# Power and Sample Size Analysis: Z test

October 17, 2012
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(This article was first published on Milano R net, and kindly contributed to R-bloggers)

### Abstract

This article provide a brief background about power and sample size analysis. Then, power and sample size analysis is computed for the Z test.
Next articles will describe power and sample size analysis for:

• one sample and two samples t test;,
• p test, chi-square test, correlation;
• one-way ANOVA;
• DOE $2^k$.

Finally, a PDF article showing both the underlying methodology and the R code here provided, will be published.

### Background

Power and sample size analysis are important tools for assessing the ability of a statistical test to detect when a null hypothesis is false, and for deciding what sample size is required for having a reasonable chance to reject a false null hypothesis.

The following four quantities have an intimate relationship:

1. sample size
2. effect size
3. significance level = P(Type I error) = probability of finding an effect that is not there
4. power = 1 – P(Type II error) = probability of finding an effect that is there

Given any three, we can determine the fourth.

### Z test

The formula for the power computation can be implemented in R, using a function like the following:

 powerZtest = function(alpha = 0.05, sigma, n, delta){ zcr = qnorm(p = 1-alpha, mean = 0, sd = 1) s = sigma/sqrt(n) power = 1 - pnorm(q = zcr, mean = (delta/s), sd = 1) return(power) }

In the same way, the function to compute the sample size can be built.

 sampleSizeZtest = function(alpha = 0.05, sigma, power, delta){ zcra=qnorm(p = 1-alpha, mean = 0, sd=1) zcrb=qnorm(p = power, mean = 0, sd = 1) n = round((((zcra+zcrb)*sigma)/delta)^2) return(n) }

The above code is provided for didactic purpose. In fact, the pwr package provide a function to perform power and sample size analysis.

 install.packages("pwr") library(pwr)

The function pwr.norm.test() computes parameters for the Z test. It accepts the four parameters see above, one of them passed as NULL. The parameter passed as NULL is determined from the others.

#### Some examples

Power at $\mu = 105$ for $H0: \mu = 100$ against $H1: \mu>100$.
$\sigma = 15$, $n = 20$, $\alpha = 0.05$

 sigma = 15 h0 = 100 ha = 105

This is the result with the self-made function:

 > powerZtest(n = 20, sigma = sigma, delta = (ha-h0)) [1] 0.438749

And here the same with the pwr.norm.test() function:

 > d = (ha - h0)/sigma > pwr.norm.test(d = d, n = 20, sig.level = 0.05, alternative = "greater")   Mean power calculation for normal distribution with known variance   d = 0.3333333 n = 20 sig.level = 0.05 power = 0.438749 alternative = greater

The sample size of the test for power equal to 0.80 can be computed using the self-made function

 > sampleSizeZtest(sigma = sigma, power = 0.8, delta = (ha-h0)) [1] 56

or with the pwr.norm.test() function:

 > pwr.norm.test(d = d, power = 0.8, sig.level = 0.05, alternative = "greater")   Mean power calculation for normal distribution with known variance   d = 0.3333333 n = 55.64302 sig.level = 0.05 power = 0.8 alternative = greater

The power function can be drawn:

 ha = seq(95, 125, l = 100) pwrTest = pwr.norm.test(d = d, n = 20, sig.level = 0.05, alternative = "greater")$power plot(d, pwrTest, type = "l", ylim = c(0, 1)) View (and download) the full code: ?Download powerZtest.R  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40  ### Self-made functions to perform power and sample size analysis powerZtest = function(alpha = 0.05, sigma, n, delta){ zcr = qnorm(p = 1-alpha, mean = 0, sd = 1) s = sigma/sqrt(n) power = 1 - pnorm(q = zcr, mean = (delta/s), sd = 1) return(power) } sampleSizeZtest = function(alpha = 0.05, sigma, power, delta){ zcra=qnorm(p = 1-alpha, mean = 0, sd=1) zcrb=qnorm(p = power, mean = 0, sd = 1) n = round((((zcra+zcrb)*sigma)/delta)^2) return(n) } ### Load pwr package to perform power and sample size analysis library(pwr) ### Data sigma = 15 h0 = 100 ha = 105 ### Power analysis # Using the self-made function powerZtest(n = 20, sigma = sigma, delta = (ha-h0)) # Using the pwr package pwr.norm.test(d = (ha - h0)/sigma, n = 20, sig.level = 0.05, alternative = "greater") ### Sample size analysis # Using the self-made function sampleSizeZtest(sigma = sigma, power = 0.8, delta = (ha-h0)) # Using the pwr package pwr.norm.test(d = (ha - h0)/sigma, power = 0.8, sig.level = 0.05, alternative = "greater") ### Power function for the two-sided alternative ha = seq(95, 125, l = 100) d = (ha - h0)/sigma pwrTest = pwr.norm.test(d = d, n = 20, sig.level = 0.05, alternative = "greater")$power plot(d, pwrTest, type = "l", ylim = c(0, 1))

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