It is important not to overcalculate sample sizes for clinical trials due to economic, ethical, and scientific reasons. Kang and Kim (2014) investigated the accuracy of a well-known sample size calculation formula based on the approximate power for continuous endpoints in equivalence trials, which has been widely used for Development of Biosimilar Products. They concluded that this formula is overly conservative and that sample size should be calculated based on an exact power. This paper extends these results to binary endpoints for three popular metrics: the risk difference, the log of the relative risk, and the log of the odds ratio. We conclude that the sample size formulae based on the approximate power for binary endpoints in equivalence trials are overly conservative. In many cases, sample sizes to achieve 80% power based on approximate powers have 90% exact power. We propose that sample size should be computed numerically based on the exact power.
Many best-selling biological products are set to lose their patents over the next few years; constantly, the assessment of biological product biosimilarity for regulatory approval has received significant attention (Chow, 2014; Chow and Liu, 2010; Chow et al., 2009; Chow et al., 2013; Hsieh et al., 2013; Kang and Chow, 2013; Li et al., 2013; US FDA, 2012; World Health Organization, 2009). It is therefore necessary to demonstrate similar qualities, efficacy, and safety for biosimilar products and renovator biological products in order to obtain regulatory approval. Consequently, the characterization of both products are examined by comparing physicochemical properties, biological activities, impurities, and stability. Immunogenicity tests, preclinical studies, and clinical trials are also conducted to demonstrate no clinically significant differences in safety and efficacy. A phase III comparative study (often designed as an equivalence study) is an important step in systematic studies.
The sample size calculation for a phase III clinical trial is important due to economic, ethical, and scientific considerations (Altman, 1980; Moher et al., 1994). This paper emphasize the drawbacks of oversized studies that calculate sample size based on approximate powers. An oversized study results in an unnecessary waste of resources with the potential to expose unnecessarily large number of subjects to potentially harmful or ineffective treatments (Altman, 1980). Therefore, it is important to compute the minimal sample size needed to achieve a pre-specified power, such as 80% or 90%.
Kang and Kim (2014) investigated the accuracy of a well-known sample size calculation formula for continuous endpoints in equivalence trials. The formula is given as (Chow et al., 2003, p.60)
to test H_{0} : |μ_{T} ? μ_{R}| ≥ δ against H_{a} : |μ_{T} ? μ_{R}| < δ, where z_{α} is the upper α quartile of the standard normal distribution (for example, z_{0.05} = 1.645); μ_{T} and μ_{R} represent the population means of primary endpoints for a biosimilar product and a renovator biological product, respectively; σ^{2} is the population variance of the primary endpoint; k is the allocation ratio; n_{T} and n_{R} are the sample sizes of a biosimilar product group and the renovator biological product group, respectively. Kang and Kim (2014) found that the sample size calculation based on (
The primary endpoint is often binary; therefore, it is important to investigate the accuracy of sample size calculation formulae for binary endpoints in equivalence trials. This paper extends the results of Kang and Kim (2014) to binary endpoints for three popular metrics: the risk difference, the log of the relative risk, and the log of the odds ratio.
This paper is organized as follows. Section 2 reviews the hypotheses of equivalence trials for binary endpoints. Section 3 provides the sample size calculation formulae based on the approximate and exact powers. Section 4 numerically compares approximate powers with exact powers. Section 5 presents the conclusions.
Let X_{T} and X_{R} denote the number of events of interest from the biosimilar product group and the renovator biological product group, respectively. It is assumed that X_{T} and X_{R} follow binomial distributions B(n_{T}, p_{T}) and B(n_{R}, p_{R}), respectively. There are three popular metrics that can be used to assess the treatment effect estimated from an equivalence trial. The first is the risk difference, RD = p_{T} ? p_{R}, which is the difference between the test and control groups in proportions of outcomes. The second is the relative risk, or risk ratio (RR = p_{T}/p_{R}), which is the ratio of the rates of unfavorable events in the test and control groups. The third is the odds ratio, which is the ratio of the odds of success (or failure) of the test product relative to the control product. The characteristics of these three metrics are shown in Sinclair and Bracken (1994) and Walter (2000). In this paper, the log of the relative risk and the log of the odds ratio are investigated instead of the relative risk and the odds ratio, as the former allow metrics that are normally distributed and easier to evaluate in the analysis.
