Keywords

1 Introduction

Analysis of variance (ANOVA), also known as F-test, is one of the inferential statistical tests of hypothesis mostly applied by researchers or the data analysts to compare/determine the differences in mean between data samples that are represented in more than two independent comparison groups (usually categorical or ordinal) and a continuous dependent variable (Connelly, 2021; Sullivan, 2020). The ANOVA statistics can be regarded as an extension of the Independent Samples t-test (see: Chap. 8), that is mainly used when there are specifically two or more groups of independent variables(s) being compared against a continuous dependent variable. Therefore, ANOVA tests are only applicable when the data sample that is being analyzed is made up of more than two groups (i.e., a minimum of three) of an independent variable(s). The main aim of using the tests (ANOVA) is to statistically examine the differences (variability) that may exist within the groups of the (independent) variables being compared, as well as, among the groups that are being compared. Thus, statistically ANOVA tests determine whether the means of the three or more groups of an independent variable(s) are different taking into account the influence (usually referred to as Between-subject effect) that they may have on the dependent variable. With ANOVA, researchers or data analysts can ascertain the statistical significance of both the main effects (the variation) and their interaction (i.e., between-subjects effects) based on the significant values, usually determined through the p-values (p ≤ 0.05).

The formula for calculating ANOVA is explained as follows: it uses the F-test to determine whether the group means are equal by including the correct variances in the ratio (Connelly, 2021). In other words, the F-statistic is the ratio where:

$$F\, = \,variation \, between \, sample \, means/variation \, within \, the \, samples$$

Thus,

$${\text{F}} = {\text{MSE}}/{\text{MST}}$$

where:

F = ANOVA coefficient, MST = Mean sum of squares due to treatment, and MSE = Mean sum of squares due to error​.

There are two main types of ANOVA tests commonly used in the works of literature (Christensen, 2020; Guillén-Gámez et al., 2021; Nibrad, 2019). These are:

  • One-way ANOVA: used to compare the differences in mean between one (categorical or ordinal) independent variable and one (continuous) dependent variable, whereby the independent variable must have at least three levels, i.e., a minimum of three different groups or categories.

  • Two-way ANOVA: used to compare the differences in mean between two independent variables (with three or more multilevel) and one (continuous) dependent variable. For example, it is used for examining the effects that two factors (independent variables) may have on the population of the study (continuous dependent variable) simultaneously or all at the same time.

Other types of ANOVA statistics or multivariate analysis are also used in the existing literature or statistical analysis, such as the multivariate analysis of variance (MANOVA) (Dugard et al., 2022; Okoye et al., 2022), analysis of co-variance (ANCOVA) (Kaltenecker & Okoye, 2023; Li & Chen, 2019), multivariate analysis of co-variance (MANCOVA) (Li & Chen, 2019; Okoye et al., 2023), etc.

Just like many of the different types of parametric tests or statistical procedures; the main “assumptions” or “conditions” that are necessary for performing the ANOVA tests both for research experiments or data analytics are summarized as follows (Connelly, 2021; Sullivan, 2020)—see Chap. 6, Sect. 6.2.5:

  • Independence of cases: there should be no relationship among the observations in each group or among the groups of the variables themselves, i.e., independence of observations must hold.

  • Normality of data: there should be no significant outliers, that might have a negative effect on the ANOVA test. The dependent variable should have an approximately normal distribution for each category of the target independent variable.

  • Homogeneity of variances: the variance among the groups must hold or should be approximately equal.

  • The independent variable(s) must consist of more than two independent groups or categories, i.e., a minimum of three groups or levels.

  • The independent variable(s) must be categorical or ordinal.

  • The dependent variable must be continuous.

In the next sections of this chapter (Sects. 9.2 and 9.3); the authors will explain and demonstrate to the readers how to conduct the One-way and Two-way ANOVA tests in R. We will illustrate the different steps to performing the two tests using the following steps in R outlined in Fig. 9.1.

Fig. 9.1
A table outlines the workflow, starting with installing necessary packages, followed by data import and inspection, analysis, visualization, and interpretation of the results. Conduct assumption and ANOVA tests. Visualize results with g g plot. Interpret and explain the results.

