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EXAMPLE: MATCHING IN CASE-CONTROL STUDIES
Biostat/Epi 536 Discussion Session– November 18, 2008
The following is based on a homework assignment from Dr. Norm Breslow’s Autumn 2006 Biostat/Epi 536 course.
A case-control study of esophageal cancer was conducted in Singapore to study the association between cancer occurrence and the consumption of cigarettes and alcoholic beverages and the temperature at which various non-alcoholic hot beverages were consumed. The study involved 80 male cases of Chinese background who were individually matched to four controls on the basis of age, date of cancer diagnosis and other factors. Subjects were also classified according to their “dialect group”, an indication of their ancestral origin in China and hence of various dietary customs. The first two controls for each case were drawn from the same hospital ward, whereas the second two controls were from a general orthopedic hospital.
We begin by restricting our analysis to matched pairs , consisting of the case and the first control in each matched set. We fit a model with dialect group as the only covariate three ways: using ordinary logistic regression ignoring the matching (MODEL 1), using ordinary logistic regression with the matched set treated as a factor with 80 levels (MODEL 2), and using conditional logistic regression (MODEL 3).
QUESTION: Write the equations for MODELS 1, 2, and 3. Here, dialect group is represented by the 0/1 variable DIAL, with 1 indicating the Hohkien/Teochew dialect group and 0 indicating the Cantonese/Other dialect group.
Let pi = probability of cancer for subject i
p ji = probability of cancer for subject i in group j
MODEL 1: logit( pi ) = β 0 +β 1 DIAL i
MODEL 2: logit( pi ) = β 0 + β 2 * (SET =2) i + β 3 * (SET =3) i + ⋅⋅⋅ + β 80 * (SET =80) i +β 1 *DIAL i
MODEL 3: logit( pji ) = α j +β 1 #DIAL i
We fit each of MODELS 1, 2, and 3 using STATA and obtain the following results.
OUTPUT 1:
OUTPUT 2:
OUTPUT 3:
QUESTION: Comment on the relationship between the odds ratio estimates comparing risk of esophageal cancer between people of Hohkien/Teochew dialect and people of Cantonese dialect when ordinary logistic regression ignoring the matching and conditional logistic regression are used. What do you conclude from this?
Using the ordinary logistic regression model, odds of cancer are 3.86 times higher in people of Hohkien/Teochew dialect than in people of Cantonese dialect. However, remember that the cases and controls were matched at time of data collection so some confounding has been accounted for by the frequency matching. Using the conditional model, odds are 3.44 times higher. These estimates are not too different (about 9% decrease on the coefficient scale). However, the interpretation of the conditional OR is different – it is the OR within a matched pair, so it is adjusted for any factors (measured or unmeasured) that are matched on.
Suppose now we return to the original dataset with 4 controls per case. We fit a model with main effects for dialect group (0/1 dial), cigarette consumption (continuous cigs), Samsu wine consumption (0/1 samsu), and beverage temperature (continuous bev) using conditional logistic regression and obtain the following:
QUESTION: How would you interpret each of the coefficients in the final model?
DIAL: The log odds ratio of esophageal cancer for an individual with Hohkien/Teochew dialect compared to an individual in the same matched set with Cantonese/Other dialect, given cigarette, Samsu wine, and hot beverage consumption are the same for these two individuals. We are assuming that this log OR comparing two individuals in the same set is constant over all matched sets given the same modeled covariates.
CIGS: The log odds ratio of esophageal cancer for an individual with one unit higher cigarette consumption compared to an individual in the same matched set with one unit lower cigarette consumption, given dialect, Samsu wine consumption, and hot beverage consumption are the same for these two individuals.
SAMSU: The log odds ratio of esophageal cancer for an individual who consumed Samsu wine compared to an individual in the same matched set who did not, given dialect, cigarette use, and hot beverage consumption are the same for these two individuals.
BEV: The log odds ratio of esophageal cancer for an individual who consumed beverages one unit higher in temperature compared to an individual in the same matched set with one unit lower temperature, given dialect, cigarette use, and Samsu wine consumption are the same for these two individuals.
Finally, suppose that we fit a model with DIAL, CIGS, SAMSU, and BEV (treated as grouped linear variables) to the 1:4 matched data. We plot the “delta-Pearson” and “delta-beta” (Cook’s distance) diagnostics against the leverages, with separate symbols for cases and controls, and obtain the following:
We identify the matched sets 30, 48, and 49 as having the largest delta-Pearson and Cook’s distance values. By set, the data for the case and controls are as follows:
Set 30:
Set 48:
Set 49:
QUESTION: Based on the data above, why might these three sets be so influential?
Each has a control with high risk factor levels (#148 in 30, #237 in 48, and#243 in 49) relative to the case’s risk factor levels, which is not consistent with the OR estimate.
