• Introduction
• Perform multiple tests at once
• Concise and easily interpretable results
  • T-test
    • Additional p-value adjustment methods
  • ANOVA
• To go even further
• References

Introduction

As part of my teaching assistant position in a Belgian university, students often ask me for some help in their statistical analyses for their master’s thesis.

A frequent question is how to compare groups of patients in terms of several quantitative continuous variables. Most of us know that:

• To compare two groups, a Student’s t-test should be used¹
• To compare three groups or more, an ANOVA should be performed

These two tests are quite basic and have been extensively documented online and in statistical textbooks so the difficulty is not in how to perform these tests.

In the past, I used to do the analyses by following these 3 steps:

1. Draw boxplots illustrating the distributions by group (with the boxplot() function or thanks to the {esquisse} R Studio addin if I wanted to use the {ggplot2} package)
2. Perform a t-test or an ANOVA depending on the number of groups to compare (with the t.test() and oneway.test() functions for t-test and ANOVA, respectively)
3. Repeat steps 1 and 2 for each variable

This was feasible as long as there were only a couple of variables to test. Nonetheless, most students came to me asking to perform these kind of tests not on one or two variables, but on multiples variables. So when there were more than one variable to test, I quickly realized that I was wasting my time and that there must be a more efficient way to do the job.

Note: you must be very careful with the issue of multiple testing (also referred as multiplicity) which can arise when you perform multiple t-tests. In short, when a large number of statistical tests are performed, some will have p-values less than 0.05 purely by chance, even if all null hypotheses are in fact really true. This is known as multiplicity or multiple testing. You can tackle this problem by using the Bonferroni correction, among others. The Bonferroni correction is a simple method that allows many t-tests to be made while still assuring an overall confidence level is maintained. For this, instead of using the standard threshold of \(\alpha = 5\)% for the significance level, you can use \(\alpha = \frac{0.05}{m}\) where \(m\) is the number of t-tests. For example, if you perform 20 t-tests with a desired \(\alpha = 0.05\), the Bonferroni correction implies that you would reject the null hypothesis for each individual test when the p-value is smaller than \(\alpha = \frac{0.05}{20} = 0.0025\).

Note also that there is no universally accepted approach for dealing with the problem of multiple comparisons. Usually, you should choose a p-value adjustment measure familiar to your audience or in your field of study. The Bonferroni correction is the most common way to take this into account and is easy to implement. It is however not appropriate if you have a very large number of tests to perform (imagine you want to do 10,000 t-tests, a p-value would have to be less than \(\frac{0.05}{10000} = 0.000005\) to be significant). A more powerful method is also to adjust the false discovery rate using the Benjamini-Hochberg procedure (McDonald, 2014). This article aims at presenting a way to perform multiple t-tests and ANOVA from a technical point of view (how to implement it in R). Discussion on which adjustment method to use or whether there is a more appropriate model to fit the data is beyond the scope of this article (so be sure to understand the implications of using the code below for your own analyses). Make sure also to test the assumptions of the ANOVA before interpreting results.

Perform multiple tests at once

I thus wrote a piece of code that automated the process, by drawing boxplots and performing the tests on several variables at once. Below is the code I used, illustrating the process with the iris dataset. The Species variable has 3 levels, so let’s remove one, and then draw a boxplot and apply a t-test on all 4 continuous variables at once. Note that the continuous variables that we would like to test are variables 1 to 4 in the iris dataset.

dat <- iris

# remove one level to have only two groups
dat <- subset(dat, Species != "setosa")
dat$Species <- factor(dat$Species)

# boxplots and t-tests for the 4 variables at once
for (i in 1:4) { # variables to compare are variables 1 to 4
  boxplot(dat[, i] ~ dat$Species, # draw boxplots by group
    ylab = names(dat[i]), # rename y-axis with variable's name
    xlab = "Species"
  )
  print(t.test(dat[, i] ~ dat$Species)) # print results of t-test
}

## 
##  Welch Two Sample t-test
## 
## data:  dat[, i] by dat$Species
## t = -5.6292, df = 94.025, p-value = 1.866e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.8819731 -0.4220269
## sample estimates:
## mean in group versicolor  mean in group virginica 
##                    5.936                    6.588