Steps to conducting the ANOVA test and analysis in R

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2 One-Way ANOVA Test in R

One-way ANOVA is used when the dataset the researcher or analysts wants to investigate are made up of more than two groups of an independent variable and are statistically independent, and a continuous dependent variable. Thus, the One-way ANOVA test as the name implies, is statistically used to compare the differences in mean between one (categorical or ordinal) independent variable and one (continuous) dependent variable, whereby the independent variable must have at least three levels, i.e., a minimum of three different groups or categories.

By default, the hypothesis for testing whether there is a difference or variation in the mean of the more than two (> 2) specified groups of independent data or variable against the one dependent (usually continuous) variable is; IF the p-value of the test is less than or equal to 0.05 (p ≤ 0.05), THEN we assume that the mean of the groups of population (minimum of three) in the data sample are statistically different (i.e., varies) and that this is not by chance (H1), ELSE IF the p-value is greater than 0.05 (p > 0.05) THEN we can conclude that there is no difference in the mean of the groups and any difference observed could only occur by chance (H0).

The authors will practically demonstrate to the readers how to conduct the One-way ANOVA test in R using the anova( ), aov( ), and bartlette.test( ) functions. We will do this by using the outlined steps in Fig. 9.1.

To begin with the illustration, Open RStudio and create a new or open an existing project. Once the user have RStudio and an R Project opened, Create a new RScript and name it “OneWayANOVADemo” or any name the user chooses (see: Chaps. 1 and 2).

Now, download an example file that we will use to demonstrate the two types of ANOVA analysis (One-Way and Two-way). ***Note the users can use any dataset or format of their choice provided they are able to follow the different steps described in the code by the authors in the illustration).

As shown in Fig. 8.2, download the example data named “Diet_R.csv” from the following source (https://www.sheffield.ac.uk/mash/statistics/datasets) and save it on your local machine or computer. ***Note: the readers can also refer to the following repository (https://doi.org/10.6084/m9.figshare.24728073) where the authors have uploaded all the example files used in this book to download the file.

Once the user have downloaded the file and saved it on the computer, we can proceed to conduct the first ANOVA analysis (One-way ANOVA) in R.

# Step 1—Install and Load the required R Packages and Libraries

Install and load the following R packages and libraries (see Fig. 9.3, Step1, Lines 3 to 11) that will be used to call the different R functions, data manipulations, and graphical visualizations for the One-way ANOVA test.

Fig. 9.2
A screenshot depicts the datasets available on datasets, including data on diets and factors influencing graduate school application decisions. It offers ANOVA, descriptive stats, and recoding options for ANOVA suitability.

Example of file download for ANOVA test. (Source https://www.sheffield.ac.uk/mash/statistics/datasets)

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Fig. 9.3
37 lines of code depict code installs and loads necessary packages, imports data, inspects its structure, performs assumption tests, and prepares data for analysis. It is a 3 process step, step 2, and step 3 A assumptions test.

Steps to conducting one-way ANOVA test in R

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The syntax and code to install and load the R packages and Libraries are as follows:

6 lines of code commands install the packages tidy verse, g g p u b r, and d p l y r, and then load them into the R session using the library function.

# Step 2—Import and Inspect example dataset for Analysis

As defined in Fig. 7.3 (Step 2, Lines 13 to 18); import the dataset named “Diet_R.csv” that we have downloaded earlier, and store this in an R object named “ANOVA_Tests.data” (the users can use any name of their choice if they wish to do so).

Once the user have successfully imported the dataset in R, you will be able to view the details of the Diet_R.csv file as shown in Fig. 9.4 with 78 observations and 7 variables in the sample data.

Fig. 9.4
A screenshot exhibits R code for data analysis. It imports dataset ANOVA Tests dot data, conducts tests for assumptions, and analyzes data using ANOVA and normality tests. The dataset comprises 78 observations with seven variables.

Example of dataset imported and stored in RStudio environment as an R object

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4 lines of code depict ANOVA tests dot data to choose, attach, view, and string.