The hypotheses of equivalence for the three metrics are given as follows. For the risk difference, it is given as
where δ(> 0) is a pre-specified equivalence margin. For the log of the relative risk, it is given as
For the log of the odds ratio, it is given as
The hypotheses in (
and
The hypotheses in (
and
For the log of the odds ratio, they are given as
and
If two null hypotheses (
Section 2 introduces two one-sided hypotheses for the risk difference, the log of the relative risk, and the log of the odds ratio. Test statistics for each hypothesis can be constructed using the central limit theorem, Slutsky’s theorem, and the delta method. For the risk difference, the test statistics are given by
For the log of the relative risk, the test statistics are given by
For the log of the odds ratio, the test statistics are given by
For the risk difference, both
Kang and Kim (2014) showed that, under the alternative hypothesis H_{a} : |p_{T} ? p_{R}| < δ, the power of the test for the risk difference is given by
where Ψ is the cumulative distribution function of the standard normal distribution. The powers in (
Then we have
Therefore, the sample size to achieve power 1 ? β based on the approximate power to test the hypothesis in (
where k is an allocation ratio. The sample size calculation formula in (
and the sample size to test the hypothesis in (
Wang et al. (2002) obtained the sample size calculation formula in (
The closed forms of the sample size calculation formulae based on the approximate power in equivalence trials for binary endpoints were derived in Section 3 and given by (
Figure 1 is a graphical representation of the differences between the two powers for the risk difference when α = 5%, n_{1} = n_{2} = 100, and δ = 0.2. As the value of p_{1} ? p_{2} increases, the differences between the two curves also increase and means that the accuracy of the approximate power drops rapidly. When the value of p_{1} ? p_{2} is greater than 0.12, the approximate power drops below zero, which is unacceptable because powers should be positive. Figures 2 and 3 show similar patterns of differences between two powers for the relative risk and the odds ratio.
In order to investigate how many sample size differences are produced by two different powers, the R code was made to compute sample sizes based on exact and approximate powers. Tables 4
Kang and Kim (2014) discovered an interesting phenomenon that the sample sizes needed to achieve 80% approximate power are the same as those needed to achieve 90% exact power for a continuous endpoint. Similar phenomena are also observed for a binary endpoint. Such phenomena occur in 34 of 75 cases in Tables 4
Let n_{T} = n_{R} and
When w is so small that Ψ(w) is negligible, the exact power with the sample size to achieve 1 ? β approximate power is actually 1 ? β/2.
The proof of this theorem is the same as Theorem 1 in Kang and Kim (2014).
Some cases in which the phenomenon described in Theorem 1 occurs were chosen from Tables 4
In this paper, we studied the accuracy of sample size calculation formulae based on the approximate power for binary endpoints in equivalence trials. The risk difference, the log of the relative risk, and the log of the odds ratio were investigated. Formulae were very conservative because the two inequalities derived the closed form of the sample size calculation based on approximate power. In many practical cases, equivalence trials are planned to achieve 80% power. However, this paper shows that the sample sizes to achieve 80% approximate power often have 90% exact power. Therefore, sample size calculation based on the approximate power may produce unnecessary costs and ethical concerns.
This paper proposes that sample sizes for binary endpoints in equivalence trials should be calculated based on the exact power. The R code to calculate the sample sizes based on the exact power is available from the authors upon request.