## 
##  Welch Two Sample t-test
## 
## data:  dat[, i] by dat$Species
## t = -3.2058, df = 97.927, p-value = 0.001819
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.33028364 -0.07771636
## sample estimates:
## mean in group versicolor  mean in group virginica 
##                    2.770                    2.974

## 
##  Welch Two Sample t-test
## 
## data:  dat[, i] by dat$Species
## t = -12.604, df = 95.57, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -1.49549 -1.08851
## sample estimates:
## mean in group versicolor  mean in group virginica 
##                    4.260                    5.552

## 
##  Welch Two Sample t-test
## 
## data:  dat[, i] by dat$Species
## t = -14.625, df = 89.043, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.7951002 -0.6048998
## sample estimates:
## mean in group versicolor  mean in group virginica 
##                    1.326                    2.026

As you can see, the above piece of code draws a boxplot and then prints results of the test for each continuous variable, all at once.

At some point in the past, I even wrote code to:

1. draw a boxplot
2. test for the equality of variances (thanks to the Levene’s test)
3. depending on whether the variances were equal or unequal, the appropriate test was applied: the Welch test if the variances were unequal and the Student’s t-test in the case the variances were equal (see more details about the different versions of the t-test for two samples)
4. apply steps 1 to 3 for all continuous variables at once

I had a similar code for ANOVA in case I needed to compare more than two groups.

The code was doing the job relatively well. Indeed, thanks to this code I was able to test several variables in an automated way in the sense that it compared groups for all variables at once.

The only thing I had to change from one project to another is that I needed to modify the name of the grouping variable and the numbering of the continuous variables to test (Species and 1:4 in the above code).

Concise and easily interpretable results

library(ggpubr)

# Edit from here #
x <- which(names(dat) == "Species") # name of grouping variable
y <- which(names(dat) == "Sepal.Length" # names of variables to test
| names(dat) == "Sepal.Width"
| names(dat) == "Petal.Length"
| names(dat) == "Petal.Width")
method <- "t.test" # one of "wilcox.test" or "t.test"
paired <- FALSE # if paired make sure that in the dataframe you have first all individuals at T1, then all individuals again at T2
# Edit until here


# Edit at your own risk
for (i in y) {
  for (j in x) {
    ifelse(paired == TRUE,
      p <- ggpaired(dat,
        x = colnames(dat[j]), y = colnames(dat[i]),
        color = colnames(dat[j]), line.color = "gray", line.size = 0.4,
        palette = "npg",
        legend = "none",
        xlab = colnames(dat[j]),
        ylab = colnames(dat[i]),
        add = "jitter"
      ),
      p <- ggboxplot(dat,
        x = colnames(dat[j]), y = colnames(dat[i]),
        color = colnames(dat[j]),
        palette = "npg",
        legend = "none",
        add = "jitter"
      )
    )
    #  Add p-value
    print(p + stat_compare_means(aes(label = paste0(..method.., ", p-value = ", ..p.format..)),
      method = method,
      paired = paired,
      # group.by = NULL,
      ref.group = NULL
    ))
  }
}

raw_pvalue <- numeric(length = length(1:4))
for (i in (1:4)) {
  raw_pvalue[i] <- t.test(dat[, i] ~ dat$Species,
    paired = FALSE,
    alternative = "two.sided"
  )$p.value
}

df <- data.frame(
  Variable = names(dat[, 1:4]),
  raw_pvalue = round(raw_pvalue, 3)
)

df$Bonferroni <-
  p.adjust(df$raw_pvalue,
    method = "bonferroni"
  )
df$BH <-
  p.adjust(df$raw_pvalue,
    method = "BH"
  )
df$Holm <-
  p.adjust(df$raw_pvalue,
    method = "holm"
  )
df$Hochberg <-
  p.adjust(df$raw_pvalue,
    method = "hochberg"
  )
df$Hommel <-
  p.adjust(df$raw_pvalue,
    method = "hommel"
  )
df$BY <-
  round(p.adjust(df$raw_pvalue,
    method = "BY"
  ), 3)
df
##       Variable raw_pvalue Bonferroni    BH  Holm Hochberg Hommel    BY
## 1 Sepal.Length      0.000      0.000 0.000 0.000    0.000  0.000 0.000
## 2  Sepal.Width      0.002      0.008 0.002 0.002    0.002  0.002 0.004
## 3 Petal.Length      0.000      0.000 0.000 0.000    0.000  0.000 0.000
## 4  Petal.Width      0.000      0.000 0.000 0.000    0.000  0.000 0.000

