# 10.4 - Unmatched Case Control

## Example Section

An unmatched case-control study evaluating the association between smoking and CHD is planned.
If 30% of the population is estimated to be smokers, what is the number of study subjects (assuming an equal number of cases and controls in an unmatched study design) necessary to detect a hypothesized odds ratio of 2.0? Assume 90% power and α=0.05.

## Formula Section

Due to the design of unmatched case-control studies, where unequal sampling rates are used for the exposed and unexposed, we cannot estimate relative risks for case-control studies, and must instead estimate odds ratios.  The hypothesis we wish to test is slightly altered

$$\begin{array}{l} \mathrm{H}_{0}^{*}\colon \pi_{1}^{*}=\pi_{2}^{*} \\ \mathrm{H}_{1}^{*}\colon \pi_{1}^{*} / \pi_{2}^{*}=\lambda^{*} \end{array}$$

Where

$$\begin{array}{l} \pi_{1}^{*}=p(\text { Exposed } \mid \text { Disease })=p(\text { Exposed } \mid \text { Case }) \\ \pi_{2}^{*}=p(\text { Exposed } \mid \text { No disease })=p(\text { Exposed } \mid \text { Control }) \end{array}$$

The formulas are similar to the formula for relative risk, but with additional parameters.

\begin{aligned} n=\frac{(r+1)(1+(\lambda-1) P)^{2}}{r P^{2}(P-1)^{2}(\lambda-1)^{2}}[ & z_{\alpha} \sqrt{(r+1) p_{c}^{*}\left(1-p_{c}^{*}\right)} \\ & \left.+z_{\beta} \sqrt{\frac{\lambda P(1-P)}{[1+(\lambda-1) P]^{2}}+r P(1-P)}\right]^{2} \end{aligned}

and

$$\displaystyle{p_{c}^{*}=\frac{P}{r+1}\left(\frac{r \lambda}{1+(\lambda-1) P}+1\right)}$$

Where

• $$\mathrm{P}=$$ exposure prevalence
• $$\lambda=$$ estimated relative risk
• r = ratio of cases to controls

For our example, $$p_{c}{ }^{*}=0.3808$$ and n= 375.6 total patients - that is 188 cases and 188 controls.

These tables give requirements for a one-sided test directly. For two-sided tests, use the table corresponding to half the required significance level. Note that $$P$$ is the prevalence of the risk factor in the entire population and $$\lambda$$ is the appropriate relative risk to be tested.

(a) 5% significance, 90% power

$$P$$

$$\lambda$$ 0.010 0.050 0.100 0.200 0.300 0.400 0.500 0.700 0.900
0.10 2 318 456 224 108 70 50 40 30 38
0.20 3 206 638 316 158 104 80 66 56 88
0.30 4 546 912 458 232 160 124 106 98 176
0.40 6 676 1 348 684 356 248 200 176 172 330
0.50 10 318 2 098 1 074 566 404 332 296 306 616
0.60 17 220 3 522 1 816 974 706 588 536 576 1 206
0.70 32 570 6 698 3 476 1 890 1 390 1 174 1 088 1 206 2 612
0.80 77 686 16 052 8 382 4 614 3 438 2 944 2 764 3 146 7 012
0.90 328 374 68 156 35 786 19 922 15 020 13 006 12 354 14 400 32 892
1.10 363 666 76 090 40 352 22 918 17 630 15 574 15 096 18 316 43 550
1.20 95 332 20 020 10 664 6 112 4 744 4 228 4 134 5 102 12 340
1.30 44 334 9 342 4 998 2 888 2 260 2 032 2 002 2 510 6 166
1.40 26 044 5 506 2 958 1 722 1 358 1 230 1 222 1 554 3 870
1.50 17 376 3 684 1 986 1 166 926 846 846 1 090 2 748
1.60 12 558 2 672 1 446 854 684 628 632 826 2 106
1.80 7 618 1 630 888 532 432 400 408 546 1 420
2.00 5 230 1 124 616 374 306 288 296 404 1 074
3.00 1 754 386 218 138 120 118 126 184 522
4.00 978 220 126 84 74 76 84 130 380
5.00 664 150 88 60 56 58 66 104 316
10.00 244 60 38 30 30 34 40 70 224
20.00 108 30 20 18 20 24 30 56 190

(Tables from Woodward, M. Epidemiology Study Design and Analysis. Boca Raton: Chapman and Hall, 2013)

## Stop and Think!

What happens to the necessary sample size as:
1. Prevalence of the risk factor increases (P)?
2. Odds ratio decreases ($$\lambda$$)?
1. For many $$\lambda$$, 0.5 has the smallest sample size requirement
2. largest sample sizes with OR closest to 1; 1.1 requires greater n than 0.9