# Step 3—Conduct the tests for Assumptions and Analyze the data

To analyze the imported dataset that we stored as ANOVA_Tests.data (see Fig. 9.4). First, the authors will be conducting the different tests of assumptions, e.g., normality test and homogeneity of variance (see: Sect. 9.1), before performing the actual One-way ANOVA analysis if the dataset in question meets or satisfies the necessary assumptions or test condition for the One-way ANOVA.

The syntax and code for conducting the different tests of assumptions are presented below and highlighted in Fig. 9.3 (Step 3A, Lines 20 to 37):

5 sets of code conduct Shapiro-Wilk normality tests for three diet types, check homogeneity of variances using Bartlett's test, and convert the diet variable to a factor. These steps are essential assumptions for subsequent ANOVA tests.

Once the user have successfully run the lines of codes defined in Step 3A (Fig. 9.3, Lines 20 to 37), they will be presented with the results of the “tests for assumptions” in the Console as shown in Fig. 9.5.

Fig. 9.5
A screenshot depicts R console commands performing Shapiro-Wilk normality tests for three diet types, checking homogeneity of variances using Bartlett's test, and converting the diet variable to a factor. These steps are crucial assumptions for ANOVA tests. Highlights data weight 0.9667, 0.8763 and 0.95.

Result of tests for assumption prior to conducting the one-way ANOVA analysis in R

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As highlighted in the results in Fig. 9.5; the normality test using Shapiro–Wilk’s method shows that the distribution for majority of the different groups of “Diet” variable were normally distributed when considered against the target variable “weight6weeks”, assuming p-value of greater than 0.05, i.e., p > 0.05 and test statistics, W, of value greater than 0.5 as the threshold whereby: (weight6weeks[Diet == “1”], W=0.96677, p-value=0.5884), (weight6weeks[Diet == “2”], W=0.87631, p-value=0.004003), (weight6weeks[Diet == “3”], W=0.95941, p-value=0.3584). Therefore, from the results, we can assume or proceed to conduct the One-way ANOVA (parametric) analysis since the normality test and all the other necessary conditions are met. Moreover, it is important to mention that datasets which contains more than n > 40 samples or observations (see: Chap. 3) is considered also a scientifically acceptable sample size for conducting any type of the parametric tests in scientific research or statistical analysis purposes (Roscoe, 1975).

Furthermore, in the second test of assumption, we tested the homogeneity of variance for the two targeted/analyzed variables (weight6weeks ~ Diet) using the bartlett.test( ) function in R; whereby we assume that a value of p > 0.05 indicates “equality in variance”. As shown and highlighted in the results presented in Fig. 9.5, we can see that there are no difference in the homogeneity of variance for the two analyzed variables with p-value = 0.4784. Therefore, we accept that the assumption of equality in variance is met.

Lastly, in the third assumption (Fig. 9.5), we converted the independent variable “Diet” with 3 levels (1, 2, and 3) to a factor format to represent categorical values—see Chap. 2 for more details on Factorization in R.

With all the necessary conditions met, we can proceed to conduct the “One-way ANOVA” test using the anova( ), aov( ), and TukeyHSD( ) methods or function in R, as described in Step 3B (Fig. 9.6, Lines 39 to 52) and consequently in the outcome of the ANOVA test or results represented in Fig. 9.7.

Fig. 9.6
A screenshot presents 39 to 62 lines of R code for conducting a one-way ANOVA test using two different methods. Method 1 utilizes the function, while method 2 applies linear regression. Post-hoc analysis is performed using Tukey's H S D test to identify mean differences between groups. Visualization is done with boxplots and jitter plots.

One-way ANOVA test in R using two different types of method or approach

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Fig. 9.7
A console output depicts the results of a one-way ANOVA test in R using two different methods. Method 1 utilizes the function, while Method 2 employs linear regression. Post-hoc analysis with Tukey's H S D test identifies mean differences between groups for each method. Diet values D F, sum, square, mean, value of p r greater than F.

