# [R] Sensitivity analysis - minimum effect size detectable by a binomial test

Marc Schwartz marc_schwartz at me.com
Wed Feb 5 17:25:20 CET 2014

```On Feb 5, 2014, at 10:05 AM, Simone <miseno77 at hotmail.com> wrote:

> Hi all,
>
> I have performed a binomial test to verify if the number of males in a study is significantly different from a null hypothesis (say, H0:p of being a male= 0.5).
> For instancee:
> binom.test(10, 30, p=0.5, alternative="two.sided", conf.level=0.95)
>
>    Exact binomial test
>
> data:  10 and 30
> number of successes = 10, number of trials =
> 30, p-value = 0.09874
> alternative hypothesis: true probability of success is not equal to 0.5
> 95 percent confidence interval:
> 0.1728742 0.5281200
> sample estimates:
> probability of success
>             0.3333333
>
> This way I get the estimated proportion of males (in this case p of success) that is equal to 0.33 and an associated p-value (this is not significant at alpha=0.05 with respect to the H0:P=0.5).
>
> Now, I want to know, given a power of, say, 0.8, alpha=0.05 and the above sample size (30), what is the minimum proportion of males as low or as high (two sided) like to be significantly detected with respect to a H0 (not necessarily H0:P=0.5 - I am interested also in other null hypotheses). In other words, I would have been able to detect a significant deviation from the H0 for a given power, alpha and sample size if the proportion of males would have been more than Xhigh or less than Xlow.
>
> I have had a look at the "pwr" package but it seems to me it doesn't allow to calculate this.
> I would appreciate very much any suggestion.

Take a look at ?power.prop.test, where you can specify that one of the proportions is NULL, yielding the value you seek:

> power.prop.test(n = 30, p1 = 0.5, p2 = NULL, power = 0.8, sig.level = 0.05)

Two-sample comparison of proportions power calculation

n = 30
p1 = 0.5
p2 = 0.834231
sig.level = 0.05
power = 0.8
alternative = two.sided

NOTE: n is number in *each* group

The value for 'p2' is your high value for the detectible difference from a proportion of 0.5, given the other parameters. 1 - p2 would be your low value.

Regards,

Marc Schwartz

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