Comparing Risks

To evaluate whether an exposure is at all contributing to risk of disease, the simplest form of comparison is as follows:
x1 ~ Binomial(n1, pi1)
x2 ~ Binomial(n2, pi2)



Then, a variety of comparison can be performed with these two parameters pi1 and pi2: here is a list

Calculation of these measures can be done easily in R. The following example table can be used to illustrate:

cross-sectional data		outcome
		Diseased	Not diseased	Marginal total
exposure	Exposed	a = 23	b = 87	n1
	Unexposed	c = 8	d = 92	n2
Marginal total		m1	m2	N

Here are the relevant codes:

> # Data > a = 23 > b = 87 > c = 8 > d = 92 > n1 = a + b > n2 = c + d > m1 = a + c > m2 = b + d > p1 = a/n1 > p2 = c/n2 > N = n1 + n2 >  > # Use of package > require(epiR) > epi.2by2(a,b,c,d, method = "cross.sectional", conf.level = 0.95, verbose = T) $RR        est      se    lower   upper 1 2.613636 1.47183 1.225322 5.57494 $OR       est       se    lower    upper 1 3.04023 1.547831 1.291383 7.157442 $AR        est      se  weight   lower    upper 1 12.90909 4.73221 446.552 3.63413 22.18405 $ARp        est       se      lower    upper 1 6.761905 3.654011 -0.3998244 13.92363 $AFe         est     lower     upper 1 0.6173913 0.1838878 0.8206259 $AFp         est      lower     upper 1 0.4580645 0.04635869 0.6920288 $chisq   test.statistic df     p.value 1        6.93727  1 0.008441788 > epi.2by2(a,b,c,d, method = "cross.sectional", conf.level = 0.95, verbose = F)              Disease +    Disease -      Total        Prevalence *        Odds Exposed +           23           87        110                20.9       0.264 Exposed -            8           92        100                 8.0       0.087 Total               31          179        210                14.8       0.173 Point estimates and 95 % CIs: --------------------------------------------------------- Prevalence ratio                             2.61 (1.23, 5.57) Odds ratio                                   3.04 (1.29, 7.16) Attrib prevalence *                          12.91 (3.63, 22.18) Attrib prevalence in population *            6.76 (-0.4, 13.92) Attrib fraction in exposed (%)               61.74 (18.39, 82.06) Attrib fraction in population (%)            45.81 (4.64, 69.2) ---------------------------------------------------------  * Cases per 100 population units  >  > # with out any external package > RD = p1 - p2 > RD [1] 0.1290909 > v.RD = p1*(1-p1)/n1 + p2*(1-p2)/n2 > c(RD - qnorm(.975)*sqrt(v.RD), RD + qnorm(.975)*sqrt(v.RD)) [1] 0.0363413 0.2218405 > RR = p1/p2 > RR [1] 2.613636 > LRR = log(RR) > LRR [1] 0.9607425 > v.LRR = (1-p1)/(p1*n1) + (1-p2)/(p2*n2) > c(LRR - qnorm(.975)*sqrt(v.LRR), LRR + qnorm(.975)*sqrt(v.LRR)) [1] 0.2032035 1.7182815 > OR  = (p1/(1-p1)) / (p2/(1-p2)) > OR [1] 3.04023 > LOR  = log(OR) > LOR [1] 1.111933 > v.LOR = (1/(n1*p1*(1-p1))) + (1/(n2*p2*(1-p2))) > c(LOR - qnorm(.975)*sqrt(v.LOR), LOR + qnorm(.975)*sqrt(v.LOR)) [1] 0.2557135 1.9681527

Asymmetric 95% confidence limits

> c(exp(LRR - qnorm(.975)*sqrt(v.LRR)), exp(LRR + qnorm(.975)*sqrt(v.LRR))) [1] 1.225322 5.574940 > c(exp(LOR - qnorm(.975)*sqrt(v.LOR)), exp(LOR + qnorm(.975)*sqrt(v.LOR))) [1] 1.291383 7.157442

Reversing the groups, the theta2:1 estimates can be calculated:

> RR21 = p2/p1 > RR21 [1] 0.3826087 > LRR21 = log(RR21) > LRR21 [1] -0.9607425 > v.LRR21 = (1-p1)/(p1*n1) + (1-p2)/(p2*n2) > c(LRR21 - qnorm(.975)*sqrt(v.LRR21), +   LRR21 + qnorm(.975)*sqrt(v.LRR21)) [1] -1.7182815 -0.2032035 > OR21  = (p2/(1-p2)) / (p1/(1-p1)) > OR21 [1] 0.3289225 > LOR21  = log(OR21) > LOR21 [1] -1.111933 > v.LOR21 = (1/(n1*p1*(1-p1))) + (1/(n2*p2*(1-p2))) > c(LOR21 - qnorm(.975)*sqrt(v.LOR21), +   LOR21 + qnorm(.975)*sqrt(v.LOR21)) [1] -1.9681527 -0.2557135