Power and Sample Size Analysis: Z test

Posted on October 17, 2012 by Alice Bossi in R bloggers | 0 Comments

[This article was first published on Milano R net, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

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:

?View Code R
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.

?View Code R
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.

?View Code R
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 100″ />.
\sigma = 15, n = 20, \alpha = 0.05

?View Code R
sigma = 15
h0 = 100
ha = 105

This is the result with the self-made function:

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

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

?View Code R
> 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

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

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

?View Code R
> 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:

?View Code R
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))

To leave a comment for the author, please follow the link and comment on their blog: Milano R net.
R-bloggers.com offers daily e-mail updates about R news and tutorials about learning R and many other topics. Click here if you're looking to post or find an R/data-science job.
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

Copyright © 2022 | MH Corporate basic by MH Themes

Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)