Results of one-way ANOVA test in R

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Note: as defined in the introduction section (Sect. 9.1);

  • One-way ANOVA test compares the differences in mean between one independent variable (with three or more multilevel or groups) and one dependent (continuous) variable.

  • The targeted “independent” variable (x) is often a categorical or ordinal type, while the “dependent” variable (y) must be numeric.

To demonstrate the One-way ANOVA using the example dataset we called “ANOVA_Tests.data” in R (see: highlighted columns and data in Fig. 9.4).

  • We will test whether the mean of the 3 groups of Diet (the independent variable) varies, and if so, which diet was best for losing weight taking into account the “weight6weeks” (dependent) variable.

The syntax to performing this test in R is as shown in the codes below and in Fig. 9.6 (Step 3B, Lines 39 to 52).

3 set of methods. Perform ANOVA tests on weight 6 weeks data grouped by Diet. Method 1 Use a o v for ANOVA and Tukey H S D for post-hoc analysis. Method 2 Create linear model with l m, perform ANOVA, then post-hoc Tukey H S D.

As presented in Figs. 9.6 and 9.7, we conducted the One-way ANOVA test by considering the two variables (weight6weeks ~ Diet). We illustrated the ANOVA analysis using two different methods or functions in R; the aov( ) and anova( ) methods. Both methods (named Method1 and Method2, respectively) produced the same results (see Fig. 9.7) and are explained in detail in the Step 5 in the later part of this section.

# Step 4—Plot and visualize the mean differences for the results or data

As illustrated in Fig. 9.8 (Step 4, Lines 55 to 62), the authors used the ggplot( ) function in R to visualize the mean differences between the 3 groups of “Diet” (1, 2 and 3) representing the independent variable by taking into account the “weight6weeks” (dependent variable) as contained in the analyzed data “ANOVA_Tests.data”.

Fig. 9.8
A script conducts One-Way ANOVA on weight 6 weeks data grouped by Diet. It identifies group mean differences using Tukey H S D. Visualizes mean differences via boxplot and jitter.

Plot of the mean difference for the 3 groups of the independent variable versus the dependent variable in R using the ggplot( ) function

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The syntax to plot the mean or results of the analyzed variables is as shown in the code below, and the resultant graph represented in Fig. 9.8.

4 lines of code depict g g plot 2 code creates a boxplot and adds jitter for each Diet group against weight 6 weeks data, filled by Diet. The jitter adds data points to avoid overlap.

# Step 5—Results Interpretation (One-way ANOVA)

The last step for One-way ANOVA analysis is to interpret and understand the results of the test.

By default, the hypothesis for conducting the test (One-way ANOVA) is; IF the p-value of the test result is less than or equal to 0.05 (p ≤ 0.05), THEN we assume that the mean of the group (minimum of three levels or categories) of population in the data sample are statistically different (varies) and that this is not by chance (H1), ELSE IF the p-value is greater than 0.05 (p > 0.05) THEN we can conclude that there is no difference in the mean of the groups and any difference observed could only occur by chance (H0).

R code creates a linear model one-way model using l m with weight 6 weeks as the response variable and Diet as the predictor variable from ANOVA Tests dot data. Then, ANOVA of the one-way model performs ANOVA, depicts an analysis of the variance table with D f degrees of freedom, sum of squares, mean square, F value, and p-value for diet and residuals.

As shown in the result of the test presented above (see: Fig. 9.7); the different component or meaning of the One-way ANOVA test and outcome can be explained as a list containing the following:

  • Statistics: F = 0.1834 which signifies the ratio or value of the analysis of variance test.

  • p-value: p-value = 0.8328 is the p-value or significance levels of the test.

Statistically, as we can see from the results, the p-value (p=0.8328) is greater than the defined or acceptable significance levels (p ≤ 0.05). Therefore, we can statistically conclude that there is no difference between the means of effect of the different groups of “Diet” after the 6 weeks of intervention considering the “weight6weeks” variable.

Also, to confirm the results of the One-way ANOVA test, a good practice by the researchers or statisticians is to check where the significant differences lies (if there was any).

