How to create an Interaction Plot in R

Introduction to Interaction Plots and ANOVA

When analyzing data where a single outcome is influenced by two categorical independent variables, the two-way ANOVA (Analysis of Variance) is the standard statistical test. This technique determines whether the means of multiple groups differ significantly, based on the stratification provided by two distinct factors. For instance, a researcher might use this approach to evaluate how Factors A and B individually influence a specific outcome measurement, known as the response variable.

While the primary focus of ANOVA often lies in assessing the main effects of each factor, a crucial consideration is the potential presence of an interaction effect. An interaction occurs when the effect of one factor on the response variable depends on the level of the other factor. Ignoring a significant interaction can lead to misleading conclusions about the main effects, rendering their interpretation incomplete or even incorrect.

Consider a practical example: evaluating the impact of exercise intensity (Factor 1) and gender (Factor 2) on weight loss (the response variable). While both exercise and gender might independently influence weight loss, an interaction effect suggests that the benefit derived from a specific exercise intensity level (e.g., intense exercise) is substantially different for males compared to females. It is possible that intense exercise yields dramatically higher weight loss for one gender, but only marginally higher loss for the other.

Visualizing this complex relationship is best achieved using an interaction plot. This visualization specifically displays the mean values of the response variable (Y-axis) relative to the levels of the first factor (X-axis), with separate lines (traces) dedicated to representing each level of the second factor. The pattern and slope of these lines immediately reveal the nature and magnitude of the interaction between the two independent variables.

This detailed guide explains how to properly set up, execute, and interpret an interaction plot in R, confirming the results against a formal statistical test.

Interpreting Interaction Plots: Parallel vs. Intersecting Lines

The visual analysis of an interaction plot is based on observing the relationship between the lines representing the different groups (levels of the second factor). This visual interpretation provides immediate insight into the statistical significance and practical relevance of the observed interaction effect. The two primary patterns to watch for are parallel lines and non-parallel (intersecting or converging/diverging) lines.

If the lines corresponding to the different levels of the trace variable are approximately parallel across the x-axis, it suggests that the effect of the primary factor (x-axis) is consistent across all levels of the secondary factor. Statistically, parallel lines indicate a lack of a significant interaction effect, meaning that the factors operate independently on the response variable. In such a scenario, the main effects of the ANOVA model can generally be interpreted without qualification.

Conversely, if the lines exhibit patterns of intersection, convergence, or divergence, it is a strong visual indicator of a significant interaction effect. Intersecting lines imply that the effect of the primary factor changes direction or magnitude depending on which level of the secondary factor is being considered. For instance, if one line slopes steeply upward while another remains flat or slopes downward, it confirms that the factors are interdependent in their impact on the response variable. This visual evidence should always be validated by the corresponding statistical significance reported in the ANOVA output, specifically the p-value for the interaction term.

Detailed Components of the interaction.plot() Function

The core function used in R for generating these visualizations is interaction.plot(). To ensure the plot accurately reflects the intended experimental design, it is essential to correctly specify the function’s arguments. Unlike plotting functions that automatically infer roles, this function requires explicit assignment of the three main variable types.

The primary arguments required for successful execution include:

• x.factor: This specifies the categorical variable that will be displayed along the horizontal axis. In a factorial design, this is typically one of the two independent factors, such as “Exercise Intensity.”

• trace.factor: This defines the second categorical independent variable, whose levels are represented by distinct lines (traces) on the plot. For example, “Gender” might be assigned to this argument.

• response: This is the quantitative dependent variable, whose aggregated values (means or medians) are plotted on the vertical axis (Y-axis). In our example, this would be “Weight Loss.”

Furthermore, the function offers powerful optional arguments that allow for customization and clarity:

• fun: Specifies the function used to aggregate the response variable values at each combination of factor levels. By default, this is the mean, but using median is often recommended if the data contains outliers or is not normally distributed.

• ylab and xlab: Used to provide clear, descriptive labels for the Y and X axes, respectively, greatly enhancing the plot’s readability.

• col, lty, and lwd: Control the visual aesthetics of the plot, including the color of the lines (col), the line type (lty, e.g., solid or dashed), and the line width (lwd). Customizing these parameters ensures differentiation between the trace factors.

Case Study: Evaluating Factors Influencing Weight Loss

To illustrate the complete process of detecting and visualizing interaction effects, we will explore a common scenario in health research. Suppose a team of researchers aims to rigorously determine whether the intervention factors of exercise intensity and gender jointly influence the outcome of weight loss over a standardized period. This experiment follows a classic two-way ANOVA design.

The experimental setup involves recruiting 60 participants: 30 men and 30 women. These participants are then randomly assigned 10 of each gender to one of three exercise programs: None, Light exercise, or Intense exercise. The study duration is one month, after which the net weight loss (in kilograms) is recorded for every individual. This design allows for the calculation of the main effects of exercise and gender, as well as their crucial interaction effect.

