10.3 - Compare Two Proportions

Example Section

Suppose the rate of disease in an unexposed population is 10/100 person-years. You hypothesize an exposure has a relative risk of 2.0. How many persons must you enroll assuming half are exposed and half are unexposed to detect this increased risk, with a one-sided alpha of 0.05 and power of 90%?

Formula Section

We are interested in testing the following hypothesis:

$$\begin{array}{l} \mathrm{H}_{0}\colon \pi_{1}=\pi_{2} \\ \mathrm{H}_{1}\colon \pi_{1}-\pi_{2}=\delta \end{array}$$

But it is usually more convenient to consider the ratio (i.e. relative risk = λ), so we can consider this hypothesis:

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

The formulas needed to calculate the total sample size are:

$$\displaystyle{n=\frac{r+1}{r(\lambda-1)^{2} \pi^{2}}\left[z_{\alpha} \sqrt{(r+1) p_{c}\left(1-p_{c}\right)}+z_{\beta} \sqrt{\lambda \pi(1-\lambda \pi)+r \pi(1-\pi)}\right]^{2}}$$,

and

$$\displaystyle{p_{c}=\frac{\pi(r \lambda+1)}{r+1}}$$

where

$$\pi=\pi_{2}$$ is the proportion in the reference group

$$\mathrm{r}=\mathrm{n}_{1} / \mathrm{n}_{2}$$ (ratio of sample sizes in each group)

$$p_{0}=$$ the common proportion over the two groups

When r = 1 (equal-sized groups), the formula above reduces to:

$$p_{c}=\dfrac{\pi(\lambda+1)}{2}=\dfrac{\pi_{1}+\pi_{2}}{2}$$

From the formula, we can calculate that n=433 total, thus n=217 per group.

The table below can also be used to estimate the necessary sample size.  For the column with $$\pi$$=0.10, with $$\lambda$$=2.0, we see that n=448 total, with n=224 per group.  Approximately the same as from the formula.

Sample Size statement:  A sample size of n=217 per group (total of 434) is needed to detect an increased risk of disease (relative risk=2.0) when the proportions are 10% in the unexposed and 20% in the exposed groups, using a two group chi-squared test with one-sided alpha of 0.05 and 90% power.

Table B.9. Total sample size requirements (for the two groups combined) for testing the ratio
of two proportions (relative risk) with equal numbers in each group.

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 $$\pi$$ is the proportion for the reference group (the denominator) and $$\lambda$$ is the relative risk to be tested.

(a) 5% significance, 90% power

$$\pi$$

$$\lambda$$ 0.001 0.005 0.010 0.050 0.100 0.150 0.200 0.500 0.900
0.10 23 244 4 636 2 310 488 216 138 100 30 8
0.20 32 090 6 398 3 188 618 298 190 136 40 10
0.30 45 406 9 052 4 508 874 418 268 192 56 14
0.40 66 554 13 268 6 606 1 278 612 390 278 78 18
0.50 102 678 20 466 10 190 1 968 940 598 426 118 26
0.60 171 126 34 104 16 976 3 274 1 562 990 706 192 38
0.70 323 228 64 410 32 058 6 176 2 940 1 862 1 322 352 62
0.80 770 020 153 422 76 348 14 688 6 980 4 412 3 128 814 126
0.90 3 251 102 647 690 322 264 61 924 29 380 18 534 13 110 3 336 450
1.10 3 593 120 715 666 355 984 68 240 32 272 20 282 14 288 3 496 292
1.20 941 030 187 410 93 208 17 846 8 426 5 286 3 716 890
1.30 437 234 87 068 43 298 8 280 3 904 2 444 1 714 402
1.40 256 630 51 098 25 406 4 854 2 284 1 428 1 000 228
1.50 171 082 34 062 16 934 3 232 1 518 948 662 148
1.60 123 556 24 596 12 226 2 330 1 094 680 474 104
1.80 74 842 14 896 7 402 1 408 658 408 284 58
2.00 51 318 10 212 5 074 962 448 278 192
3.00 17 102 3 400 1 688 316 146 88 60
4.00 9 498 1 886 934 174 78 46 30
5.00 6 419 1 272 630 116 52 30
10.00 2 318 458 226 40
20.00 992 194 94

(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. Incidence rate increase $$(\pi)$$?
2. Relative risk decreases $$(\lambda)$$?
3. How would you use this table to determine sample size for 'protective' effects (i.e., nutritional components or medical procedures which prevent a negative outcome), as opposed to an increased risk?
4. What is the minimal detectable relative risk if you had funds for 1000 subjects?
1. n decreases
2. Largest n is closest to l
3. Protective effects would be those with $$\lambda \lt 1$$
4. With a background rate of 10/100 and 1000 subjects, a relative risk of about 1.65 could be detected.