To show the readers how to carry out this post-hoc test in R supposing we found any significant difference which the authors will be explaining more in detail in other chapters of this book; we conducted a post-hoc test using the TukeyHSD( ) method by comparing the individual groups of diet against each other (see Fig. 9.7).

The code executes Tukey multiple comparisons of means for Diet groups derived from one way model, using a 95% confidence level. It reports mean differences, confidence intervals, and adjusted p-values for each comparison.

As seen in the results above (see Fig. 9.7), we can see that there were no differences found between the subjects or comparisons for the 3 groups (group 2–1, p=0.8265896; group 3–1, p=0.9023447; group 3–2, p=0.9857412), respectively. Thus, also confirming the results of the One-way ANOVA analysis we have explained earlier in this section.

3 Two-Way ANOVA Test in R

Two-way ANOVA is used when the dataset the researchers or analyst wants to analyze consists of “two independent variables” (with more than two groups) that are statistically independent. Unlike the One-way ANOVA that considers only one independent variable, the Two-way ANOVA is applied to compare the effects or differences in mean between two independent variables (categorical or ordinal) against one dependent (continuous) variable, whereby the independent variables must have at least three levels, i.e., a minimum of three different groups or categories.

***It is also noteworthy to mention that ANOVA tests can be performed for independent variables with two groups (although it is best recommended to use the Independent Samples t-test in this type of scenario)***.

By default, the hypothesis for testing whether there is a difference or variation in the mean of two specified groups of independent data samples (with three or more levels) against one dependent (usually continuous) variable is; IF the p-value of the test is less than or equal to 0.05 (p ≤ 0.05), THEN we can assume that the impact or mean effect of the groups (usually minimum of three groups) of population in the data sample are statistically different (varies) and that this is not by chance (H1), ELSE IF the p-value is greater than 0.05 (p > 0.05) THEN we can say that there is no effect or difference in the mean of the groups of variables and any difference observed could only occur by chance (H0).

Let’s continue to use the Diet_R.csv data we imported earlier and stored as an object we called “ANOVA_Tests.data” in R (see: Fig. 9.4) to illustrate how to perform the Two-way ANOVA using the anova( ), aov( ) and leveneTest( ) functions in R. We will do this using the same steps we have previously outlined in Fig. 9.1. ***Users can refer to the following repository to download the example file if they need to: https://doi.org/10.6084/m9.figshare.24728073.

To begin, Create a new R Script in the current R project (this can be done by using the file menu option, see also Chaps. 1 and 2) and name it as “TwoWayANOVADemo”.

# Step 1—Install and Load the required R Packages and Libraries

Load the following R libraries (Fig. 9.9, Step1, Lines 3 to 7), that we will be using to call the different R functions, data manipulations, and graphical visualizations for the Two-way ANOVA analysis.

Fig. 9.9
32 lines of R script performs a two-way ANOVA, including loading libraries, inspecting and cleaning data, testing assumptions converting factors, normality via Shapiro-Wilk, homogeneity via Levene's test, and conducting the ANOVA using ANOVA tests dot data.

Steps to performing the two-way ANOVA test in R

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Note: we did not need to repeat or re-install the required R packages again as this has already been previously installed in RStudio in the previous example in Sect. 9.2. However, if the user have directly visited this particular section for the first time or have previously exited or reinstalled R, then they may require to install or re-install the necessary R packages listed below again (see Chap. 2, Sect. 2.6 on how to install the R packages in RStudio).

The syntax and code to run/load the required R Libraries are as follows:

3 lines of code reads library of tidy verse, library of g g p u g r, and library of d p l y r.

***Note: for the leveneTest of assumption for homogeneity in variances (see Fig. 9.9, Step 3A, Assump: 3) by using the leveneTest( ) function, the user may also need or require to install the following additional highlighted R packages and libraries if they should encounter an error depending on the updated version of software installed.

8 lines of R script installs and loads the necessary packages for data analysis. r stat i x, car, car data, tidy verse, g g pub r, and d p l y r.