Risk difference: the exact and approximate powers (α = 0.05)
n_{1} = n_{2} | δ | p_{1} | p_{2} | p_{1} ? p_{2} | Exact | Approx |
---|---|---|---|---|---|---|
100 | 0.1 | 0.052 | 0.05 | 0.002 | 0.8827 | 0.8677 |
0.1 | 0.054 | 0.05 | 0.004 | 0.8734 | 0.8422 | |
0.1 | 0.056 | 0.05 | 0.006 | 0.8625 | 0.8139 | |
0.1 | 0.058 | 0.05 | 0.008 | 0.8500 | 0.7827 | |
0.1 | 0.060 | 0.05 | 0.010 | 0.8358 | 0.7487 | |
100 | 0.2 | 0.120 | 0.1 | 0.020 | 0.9919 | 0.9847 |
0.2 | 0.140 | 0.1 | 0.040 | 0.9672 | 0.9347 | |
0.2 | 0.160 | 0.1 | 0.060 | 0.9049 | 0.8100 | |
0.2 | 0.180 | 0.1 | 0.080 | 0.7930 | 0.5861 | |
0.2 | 0.200 | 0.1 | 0.100 | 0.6388 | 0.2775 | |
100 | 0.25 | 0.220 | 0.2 | 0.020 | 0.9894 | 0.9812 |
0.25 | 0.240 | 0.2 | 0.040 | 0.9736 | 0.9481 | |
0.25 | 0.260 | 0.2 | 0.060 | 0.9399 | 0.8802 | |
0.25 | 0.280 | 0.2 | 0.080 | 0.8814 | 0.7629 | |
0.25 | 0.300 | 0.2 | 0.100 | 0.7942 | 0.5884 | |
100 | 0.25 | 0.320 | 0.3 | 0.020 | 0.9629 | 0.9389 |
0.25 | 0.340 | 0.3 | 0.040 | 0.9355 | 0.8768 | |
0.25 | 0.360 | 0.3 | 0.060 | 0.8872 | 0.7769 | |
0.25 | 0.380 | 0.3 | 0.080 | 0.8159 | 0.6329 | |
0.25 | 0.400 | 0.3 | 0.100 | 0.7226 | 0.4456 | |
100 | 0.3 | 0.430 | 0.4 | 0.030 | 0.9862 | 0.9744 |
0.3 | 0.460 | 0.4 | 0.060 | 0.9630 | 0.9264 | |
0.3 | 0.490 | 0.4 | 0.090 | 0.9123 | 0.8247 | |
0.3 | 0.520 | 0.4 | 0.120 | 0.8232 | 0.6464 | |
0.3 | 0.550 | 0.4 | 0.150 | 0.6927 | 0.3854 |
Relative risk: the exact and approximate powers (α = 0.05)
n_{1} = n_{2} | δ | p_{1} | p_{2} | p_{1}/p_{2} | Exact | Approx |
---|---|---|---|---|---|---|
100 | 0.8 | 0.32 | 0.3 | 1.067 | 0.9598 | 0.9340 |
0.8 | 0.34 | 0.3 | 1.133 | 0.9450 | 0.8946 | |
0.8 | 0.36 | 0.3 | 1.200 | 0.9188 | 0.8389 | |
0.8 | 0.38 | 0.3 | 1.267 | 0.8819 | 0.7642 | |
0.8 | 0.40 | 0.3 | 1.333 | 0.8344 | 0.6689 | |
100 | 0.6 | 0.42 | 0.4 | 1.050 | 0.9308 | 0.8910 |
0.6 | 0.44 | 0.4 | 1.100 | 0.9114 | 0.8343 | |
0.6 | 0.46 | 0.4 | 1.150 | 0.8769 | 0.7579 | |
0.6 | 0.48 | 0.4 | 1.200 | 0.8293 | 0.6598 | |
0.6 | 0.50 | 0.4 | 1.250 | 0.7697 | 0.5398 | |
100 | 0.5 | 0.52 | 0.5 | 1.040 | 0.9409 | 0.9066 |
0.5 | 0.54 | 0.5 | 1.080 | 0.9236 | 0.8568 | |
0.5 | 0.56 | 0.5 | 1.120 | 0.8925 | 0.7882 | |
0.5 | 0.58 | 0.5 | 1.160 | 0.8486 | 0.6983 | |
0.5 | 0.60 | 0.5 | 1.200 | 0.7926 | 0.5854 | |
100 | 0.4 | 0.62 | 0.6 | 1.033 | 0.9308 | 0.8907 |
0.4 | 0.64 | 0.6 | 1.067 | 0.9109 | 0.8327 | |
0.4 | 0.66 | 0.6 | 1.100 | 0.8747 | 0.7531 | |
0.4 | 0.68 | 0.6 | 1.133 | 0.8239 | 0.6488 | |
