One-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.

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

 

F	M
2.15	2.55
2.95	2.20
2.70	2.55
1.85	2.60

Figure 1: Data should be in long format (left) where each row is an observation and each column is a variable. This is the preferred format for most statistical software. Wide format (right) is also common, but may require additional steps to analyse or visualise in some instances.

Data

For this cheatsheet we will use data from the possums dataset used in BIOL2022 labs.

About

The one-sample t-test is used to determine whether the mean of a single sample \(y\) is significantly different from a known or hypothesised population mean (\(\mu\)). Examples:

• Is the mean weight of canned tuna significantly different from what was stated on the label (400 g)?
• Is the mean height of a sample of male students significantly different from the national average height (175.6 cm)?
• Is the mean number of kittens in a litter significantly different from 4?

Modelling

Is the mean body weight of possums (BW) significantly different from 3.5 kg?

The simplified model for the mathematically-adverse individual is \[\color{olive}\text{body weight} \sim 3.5\] which translates to “the body weight of possums is around 3.5 kg”. The statistical model is \[\color{red}\text{body weight} = \beta_0 + \epsilon\] where \(\beta_0\) is the hypothesised population mean and \(\epsilon\) is the error term.

Preparing the data

Extract only the variable of interest from the dataset using select() from the dplyr package – BW. Assign the variable to a new object – bw in this case.

library(dplyr)
library(readxl)
possums <- read_excel("possums.xlsx", sheet = 2) # import
bw <- select(possums, BW) # select variable

Your own data should be in a similar format.

Analytical approaches

The traditional approach to the one-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.

t.test() function

Methods reporting

A one-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).

Perform the analysis

t.test(bw, mu = 3.5)

Check assumption(s)

Normality

Any combination of one or more of the following checks can be used to assess normality:

• Histogram: hist(bw$BW)
• Q-Q plot: qqnorm(bw$BW)
• Shapiro-Wilk test: shapiro.test(bw$BW)

Include the appropriate description in your methods section.

The normality of body weight was assessed using [insert method(s)].

How to report results

The mean body weight of possums was significantly different from 3.5 kg (t₁₉ = -10.3, 95% CI [2.3, 2.7], p < 0.001).

lm() function

Methods reporting

A general linear model was used to determine whether the mean body weight of possums was significantly different from 3.5 kg. This was computed using the lm() function in R version 4.4.0 (R Core Team, 2024).

Perform the analysis

For a one-sample t-test, the formula needs to be specified as y - µ ~ 1 where y is the variable of interest and µ is the hypothesised value that is being tested. The 1 indicates that the model has an intercept only i.e. we are testing whether the mean difference is significantly different from 0.

fit <- lm((BW - 3.5) ~ 1, data = bw)
summary(fit)

Check assumption(s)

Normality

With a GLM, normality can be assessed using the residuals of the model. The following checks can be used:

• Histogram: hist(residuals(fit))
• Q-Q plot: qqnorm(residuals(fit))
• Shapiro-Wilk test: shapiro.test(residuals(fit))

How to report results

There is evidence to suggest that the mean body weight of possums was significantly different from 3.5 kg (GLM, t₁₉ = -10.3, p < 0.001).

Exercise(s)

Download the penguins dataset (from below if you are reading this in HTML), or load the dataset from the palmerpenguins package. Perform a one-sample t-test to determine whether the mean flipper length of penguins is significantly different from 200 mm.