# Step 2—Inspect the example dataset for Analysis

Since we have already imported the “Diet_R.csv” and stored the example data file in an R object named “ANOVA_Tests.data” (Fig. 9.3) in the previous example in Sect. 9.2, the users do not need to import the data again. Rather, as shown in Fig. 9.9 (Step 2, Lines 9 to 12) you can view the dataset to inspect the different variables and confirm the items or variables we will be using to conduct the Two-way ANOVA test.

The code to do this is as shown below (see Fig. 9.9, Step 2, Lines 9 to 12).

2 lines of code read view of ANOVA test dot data. String of ANOVA tests dot data.

***Note: In the event that the reader has exited or closed RStudio and returned back to this current section at a later time, or directly visited this section of the book or example, then the user would need to use the following code to attach the example file or re-read the data again, as the case may be:

R script reads a C S V file into ANOVA Tests dot data 2, attaches it for easier variable access, and displays the data in a viewer.

Also, one important data cleaning task that the authors would like to bring the readers’ attention to and to illustrate, which is a good practice in scientific research and statistics, is to remove the incomplete rows or data with NA otherwise referred to as empty cells (see Fig. 9.4). The incomplete datasets (NA) can be removed by using the na.omit( ) function in R. Moreover, the reason for cleaning this dataset is because we will be including the “gender” variable (see Fig. 9.4) in our analysis in this particular example or section.

The syntax to remove the NAs or empty cells is as shown in the code below (Fig. 9.9, Step 2, Lines 14 to 16).

3 lines of code depict data cleaning to remove N A, ANOVA Tests dot data 2, n a dot omit of ANOVA Tests dot data, s t r of ANOVA Tests dot data 2.

***Note: as you can see, when the user have successfully run the codes, a new set of data “without the NAs” will be created, and we stored this new dataset in an R object we called “ANOVA_Tests.data2”.

Now we can proceed to conduct the next steps in the Two-way ANOVA analysis using the new cleaned data (ANOVA_Tests.data2).

# Step 3—Conduct tests for Assumptions and Analyze the data

As a necessary procedure, as shown in Fig. 9.9 (Step 3A, Lines 18 to 32), we will conduct the different tests of assumptions (i.e., check the variable types and format, normality test, and homogeneity of variances) before performing the Two-way ANOVA test.

The code to conduct the different tests of assumptions is presented below (see Fig. 9.9, Step 3A):

3 sets of code prepares data for two-way ANOVA by converting independent variables to factors, checks normality assumptions with Shapiro-Wilk test, and assesses homogeneity of variances using Levene's Test.

Once the user have successfully run the codes as defined in the Step 3A above (Fig. 9.9, Lines 18 to 32), you will be presented with the results of the “tests for assumptions” in the Console in R as shown in Fig. 9.10.

Fig. 9.10
Multiple line script ensures independent variables are factors, checks normality assumption with Shapiro-Wilk test p-value = 0.164, and assesses homogeneity of variances using Levene's Test p-value = 0.6012. It highlights gender factors and diet factors.

Results of the different tests of assumptions prior to conducting the two-way ANOVA test in R

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As gathered in Fig. 9.10, in Assmp1: the authors have converted (factored) and ensured that the two Independent variables “Diet” and “gender” that we will be analyzing or using to illustrate the Two-way ANOVA analysis are stored or recognized as a Factor (categorical variable) in R.

Also, we conducted a normality test in Assmp2 by using Shapiro–Wilk’s method to check the distribution of the data or targeted variables that we will be using to build the model. By assuming p-value of > 0.05 and test statistics value greater than 0.5 as the acceptable threshold. We can see that the distribution of the variables is normal with test result of 0.976, and p-value=0.164.

Lastly, in Assmp3: the authors tested the homogeneity of variance for the selected variables using the leveneTest( ) function, whereby we assume that a value of p > 0.05 indicates “equality in variance”. Consequentially, as highlighted in the third assumption (Assmp3) in Fig. 9.10, we can see that there is no difference in variance for the analyzed variables with p-value=0.6012.

Therefore, we can accept that all the necessary conditions to perform the Two-way ANOVA test are met.

