The paired ttest is commonly used. It compares the means of two populations of paired observations by testing if the difference between pairs is statistically different from zero.
Appropriate data
• Twosample data. That is, one measurement variable in two groups or samples
• Dependent variable is interval/ratio, and is continuous
• Independent variable is a factor with two levels. That is, two groups
• Data are paired. That is, the measurement for each observation in one group can be paired logically or by subject to a measurement in the other group
• The distribution of the difference of paired measurements is normally distributed
• Moderate skewness is permissible if the data distribution is unimodal without outliers
Hypotheses
• Null hypothesis: The population mean of the differences between paired observations is equal to zero.
• Alternative hypothesis (twosided): The population mean of the differences between paired observations is not equal to zero.
Interpretation
Reporting significant results as “Mean of variable Y for group A was different than that for group B.” or “Variable Y increased from before to after” is acceptable.
Other notes and alternative tests
• The nonparametric analogue for this test is the twosample paired ranksum test.
• Power analysis for the paired ttest can be found at Mangiafico (2015) in the “References” section.
Packages used in this chapter
The packages used in this chapter include:
• psych
• rcompanion
• lsr
The following commands will install these packages if they are not already installed:
if(!require(psych)){install.packages("psych")}
if(!require(rcompanion)){install.packages("rcompanion")}
if(!require(lsr)){install.packages("lsr")}
Paired ttest example
In the following example, Dumbland Extension had adult students fill out a financial literacy knowledge questionnaire both before and after completing a home financial management workshop. Each student’s score before and after was paired by student.
Note in the following data that the students’ names are repeated, so that there is a before score for student a and an after score for student a.
Since the data is in long form, we’ll order by Time, then Student to be sure the first observation for Before is student a and the first observation for After is student a, and so on.
Input = ("
Time Student Score
Before a 65
Before b 75
Before c 86
Before d 69
Before e 60
Before f 81
Before g 88
Before h 53
Before i 75
Before j 73
After a 77
After b 98
After c 92
After d 77
After e 65
After f 77
After g 100
After h 73
After i 93
After j 75
")
Data = read.table(textConnection(Input),header=TRUE)
### Order data by Time and Student
Data = Data[order(Time, Student),]
### Check the data frame
library(psych)
headTail(Data)
str(Data)
summary(Data)
### Remove unnecessary objects
rm(Input)
Check the number of paired observations
It is helpful to create a table of the counts of observations to be sure that there is one observation for each student for each time period.
xtabs(~ Student + Time,
data = Data)
Time
Student After Before
a 1 1
b 1 1
c 1 1
d 1 1
e 1 1
f 1 1
g 1 1
h 1 1
i 1 1
j 1 1
Histogram of difference data
A histogram with a normal curve imposed will be used to check if the paired differences between the two populations is approximately normal in distribution.
First, two new variables, Before and After, are created by
extracting the values of Score for observations with the Time
variable equal to Before or After, respectively.
Note that for this code to make sense, the first observation for Before is student a and the first observation for After is student a, and so on.
Before = Data$Score[Data$Time=="Before"]
After = Data$Score[Data$Time=="After"]
Difference = After  Before
x = Difference
library(rcompanion)
plotNormalHistogram(x,
xlab="Difference (After  Before)")
Plot the paired data
Scatter plot with onetoone line
Paired data can visualized with a scatter plot of the paired cases. In the plot below, points that fall above and to the left of the blue line indicate cases for which the value for After was greater than for Before.
Note that the points in the plot are jittered slightly so that points that would fall directly on top of one another can be seen.
First, two new variables, Before and After, are created by extracting the values of Score for observations with the Time variable equal to Before or After, respectively.
Note that for this code to make sense, the first observation for Before is student a and the first observation for After is student a, and so on.
A variable Names is also created for point labels.
Before = Data$Score[Data$Time=="Before"]
After = Data$Score[Data$Time=="After"]
Names = Data$Student[Data$Time=="Before"]
plot(Before, jitter(After), # jitter offsets
points so you can see them all
pch = 16, # shape of points
cex = 1.0, # size of points
xlim=c(50, 110), # limits of xaxis
ylim=c(50, 110), # limits of yaxis
xlab="Before", # label
for xaxis
ylab="After") # label
for yaxis
text(Before, After, labels=Names, # Label location
and text
pos=3, cex=1.0) #
Label text position and size
abline(0, 1, col="blue", lwd=2) #
line with intercept of 0 and slope of 1
Bar plot of differences
Paired data can also be visualized with a bar chart of differences. In the plot below, bars with a value greater than zero indicate cases for which the value for After was greater than for Before.
New variables are first created for Before, After, and their Difference.
Note that for this code to make sense, the first observation for Before is student a and the first observation for After is student a, and so on.
A variable Names is also created for bar labels.
