# 5.4: Size to give adequate power

• Peter G. Smith, Richard H. Morrow, and David A. Ross
• London School of Hygiene & Tropical Medicine & The Johns Hopkins Bloomberg School of Public Health via Oxford University Press

The alternative approach to setting trial size is based upon selecting the trial size to achieve a specified power. In order to do this, the following must be specified:

1. What size of difference, D, between the two groups would be of clinical or public health importance? The trial size will be chosen so it would have a good chance of detecting this size of true difference, i.e. there would be a good chance of obtaining a statistically significant result, thus concluding that there is a real difference between the two trial arms. D is the true difference between the two groups, not the estimated difference as measured in the trial. Very small differences are generally of no public health importance, and it would not be of concern if they were not detected in the trial. The general principle, in most cases, is to choose D to be the minimum difference which would be of public health relevance and therefore be important to detect in a trial. Note that ‘detecting’ D means that a significant difference is obtained, indicating that there is some difference between the two groups. This does not mean that the difference is estimated precisely. To ensure a precise estimate is obtained, the approach of Section 3 should be used.
2. Having specified D, the investigators must decide how confident they wish to be of obtaining a significant result if this were the true difference between the groups. In other words, the power is set for this value of D. Note that, if the true difference between the groups is actually larger than D, the power of the trial will be larger than the value set. The required power is specified in the calculations by choosing the corresponding value of z2,z2,as shown in Table 5.1. Commonly chosen values for the power are 80%, 90%, and 95%, the corresponding values of z2z2being 0.84, 1.28, and 1.64. It would generally be regarded as unsatisfactory to proceed with a trial with a power of less than 70% for the primary outcome, because that means that one would have a more than 30% chance of ‘missing’ a true difference of D.
3. The significance level must also be specified for the comparison of the two groups under study. This is entered into the calculations in terms of the parameterz1z1The commonest choice for the required p-value is 0.05, corresponding to az1z1of 1.96. Alternative values might be 0.01 or 0.001, corresponding toz1z1values of 2.58 or 3.29, respectively. It is assumed throughout this chapter that two-sided significance tests are to be used (see Chapter 21, Section 2.3). A significance level of 0.05 is assumed in the numerical examples, unless otherwise stated.
4. In addition, certain additional information must be specified, which varies according to the type of measure being examined. This may be a rough estimate of the rates or proportions that are expected, or an estimate of the standard deviation for a quantitative variable. Note that, if these quantities were known exactly, no trial would be needed! Only rough estimates are required.

Having specified these values, the formulae or tables given in Sections 4.1 to 4.3 can be used to calculate the required trial size.

It is often useful, however, to proceed in the opposite direction, i.e. to explore the power that would be achieved for a range of possible trial sizes and for a range of possible values of the true difference D. This enables the construction of power curves, as illustrated in Figure 5.1. Formulae for this approach are also given in Sections 4.1 to 4.3.

Table 5.1 Relationship between $$z_2$$ and % power (numbers in the body of the table show power corresponding to each value of $$z_2$$

First decimal place of $$z_2$$
$$z_2$$ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
−3.0 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0
−2.0 2.3 1.8 1.4 1.1 0.8 0.6 0.5 0.3 0.3 0.2
−1.0 15.9 13.6 11.5 9.7 8.1 6.7 5.5 4.5 3.6 2.9
−0.0 50.0 46.0 42.1 38.2 34.5 30.9 27.4 24.2 21.2 18.4
+0.0 50.0 54.0 57.9 61.8 65.5 69.1 72.6 75.8 78.8 81.6
+1.0 84.1 86.4 88.5 90.3 91.9 93.3 94.5 95.5 96.4 97.1
+2.0 97.7 98.2 98.6 98.9 99.2 99.4 99.5 99.7 99.7 99.8
+3.0 99.9 99.9 99.9 100.0 100.0 100.0 100.0 100.0 100.0 100.0

Note: for example, $$z_2=−0.7$$ corresponds to a power of 24.2%.

## 4.1 Comparison of proportions

The trial size required in each group to detect a specified difference $$D=p_1−p_2$$, with power specified by z2z2and significance level specified by $$z_1$$, is given by:

$n=[(z_1+z_2)^22p(1−p)]/(p_1−p_2)^2$

where p is the average of $$p_1$$and $$p_2$$.For 90% power and significance at $$p<0.05,$$this simplifies to:

$n=[21p(1−p)]/(p_1−p_2)^2.$

Table 5.2 shows the required trial size for a range of values of p1p1and p2p2for 80%, 90%, or 95% power.

