digits presented, not because the estimators produce different values from the same data.
  3. (p. 52, lines 17-18) I think the odds ratio is a chief focus in logistic regression due to the form of the logistic regression model, which makes the odds ratio a convenient thing to estimate, more than "due to its ease of interpretation."
  4. (p. 52) Of course the sampling distribution of the odds ratio estimator is only approximately normal for large sample sizes. (H&L have a habit of stating that distributions are normal or chi-square when in fact these are the asymptotic sampling distributions, and not the actual sampling distributions for any sample sizes.)
  5. (p. 58) Can you get the maximum likelihood estimates presented in Table 3.7 from the data presented in Table 3.5 on p. 56 without using software or an iterative method?
  6. (p. 59, line 7) I guess CIE stands for confidence interval estimate, although adding estimate to confidence interval seems unnecessary.
  7. (p. 65, 3rd full paragraph) Even if the age distribution is the same for the two groups, unless one was content to simply report a point estimate and not report test results and give confidence intervals, age should still be used in order to reduce standard error. (For example, using age to create matched pairs should reduce the width of a 95% confidence interval for the difference of the means.)
  8. (p. 68, middle portion of page) Keep in mind that here the wi are not weights, but rather the logits for the two groups. The difference in the logits is the log of the odds ratio, and so exponentiating this difference gives the odds ratio.
  9. (p. 69, first paragraph) Hopefully Ch. 4 will make this clearer!
  10. (p. 69, lines 21-22) The phrase "The adjustment is statistical" seems screwy to me.
  11. (p. 70) Note that a confounder is any covariate which needs to be included in the model in order to adjust for its effect, since it influences the binary response --- so in order to properly measure or assess the effect of a certain risk factor, confounding variables need to be adjusted for. A confounder doesn't have to interact with the risk factor --- they can both effect the outcome in an additive sense, and in this case, once an adjustment is made to account for the cofounder, the odds ratio associated with a risk factor is constant (assuming no other covariate is an effect modifier). If a covariate interacts with a certain risk factor, it is called an effect modifier. If an effect modifier is present, the odds ratio associated with a risk factor is not constant, but is instead a function of the effect modifier. It seems to me that the set of effect modifiers is a subset of the set of confounders. Am I right? I don't think I am! Based on bottom of p. 73, perhaps if the distribution of an explanatory variable is the same for both groups corresponding to a binary risk factor (and so is not associated with the risk factor), then the variable isn't considered to be a confounder (although it could still be that one should use the variable in the model since it may be useful in predicting the response variable). (By omitting a confounder from the model, you bias the assessment of a risk factor. By omitting a variable not associated with the risk factor, you don't bias its assessment, but at the same time you may not may the best model because the omitted variable could be useful for predicting the response.)
  12. (p. 72 (line 26) & p. 73 (line 10)) The percent decrease values given are not correct --- it should be 32% on p. 72 and 47% on p. 73. Doing it H&L's way, a change from 1.00 to 0.10 would be a 900% decrease (whereas I would call it a 90% decrease), and a change from 1.00 to 0.00 wouldn't even be well defined (whereas I would call it a 100% decrease).
  13. (p. 77) Note that H&L want to include the interaction with AGE in the model even though Model 2 isn't significantly better than Model 1, and Model 3 isn't significantly better than Model 2. (Comparing Model 3 to Model 1 with a likelihood ratio test results in a p-value of 0.078.)
  14. (p. 78, Fig. 3.3) Note that the plot shows the estimated logits, which are guaranteed to be linear. We don't have that the data itself exhibits such a strong pattern. So it's not the linearity of the plotted patterns that's notabale --- rather it's the fact that the slopes differ so much.
  15. (p. 79) Note that the confidence intervals contain 1 for ages 15 and 20, but not for 25 and 30. Even though the p-values from p. 77 don't strongly indicate that AGE should be in the model, when AGE and its interaction with the risk factor are included, the odds ratio associated with the risk factor seems to vary appreciably with AGE. Going back to the p-values on p. 77, what guidelines should be followed with regard to including terms in the model?
  16. (p. 80) To refer to the Mantel-Haenszel estimator as "a weighted average of the stratum specific odds ratios" doesn't seem right, since weights are used in both the numerator and the denominator --- it's not a linear combination of the stratum specific odds ratios. Note that the form of the weighting may appear odd if you give it insufficient thought, since the weights are the inverse strata sizes --- it may appear that the larger strata contribute less to the overall estimate, but this isn't really the case since the ai, bi, ci, and di values will be collectively larger with the larger strata sizes. If we use sample proportions instead of cell counts in the numerator and denominator, the weights would be Ni instead of 1/Ni.
