This article describes how to do a **one-sample t-test in R** (or in *Rstudio*). You will learn how to:

*Perform a one-sample 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 one-sample t-test**Add p-values and significance levels to a plot**Calculate and report the one-sample 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

Demo dataset: `mice`

[in datarium package]. Contains the weight of 10 mice:

```
# Load and inspect the data
data(mice, package = "datarium")
head(mice, 3)
```

```
## # A tibble: 3 x 2
## name weight
## <chr> <dbl>
## 1 M_1 18.9
## 2 M_2 19.5
## 3 M_3 23.1
```

We want to know, whether the average weight of the mice differs from 25g (two-tailed test)?

## Summary statistics

Compute some summary statistics: count (number of subjects), mean and sd (standard deviation)

`mice %>% get_summary_stats(weight, type = "mean_sd")`

```
## # A tibble: 1 x 4
## variable n mean sd
## <chr> <dbl> <dbl> <dbl>
## 1 weight 10 20.1 1.90
```

## Calculation

### Using the R base function

```
# One-sample t-test
res <- t.test(mice$weight, mu = 25)
# Printing the results
res
```

```
##
## One Sample t-test
##
## data: mice$weight
## t = -8, df = 9, p-value = 2e-05
## alternative hypothesis: true mean is not equal to 25
## 95 percent confidence interval:
## 18.8 21.5
## sample estimates:
## mean of x
## 20.1
```

In the result above :

`t`

is the t-test statistic value (t = -8.105),`df`

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

is the significance level of the t-test (p-value = 1.99510^{-5}).`conf.int`

is the confidence interval of the mean at 95% (conf.int = [18.7835, 21.4965]);`sample estimates`

is the mean value of the sample (mean = 20.14).

### 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 <- mice %>% t_test(weight ~ 1, mu = 25)
stat.test
```

```
## # A tibble: 1 x 7
## .y. group1 group2 n statistic df p
## * <chr> <chr> <chr> <int> <dbl> <dbl> <dbl>
## 1 weight 1 null model 10 -8.10 9 0.00002
```

The results above show the following components:

`.y.`

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

: generally, the compared groups in the pairwise tests. Here, we have null model (one-sample test).`statistic`

: test statistic (t-value) used to compute the p-value.`df`

: degrees of freedom.`p`

: p-value.

You can obtain a detailed result by specifying the option `detailed = TRUE`

in the function `t_test()`

.

`mice %>% t_test(weight ~ 1, mu = 25, detailed = TRUE)`

```
## # A tibble: 1 x 12
## estimate .y. group1 group2 n statistic p df conf.low conf.high method alternative
## * <dbl> <chr> <chr> <chr> <int> <dbl> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 20.1 weight 1 null model 10 -8.10 0.00002 9 18.8 21.5 T-test two.sided
```

## Interpretation

The p-value of the test is 210^{-5}, which is less than the significance level alpha = 0.05. We can conclude that the mean weight of the mice is significantly different from 25g with a **p-value** = 210^{-5}.

## Effect size

To calculate an effect size, called `Cohen's d`

, for the one-sample t-test you need to divide the mean difference by the standard deviation of the difference, as shown below. Note that, here: `sd(x-mu) = sd(x)`

.

**Cohen’s d formula**:

\[

d = \frac{m-\mu}{s}

\]

- \(m\) is the sample mean
- \(s\) is the sample standard deviation with \(n-1\) degrees of freedom
- \(\mu\) is the theoretical mean against which the mean of our sample is compared (default value is mu = 0).

**Calculation**:

`mice %>% cohens_d(weight ~ 1, mu = 25)`

```
## # A tibble: 1 x 6
## .y. group1 group2 effsize n magnitude
## * <chr> <chr> <chr> <dbl> <int> <ord>
## 1 weight 1 null model 10.6 10 large
```

Recall that, t-test conventional effect sizes, proposed by Cohen J. (1998), are: 0.2 (small effect), 0.5 (moderate effect) and 0.8 (large effect) (Cohen 1998). As the effect size, d, is 2.56 you can conclude that there is a large effect.

## Reporting

We could report the result as follow:

A one-sample t-test was computed to determine whether the recruited mice average weight was different to the population normal mean weight (25g).

The measured mice mean weight (20.14 +/- 1.94) was statistically significantly lower than the population normal mean weight 25 (`t(9) = -8.1, p < 0.0001, d = 2.56`

); where t(9) is shorthand notation for a t-statistic that has 9 degrees of freedom.

The results can be visualized using either a box plot or a density plot.

### Box Plot

Create a boxplot to visualize the distribution of mice weights. Add also jittered points to show individual observations. The big dot represents the mean point.

```
# Create the box-plot
bxp <- ggboxplot(
mice$weight, width = 0.5, add = c("mean", "jitter"),
ylab = "Weight (g)", xlab = FALSE
)
# Add significance levels
bxp + labs(subtitle = get_test_label(stat.test, detailed = TRUE))
```

### Density plot

Create a density plot with p-value:

- Red line corresponds to the observed mean
- Blue line corresponds to the theoretical mean

```
ggdensity(mice, x = "weight", rug = TRUE, fill = "lightgray") +
scale_x_continuous(limits = c(15, 27)) +
stat_central_tendency(type = "mean", color = "red", linetype = "dashed") +
geom_vline(xintercept = 25, color = "blue", linetype = "dashed") +
labs(subtitle = get_test_label(stat.test, detailed = TRUE))
```

## Summary

This article shows how to perform the one-sample 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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