With all assumptions met, we can now proceed to conduct the “Two-way ANOVA” analysis using the anova( ), aov( ), and TukeyHSD( ) methods as defined in Fig. 9.11 (Step 3B, Lines 35 to 49), and the results of the Two-way ANOVA test reported in Figs. 9.12a and b.

Fig. 9.11
56 lines of script conducts a two-way ANOVA on weight 6 weeks data grouped by Diet and gender, utilizing two methods. Method 1 with a o v and Method 2 with l m. Post-hoc analysis is done using Tukey's H S D test for both methods, followed by visualization of mean differences using g g line.

Conducting Two-way ANOVA analysis in R using two different methods

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Fig. 9.12
2 sets of screenshots depict the results of two-way ANOVA analyses with post-hoc Tukey H S D tests for Method 1 and Method 2. Both methods analyze the effects of Diet and gender on weight 6 weeks of data. Results include mean differences between groups with adjusted p-values.

a Result of two-way ANOVA (Method1) test in R with Post-Hoc test. b Result of two-way ANOVA (Method2) test in R with Post-Hoc test

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As defined earlier in the introduction section (Sect. 9.1);

  • Two-way ANOVA is applied to compare the differences in mean between two independent variables and one dependent variable, whereby the independent variable(s) must have at least three or more levels or groups.

  • The targeted “independent” variable (x) is often a categorical or ordinal type, while the “dependent” variable (y) must be numeric.

To illustrate the Two-way ANOVA using the cleaned example dataset (Diet_R.csv) which we have stored as “ANOVA_Tests.data2” in R.

  • We will test whether the weight lost after 6 weeks (“weight6weeks”) by the participants was influenced by the “diet” and “gender” variables. In other words, we will check the effect that the “diet” and “gender” variables (the independent variables) have on weight lost after 6 weeks (“weight6weeks”) (dependent variable), and if so, where the differences may lie across the data.

The syntax for conducting this above test in R is as shown in the codes below (Fig. 9.11, Step 3B, Lines 35 to 49).

3 sets of scripts depict method 1 conducts two-way ANOVA with a o v, and summarizes results. Method 2 fits linear model with interaction terms, and displays ANOVA table. Post-hoc Tukey H S D tests were performed for both methods to identify group mean differences.

As shown in Figs. 9.11, 9.12a, and b, we conducted the Two-way ANOVA analysis by considering the following variables (weight6weeks ~ Diet * gender). We illustrated this using two different ways or methods in R. As we can see, both methods (defined as Method1 and Method2) tend to produce the same results as shown in Figs. 9.12a and b, respectively. The results are explained in detail in the subsequent Step 5 in this section.

# Step 4—Plot and visualize the mean differences for the ANOVA model

As shown in Fig. 9.13 (Step 4, Lines 52 to 56), we used the ggline( ) function in R to visualize the mean differences that exist between the different groups of variables in the ANOVA model.

Fig. 9.13
A screenshot depicts code performs two-way ANOVA with Method 1 using an a o v function and Method 2 using the l m function. Post-hoc Tukey H S D tests are conducted for both methods and summary statistics for Method 3 are provided. A scattered chart on the right side depicts weight 6 weeks versus diet.

Plot of the mean differences between the different groups of variables in the ANOVA model using the ggline( ) function in R

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The code for the ANOVA model is shown below, and the result presented in the graph in Fig. 9.13.

A code generates a line plot g g line using data from ANOVA Tests dot data 2, with diet on the x-axis and weight 6 weeks on the y-axis. Points are colored by gender, and mean and standard error bars are added, with navy blue and dark red color palettes.

# Step 5—Two-way ANOVA Results Interpretation

The final step for the Two-way ANOVA analysis is to interpret and understand the results of the test.

By default, the hypothesis for conducting the test (Two-way ANOVA) is; IF the p-value of the test is less than or equal to 0.05 (p ≤ 0.05), THEN we can assume that the mean of the groups (minimum of three levels or groups) of variables or population (of which are two independent variables) in the data are statistically different (varies) and that this is not by chance (H1), ELSE IF the p-value is greater than 0.05 (p > 0.05) THEN we can conclude that there is no difference in the mean of the analyzed group of variables and any difference observed could only occur by chance (H0).