  17. (p. 81) The log of the logit-based summary estimator given by (3.21) is a weighted average of the stratum specific log-odds ratios, but the actual estimator given by (3.21) is not a weighted average --- it has more of a (weighted) geometric mean flavor.
  18. (p. 81) Do you know why it's good to use the inverse variances as the weights in (3.21), as opposed to say the inverse standard errors? (It can be shown (using the method of Lagrange multipliers is one way to attack it) that using the inverse variances as weights minimizes the variance of the sum inside of the brackets.)
  19. (p. 81, line 7) The word estimator isn't proper here --- it should be estimate. Similarly, on line 16 of p. 85, it should be estimate. I believe in other places, H&L use estimate where estimator would be better. For example, they refer to the standard error of an estimate. An estimator, being a random variable, can have a standard error, but an estimate, being just a nonvariable number, cannot. When people refer to the standard error of an estimate, it would typically be better to use the phrase estimated standard error associated with the estimate.
  20. (p. 82) Note that it's only proper to employ the estimators being considered if the odds ratio is assumed to be constant. In such a case, you might wonder what's wrong with simply pooling and estimating the common odds ratio from a single 2 by 2 table. But it's possible to have the odds ratio from each 2 by 2 table be the exact same value, and when these tables are pooled, the estimated odds ratio can be a different value.
  21. (p. 82) The use of the word "Thus" in the last sentence on the page doesn't seem proper, since it doesn't follow from the immediately preceding sentence that one should reject for large values of the test statistic.
  22. (p. 83) The sentence after (3.23) should have and is inserted before "given by the following formula" since the quantity, and not the quadratic equation, is given by (3.24).
  23. (p. 83) StatXact has the Mantel-Haenszel procedures for estimating the common odds ratio (both point and interval estimates) and testing the null hypothesis that it equals 1 against the general alternative (it does exact versions of these things), and it has the Breslow-Day test of the null hypothesis of a common odds ratio against the general alternative. The Breslow-Day statistic used doesn't seem to include the correction term that improves the approximation, but that's of little concern because StatXact also does an exact test (Zelen's test). If one wants to focus on the relative risk instead of the odds ratio, StatXact has some pertinent procedures.
  24. (p. 85, line 2) Here the term saturated model is being used in the way that Dr. Bolstein was using it during our June 6 meeting. With the three tables of p. 81, we just have 6 sample proportions, and so 6 parameters can achive a perfect fit of the data in the tables. If SMOKE and RACE are the only covariates available, no model can fit better than this 6 parameter model including the interaction terms --- the model is saturated in that the best possible fit is achieved and there are no other terms to add to improve the fit. However, the likelihood associated with this model is not 1, and this is in contradiction of H&L's use of the term saturated model on p. 13. (On p. 13, the saturated model has to include variables not available in the data set, since in order to have a likelihood if 1 all of the logistic probabilities have to be 1. If observations having the same covariate values have different outcomes, as is the case in the data of p. 81, then in order to have each observation have a logistic probability of 1, other unobserved variables have to be considered.) Perhaps it would have been better to use the term ideal model on p. 13 to refer to the model corresponding to a likelihood of 1.
  25. (p. 85, line 5) It should be Table 3.18 (not 3.17).
  26. (p. 85, lines 9-10) The specific version of a Mantel-Haenszel test referred to here has not yet been described in the book.
  27. (pp. 85-86) The confidence bands considered are pointwise correct as opposed to being such that the entire logit function is contained with a probability of at least 0.95. Question: Is the term confidence band used for both pointwise coverage and overall coverage situations? If so, what words can be added to make it clearer which type of band is being referred to?
  28. (p. 86, last two lines) H&L point out that the band is "narrowest near the mean weight of LWT, approximately 130 pounds." Using (2.7) on p. 41, and Table 2.4 on p. 42, calculus can be used to show the estimated standard error is smallest for a value of LWT of about 127. (Note, that the form of the logit under consideration is rather simple, including just the intercept and only one product of a coefficient and variable. This makes the minimization of (2.7) rather simple.) Is there a reason why the band will be narrowest near the mean of the variable in a case like this, or does it just turn out that way?