0.4 | 0.70 | 0.6 | 1.167 | 0.7593 | 0.5188 | |
100 | 0.4 | 0.72 | 0.7 | 1.029 | 0.9922 | 0.9864 |
0.4 | 0.74 | 0.7 | 1.057 | 0.9877 | 0.9759 | |
0.4 | 0.76 | 0.7 | 1.086 | 0.9792 | 0.9585 | |
0.4 | 0.78 | 0.7 | 1.114 | 0.9653 | 0.9307 | |
0.4 | 0.80 | 0.7 | 1.143 | 0.9441 | 0.8882 |
Odd ratio: the exact and approximate powers (α = 0.05)
n_{1} = n_{2} | δ | p_{1} | p_{2} | Exact | Approx | |
---|---|---|---|---|---|---|
100 | 1.0 | 0.42 | 0.4 | 1.086 | 0.9218 | 0.8776 |
1.0 | 0.43 | 0.4 | 1.132 | 0.9086 | 0.8406 | |
1.0 | 0.44 | 0.4 | 1.179 | 0.8899 | 0.7956 | |
1.0 | 0.45 | 0.4 | 1.227 | 0.8657 | 0.7419 | |
1.0 | 0.46 | 0.4 | 1.278 | 0.8361 | 0.6791 | |
100 | 1.0 | 0.52 | 0.5 | 1.083 | 0.9310 | 0.8918 |
1.0 | 0.53 | 0.5 | 1.128 | 0.9179 | 0.8566 | |
1.0 | 0.54 | 0.5 | 1.174 | 0.8993 | 0.8128 | |
1.0 | 0.55 | 0.5 | 1.222 | 0.8750 | 0.7596 | |
1.0 | 0.56 | 0.5 | 1.273 | 0.8449 | 0.6962 | |
100 | 1.0 | 0.62 | 0.6 | 1.088 | 0.9167 | 0.8698 |
1.0 | 0.63 | 0.6 | 1.135 | 0.8999 | 0.8255 | |
1.0 | 0.64 | 0.6 | 1.185 | 0.8762 | 0.7703 | |
1.0 | 0.65 | 0.6 | 1.238 | 0.8453 | 0.7029 | |
1.0 | 0.66 | 0.6 | 1.294 | 0.8071 | 0.6225 | |
100 | 1.2 | 0.73 | 0.7 | 1.159 | 0.9525 | 0.9131 |
1.2 | 0.74 | 0.7 | 1.220 | 0.9341 | 0.8735 | |
1.2 | 0.75 | 0.7 | 1.286 | 0.9083 | 0.8201 | |
1.2 | 0.76 | 0.7 | 1.357 | 0.8739 | 0.7500 | |
1.2 | 0.77 | 0.7 | 1.435 | 0.8297 | 0.6608 | |
100 | 1.4 | 0.82 | 0.8 | 1.139 | 0.9648 | 0.9391 |
1.4 | 0.83 | 0.8 | 1.221 | 0.9467 | 0.8996 | |
1.4 | 0.84 | 0.8 | 1.313 | 0.9178 | 0.8397 | |
1.4 | 0.85 | 0.8 | 1.417 | 0.8750 | 0.7526 | |
1.4 | 0.86 | 0.8 | 1.536 | 0.8152 | 0.6320 |
Risk difference: sample size calculations based on exact and approximate powers (α = 0.05)
δ | p_{1} | p_{2} | p_{1} ? p_{2} | Power | Exact | Approx | Power | Exact | Approx |
---|---|---|---|---|---|---|---|---|---|
0.1 | 0.052 | 0.05 | 0.002 | 80% | 84 | 87 | 90% | 105 | 110 |
0.1 | 0.054 | 0.05 | 0.004 | 85 | 92 | 108 | 116 | ||
0.1 | 0.056 | 0.05 | 0.006 | 88 | 98 | 111 | 123 | ||
0.1 | 0.058 | 0.05 | 0.008 | 90 | 104 | 115 | 131 | ||
0.1 | 0.060 | 0.05 | 0.010 | 93 | 110 | 119 | 139 | ||
0.2 | 0.120 | 0.1 | 0.020 | 80% | 44 | 52 | 90% | 56 | 66 |
0.2 | 0.140 | 0.1 | 0.040 | 54 | 71 | 72 | 89 | ||
0.2 | 0.160 | 0.1 | 0.060 | 72 | 99 | 99 | 124 | ||
0.2 | 0.180 | 0.1 | 0.080 | 103 | 142 | 142 | 179 | ||
0.2 | 0.200 | 0.1 | 0.100 | 155 | 215 | 215 | 271 | ||
