# What is paired samples t-test?

The paired samples t-test is used to compare the means between two related groups of samples. In this case, you have two values (i.e., pair of values) for the same samples. This article describes how to compute paired samples t-test using R software.

As an example of data, 20 mice received a treatment X during 3 months. We want to know whether the treatment X has an impact on the weight of the mice.

To answer to this question, the weight of the 20 mice has been measured before and after the treatment. This gives us 20 sets of values before treatment and 20 sets of values after treatment from measuring twice the weight of the same mice.

In such situations, paired t-test can be used to compare the mean weights before and after treatment.

Paired t-test analysis is performed as follow:

1. Calculate the difference ($$d$$) between each pair of value
2. Compute the mean ($$m$$) and the standard deviation ($$s$$) of $$d$$
3. Compare the average difference to 0. If there is any significant difference between the two pairs of samples, then the mean of d ($$m$$) is expected to be far from 0.

Paired t-test can be used only when the difference $$d$$ is normally distributed. This can be checked using Shapiro-Wilk test.

# Research questions and statistical hypotheses

Typical research questions are:

1. whether the mean difference ($$m$$) is equal to 0?
2. whether the mean difference ($$m$$) is less than 0?
3. whether the mean difference ($$m$$) is greather than 0?

In statistics, we can define the corresponding null hypothesis ($$H_0$$) as follow:

1. $$H_0: m = 0$$
2. $$H_0: m \leq 0$$
3. $$H_0: m \geq 0$$

The corresponding alternative hypotheses ($$H_a$$) are as follow:

1. $$H_a: m \ne 0$$ (different)
2. $$H_a: m > 0$$ (greater)
3. $$H_a: m < 0$$ (less)

Note that:

• Hypotheses 1) are called two-tailed tests
• Hypotheses 2) and 3) are called one-tailed tests

# Formula of paired samples t-test

t-test statistisc value can be calculated using the following formula:

$t = \frac{m}{s/\sqrt{n}}$

where,

• m is the mean differences
• n is the sample size (i.e., size of d).
• s is the standard deviation of d

We can compute the p-value corresponding to the absolute value of the t-test statistics (|t|) for the degrees of freedom (df): $$df = n - 1$$.

If the p-value is inferior or equal to 0.05, we can conclude that the difference between the two paired samples are significantly different.

# Visualize your data and compute paired t-test in R

## R function to compute paired t-test

To perform paired samples t-test comparing the means of two paired samples (x & y), the R function t.test() can be used as follow:

t.test(x, y, paired = TRUE, alternative = "two.sided")

• x,y: numeric vectors
• paired: a logical value specifying that we want to compute a paired t-test
• alternative: the alternative hypothesis. Allowed value is one of “two.sided” (default), “greater” or “less”.

## Import your data into R

1. Prepare your data as specified here: Best practices for preparing your data set for R

2. Save your data in an external .txt tab or .csv files

3. Import your data into R as follow:

# If .txt tab file, use this
my_data <- read.delim(file.choose())
# Or, if .csv file, use this
my_data <- read.csv(file.choose())

Here, we’ll use an example data set, which contains the weight of 10 mice before and after the treatment.

# Data in two numeric vectors
# ++++++++++++++++++++++++++
# Weight of the mice before treatment
before <-c(200.1, 190.9, 192.7, 213, 241.4, 196.9, 172.2, 185.5, 205.2, 193.7)
# Weight of the mice after treatment
after <-c(392.9, 393.2, 345.1, 393, 434, 427.9, 422, 383.9, 392.3, 352.2)
# Create a data frame
my_data <- data.frame(
group = rep(c("before", "after"), each = 10),
weight = c(before,  after)
)

We want to know, if there is any significant difference in the mean weights after treatment?

## Check your data

# Print all data
print(my_data)
    group weight
1  before  200.1
2  before  190.9
3  before  192.7
4  before  213.0
5  before  241.4
6  before  196.9
7  before  172.2
8  before  185.5
9  before  205.2
10 before  193.7
11  after  392.9
12  after  393.2
13  after  345.1
14  after  393.0
15  after  434.0
16  after  427.9
17  after  422.0
18  after  383.9
19  after  392.3
20  after  352.2

Compute summary statistics (mean and sd) by groups using the dplyr package.

• To install dplyr package, type this:
install.packages("dplyr")
• Compute summary statistics by groups:
library("dplyr")
group_by(my_data, group) %>%
summarise(
count = n(),
mean = mean(weight, na.rm = TRUE),
sd = sd(weight, na.rm = TRUE)
)
Source: local data frame [2 x 4]
group count   mean       sd
(fctr) (int)  (dbl)    (dbl)
1  after    10 393.65 29.39801
2 before    10 199.16 18.47354

## Visualize your data using box plots

To use R base graphs read this: R base graphs. Here, we’ll use the ggpubr R package for an easy ggplot2-based data visualization.

