A one-way ANOVA is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups.

This tutorial provides a complete guide on how to interpret the results of a one-way ANOVA in R.

**Step 1: Create the Data**

Suppose we want to determine if three different workout programs lead to different average weight loss in individuals.

To test this, we recruit 90 people to participate in an experiment in which we randomly assign 30 people to follow either program A, program B, or program C for one month.

The following code creates the data frame we’ll be working with:

#make this example reproducible set.seed(0) #create data frame data <- data.frame(program = rep(c('A', 'B', 'C'), each = 30), weight_loss = c(runif(30, 0, 3), runif(30, 0, 5), runif(30, 1, 7))) #view first six rows of data frame head(data) program weight_loss 1 A 2.6900916 2 A 0.7965260 3 A 1.1163717 4 A 1.7185601 5 A 2.7246234 6 A 0.6050458

**Step 2: Perform the ANOVA**

Next, we’ll use the **aov()** command to perform a one-way ANOVA:

#fit one-way ANOVA model model <- aov(weight_loss ~ program, data = data)

**Step 3: Interpret the ANOVA Results**

Next, we’ll use the **summary()** command to view the results of the one-way ANOVA:

#view summary of one-way ANOVA model summary(model) Df Sum Sq Mean Sq F value Pr(>F) program 2 98.93 49.46 30.83 7.55e-11 *** Residuals 87 139.57 1.60 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Here’s how to interpret every value in the output:

**Df program:** The degrees of freedom for the variable *program*. This is calculated as #groups -1. In this case, there were 3 different workout programs, so this value is: 3-1 = **2**.

**Df Residuals:** The degrees of freedom for the residuals. This is calculated as #total observations – # groups. In this case, there were 90 observations and 3 groups, so this value is: 90 -3 = **87**.

**Sum Sq program:** The sum of squares associated with the variable *program*. This value is **98.93**.

**Sum Sq Residuals:** The sum of squares associated with the residuals or “errors.” This value is **139.57**.

**Mean Sq. Program:** The mean sum of squares associated with program. This is calculated as Sum Sq. program / Df program. In this case, this is calculated as: 98.93 / 2 = **49.46**.

**Mean Sq. Residuals:** The mean sum of squares associated with the residuals. This is calculated as Sum Sq. residuals / Df residuals. In this case, this is calculated as: 139.57 / 87 = **1.60**.

**F Value:** The overall F-statistic of the ANOVA model. This is calculated as Mean Sq. program / Mean sq. Residuals. In this case, it is calculated as: 49.46 / 1.60 = **30.83**.

**Pr(>F):** The p-value associated with the F-statistic with numerator df = 2 and denominator df = 87. In this case, the p-value is **7.552e-11**, which is an extremely tiny number.

The most important value in the entire output is the p-value because this tells us whether there is a significant difference in the mean values between the three groups.

Recall that a one-way ANOVA uses the following null and alternative hypotheses:

**H**All group means are equal._{0 }(null hypothesis):**H**At least one group mean is different from the rest._{A }(alternative hypothesis):

Since the p-value in our ANOVA table (.7552e-11) is less than .05, we have sufficient evidence to reject the null hypothesis.

This means we have sufficient evidence to say that the mean weight loss experienced by the individuals is not equal between the three workout programs.

**Step 4: Perform Post-Hoc Tests (If Necessary)**

If the p-value in the ANOVA output is less than .05, we reject the null hypothesis. This tells us that the mean value between each group is not equal. However, it doesn’t tell us *which* groups differ from each other.

In order to find this out, we must perform a post hoc test. In R, we can use the **TukeyHSD()** function to do so:

#perform Tukey post-hoc test TukeyHSD(model) $program diff lwr upr p adj B-A 0.9777414 0.1979466 1.757536 0.0100545 C-A 2.5454024 1.7656076 3.325197 0.0000000 C-B 1.5676610 0.7878662 2.347456 0.0000199

Here’s how to interpret the results:

- The adjusted p-value for the mean difference between group A and B is
**.0100545**. - The adjusted p-value for the mean difference between group A and C is
**.0000000**. - The adjusted p-value for the mean difference between group B and C is
**.0000199**.

Since each of the adjusted p-values is less than .05, we can conclude that there is a significant difference in mean weight loss between *each* group.

**Additional Resources**

Introduction to the One-Way ANOVA

How to Check ANOVA Assumptions

How to Perform a One-Way ANOVA By Hand

One-Way ANOVA Calculator

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