T-Test Essentials: Definition, Formula and 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.

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

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))