The proof can be presented briefly as follows. Denoting by
l 1 ¼ 1 c 1 ð 0 al 1 a 1 Þ
l 0 ¼ 1 c 0 ð 0 al 0 a 1 Þthe exposure probabilities for cases and controls, respectively, the probability
of observing a case–control pair with only the case exposed isl 1 c 0 , while that
of observing a pair where only the control is exposed isc 1 l 0. Hence the con-
ditional probability of observing a pair of the former variety, given that it is
discordant, is
P¼
l 1 c 0
l 1 c 0 þc 1 l 0¼l 1 c 0 =c 1 l 0
l 1 c 0 =c 1 l 0 þ 1¼ðl 1 =c 1 Þ=ðl 0 =c 0 Þ
ðl 1 =c 1 Þ=ðl 0 =c 0 Þþ 1¼y
yþ 1a function of the odds ratioyonly.
3.6 NOTES ON COMPUTATIONS
In Sections 1.4 and 2.5 we covered basic techniques for Microsoft’s Excel: how
to open/form a spreadsheet, save it, retrieve it, and perform certain descriptive
statistical tasks. Topics included data-entry steps, such asselect and drag, use of
formula bar, bar and pie charts, histograms, calculations of descritive statistics
such as mean and standard deviation, and calculation of a coe‰cient of corre-
lation. In this short section we focus on probability models related to the cal-
culation of areas under density curves, especially normal curves andtcurves.
Normal Curves The first two steps are the same as in obtaining descriptive
statistics (but no data are needed now): (1) click thepaste function icon, f*, and
(2) clickStatistical. Among the functions available, two are related tonormal
curves: NORMDIST and NORMINV. Excel provides needed information
for any normal distribution, not just the standard normal distribution as in
Appendix B. Upon selecting either one of the two functions above, a box
appears asking you to provide (1) the meanm, (2) the standard deviations, and
(3) in the last row, markedcumulative, to enterTRUE(there is a choice
FALSE, but you do not need that). The answer will appear in apreselected
cell.
140 PROBABILITY AND PROBABILITY MODELS