The following steps outline the necessary procedures within R: first, generating a synthetic data structure that mirrors the experimental outcomes; second, fitting the statistical ANOVA model to test for significance; and finally, generating the interaction plot to visually confirm and interpret the findings derived from the hypothesis test between exercise and gender.

Step 1: Preparing the Data Frame in R

The initial phase involves constructing a suitable data structure to hold the experimental results. In R, this is typically a data frame, which organizes the data into columns representing variables and rows representing individual observations. For reproducibility and consistent results, we start by setting a seed value.

The data frame must include three columns corresponding to our factors and response variable: gender (a nominal factor), exercise (an ordinal factor), and weight_loss (a continuous numerical variable). The code below uses random uniform distribution functions (runif) to simulate realistic weight loss data corresponding to the three experimental conditions for each gender group, aligning with the 10 observations per subgroup established in the study design.

The following code snippet demonstrates the creation and structure verification of the necessary data frame:

#make this example reproducible
set.seed(10)

#create data frame
data <- data.frame(gender = rep(c("Male", "Female"), each = 30),
                   exercise = rep(c("None", "Light", "Intense"), each = 10, times = 2),
                   weight_loss = c(runif(10, -3, 3), runif(10, 0, 5), runif(10, 5, 9),
                                   runif(10, -4, 2), runif(10, 0, 3), runif(10, 3, 8)))

#view first six rows of data frame
head(data)

  gender exercise weight_loss
1   Male     None  0.04486922
2   Male     None -1.15938896
3   Male     None -0.43855400
4   Male     None  1.15861249
5   Male     None -2.48918419
6   Male     None -1.64738030

Step 2: Fitting the Two-Way ANOVA Model and Interpreting Results

After successfully preparing the data, the next critical step is to fit the two-way ANOVA model using the aov() function in R. This function tests the null hypothesis that the population means are equal across the levels of the factors. The model formula specifies weight_loss as the response variable, and gender * exercise indicates that we are interested in the main effects of gender and exercise, as well as their multiplicative interaction effect.

Fitting the model and examining the summary output provides the statistical evidence regarding the significance of the main and interaction effects. The Summary of the ANOVA table is structured to present the degrees of freedom (Df), sum of squares (Sum Sq), mean squares (Mean Sq), the F statistic (F value), and the critical p-value (Pr(>F)) for each tested term.

#fit the two-way ANOVA model
model <- aov(weight_loss ~ gender * exercise, data = data)

#view the model output
summary(model)

# Df Sum Sq Mean Sq F value Pr(>F)    
#gender           1   15.8   15.80  11.197 0.0015 ** 
#exercise         2  505.6  252.78 179.087 <2e-16 ***
#gender:exercise  2   13.0    6.51   4.615 0.0141 *  
#Residuals       54   76.2    1.41                   
#---
#Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interpreting this output is crucial. We observe that both gender and exercise show statistically significant main effects (very low p-values). Most importantly, the row labeled gender:exercise represents the interaction term. With a p-value of 0.0141, which is less than the conventional alpha level of 0.05, we confirm that the interaction effect between exercise intensity and gender is statistically significant. This significance confirms that the effect of exercise on weight loss is not constant across both gender groups, necessitating a detailed visual examination using the interaction plot.

Step 3: Visualizing the Interaction Effect

The final step involves translating the statistical significance found in the ANOVA table into an intuitive visual format using the interaction.plot() function. Since the ANOVA demonstrated a significant interaction, the plot is necessary to understand the specific nature of this effect—that is, where and how the groups differ.

The following code generates the plot, explicitly defining data$exercise as the primary x-axis factor, data$gender as the trace factor (creating separate lines), and data$weight_loss as the response variable. We use fun = median for a robust measure of central tendency and apply customized labels and colors to ensure maximum clarity and professionalism in the final output.

interaction.plot(x.factor = data$exercise, #x-axis variable
                 trace.factor = data$gender, #variable for lines
                 response = data$weight_loss, #y-axis variable
                 fun = median, #metric to plot
                 ylab = "Weight Loss",
                 xlab = "Exercise Intensity",
                 col = c("pink", "blue"),
                 lty = 1, #line type
                 lwd = 2, #line width
                 trace.label = "Gender")

In general, if the two lines on the interaction plot are parallel, then there is no interaction effect. However, if the lines intersect, converge, or diverge significantly, an interaction effect is highly probable.

We can clearly see in this plot that the lines representing the median weight loss for males and females intersect between the “Light” and “Intense” exercise conditions. This distinct non-parallel pattern strongly indicates a significant interaction effect between the variables of exercise intensity and gender, confirming the statistical result (p = 0.0141) obtained from the ANOVA model.

This matches the fact that the p-value in the output of the ANOVA table was statistically significant for the interaction term in the ANOVA model.

A Comprehensive Guide to Conducting a One-Way ANOVA in R
Mastering the Execution of a Two-Way ANOVA in R