Key to Odd Numbered Problems for Unit 14 (Case-Control Studies)

(14.1) STUDY QUESTIONS
(A) In case-control studies, subjects are selected for study based on their disease status. In cohort studies, subjects are selected based on their exposure status or based on some other criterion.
(B) risk (or incidence) . . . relative risk (or, more specifically, rate ratio)
(C) OR; "OR hat"
(D) natural logarithmic
(E) H₀: OR = 1, H₁ OR not = 1
(F) When an expected frequency in the 2-by-2 table is less than 5.
(G) concordant . . . discordant
(H) True
(I) McNemar's chi-square
(J) Wald

(14.3) DOLL1950

2-by-2 Table

          Cases   Controls 
Smoke+      647       622    1269
Smoke-        2        27      29
            649       649    1298

OR^ = 647*27 / 622*2 = 14.04 ; lnOR^ = 2.6419; se_lnOR^ = 0.7350; 95% CI for lnOR = (1.2013, 4.0825); 95% CI for the OR = (3.32, 59.29)

Interpretation: This important, early study of smoking and lung cancer found a strong, significant, positive association between smoking and lung cancer. The odds ratio of 14 suggests that smokers experience lung cancer 14 times more often than non-smokers. It also suggest that the effect of the exposure (smoking) is to increase risk by 1300%.

(14.5) IUD

Estimates: odds ratio = 2.29 (95% CI: 1.76, 2.98)

Test: H₀: OR = 1 vs. H₁: OR not = 1
a = .01

Observed counts:
        INFERT+    INFERT-
IUD+       89       640
IUD-      194      3193
          283      3833 

Expected counts:
        INFERT+    INFERT-
IUD+      50.09     678.91
IUD-     232.91    3157.09

Chi-square, Yates = 38.24, df = 1
p < .001
Reject H₀

These data suggest a significant increase in the risk of infertility with IUD use. (IUD use increased risk by about 130%.

(14.7) ASBESTOS

(A) Lung cancer and smoking

        LungCA+    LungCA-
SMOKE+   80          100
SMOKE-   15           90

OR^ = 4.8 (95% confidence limits for OR: 2.6, 8.9).
This suggests a significant, positive association between smoking and lung cancer ( 380% increase in risk).

(B) Lung cancer and asbestos exposure

        LungCA+    LungCA-
ASBES+   80          38
ASBES-   15         152

OR^ = 21.3 (95% CI for OR: 11.1, 41.1).
This suggests a very large increase in risk (2000% increase).

(C) Lung cancer and asbestos exposure stratified by smoking status

In smokers, the odds ratio is 60.0 (95% CI: 21.4, 168.0)
In non-smokers, the odds ratio is 2.0 (95% CI: 0.6, 6.6)
This suggests an interaction between asbestos and smoking.

(14.9) OR^ = 45 / 24 = 1.9
se_lnOR^ = sqrt(1/45 + 1/24) = 0.2528
95% CI for OR = 1.1, 3.1

H₀: OR = 1 vs. H₁: OR not = 1, a = .05, Chi-sq_McN = 6.39, df = 1, .01 < p < .025, reject H₀

Statistically significant OR of 1.9 (95% CI: 1.1, 3.1)

(14.11) LILIENFELDp220.

(A) Matched-Pair Analysis - This is the proper way to do the analysis.

For thrombotic stroke
OR^ = 44 / 5 = 8.8
95% CI for OR = e^{ln(8.8)�(1.96)(0.4714)} = e^{1.2499, 3.0996} = (3.49, 22.19)

For hemorrhagic stroke
OR^ = 30 / 13 = 2.3
95% CI for OR = e^{ln(2.308)�(1.96)(0.3321)} = e^{0.1852, 1.487} = (1.20, 4.42)

OC use associated with a large increase in risk, especially for thrombotic stroke.

(B) Unmatched analysis. This is the improper way to do the analysis, since there was pair-matching in the sample.

For thrombotic stroke
OR = 4.0 (95% CI: 1.2, 4.7)

For hemorrhagic stroke
OR = 1.6 (95% CI: 1.1, 2.4)

These risk estimates, although still elevated, are much lower than those of the matched analysis. In my view, they represent underestimates of risk.