• Install the latest version of ggpubr from GitHub as follow (recommended):
# Install
if(!require(devtools)) install.packages("devtools")
devtools::install_github("kassambara/ggpubr")
• Or, install from CRAN as follow:
install.packages("ggpubr")
• Visualize your data:
# Plot weight by group and color by group
library("ggpubr")
ggboxplot(my_data, x = "group", y = "weight",
color = "group", palette = c("#00AFBB", "#E7B800"),
order = c("before", "after"),
ylab = "Weight", xlab = "Groups")

Box plots show you the increase, but lose the paired information. You can use the function plot.paired() [in pairedData package] to plot paired data (“before - after” plot).

• Install pairedData package:
install.packages("PairedData")
• Plot paired data:
# Subset weight data before treatment
before <- subset(my_data,  group == "before", weight,
drop = TRUE)
# subset weight data after treatment
after <- subset(my_data,  group == "after", weight,
drop = TRUE)
# Plot paired data
library(PairedData)
pd <- paired(before, after)
plot(pd, type = "profile") + theme_bw()

## Preleminary test to check paired t-test assumptions

Assumption 1: Are the two samples paired?

Yes, since the data have been collected from measuring twice the weight of the same mice.

Assumption 2: Is this a large sample?

No, because n < 30. Since the sample size is not large enough (less than 30), we need to check whether the differences of the pairs follow a normal distribution.

How to check the normality?

Use Shapiro-Wilk normality test as described at: Normality Test in R.

• Null hypothesis: the data are normally distributed
• Alternative hypothesis: the data are not normally distributed
# compute the difference
d <- with(my_data,
weight[group == "before"] - weight[group == "after"])
# Shapiro-Wilk normality test for the differences
shapiro.test(d) # => p-value = 0.6141

From the output, the p-value is greater than the significance level 0.05 implying that the distribution of the differences (d) are not significantly different from normal distribution. In other words, we can assume the normality.

Note that, if the data are not normally distributed, it’s recommended to use the non parametric paired two-samples Wilcoxon test.

## Compute paired samples t-test

Question : Is there any significant changes in the weights of mice after treatment?

1) Compute paired t-test - Method 1: The data are saved in two different numeric vectors.

# Compute t-test
res <- t.test(before, after, paired = TRUE)
res

Paired t-test
data:  before and after
t = -20.883, df = 9, p-value = 6.2e-09
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-215.5581 -173.4219
sample estimates:
mean of the differences
-194.49 

2) Compute paired t-test - Method 2: The data are saved in a data frame.

# Compute t-test
res <- t.test(weight ~ group, data = my_data, paired = TRUE)
res

Paired t-test
data:  weight by group
t = 20.883, df = 9, p-value = 6.2e-09
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
173.4219 215.5581
sample estimates:
mean of the differences
194.49 

As you can see, the two methods give the same results.

In the result above :

• t is the t-test statistic value (t = 20.88),
• df is the degrees of freedom (df= 9),
• p-value is the significance level of the t-test (p-value = 6.210^{-9}).
• conf.int is the confidence interval (conf.int) of the mean differences at 95% is also shown (conf.int= [173.42, 215.56])
• sample estimates is the mean differences between pairs (mean = 194.49).

Note that:

• if you want to test whether the average weight before treatment is less than the average weight after treatment, type this:
t.test(weight ~ group, data = my_data, paired = TRUE,
alternative = "less")
• Or, if you want to test whether the average weight before treatment is greater than the average weight after treatment, type this
t.test(weight ~ group, data = my_data, paired = TRUE,
alternative = "greater")

## Interpretation of the result

The p-value of the test is 6.210^{-9}, which is less than the significance level alpha = 0.05. We can then reject null hypothesis and conclude that the average weight of the mice before treatment is significantly different from the average weight after treatment with a p-value = 6.210^{-9}.

## Access to the values returned by t.test() function

The result of t.test() function is a list containing the following components:

• statistic: the value of the t test statistics
• parameter: the degrees of freedom for the t test statistics
• p.value: the p-value for the test
• conf.int: a confidence interval for the mean appropriate to the specified alternative hypothesis.
• estimate: the means of the two groups being compared (in the case of independent t test) or difference in means (in the case of paired t test).

The format of the R code to use for getting these values is as follow:

# printing the p-value
res$p.value [1] 6.200298e-09 # printing the mean res$estimate
mean of the differences
194.49 
# printing the confidence interval
res\$conf.int
[1] 173.4219 215.5581
attr(,"conf.level")
[1] 0.95

# Online paired t-test calculator

You can perform paired-samples t-test, online, without any installation by clicking the following link:

# See also

Paired Samples Wilcoxon Test (non-parametric)

# Infos

This analysis has been performed using R software (ver. 3.2.4).

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