T-Test Essentials: Definition, Formula and Calculation

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

Identify outliers

Outliers can be easily identified using boxplot methods, implemented in the R function identify_outliers() [rstatix package].

mice %>% identify_outliers(weight)

## [1] name       weight     is.outlier is.extreme
## <0 rows> (or 0-length row.names)

Check normality assumption

The normality assumption can be checked by computing the Shapiro-Wilk test. If the data is normally distributed, the p-value should be greater than 0.05.

mice %>% shapiro_test(weight)

## # A tibble: 1 x 3
##   variable statistic     p
##   <chr>        <dbl> <dbl>
## 1 weight       0.923 0.382

You can also create a QQ plot of the weight data. QQ plot draws the correlation between a given data and the normal distribution.

ggqqplot(mice, x = "weight")

Demo data

Demo dataset: genderweight [in datarium package] containing the weight of 40 individuals (20 women and 20 men).

Load the data and show some random rows by groups:

# Load the data
data("genderweight", package = "datarium")
# Show a sample of the data by group
set.seed(123)
genderweight %>% sample_n_by(group, size = 2)

## # A tibble: 4 x 3
##   id    group weight
##   <fct> <fct>  <dbl>
## 1 6     F       65.0
## 2 15    F       65.9
## 3 29    M       88.9
## 4 37    M       77.0

Identify outliers by groups

genderweight %>%
  group_by(group) %>%
  identify_outliers(weight)

## # A tibble: 2 x 5
##   group id    weight is.outlier is.extreme
##   <fct> <fct>  <dbl> <lgl>      <lgl>     
## 1 F     20      68.8 TRUE       FALSE     
## 2 M     31      95.1 TRUE       FALSE

Check normality by groups

# Compute Shapiro wilk test by goups
data(genderweight, package = "datarium")
genderweight %>%
  group_by(group) %>%
  shapiro_test(weight)

## # A tibble: 2 x 4
##   group variable statistic     p
##   <fct> <chr>        <dbl> <dbl>
## 1 F     weight       0.938 0.224
## 2 M     weight       0.986 0.989

# Draw a qq plot by group
ggqqplot(genderweight, x = "weight", facet.by = "group")

Check the equality of variances

This can be done using the Levene’s test. If the variances of groups are equal, the p-value should be greater than 0.05.

genderweight %>% levene_test(weight ~ group)

## # A tibble: 1 x 4
##     df1   df2 statistic      p
##   <int> <int>     <dbl>  <dbl>
## 1     1    38      6.12 0.0180

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

First, start by computing the difference between groups:

mice2 <- mice2 %>% mutate(differences = before - after)
head(mice2, 3)

##   id before after differences
## 1  1    187   430        -242
## 2  2    194   404        -210
## 3  3    232   406        -174

Identify outliers

mice2 %>% identify_outliers(differences)

## [1] id          before      after       differences is.outlier  is.extreme 
## <0 rows> (or 0-length row.names)

Check normality assumption

# Shapiro-Wilk normality test for the differences
mice2 %>% shapiro_test(differences) 

## # A tibble: 1 x 3
##   variable    statistic     p
##   <chr>           <dbl> <dbl>
## 1 differences     0.968 0.867

# QQ plot for the difference
ggqqplot(mice2, "differences")