| Sex | BW |
|---|---|
| F | 2.15 |
| M | 2.55 |
| F | 2.95 |
| F | 2.70 |
| M | 2.20 |
| F | 1.85 |
| M | 2.55 |
| M | 2.60 |
Two-sample t-test in R
Cheatsheet
License
This work was developed using resources that are available under a Creative Commons Attribution 4.0 International License, made available on the SOLES Open Educational Resources repository by the School of Life and Environmental Sciences, The University of Sydney.
Assumed knowledge
- You know how to install and load packages in R.
- You know how to import data into R.
- You recognise data frames and vectors.
Data structure
The data should be in a long format (also known as tidy data), where each row is an observation and each column is a variable (Figure 1). If your data is not already structured this way, reshape it manually in a spreadsheet program or in R using the pivot_longer() function from the tidyr package.
| F | M |
|---|---|
| 2.15 | 2.55 |
| 2.95 | 2.20 |
| 2.70 | 2.55 |
| 1.85 | 2.60 |
Data
For this cheatsheet we will use data from the penguins dataset from the palmerpenguins package. You may need to install this package:
install.packages("palmerpenguins")About
The two-sample t-test is used to determine whether the means of two independent groups are significantly different from each other. Examples:
- Comparing plant growth: Is the mean height of plants in a shaded area significantly different from those in a sunlit area?
- Species abundance: Are the mean numbers of a specific bird species found in two different forest types significantly different?
- Soil nutrient levels: Is the mean nitrogen concentration in soil samples from an agricultural field significantly different from that in a nearby natural forest?
Modelling
The model used in this cheatsheet is based on the penguins dataset.
Is the bill length of penguins significantly different between male and female penguins?
The simplified model for the mathematically-adverse individual is \[\color{olive}\text{bill length} \sim \text{sex}\] which translates to “the bill length of penguins is influenced by sex”. The statistical model is \[\color{red}\text{bill length} = \beta_0 + \beta_1 \cdot sex + \epsilon\] where \(\beta_0\) is the mean of the first group in the sex variable (in this case, female), \(\beta_1\) is the difference in means between the two groups, and \(\epsilon\) is the error term.
Preparing the data
Extract only the variables of interest from the dataset using select() from the dplyr package. This makes it easier to work with the data, especially if cleaning is required. In this case, there are NA values in the sex column, so we remove them using drop_na(sex).
library(palmerpenguins)
library(dplyr)
data(penguins)
df <- penguins |>
select(bill_length_mm, sex) |>
drop_na(sex)Your own data should be in a similar format.
Analytical approaches
The traditional approach to the two-sample t-test is to use the t.test() function in R, while the modern approach is to use a general linear model (GLM) with the lm() or glm() functions.
Methods reporting
Assumption checks for normality and homogeneity of variance were performed performed using the [insert check(s)] and [insert check(s)], respectively. Both groups were found to be normally distributed and have equal variance, so a two-sample t-test was used to determine whether the mean body weight of possums was significantly different from 3.5 kg. This was computed using the
t.test()function in R version 4.4.0 (R Core Team, 2024).
Check assumption(s)
In the traditional approach, assumptions of normality and homogeneity of variance are checked using the raw data and can be done before performing the t-test.
Note
Assumption checks are not required to be reported in the results, but it is good practice to include them in the methods section and indicate how they were checked.
Normality
Any of the following methods can be used to check for normality. It is important to note that using raw data to check for assumptions requires each group (i.e. sex) to be checked separately.
The histogram shows that the data is not normally distributed as it is multimodal.
library(ggplot2)
ggplot(df, aes(x = bill_length_mm)) +
geom_histogram(binwidth = 1) +
facet_wrap( ~ sex) +
theme_classic()
The Q-Q plot shows that the data is not normally distributed as many points do not fall on the line. The S-shaped curvature indicates underdispersion.
ggplot(df, aes(sample = bill_length_mm)) +
stat_qq() +
stat_qq_line() +
facet_wrap( ~ sex) +
theme_classic()
This test is not recommended for large sample sizes (> 50) as it is sensitive to even the smallest deviations from normality. Nevertheless, if you choose to use it, the results show that the data is not normally distributed in either group since the p-value is less than 0.05.
df |>
filter(sex == "male") |>
pull(bill_length_mm) |>
shapiro.test()
df |>
filter(sex == "female") |>
pull(bill_length_mm) |>
shapiro.test()Equal variance
Eyeballing a boxplot of the data can be used to check for homogeneity of variance. The var.test() and leveneTest() functions can also be used to check for homogeneity of variance, with the latter being more robust. In general, it is preferable to not rely on the results of these formal tests if the samples are not normally distributed.
#
var.test(bill_length_mm ~ sex, df)
# Levene's test
library(car)
leveneTest(bill_length_mm ~ sex, df)Perform the analysis
These days, irregardless of whether the data meets the assumptions of equal variance, or not, the Welch’s t-test is recommended since it will provide almost identical results to the traditional t-test if equal variance is met anyway.
t.test(bill_length_mm ~ sex, data = df)How to report results
The mean bill length of penquins was significantly different between males and females (Welch’s t329 = -6.67, p < 0.001). Mean male bill length (45.9 mm) is higher than mean female bill length (42.1 mm).
Methods reporting
A general linear model was used to determine whether the bill length of penguins is significantly different between the two sexes by fitting the model
bill length ~ sex. This was computed using thelm()function in R version 4.4.0 (R Core Team, 2024). The assumptions of normality and homogeneity of variance were checked visually using the residuals of the model. Normality was not met but the analysis was still performed using a GLM as the sample size was large enough to assume that the sampling distribution of the mean is approximately normal.
Perform the analysis
fit <- lm(bill_length_mm ~ sex, data = df)
summary(fit)Check assumption(s)
With a GLM, assumptions of normality and homogeneity of variance are checked using the residuals of the model. A plot() function can be used to check these assumptions.
par(mfrow = c(2, 2))
plot(fit)
How to report results
There is evidence to suggest that the mean bill length of penguins is significantly different between sexes (GLM, F~1, 331~ = 44.5, p < 0.001). Male penguins had a higher mean bill length than females (mean difference = 3.76 mm).