Paired_Sample_t-Test
Priyank Goyal
03/04/2020
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.
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:
Calculate the difference (d) between each pair of value Compute the mean (m) and the standard deviation (s) of d 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.
R Function
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”.
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 352.2Compute summary statistics (mean and sd) by groups using the dplyr package.
library(dplyr)## Warning: package 'dplyr' was built under R version 3.6.3##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionCompute summary by groups
group_by(my_data, group) %>%
summarise(
count = n(),
mean = mean(weight, na.rm = TRUE),
sd = sd(weight, na.rm = TRUE)
)## # A tibble: 2 x 4
## group count mean sd
## <fct> <int> <dbl> <dbl>
## 1 after 10 394. 29.4
## 2 before 10 199. 18.5Visualize your data using Box Plot
library(ggpubr)## Warning: package 'ggpubr' was built under R version 3.6.2## Loading required package: ggplot2## Registered S3 methods overwritten by 'ggplot2':
## method from
## [.quosures rlang
## c.quosures rlang
## print.quosures rlang## Loading required package: magrittrVisualize your data
library("ggpubr")
ggboxplot(my_data, x = "group", y = "weight",
color = "group", palette = c("#00AFBB", "#E7B800"),
order = c("before", "after"),
ylab = "Weight", xlab = "Groups")
Preliminary 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)##
## Shapiro-Wilk normality test
##
## data: d
## W = 0.94536, p-value = 0.6141From 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 sample t-test
Question : Is there any significant changes in the weights of mice after treatment?
- 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- 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
## 194.49As 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 Results
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).
# 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