To calculate the power of a trial of specified size, calculate as follows, and refer the value of $$z_2$$to Table 5.1.

$z2=(√ \{n/[2p(1−p)]\})(|p_1−p_2|)−z1.$

Example: assume that the spleen rate in the control group of the mosquito-net trial is around 40%. To have very high power (say 95%) of detecting a significant effect if the intervention reduces the spleen rate to 30% (so that $$p=0.35$$ the number of children required in each group is given by:

$n=[(1.96+1.64)^2(2×0.35×0.65)]/(0.3−0.4)^2=590.$

If the true risk ratio is R and we wish to power the trial, such that the lower confidence limit on the risk ratio will be greater than or equal to RLRLwhere RLRLis the lowest acceptable efficacy (say, for whether or not to implement the intervention in a public health system, i.e. we need to be sure that the efficacy is at least $$R_L$$), the required sample size is:

$n=(z_1+z_2)^2[(1−p_1)/(p_1)+(1−p_2)/(p_1)]/[\log_e(R/R_L)]^2.$

Table 5.2 Sample size requirements for comparison of proportions

Smaller prop. $$p_1$$ Difference $$D=p_2−p_1$$
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
0.05 435 141 76 50 36 28 22 18 15 13 11 10
583 189 102 67 48 37 30 25 21 18 15 13
719 233 126 83 60 46 37 30 26 22 19 16
0.10 686 200 101 63 44 33 26 21 17 14 12 10
919 268 135 84 59 44 34 28 23 19 16 14
1134 330 166 104 72 54 42 34 28 24 20 17
0.15 906 251 122 74 50 37 28 22 18 15 13 10
1212 336 163 98 67 49 38 30 24 20 17 14
1497 415 201 122 83 60 46 37 30 25 21 18
0.20 1094 294 139 82 55 40 30 24 19 16 13 11
1464 394 186 110 74 53 40 31 25 21 17 15
1808 486 230 136 91 66 50 39 31 26 21 18
0.25 1250 329 153 89 59 42 31 24 19 16 13 11
1674 441 205 119 79 56 42 32 26 21 17 14
2067 544 253 147 97 69 52 40 32 26 21 18
0.30 1376 357 163 94 61 43 32 24 19 16 13 10
1842 478 219 126 82 58 43 33 26 21 17 14
2274 590 270 156 101 71 53 40 32 26 21 17
0.35 1470 376 170 97 63 44 32 24 19 15 12 10
1968 504 228 130 84 58 43 32 25 20 16 13
2430 622 282 160 103 72 53 40 31 25 20 16
0.40 1533 388 174 98 63 43 31 24 18 14 11
2052 520 233 131 84 58 42 31 24 19 15
2534 642 287 162 103 71 52 39 30 24 19
0.45 1564 392 174 97 61 42 30 22 17 13
2094 525 233 130 82 56 40 30 23 18
2586 648 287 160 101 69 50 37 28 22
0.50 1564 388 170 94 59 40 28 21 15
2094 520 228 126 79 53 38 28 21
2586 642 282 156 97 66 46 34 26
0.55 1533 376 163 89 55 37 26 18
2052 504 219 119 74 49 34 25
2534 622 270 147 91 60 42 30
0.60 1470 357 153 82 50 33 22
1968 478 205 110 67 44 30
2430 590 253 136 83 54 37
0.65 1376 329 139 73 44 28
1842 441 186 98 59 37
2274 544 230 121 72 46
0.70 1250 294 122 63 36
1674 394 163 84 48
2067 486 201 104 60
0.75 1094 251 101 50
1464 336 135 67
1808 415 166 83
0.80 906 200 76
1212 268 102
1497 330 126
0.85 686 141
919 189
1134 233
0.90 435
583
719

Shown in the body of the table are the sample sizes required in each group to give the specified power.**Upper figure: power, 80%; middle figure: power, 90%; lower figure: power, 95%. Using a two-sided significance test withp<0.05.p<0.05.The two groups are assumed to be of equal size.