t_table <- function(data, dvs, iv,
                    var_equal = TRUE,
                    p_adj = "none",
                    alpha = 0.05,
                    paired = FALSE,
                    wilcoxon = FALSE) {
  if (!inherits(data, "data.frame")) {
    stop("data must be a data.frame")
  }

  if (!all(c(dvs, iv) %in% names(data))) {
    stop("at least one column given in dvs and iv are not in the data")
  }

  if (!all(sapply(data[, dvs], is.numeric))) {
    stop("all dvs must be numeric")
  }

  if (length(unique(na.omit(data[[iv]]))) != 2) {
    stop("independent variable must only have two unique values")
  }

  out <- lapply(dvs, function(x) {
    if (paired == FALSE & wilcoxon == FALSE) {
      tres <- t.test(data[[x]] ~ data[[iv]], var.equal = var_equal)
    }

    else if (paired == FALSE & wilcoxon == TRUE) {
      tres <- wilcox.test(data[[x]] ~ data[[iv]])
    }

    else if (paired == TRUE & wilcoxon == FALSE) {
      tres <- t.test(data[[x]] ~ data[[iv]],
        var.equal = var_equal,
        paired = TRUE
      )
    }

    else {
      tres <- wilcox.test(data[[x]] ~ data[[iv]],
        paired = TRUE
      )
    }

    c(
      p_value = tres$p.value
    )
  })

  out <- as.data.frame(do.call(rbind, out))
  out <- cbind(variable = dvs, out)
  names(out) <- gsub("[^0-9A-Za-z_]", "", names(out))

  out$p_value <- ifelse(out$p_value < 0.001,
    "<0.001",
    round(p.adjust(out$p_value, p_adj), 3)
  )
  out$conclusion <- ifelse(out$p_value < alpha,
    paste0("Reject H0 at ", alpha * 100, "%"),
    paste0("Do not reject H0 at ", alpha * 100, "%")
  )

  return(out)
}

result <- t_table(
  data = dat,
  c("Sepal.Length", "Sepal.Width", "Petal.Length", "Petal.Width"),
  "Species"
)

result
##       variable p_value      conclusion
## 1 Sepal.Length  <0.001 Reject H0 at 5%
## 2  Sepal.Width   0.002 Reject H0 at 5%
## 3 Petal.Length  <0.001 Reject H0 at 5%
## 4  Petal.Width  <0.001 Reject H0 at 5%

result <- t_table(
  data = dat,
  c("Sepal.Length", "Sepal.Width", "Petal.Length", "Petal.Width"),
  "Species",
  p_adj = "holm"
)

result
##       variable p_value      conclusion
## 1 Sepal.Length  <0.001 Reject H0 at 5%
## 2  Sepal.Width   0.002 Reject H0 at 5%
## 3 Petal.Length  <0.001 Reject H0 at 5%
## 4  Petal.Width  <0.001 Reject H0 at 5%

dat <- iris

# Edit from here
x <- which(names(dat) == "Species") # name of grouping variable
y <- which(names(dat) == "Sepal.Length" # names of variables to test
| names(dat) == "Sepal.Width"
| names(dat) == "Petal.Length"
| names(dat) == "Petal.Width")
method1 <- "anova" # one of "anova" or "kruskal.test"
method2 <- "t.test" # one of "wilcox.test" or "t.test"
my_comparisons <- list(c("setosa", "versicolor"), c("setosa", "virginica"), c("versicolor", "virginica")) # comparisons for post-hoc tests
# Edit until here


# Edit at your own risk
for (i in y) {
  for (j in x) {
    p <- ggboxplot(dat,
      x = colnames(dat[j]), y = colnames(dat[i]),
      color = colnames(dat[j]),
      legend = "none",
      palette = "npg",
      add = "jitter"
    )
    print(
      p + stat_compare_means(aes(label = paste0(..method.., ", p-value = ", ..p.format..)),
        method = method1, label.y = max(dat[, i], na.rm = TRUE)
      )
      + stat_compare_means(comparisons = my_comparisons, method = method2, label = "p.format") # remove if p-value of ANOVA or Kruskal-Wallis test >= alpha
    )
  }
}