A code fits a two-way linear model l m with weight 6 weeks as the response variable and Diet and gender as predictors, including their interaction. The subsequent ANOVA function provides an analysis of the variance table, depicting each predictor's significance and their interaction.

As shown in the outcome of the Two-way ANOVA tests, with the same similar results observed for the method 1 and method 2 (see: Fig. 9.12a and b); statistically, we can see that the weight lost by the participants after 6 weeks “weight6weeks” was not influenced by the Diet (p=0.2745). Also, the “weight6weeks” was not influenced by the combination of the “Diet” and “gender” factors (Diet:gender) with p-value greater than the significance levels of p ≤ 0.05 (i.e., p-value=0.7575). However, we can see also that even though the combination of the variables (Diet:gender) does not have any significant effect on weight lost after 6 weeks, there were differences in mean (variation) for the genders (1 = male, 0 = female) variables when taking into account the weight lost after 6 weeks “weight6weeks” with p-value = 1.111e-13 (p ≤ 0.05).

Therefore, it will be necessary and important to further conduct a post-hoc test, as shown below, to determine where the significant differences lies (see: Figs. 9.12a and b).

A table depicts the differences in means between various groups defined by the combination of the diet and gender factors. The columns represent the difference, lower and upper confidence interval bounds, and adjusted p-values for each comparison.

As reported in the pairwise multiple comparisons test by using the TukeyHSD( ) method or function in R, we can see that most of the significant differences (p ≤ 0.05) observed for the between-subjects effects were found mainly for the female gender group (0).

Consequently, we can statistically conclude that the mean of weight lost after the 6 weeks (“weight6weeks”) by the participants varies by gender with p-value=1.111e-13 (p ≤ 0.05) but not influenced by Diet (p=0.2745).

***Useful Tips:

  • The researchers or analysts can also analyze more than two independent variables. This is known as N-Way ANOVA, whereby N represents the number of independent variables the researcher or data analysts are testing against the one dependent (response) variable. For instance, in our example data (Fig. 9.4), the users can simultaneously analyze the influence or effects that the Diet, Gender, Age group, etc. have on the “weight6weeks” variable.

4 Summary

In this chapter, the authors practically demonstrate in detail how to perform the most commonly used type of ANOVA tests (One-way and Two-way) in R.

In Sect. 9.2, it illustrates how to perform the One-way ANOVA test, while in Sect. 9.3 it looked at how to conduct the Two-way ANOVA analysis or test.

The authors also covered how to graphically plot the mean differences or results of the ANOVA tests in R in this chapter, and then subsequently discussed how to interpret and understand the results of the tests in R.

In summary, the main topics and contents covered in this chapter includes:

  • ANOVA (analysis of variance) is a statistical test of variance as the name implies or hypothesis used to compare the differences in means of data samples that are represented in more than two independent comparison groups or multilevel for the independent variable(s) (usually categorical or ordinal) and a continuous dependent variable.

When choosing whether to conduct a One-way or Two-way ANOVA test? The researcher or data analyst should:

  • Perform the “One-way ANOVA” if the groups come from one independent variable (with a minimum of three groups) usually measured as categorical or ordinal values, and one dependent variable (continuous).

  • Perform the “Two-way ANOVA” if the targeted groups come from two independent variables (with a minimum of three groups) usually measured as categorical or ordinal values, and one dependent variable (continuous).

  • In either case (One-way or Two-way), the targeted “independent” variable (x) is often a categorical or ordinal type, while the “dependent” variable (y) must be numeric.

Other types of the ANOVA statistics or “multivariate analysis” as they are called are also used in the existing literature or statistical analysis, such as the multivariate analysis of variance (MANOVA) (Dugard et al., 2022; Okoye et al., 2022), analysis of co-variance (ANCOVA) (Kaltenecker & Okoye, 2023; Li & Chen, 2019), multivariate analysis of co-variance (MANCOVA) (Li & Chen, 2019; Okoye et al., 2023), etc.