Example R programs and commands
6. Use, power and sample size in Student t tests
# All lines preceded by the "#" character are my comments.
# All other left-justified lines are my input.
# All other indented lines are the R program output.
# Perform a one-sample, two-sided t-test of H_0: mu=1.7 on data vector x:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); t.test(x,mu=1.7)
One Sample t-test
data: x
t = 2.3334, df = 11, p-value = 0.03963
alternative hypothesis: true mean is not equal to 1.7
95 percent confidence interval:
1.750127 3.416539
sample estimates:
mean of x
2.583333
# Perform a one-sample, one-sided t-test of H_A: mu>1.7 on data vector x:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); t.test(x,mu=1.7, alternative="greater")
One Sample t-test
data: x
t = 2.3334, df = 11, p-value = 0.01982
alternative hypothesis: true mean is greater than 1.7
95 percent confidence interval:
1.903482 Inf
sample estimates:
mean of x
2.583333
# Perform a two-sample, two-sided t-test of H_0:mu_x=mu_y on data vectors x,y
# of unequal length, assuming both populations have the same variance:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); y<-c(3,6,4,5,7,5,6); t.test(x,y, var.equal=TRUE)
Two Sample t-test
data: x and y
t = -4.0666, df = 17, p-value = 0.0008027
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-3.887451 -1.231596
sample estimates:
mean of x mean of y
2.583333 5.142857
# ...and using Welch's correction for not assuming equal variance:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); y<-c(3,6,4,5,7,5,6); t.test(x,y, var.equal=FALSE)
Welch Two Sample t-test
data: x and y
t = -4.0378, df = 12.415, p-value = 0.001540
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-3.935537 -1.183511
sample estimates:
mean of x mean of y
2.583333 5.142857
# Perform a two-sample, one-sided t-test of H_A:mu_x-mu_y < -1.5 on data vectors x,y
# of unequal length, assuming both populations have the same variance:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); y<-c(3,6,4,5,7,5,6);
t.test(x,y, mu= -1.5, alternative="less", var.equal=TRUE)
Two Sample t-test
data: x and y
t = -1.6834, df = 17, p-value = 0.05529
alternative hypothesis: true difference in means is less than -1.5
95 percent confidence interval:
-Inf -1.464607
sample estimates:
mean of x mean of y
2.583333 5.142857
# ...and using Welch's correction for not assuming equal variance:
t.test(x,y, mu= -1.5,alternative="less", var.equal=FALSE)
Welch Two Sample t-test
data: x and y
t = -1.6715, df = 12.415, p-value = 0.05981
alternative hypothesis: true difference in means is less than -1.5
95 percent confidence interval:
-Inf -1.432897
sample estimates:
mean of x mean of y
2.583333 5.142857
# Perform a paired-sample, one-sided t-test of H_A:mu_d < -1.5 on data vectors x,y
# of equal length:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); y<-c(3,6,4,2,6,5,5,7,5,6,4,6);
t.test(x,y, mu= -1.5, paired=TRUE, alternative="less")
Paired t-test
data: x and y
t = -2.011, df = 11, p-value = 0.03474
alternative hypothesis: true difference in means is less than -1.5
95 percent confidence interval:
-Inf -1.589141
sample estimates:
mean of the differences
-2.333333
# Perform a paired-sample, two-sided t-test of H_0:mu_d = 0 on data vectors x,y
# of equal length:
x<-c(1,4,2,2,3,1,5,3,4,2,1,3); y<-c(3,6,4,2,6,5,5,7,5,6,4,6);
t.test(x,y, paired=TRUE) # mu=0 and alternative="two.sided" are defaults
Paired t-test
data: x and y
t = -5.6308, df = 11, p-value = 0.0001531
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-3.245395 -1.421272
sample estimates:
mean of the differences
-2.333333
# Find the power in a two-sided two-sample Student t test with n1=n2=n=33
# samples per population, a true difference |mu1-mu2|=1.7, equal standard
# deviations s1=s2=2.1, using a Type I error probability of alpha=0.05:
power.t.test(n=33,delta=1.7,sd=2.1, sig.level=0.05, power=NULL)
Two-sample t test power calculation
n = 33
delta = 1.7
sd = 2.1
sig.level = 0.05
power = 0.8994822
alternative = two.sided
NOTE: n is number in *each* group
# Find the minimum sample size, if the desired power level is 95%:
power.t.test(n=NULL,delta=1.7,sd=2.1, sig.level=0.05, power=0.95)
Two-sample t test power calculation
n = 40.64486
delta = 1.7
sd = 2.1
sig.level = 0.05
power = 0.95
alternative = two.sided
NOTE: n is number in *each* group
# Find the minimum detectable difference with 50 samples, same distribution
# parameters, alpha=0.05 and power=0.95:
power.t.test(n=50,delta=NULL,sd=2.1, sig.level=0.05, power=0.95)
Two-sample t test power calculation
n = 50
delta = 1.529104
sd = 2.1
sig.level = 0.05
power = 0.95
alternative = two.sided
NOTE: n is number in *each* group
# Given d, find the the number n of samples needed from an N(mu,sigma)
# probability space so that the average m of the n samples makes a 95%
# confidence interval [m-d,m+d] for mu:
variance <- 12.8; sigma <- sqrt(variance) # example variance and its s.d.
conf <- 0.95; sig <- 1-conf # 95% confidence is significance level 5%
err <- 2.0; # allowed error; we want to estimate mu within err
power.t.test(n=NULL, # this is computed and returned
sig.level=sig, # 1 - confidence
delta=err, # half-width of conf. interval
sd=sigma, # estimated standard deviation
power=0.50, # fixed value!
type="one.sample") # required flag for non-default
# This "magic" value power=0.50 is what it takes to get the right value of n