Brief solution to Problem on matching,
Epidemiology Course VHM 811 at AVC - Fall Semester 2008

1. As a start, we compute risks for the four population groups in Table 10-1 (table numbers refer to Rothman & Greenland, 1998):

   Table 10-1	Males		Females
   	Exposed	Unexposed	Exposed	Unexposed
   No. cases in 1 year	4500	50	100	90
   Total population	900,000	100,000	100,000	900,000
   Risk (%)	0.5	0.05	0.1	0.01

   The table shows confirms that: within sex, exposed have 10 times the risk of unexposed, and within exposure levels, males have five times the risk of females. The crude risk for exposed subjects is (4500+100)/1000000=0.46%, and the crude risk for unexposed subjects is (50+90)/1000000=0.014%, giving a risk ratio of 32.9.

2. The proportion of males among exposed subjects is 9/10=0.9, and among 100,000 exposed subjects we therefore expect 90,000 males and 10,000 females. When matching the sex distribution for the unexposed subjects to the exposed subjects, we try to achieve the same distribution of males and females, i.e., 90,000 males and 10,000 females in the unexposed group as well. The expected numbers of cases in the four groups are given in Table 10-2 below. They were computed from the risk ratios in Table 10-1, e.g. for exposed males: 0.005*90000=450.

   Table 10-2	Males		Females
   	Exposed	Unexposed	Exposed	Unexposed
   Expected cases	450	45	10	1
   Total study	90,000	90,000	10,000	10,000

   We immediately see that the risk ratio equals 10 within each sex, and the crude risk ratio is (450+10)/(90000+10000) / (45+1)/(90000+10000) = 10. Therefore, there is no confounding by sex.

3. The total number of cases is 4740. The number of male cases is 4550, and when matching the sex distribution for the controls to the cases, we try to achieve the same distribution of males and females, i.e., 4550 males and 190 females. The expected numbers of exposed and unexposed subjects in the four groups are given in Table 10-4 below. The numbers for the cases we had already, in Table 10-1. For the controls, the distribution of exposed and unexposed subjects within sex in the population from Table 10-1 were used, e.g. for exposed males: 4550*9/10=4095.

   Table 10-4	Males			Females
   	Exposed	Unexposed	Total	Exposed	Unexposed	Total
   Cases	4500	50	4550	100	90	190
   Controls	4095	455	4550	19	171	190

   Within both males and females, the (expected) odds-ratio equals 10 (e.g., (4500*455)/(50*4095)=10). However, the crude odds-ratio is (4600*626)/(140*4114)=5.0. This shows that an analysis stratified on sex will produce the correct odds-ratio but the unstratified analysis will not. Note that the bias introduced is in the opposite direction of the confounding bias in the population.

   The selection bias for the exposure is seen from the proportion of exposed subjects among the selected controls being (4095+19)/4740=0.868, whereas the population proportion is 0.5. Note that the proportion among the cases is 4600/4740=0.97, so the proportion among the controls is indeed between these two numbers.

4. The desired sex distribution within controls is the same as within cases, i.e. a proportion of 4550/4740=0.96 males. Among 1000 cases this corresponds to 960 males and 40 females. We therefore aim for 960 males and 40 females among the controls as well. The expected numbers for these samples are given in the table.

   Sample of 1000 cases/controls	Males			Females
   	Exposed	Unexposed	Total	Exposed	Unexposed	Total
   Cases	949.45	10.55	960	21.05	18.95	40
   Controls	864	96	960	4	36	40

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   Henrik Stryhn (hstryhn@upei.ca) 2008-11-18