Week 26: The Single-Factor Between-Subjects Analysis of Variance

We’re back to parametric tests this week with the single-factor between-subjects analysis of variance (ANOVA)!

When Would You Use It?
The single-factor between-subjects ANOVA is a parametric tests used to determine if, in a set of k (k  ≥ 2) independent samples, at least two of the samples represent populations with different mean values.

What Type of Data?
The single-factor between-subjects ANOVA requires interval or ratio data.

Test Assumptions

  • Each sample of subjects has been randomly chosen from the population it represents.
  • For each sample, the distribution of the data in the underlying population is normal.
  • The variances of the k underlying populations are equal (homogeneity of variances).

Test Process
Step 1: Formulate the null and alternative hypotheses. The null hypothesis claims that the k population means are equal. The alternative hypothesis claims that at least two of the k population means are different.

Step 2: Compute the test statistic, an F-value. To do so, calculate the following sums of squares values for between-groups (SSB) and within-groups (SSW):


Then compute the mean squared difference scores for between-groups (MSG) and within-groups (MSE):


Finally, compute the F statistic by calculating the ratio:


Step 3: Obtain the p-value associated with the calculated F statistic. The p-value indicates the probability of a ratio of MSB to MSW equal to or larger than the observed ratio in the F statistic, under the assumption that the null hypothesis is true. Unless you have software, it probably isn’t possible to calculate the exact p-value of your F statistic. Instead, you can use an F table (such as this one) to obtain the critical F value for a prespecified α-level. To use this table, first determine the α-level. Find the degrees of freedom for the numerator (or MSB; the df are explained below) and locate the corresponding column on the table. Then find the degrees of freedom for the denominator (or MSE; the df are explained below) and locate the corresponding set of rows on the table. Find the row specific to your α-level. The value at the intersection of the row and column is your critical F value.

Step 4: Determine the conclusion. If the p-value is larger than the prespecified α-level (or the calculated F statistic is larger than the critical F value), fail to reject the null hypothesis (that is, retain the claim that the population means are all equal). If the p-value is smaller than the prespecified α-level, reject the null hypothesis in favor of the alternative.

The example I want to look at today comes from a previous semester’s STAT 217 grades. This particular section of 217 had four labs associated with it. I wanted to determine if the average final grade was different for any one lab compared to the others. Here, n = 109 and let α = 0.05.

H0: µlab1 = µlab2 = µlab3 = µlab4
Ha: at least one pair of means are different



For this case, the critical F value, as obtained by the table, is 2.70. Since the computed F value is smaller than the critical F value, we fail to reject H0 and conclude that the average final grade is equal across all four labs.

Example in R

x=read.table('clipboard', header=T)
             Df Sum Sq Mean Sq F value Pr(>F)
lab           3   1319   439.5   2.036  0.113
Residuals   105  22670   215.9

R will give you the exact p-value of your F statistic; in this case, p-value = 0.113.

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