## 4.2 Comparison of incidence rates

For a specified difference $$D=r_1−r_2$$ and values of $$z_1$$ and $$z_2$$representing the required significance level and power, the required number of person-years in each group is given by:

$y=[(z_1+z_2)^2(r_1+r_2)]/(r_1−r_2)^2$

where $$r_1$$ and $$r_2$$are the expected rates per person-year in the two groups.A rough estimate of the average of the two rates is therefore required, i.e. $$[(r_1+r_2)/2]$$For 90% power and significance at p<0.05p<0.05, this formula simplifies to:

$y=[10.5(r_1+r_2)]/(r_1−r_2)^2.$

An alternative, but equivalent, formula gives the number of events required in group 2, the control group, in terms of the rate ratio R, for which the specified power is required:

$e_2=[(z_1+z_2)^2(1+R)]/(1−R)^2.$

This formula was used to construct Table 5.3, which shows the number of events needed in group 2 to detect a rate ratio of R with 80%, 90%, or 95% power. The total number of events needed in both groups can be calculated as $$e_2(1+R)$$.

Since this can be computed without specifying the assumed rates in the two trial groups, this provides a particularly helpful approach when the rates are uncertain. Thus, in an endpoint-driven trial, we can specify the number of events that need to be observed to reach the required power, after which recruitment or follow-up may be terminated.

To calculate the power for a given trial size, compute:

$z_2=\{√[n/(r_1+r_2)]\}(|r_1−r_2|)−z_1$

where |r1−r2||r1−r2|is the absolute value of the difference between the two rates.

Refer the resulting value of $$z_2$$ to Table 5.1 to determine the power of the trial.

Example: Assume, in the mosquito-net trial, that the death rate from malaria in the control group is 10/1000 child-years, so that $$r_2=0.010.$$Eighty per cent power is wanted to detect a significant effect if the true rate in children with bed-nets is reduced by 70% to $$r_1=0.003.$$The number of child-years of observation required in each group is given by:

$y=[(1.96+0.84)^2(0.003+0.010)]/(−0.007)^2=2080.$

The power curves shown in Figure 5.1 were constructed using the same assumption concerning the death rate in controls. For example, with $$y=2000$$ and a rate ratio of $$R=0.7$$(corresponding to a death rate of 7 per 1000 child-years in the intervention group), giving a power of 18% (Table 5.1):

$z_2=\{√[2000/(0.007+0.010)]\}(|0.007−0.010|)−1.96=−0.93.$

These formulae are used to ensure that there is a high probability of rejecting the null hypothesis if the true effect is of the assumed size. However, this may still mean that the lower confidence limit for the effect size is close to the null, and this may provide insufficient evidence to recommend widespread adoption of the intervention. A larger sample size will be needed to ensure that the lower confidence limit exceeds a given value.Suppose the assumed value of the rate ratio is R and that we wish to power the trial so that there is a high probability that the CI excludes a value RLRLcorresponding to the lower limit of efficacy desired. Then the required sample size is given by the formula:

$y=(z_1+z_2)^2(1/r_1+1/r_2)/[\log_e(R/R_L)]^2.$

Example: In the mosquito-net trial, we found that 2080 child-years were required in each trial group to reject the null hypothesis with 80% power if the true rate ratio R was 0.3, corresponding to an efficacy of 70%. Now suppose we wish to ensure that there is an 80% chance that the lower 95% CI for the efficacy exceeds 30%, corresponding to $$R_L=0.7.$$ Applying the formula, we obtain the following, demonstrating the substantial increase in sample size that this would necessitate:

$y=(1.96+0.84)^2(1/0.010+1/0.003)/[\log_e(0.3/0.7)]^2=4732.$

Table 5.3 Sample size requirements for comparison of rates

Relative rate R * Expected events in group 2 to give+
80% power 90% power 95% power
0.1 10.6 14.3 17.6
0.2 14.7 19.7 24.3
0.3 20.8 27.9 34.4
0.4 30.5 40.8 50.4
0.5 47.0 63.0 77.8
0.6 78.4 105.0 129.6
0.7 148.1 198.3 244.8
0.8 352.8 472.4 583.2
0.9 1489.6 1994.5 2462.4
1.1 1646.4 2204.5 2721.6
1.2 431.2 577.4 712.8
1.4 117.6 157.5 194.4
1.6 56.6 75.8 93.6
1.8 34.3 45.9 56.7
2.0 23.5 31.5 38.9
2.5 12.2 16.3 20.2
3.0 7.8 10.5 13.0
5.0 2.9 3.9 4.9
10.0 1.1 1.4 1.8

Numbers in the body of the table are expected number of events required in group 2 to give specified power if relative rate in group 1 is R.

*R, ratio of incidence rate in group 1 to incidence rate in group 2.

