This article describes how to do a **paired t-test in R** (or in *Rstudio*). Note that the paired t-test is also referred as *dependent t-test*, *related samples t-test*, *matched pairs t test* or *paired sample t test*.

You will learn how to:

*Perform the paired t-test in R*using the following functions :`t_test()`

[rstatix package]: the result is a data frame for easy plotting using the`ggpubr`

package.`t.test()`

[stats package]: R base function.

*Interpret and report the paired t-test**Add p-values and significance levels to a plot**Calculate and report the paired t-test effect size*using*Cohen’s d*. The`d`

statistic redefines the difference in means as the number of standard deviations that separates those means. T-test conventional effect sizes, proposed by Cohen, are: 0.2 (small effect), 0.5 (moderate effect) and 0.8 (large effect) (Cohen 1998).

Contents:

#### Related Book

Practical Statistics in R II - Comparing Groups: Numerical Variables## Prerequisites

Make sure you have installed the following R packages:

`tidyverse`

for data manipulation and visualization`ggpubr`

for creating easily publication ready plots`rstatix`

provides pipe-friendly R functions for easy statistical analyses.`datarium`

: contains required data sets for this chapter.

Start by loading the following required packages:

```
library(tidyverse)
library(ggpubr)
library(rstatix)
```

## Demo data

Here, we’ll use a demo dataset `mice2`

[datarium package], which contains the weight of 10 mice before and after the treatment.

```
# Wide format
data("mice2", package = "datarium")
head(mice2, 3)
```

```
## id before after
## 1 1 187 430
## 2 2 194 404
## 3 3 232 406
```

```
# Transform into long data:
# gather the before and after values in the same column
mice2.long <- mice2 %>%
gather(key = "group", value = "weight", before, after)
head(mice2.long, 3)
```

```
## id group weight
## 1 1 before 187
## 2 2 before 194
## 3 3 before 232
```

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

## Summary statistics

Compute some summary statistics (mean and sd) by groups:

```
mice2.long %>%
group_by(group) %>%
get_summary_stats(weight, type = "mean_sd")
```

```
## # A tibble: 2 x 5
## group variable n mean sd
## <chr> <chr> <dbl> <dbl> <dbl>
## 1 after weight 10 400. 30.1
## 2 before weight 10 201. 20.0
```

## Calculation

### Using the R base function

There are two options for computing the independent t-test depending whether the two groups data are saved either in two different vectors or in a data frame.

**Option 1**. The data are saved in two different numeric vectors:

```
# Save the data in two different vector
before <- mice2$before
after <- mice2$after
# Compute t-test
res <- t.test(before, after, paired = TRUE)
res
```

```
##
## Paired t-test
##
## data: before and after
## t = -30, df = 9, p-value = 1e-09
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -217 -182
## sample estimates:
## mean of the differences
## -199
```

**Option 2**. The data are saved in a data frame.

```
# Compute t-test
res <- t.test(weight ~ group, data = mice2.long, paired = TRUE)
res
```

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

In the result above :

`t`

is the t-test statistic value (t = -25.55),`df`

is the degrees of freedom (df= 9),`p-value`

is the significance level of the t-test (p-value = 1.03910^{-9}).`conf.int`

is the confidence interval of the mean of the differences at 95% (conf.int = [-217.1442, -181.8158]);`sample estimates`

is the mean of the differences (mean = -199.48).

### Using the rstatix package

We’ll use the pipe-friendly `t_test()`

function [rstatix package], a wrapper around the R base function `t.test()`

. The results can be easily added to a plot using the `ggpubr`

R package.

```
stat.test <- mice2.long %>%
t_test(weight ~ group, paired = TRUE) %>%
add_significance()
stat.test
```

```
## # A tibble: 1 x 9
## .y. group1 group2 n1 n2 statistic df p p.signif
## <chr> <chr> <chr> <int> <int> <dbl> <dbl> <dbl> <chr>
## 1 weight after before 10 10 25.5 9 0.00000000104 ****
```

The results above show the following components:

`.y.`

: the y variable used in the test.`group1,group2`

: the compared groups in the pairwise tests.`statistic`

: Test statistic used to compute the p-value.`df`

: degrees of freedom.`p`

: p-value.

Note that, you can obtain a detailed result by specifying the option `detailed = TRUE`

.

```
mice2.long %>%
t_test(weight ~ group, paired = TRUE, detailed = TRUE) %>%
add_significance()
```

```
## # A tibble: 1 x 14
## estimate .y. group1 group2 n1 n2 statistic p df conf.low conf.high method alternative p.signif
## <dbl> <chr> <chr> <chr> <int> <int> <dbl> <dbl> <dbl> <dbl> <dbl> <chr> <chr> <chr>
## 1 199. weight after before 10 10 25.5 0.00000000104 9 182. 217. T-test two.sided ****
```

## Interpretation

The p-value of the test is 1.0410^{-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** = 1.0410^{-9}.

## Effect size

The effect size for a paired-samples t-test can be calculated by dividing the mean difference by the standard deviation of the difference, as shown below.

**Cohen’s d formula**:

\[

d = \frac{mean_D}{SD_D}

\]

Where `D`

is the differences of the paired samples values.

**Calculation**:

`mice2.long %>% cohens_d(weight ~ group, paired = TRUE)`

```
## # A tibble: 1 x 7
## .y. group1 group2 effsize n1 n2 magnitude
## * <chr> <chr> <chr> <dbl> <int> <int> <ord>
## 1 weight after before 8.08 10 10 large
```

There is a large effect size, Cohen’s d = 8.07.

## Report

We could report the result as follow: The average weight of mice was significantly increased after treatment, t(9) = 25.5, p < 0.0001, d = 8.07.

Visualize the results:

```
# Create a box plot
bxp <- ggpaired(mice2.long, x = "group", y = "weight",
order = c("before", "after"),
ylab = "Weight", xlab = "Groups")
# Add p-value and significance levels
stat.test <- stat.test %>% add_xy_position(x = "group")
bxp +
stat_pvalue_manual(stat.test, tip.length = 0) +
labs(subtitle = get_test_label(stat.test, detailed= TRUE))
```

## Summary

This article shows how to perform the paired t-test in R/Rstudio using two different ways: the R base function `t.test()`

and the `t_test()`

function in the rstatix package. We also describe how to interpret and report the t-test results.

## References

Cohen, J. 1998. *Statistical Power Analysis for the Behavioral Sciences*. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates.

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