0.25 | 0.220 | 0.2 | 0.020 | 80% | 47 | 54 | 90% | 60 | 68 |
0.25 | 0.240 | 0.2 | 0.040 | 52 | 67 | 69 | 85 | ||
0.25 | 0.260 | 0.2 | 0.060 | 62 | 84 | 84 | 106 | ||
0.25 | 0.280 | 0.2 | 0.080 | 78 | 108 | 108 | 136 | ||
0.25 | 0.300 | 0.2 | 0.100 | 102 | 141 | 141 | 178 | ||
0.25 | 0.320 | 0.3 | 0.020 | 80% | 61 | 70 | 90% | 77 | 88 |
0.25 | 0.340 | 0.3 | 0.040 | 66 | 85 | 87 | 107 | ||
0.25 | 0.360 | 0.3 | 0.060 | 77 | 105 | 105 | 133 | ||
0.25 | 0.380 | 0.3 | 0.080 | 96 | 133 | 133 | 167 | ||
0.25 | 0.400 | 0.3 | 0.100 | 124 | 172 | 172 | 217 | ||
0.3 | 0.430 | 0.4 | 0.030 | 80% | 48 | 57 | 90% | 62 | 73 |
0.3 | 0.460 | 0.4 | 0.060 | 55 | 73 | 74 | 92 | ||
0.3 | 0.490 | 0.4 | 0.090 | 69 | 96 | 96 | 121 | ||
0.3 | 0.520 | 0.4 | 0.120 | 94 | 130 | 130 | 164 | ||
0.3 | 0.550 | 0.4 | 0.150 | 134 | 186 | 186 | 235 |
Relative risk: sample size calculations based on exact and approximate powers (α = 0.05)
δ | p_{1} | p_{2} | p_{1}/p_{2} | Power | Exact | Approx | Power | Exact | Approx |
---|---|---|---|---|---|---|---|---|---|
0.8 | 0.32 | 0.3 | 1.067 | 80% | 62 | 71 | 90% | 79 | 90 |
0.8 | 0.34 | 0.3 | 1.133 | 64 | 81 | 83 | 102 | ||
0.8 | 0.36 | 0.3 | 1.200 | 69 | 93 | 93 | 117 | ||
0.8 | 0.38 | 0.3 | 1.267 | 78 | 107 | 107 | 136 | ||
0.8 | 0.40 | 0.3 | 1.333 | 91 | 126 | 126 | 159 | ||
0.6 | 0.42 | 0.4 | 1.050 | 80% | 71 | 82 | 90% | 90 | 103 |
0.6 | 0.44 | 0.4 | 1.100 | 73 | 94 | 96 | 118 | ||
0.6 | 0.46 | 0.4 | 1.150 | 80 | 109 | 109 | 137 | ||
0.6 | 0.48 | 0.4 | 1.200 | 92 | 127 | 127 | 161 | ||
0.6 | 0.50 | 0.4 | 1.250 | 109 | 151 | 151 | 191 | ||
0.5 | 0.52 | 0.5 | 1.040 | 80% | 68 | 78 | 90% | 87 | 99 |
0.5 | 0.54 | 0.5 | 1.080 | 70 | 89 | 92 | 112 | ||
0.5 | 0.56 | 0.5 | 1.120 | 76 | 103 | 103 | 130 | ||
0.5 | 0.58 | 0.5 | 1.160 | 87 | 120 | 120 | 151 | ||
0.5 | 0.60 | 0.5 | 1.200 | 103 | 142 | 142 | 179 | ||
0.4 | 0.62 | 0.6 | 1.033 | 80% | 71 | 82 | 90% | 90 | 103 |
0.4 | 0.64 | 0.6 | 1.067 | 74 | 94 | 97 | 119 | ||
0.4 | 0.66 | 0.6 | 1.100 | 81 | 110 | 110 | 138 | ||
0.4 | 0.68 | 0.6 | 1.133 | 94 | 129 | 129 | 163 | ||
0.4 | 0.70 | 0.6 | 1.167 | 113 | 156 | 156 | 197 | ||
0.4 | 0.72 | 0.7 | 1.029 | 80% | 45 | 51 | 90% | 57 | 64 |
0.4 | 0.74 | 0.7 | 1.057 | 46 | 57 | 59 | 72 | ||
0.4 | 0.76 | 0.7 | 1.086 | 48 | 64 | 64 | 80 | ||
0.4 | 0.78 | 0.7 | 1.114 | 53 | 72 | 72 | 91 | ||
0.4 | 0.80 | 0.7 | 1.143 | 60 | 82 | 82 | 104 |
Odd ratio: sample size calculations based on exact and approximate powers (α = 0.05)
