# bpower: Power and Sample Size for Two-Sample Binomial Test In Hmisc: Harrell Miscellaneous

## Description

Uses method of Fleiss, Tytun, and Ury (but without the continuity correction) to estimate the power (or the sample size to achieve a given power) of a two-sided test for the difference in two proportions. The two sample sizes are allowed to be unequal, but for `bsamsize` you must specify the fraction of observations in group 1. For power calculations, one probability (`p1`) must be given, and either the other probability (`p2`), an `odds.ratio`, or a `percent.reduction` must be given. For `bpower` or `bsamsize`, any or all of the arguments may be vectors, in which case they return a vector of powers or sample sizes. All vector arguments must have the same length.

Given `p1, p2`, `ballocation` uses the method of Brittain and Schlesselman to compute the optimal fraction of observations to be placed in group 1 that either (1) minimize the variance of the difference in two proportions, (2) minimize the variance of the ratio of the two proportions, (3) minimize the variance of the log odds ratio, or (4) maximize the power of the 2-tailed test for differences. For (4) the total sample size must be given, or the fraction optimizing the power is not returned. The fraction for (3) is one minus the fraction for (1).

`bpower.sim` estimates power by simulations, in minimal time. By using `bpower.sim` you can see that the formulas without any continuity correction are quite accurate, and that the power of a continuity-corrected test is significantly lower. That's why no continuity corrections are implemented here.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```bpower(p1, p2, odds.ratio, percent.reduction, n, n1, n2, alpha=0.05) bsamsize(p1, p2, fraction=.5, alpha=.05, power=.8) ballocation(p1, p2, n, alpha=.05) bpower.sim(p1, p2, odds.ratio, percent.reduction, n, n1, n2, alpha=0.05, nsim=10000) ```

## Arguments

 `p1` population probability in the group 1 `p2` probability for group 2 `odds.ratio` `percent.reduction` `n` total sample size over the two groups. If you omit this for `ballocation`, the `fraction` which optimizes power will not be returned. `n1` `n2` the individual group sample sizes. For `bpower`, if `n` is given, `n1` and `n2` are set to `n/2`. `alpha` type I error `fraction` fraction of observations in group 1 `power` the desired probability of detecting a difference `nsim` number of simulations of binomial responses

## Details

For `bpower.sim`, all arguments must be of length one.

## Value

for `bpower`, the power estimate; for `bsamsize`, a vector containing the sample sizes in the two groups; for `ballocation`, a vector with 4 fractions of observations allocated to group 1, optimizing the four criteria mentioned above. For `bpower.sim`, a vector with three elements is returned, corresponding to the simulated power and its lower and upper 0.95 confidence limits.

## AUTHOR

Frank Harrell

Department of Biostatistics

Vanderbilt University

## References

Fleiss JL, Tytun A, Ury HK (1980): A simple approximation for calculating sample sizes for comparing independent proportions. Biometrics 36:343–6.

Brittain E, Schlesselman JJ (1982): Optimal allocation for the comparison of proportions. Biometrics 38:1003–9.

Gordon I, Watson R (1996): The myth of continuity-corrected sample size formulae. Biometrics 52:71–6.

`samplesize.bin`, `chisq.test`, `binconf`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32``` ```bpower(.1, odds.ratio=.9, n=1000, alpha=c(.01,.05)) bpower.sim(.1, odds.ratio=.9, n=1000) bsamsize(.1, .05, power=.95) ballocation(.1, .5, n=100) # Plot power vs. n for various odds ratios (base prob.=.1) n <- seq(10, 1000, by=10) OR <- seq(.2,.9,by=.1) plot(0, 0, xlim=range(n), ylim=c(0,1), xlab="n", ylab="Power", type="n") for(or in OR) { lines(n, bpower(.1, odds.ratio=or, n=n)) text(350, bpower(.1, odds.ratio=or, n=350)-.02, format(or)) } # Another way to plot the same curves, but letting labcurve do the # work, including labeling each curve at points of maximum separation pow <- lapply(OR, function(or,n)list(x=n,y=bpower(p1=.1,odds.ratio=or,n=n)), n=n) names(pow) <- format(OR) labcurve(pow, pl=TRUE, xlab='n', ylab='Power') # Contour graph for various probabilities of outcome in the control # group, fixing the odds ratio at .8 ([p2/(1-p2) / p1/(1-p1)] = .8) # n is varied also p1 <- seq(.01,.99,by=.01) n <- seq(100,5000,by=250) pow <- outer(p1, n, function(p1,n) bpower(p1, n=n, odds.ratio=.8)) # This forms a length(p1)*length(n) matrix of power estimates contour(p1, n, pow) ```

### Example output

```Loading required package: lattice

Attaching package: ‘Hmisc’

The following objects are masked from ‘package:base’:

format.pval, units

Power1     Power2
0.01953539 0.07780432
Power      Lower      Upper
0.08120000 0.07584642 0.08655358
n1       n2
718.2381 718.2381
fraction.group1.min.var.diff   fraction.group1.min.var.ratio
0.3750000                       0.7500000
fraction.group1.min.var.logodds       fraction.group1.max.power
0.6250000                       0.4745255
```

Hmisc documentation built on Feb. 28, 2021, 9:05 a.m.