Example 9-2 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? \(\alpha=0.05\) and 90% power.
Here are the hypotheses:
Null hypothesis
\(H_0\colon \text{Incidence}_{(Unexposed)} \le \text{Incidence}_{(Exposed)}\)
Alternative hypothesis
\(H_A\colon \text{Incidence}_{(Unexposed)} \le \text{Incidence}_{(Exposed)}=\lambda\)
Where:
\(\lambda \gt 0\)
\(\text{Incidence}_{(Exposed)}=p(\text{Disease|Exposed})\)
\(\text{Incidence}_{(Unexposed)}=p(\text{Disease|Not Exposed})\)
and the resulting formula:
\(n=\dfrac{r+1}{r(\lambda -1)^{2}\pi^{2} }\left [ z_{\alpha }\sqrt{(r+1)p_{c}(1-p_{c})}+z_{\beta }\sqrt{\lambda \pi (1-\lambda \pi)+r\pi(1-\pi )} \right ]^{2}\)
where \(\pi=\pi_2\) is the proportion in the reference group and \(p_c\) is the common proportion over the two groups, which is estimated as:
\(p_{c}=\dfrac{\pi (r\lambda +1)}{r+1}\)
When r = 1 (equal-sized groups), the formula above reduces to:
\(p_{c}=\dfrac{\pi (\lambda +1)}{2}=\dfrac{\pi_{1}+\pi_{2} }{2}\)
Let's take a look at tabulated results:
Table B.9. 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 |
Click the button below to find sample size for detecting RR of 2 under conditions above.
Table B.9. 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 |
Try It! Section
- Incidence rate increase \((\pi)\)?
- Relative risk decreases \((\lambda)\)?
- 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?
- What is the minimal detectable relative risk if you had funds for 1000 subjects?
- n decreases
- Largest n is closest to l
- Protective effects would be those with \(\lambda \lt 1\)
- With a background rate of 10/100 and 1000 subjects, a relative risk of about 1.65 could be detected.