Before = Data$Score[Data$Time=="Before"]
After = Data$Score[Data$Time=="After"]
Difference = After  Before
Names = Data$Student[Data$Time=="Before"]
barplot(Difference, #
variable to plot
col="dark gray", # color of bars
xlab="Observation", # xaxis label
ylab="Difference (After – Before)", # yaxis label
names.arg=Names)
# labels for bars
Paired ttest
Note that the output shows the pvalue for the test, and the simple difference in the means for the two groups.
Note that for this test to be conducted correctly, the first observation for Before is student a and the first observation for After is student a, and so on.
t.test(Score ~ Time,
data = Data,
paired = TRUE)
Paired ttest
t = 3.8084, df = 9, pvalue = 0.004163
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
4.141247 16.258753
sample estimates:
mean of the differences
10.2
Effect size
Cohen’s d can be used as an effect size statistic for a paired ttest. It is calculated as the difference between the means of each group, all divided by the standard deviation of the data. The standard deviation used could be calculated from the differences between observations, or, for observations across two times, the observations in the “before” group.
It ranges from 0 to infinity, with 0 indicating no effect where the means are equal. In some versions, Cohen’s d can be positive or negative depending on which mean is greater.
A Cohen’s d of 0.5 suggests that the means differ by onehalf the standard deviation of the data. A Cohen’s d of 1.0 suggests that the means differ by one standard deviation of the data.
Interpretation of Cohen’s d
Interpretation of effect sizes necessarily varies by discipline and the expectations of the experiment, but for behavioral studies, the guidelines proposed by Cohen (1988) are sometimes followed. They should not be considered universal.

Small

Medium 
Large 
Cohen’s d 
0.2 – < 0.5 
0.5 – < 0.8 
≥ 0.8 
____________________________
Source: Cohen (1988).
Cohen’s d for paired ttest
Standard deviation calculated from differences in observations
Note that for this code to make sense, the first observation for Before is student a and the first observation for After is student a, and so on.
library(lsr)
cohensD(Score ~ Time,
data = Data,
method = "paired")
[1] 1.204314
Or we can calculate the value manually.
Before = Data$Score[Data$Time=="Before"]
After = Data$Score[Data$Time=="After"]
Difference = After – Before
mean(Before)
[1] 72.5
mean(After)
[1] 82.7
mean(Before)  mean(After)
[1] 10.2
( mean(Before)  mean(After) ) / sd(Difference)
[1] 1.204314
Standard deviation calculated from the “before” observations
Note that for this code to make sense, the “before” observations must correspond to the second level of the Time variable, corresponding to y.sd in the function.
library(lsr)
cohensD(Score ~ Time,
data = Data,
method = "y.sd")
[1] 0.9174268
( mean(Before)  mean(After) ) / sd(Before)
[1] 0.9174268
Optional readings
“Paired t–test” in McDonald, J.H. 2014. Handbook of Biological Statistics. www.biostathandbook.com/pairedttest.html.
References
“Paired t–test” in Mangiafico, S.S. 2015. An R Companion for the Handbook of Biological Statistics, version 1.09. rcompanion.org/rcompanion/d_09.html.
Exercises Q
1. Consider the Dumbland Extension data. Report for each
answer How you know, when appropriate, by reporting the values of
the statistic you are using or other information you used.
a. What was the mean of the differences in score before and
after the training?
b. Was this an increase or a decrease?
c. Is the data distribution for the paired differences
reasonably normal?
d. Was the mean score significantly different before and after
the training?
e. What do you conclude practically? As appropriate, report
the means before and after, the mean difference, the effect size and interpretation,
whether the difference is large relative to the scoring system, anything
notable on plots, and your practical conclusions.
2. Residential properties in Dougal County rarely need phosphorus for good
turfgrass growth. As part of an extension education program, Early and Rusty
Cuyler asked homeowners to report their phosphorus fertilizer use, in pounds of
P_{2}O_{5} per acre, before the program and then one year
later.
Date Homeowner P2O5
'20140101' a 0.81
'20140101' b 0.86
'20140101' c 0.79
'20140101' d 0.59
'20140101' e 0.71
'20140101' f 0.88
'20140101' g 0.63
'20140101' h 0.72
'20140101' i 0.76
'20140101' j 0.58
'20150101' a 0.67
'20150101' b 0.83
'20150101' c 0.81
'20150101' d 0.50
'20150101' e 0.71
'20150101' f 0.72
'20150101' g 0.67
'20150101' h 0.67
'20150101' i 0.48
'20150101' j 0.68
For each of the following, answer the question, and show the output from the analyses you used to answer the question.
a. What was mean of the differences in P_{2}O_{5}
before and after the training?
b. Is this an increase or a decrease?
c. Is the data
distribution for the paired differences reasonably normal?
d. Was the mean P_{2}O_{5 }use significantly
different before and after the training?
e. What do you conclude practically? As appropriate, report the means before and after, the mean difference, the effect size and interpretation, whether the difference is large relative to the values, anything notable on plots, and your practical conclusions.