To go even further

As we have seen, these two improved R routines allow to:

1. Perform t-tests and ANOVA on a small or large number of variables with only minor changes to the code. I basically only have to replace the variable names and the name of the test I want to use. It takes almost the same time to test one or several variables so it is quite an improvement compared to testing one variable at a time
2. Share test results in a much proper and cleaner way. This is possible thanks to a graph showing the observations by group and the p-value of the appropriate test included on this graph. This is particularily important when communicating results to a wider audience or to people from diverse backgrounds.

However, like most of my R routines, these two pieces of code are still a work in progress. Below are some additional features I have been thinking of and which could be added in the future to make the process of comparing two or more groups even more optimal:

• Add the possibility to select variables by their numbering in the dataframe. For the moment it is only possible to do it via their names. This will allow to automate the process even further because instead of typing all variable names one by one, we could simply type 4:25 (to test variables 4 to 25 for instance).
• Add the possibility to choose a p-value adjustment method. Currently, raw p-values are displayed in the graphs and I manually adjust them afterwards.
• When comparing more than two groups, it is only possible to apply an ANOVA or Kruskal-Wallis test at the moment. A major improvement would be to add the possibility to perform a repeated measures ANOVA (i.e., an ANOVA when the samples are dependent). It is currently already possible to do a t-test with two paired samples, but it is not yet possible to do the same with more than two groups.
• Another less important (yet still nice) feature when comparing more than 2 groups would be to automatically apply post-hoc tests only in the case where the null hypothesis of the ANOVA or Kruskal-Wallis test is rejected (so when there is at least one group different from the others, because if the null hypothesis of equal groups is not rejected we do not apply a post-hoc test). At the present time, I manually add or remove the code that displays the p-values of post-hoc tests depending on the global p-value of the ANOVA or Kruskal-Wallis test.

I will try to add these features in the future, or I would be glad to help if the author of the {ggpubr} package needs help in including these features (I hope he will see this article!).

Last but not least, the following packages may be of interest to some readers:

• If you want to report statistical results on a graph, I advise you to check the {ggstatsplot} package and in particular the ggbetweenstats() function. This function allows to compare a continuous variable across multiple groups or conditions. Note that many different statistical results are displayed on the graph, not only the name of the test and the p-value. However, it is still very convenient to be able to include tests results on a graph in order to combine the advantages of a visualization and a sound statistical analysis.

• The {compareGroups} package also provides a nice way to compare groups. It comes with a really complete Shiny app, available with:

# install.packages("compareGroups")
library(compareGroups)
cGroupsWUI()

Thanks for reading. I hope this article will help you to perform t-tests and ANOVA for multiple variables at once and make the results more easily readable and interpretable by nonscientists. Learn more about the t-test to compare two samples, or the ANOVA to compare 3 samples or more.

As always, if you have a question or a suggestion related to the topic covered in this article, please add it as a comment so other readers can benefit from the discussion.

References

McDonald, J.H. 2014. Handbook of Biological Statistics (3rd ed.). Sparky House Publishing, Baltimore, Maryland.

1. In theory, an ANOVA can also be used to compare two groups as it will give the same results compared to a Student’s t-test, but in practice we use the Student’s t-test to compare two samples and the ANOVA to compare three samples or more.↩︎

2. Do not forget to separate the variables you want to test with |.↩︎

3. Post-hoc test is only the name used to refer to a specific type of statistical tests. Post-hoc test includes, among others, the Tukey HSD test, the Bonferroni correction, Dunnett’s test. Even if an ANOVA or a Kruskal-Wallis test can determine whether there is at least one group that is different from the others, it does not allow us to conclude which are different from each other. For this purpose, there are post-hoc tests that compare all groups two by two to determine which ones are different, after adjusting for multiple comparisons. Concretely, post-hoc tests are performed to each possible pair of groups after an ANOVA or a Kruskal-Wallis test has shown that there is at least one group which is different (hence “post” in the name of this type of test). The null and alternative hypotheses and the interpretations of these tests are similar to a Student’s t-test for two samples.↩︎