Sample Size

The following four quantities have an intimate relationship:

• sample size
• effect size
• significance level = P(Type I error) = probability of finding an effect that is not there
• power = 1 - P(Type II error) = probability of finding an effect that is there

Given any three, we can determine the fourth.

Tests of Proportions

library(pwr)

## Warning: package 'pwr' was built under R version 4.1.2

n2 = pwr.2p2n.test(h = 0.8 , n1 = 80, sig.level=0.05,power=0.8 ) 
h = pwr.2p2n.test(n2 = 20 , n1 = 80, sig.level=0.05,power=0.8 ) 
power = pwr.2p2n.test(h = 0.8 , n1 = 80, n2 = 20, sig.level=0.05) 
cat("n2 =",n2$n2)

## n2 = 14.48425

cat("h =",h$h)

## h = 0.700393

cat("power =", power$power)

## power = 0.8925191

t-tests

power = pwr.t2n.test(n1 = 30 , n2 = 35 , d = 0.5, sig.level = 0.05)
sig_lvl = pwr.t2n.test(n1 = 30 , n2 = 35 , d = 0.8, power = 0.85, sig.level=NULL)
cat("power =",power$power)

## power = 0.5076005

cat("significance level =",sig_lvl$sig.level)

## significance level = 0.03390876

samp_size = pwr.t.test(d = 0.5 , sig.level = 0.05 , power = 0.85) 
cat("sample size =",samp_size$n)

## sample size = 72.80046

samp_size = pwr.r.test(n = , r = 0.3, sig.level = 0.05, power = 0.9) 
cat("sample size =",samp_size$n)

## sample size = 111.8068

Plot sample size curves for detecting correlations of

various sizes.

library(pwr)
par(mfrow=c(1,1))

range of correlations

r <- seq(.1,.5,.01)
nr <- length(r)

power values

p <- seq(.4,.9,.1)
np <- length(p)

obtain sample sizes

samsize <- array(numeric(nr*np), dim=c(nr,np))
for (i in 1:np){
  for (j in 1:nr){
    result <- pwr.r.test(n = NULL, r = r[j],
                         sig.level = .05, power = p[i],
                         alternative = "two.sided")
    samsize[j,i] <- ceiling(result$n)
  }
}

set up graph

xrange <- range(r)
yrange <- round(range(samsize))
colors <- rainbow(length(p))
plot(xrange, yrange, type="n",
     xlab="Correlation Coefficient (r)",
     ylab="Sample Size (n)" )
# add power curves
for (i in 1:np){
  lines(r, samsize[,i], type="l", lwd=2, col=colors[i])
}
# add annotation (grid lines, title, legend)
abline(v=0, h=seq(0,yrange[2],50), lty=2, col="grey89")
abline(h=0, v=seq(xrange[1],xrange[2],.02), lty=2,
       col="grey89")
title("Sample Size Estimation for Correlation Studies\n
  Sig=0.05 (Two-tailed)")
legend("topright", title="Power", as.character(p),
       fill=colors)

######################################################### library(pwr) # range of correlations r <- seq(.1,.5,.01) nr <- length(r)

power values

p <- seq(.4,.9,.1) np <- length(p)

obtain sample sizes

power <- array(numeric(nr*np), dim=c(nr,np)) for (i in 1:np){ for (j in 1:nr){ result <- pwr.r.test(n = 30, r = r[j], sig.level = .05, power = NULL, alternative = “two.sided”) power[j,i] <- ceiling(result$n) } }

set up graph

xrange <- range(r) yrange <- round(range(power)) colors <- rainbow(length(p)) plot(xrange, yrange, type=“n”, xlab=“Correlation Coefficient (r)”, ylab=“Sample Size (n)” )

add power curves

for (i in 1:np){ lines(r, power[,i], type=“l”, lwd=2, col=colors[i]) }

add annotation (grid lines, title, legend)

abline(v=0, h=seq(0,yrange[2],50), lty=2, col=“grey89”) abline(h=0, v=seq(xrange[1],xrange[2],.02), lty=2, col=“grey89”) title(“Sample Size Estimation for Correlation Studies Sig=0.05 (Two-tailed)”) legend(“topright”, title=“Power”, as.character(p), fill=colors)