+ Using a two-sided significance test with $$p<0.05$$.The two groups are assumed to be of equal size.

## 4.3 Comparison of means

The trial size required in each group to detect a specified difference $$D=μ_1−μ_2,$$with power specified by $$z_2$$ and the significance level specified by $$z_1,$$is given by:

$n=[(z_1+z_2)^2(σ_1^2+σ_2^2)]/(μ_1−μ_2)^2$

where $$σ_1$$ and $$σ_2$$ are the standard deviations of the outcome variable in groups 1 and 2, respectively.For 90% power and significance at $$p<0.05$$,this simplifies to:

$n=10.5(σ_1^2+σ_2^2)/(μ_1−μ_2)^2.$

To calculate the power of a trial of specified size, calculate the following, and refer the value of $$z_2$$ to Table 5.1:

$z_2=\{√[n/(σ^2_1+σ_2^2)]\}(|μ_1−μ_2|)−z_1.$

Estimates of σ1σ1and σ2σ2may be obtained from previous studies or from a pilot study. If appropriate values cannot be determined, an alternative is to dichotomize the continuous outcome variable and use the sample size formulae for comparison of proportions given in Section 4.1. This will give a conservative estimate of sample size, as it ignores some of the information, but will ensure an adequate sample size in the face of uncertainty regarding the standard deviations.

Example: In the mosquito-net trial, the mean PCV in the control group at the end of the trial is expected to be 33.0, with a standard deviation of 5.0. To have 90% power of detecting a significant effect if the intervention increases the mean PCV by 1.5, the number of children required in each group is given by:

$n = \left[ ( 1.96 + 1.28 ) ^ { 2 } \left( 5.0 ^ { 2 } + 5.0 ^ { 2 } \right) \right] / ( 1.5 ) ^ { 2 } = 233$

Suppose it turns out that only 150 children are available for study in each group. The power in these circumstances is given by the following, corresponding to a power of about 74%:

$z_2=\{√[150/(5.02+5.02)]\}(|1.5|)−1.96=0.64.$

A summary of the various formulae that have been given for calculating the trial size requirements for the comparison of two groups of equal size is given in Table 5.4.

Table 5.4 Summary of formulae for calculating trial size requirements for comparison of two groups of equal size

Type of outcome Formula Notation Section in text
A: Choosing trial size to achieve adequate precision
Proportions: $$n = \left( 1.96 / \log _ { e } f \right) ^ { 2 } \left\{ \left[ ( R + 1 ) / \left( R p _ { 2 } \right) \right] - 2 \right\}$$

n = number in each group

R = prop. in group 1/prop.

in group 2Gives 95% CI from R/f to Rf

3.1
Rates: $$e _ { 2 } = \left( 1.96 / \log _ { \mathrm { e } } f \right) ^ { 2 } [ ( R + 1 ) / R ]$$

$$e^2$$= expected events in  group 2

R = rate in group 1/rate in  group 2

Gives 95% CI from R/f to Rf

3.2
Means: $$n = ( 1.96 / f ) ^ { 2 } \left( \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } \right)$$

n = number in each group

$$σ_i=SD$$ in group i

D = mean in group 1 − mean in group 2

Gives 95% CI of $$D±f$$

3.3
B: Choosing trial size to achieve adequate power
Proportions: $$n = \left[ \left( z _ { 1 } + z _ { 2 } \right) ^ { 2 } 2 p ( 1 - p ) \right] / \left( p _ { 1 } - p _ { 2 } \right) ^ { 2 }$$

n = number in each group

$$p^i$$= proportion. in group i

p = average of p1p1and p2p2

4.1
Rates: $$y = \left[ \left( z _ { 1 } + z _ { 2 } \right) ^ { 2 } \left( r _ { 1 } + r _ { 2 } \right) \right] / \left( r _ { 1 } - r _ { 2 } \right) ^ { 2 }$$

y = person-years in each  group

$$r^i$$= rate in group i

4.2
Means: $$n = \left[ \left( z _ { 1 } + z _ { 2 } \right) ^ { 2 } \left( \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } \right) \right] / \left( \mu _ { 1 } - \mu _ { 2 } \right) ^ { 2 }$$

n = number in each group

$$σ_i=SD$$ in group i

$$μ_i$$=mean in group i

4.3

$$z_1=1.96$$ for significance at $$p<0.05$$

Power 80%, 90%, 95%

$$z_2=0.84, 1.28, 1.64$$

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