δ | p_{1} | p_{2} | Power | Exact | Approx | Power | Exact | Approx | |
---|---|---|---|---|---|---|---|---|---|
1.0 | 0.42 | 0.4 | 1.086 | 80% | 73 | 85 | 90% | 93 | 107 |
1.0 | 0.43 | 0.4 | 1.132 | 75 | 92 | 98 | 117 | ||
1.0 | 0.44 | 0.4 | 1.179 | 79 | 101 | 104 | 128 | ||
1.0 | 0.45 | 0.4 | 1.227 | 84 | 112 | 113 | 141 | ||
1.0 | 0.46 | 0.4 | 1.278 | 91 | 124 | 124 | 156 | ||
1.0 | 0.52 | 0.5 | 1.083 | 80% | 71 | 82 | 90% | 90 | 103 |
1.0 | 0.53 | 0.5 | 1.128 | 73 | 89 | 94 | 113 | ||
1.0 | 0.54 | 0.5 | 1.174 | 77 | 98 | 101 | 124 | ||
1.0 | 0.55 | 0.5 | 1.222 | 82 | 108 | 109 | 137 | ||
1.0 | 0.56 | 0.5 | 1.273 | 89 | 120 | 121 | 152 | ||
1.0 | 0.62 | 0.6 | 1.088 | 80% | 75 | 86 | 90% | 95 | 109 |
1.0 | 0.63 | 0.6 | 1.135 | 78 | 95 | 101 | 121 | ||
1.0 | 0.64 | 0.6 | 1.185 | 82 | 106 | 109 | 134 | ||
1.0 | 0.65 | 0.6 | 1.238 | 89 | 119 | 120 | 150 | ||
1.0 | 0.66 | 0.6 | 1.294 | 99 | 135 | 135 | 170 | ||
1.2 | 0.73 | 0.7 | 1.159 | 80% | 63 | 77 | 90% | 81 | 97 |
1.2 | 0.74 | 0.7 | 1.220 | 67 | 86 | 88 | 108 | ||
1.2 | 0.75 | 0.7 | 1.286 | 72 | 97 | 97 | 122 | ||
1.2 | 0.76 | 0.7 | 1.357 | 81 | 110 | 110 | 139 | ||
1.2 | 0.77 | 0.7 | 1.435 | 92 | 127 | 127 | 161 | ||
1.4 | 0.82 | 0.8 | 1.139 | 80% | 59 | 70 | 90% | 76 | 88 |
1.4 | 0.83 | 0.8 | 1.221 | 64 | 80 | 83 | 101 | ||
1.4 | 0.84 | 0.8 | 1.313 | 70 | 93 | 94 | 117 | ||
1.4 | 0.85 | 0.8 | 1.417 | 81 | 110 | 110 | 138 | ||
1.4 | 0.86 | 0.8 | 1.536 | 96 | 133 | 133 | 168 |
Further investigation on Theorem 1 (α = 0.05)
Metric | δ | p_{1} | p_{2} | Sample size with 80% approx power | Sample size with 90% exact power | w |
---|---|---|---|---|---|---|
Risk difference | 0.2 | 0.18 | 0.10 | 142 | 142 | 9.9519 × 10^{?8} |
Risk difference | 0.2 | 0.20 | 0.10 | 215 | 215 | 4.2490 × 10^{?13} |
Risk difference | 0.25 | 0.28 | 0.20 | 108 | 108 | 2.4720 × 10^{?5} |
Risk difference | 0.25 | 0.30 | 0.20 | 141 | 141 | 1.0651 × 10^{?7} |
Relative risk | 0.8 | 0.38 | 0.30 | 107 | 107 | 9.2356 × 10^{?5} |
Relative risk | 0.8 | 0.40 | 0.30 | 126 | 126 | 2.2052 × 10^{?6} |
Relative risk | 0.6 | 0.48 | 0.40 | 127 | 127 | 6.1416 × 10^{?5} |
Relative risk | 0.6 | 0.50 | 0.40 | 151 | 151 | 1.0050 × 10^{?6} |
Odd ratio | 1.2 | 0.76 | 0.70 | 110 | 110 | 5.0446 × 10^{?4} |
Odd ratio | 1.2 | 0.77 | 0.70 | 127 | 127 | 7.0076 × 10^{?5} |
Odd ratio | 1.4 | 0.85 | 0.80 | 110 | 110 | 5.9863 × 10^{?4} |
Odd ratio | 1.4 | 0.86 | 0.80 | 133 | 133 | 5.1409 × 10^{?5} |