Question

Every New Year people make resolutions to work out more often. One researcher created three interventions to help people meet their goals: #1 requires participants to write down their workout activities every day, #2 requires participants to journal about exercise every day, #3 requires participants to post their workout activities on social media. The data below show how many times the individuals in the three different groups worked out during a two-week period.

#1 Record Activity	#2 Journal about Exercise	#3 Social Media Post
6	4	6
4	3	9
4	0	6
2	1	3

What critical F value would you use to conduct your hypothesis test for this ANOVA? Use a two-tailed test with alpha = .05

		4.00
		4.26
		19.38
		3.49

Answer 1

Consider an example with three independent groups and a continuous outcome measure.

group 1 group 2 group 3

Sample Size n1 n2 n3

Sample Mean $equation_image31.gif$   $equation_image32.gif$   $equation_image33.gif$

Sample Standard Deviation S₁   S₂ S₃

The hypotheses of interest in an ANOVA are as follows:

• H₀: ?₁ = ?₂ = ?₃ ... = ?_k
• H₁: Means are not all equal.

where k = the number of independent comparison groups.

In this example, the hypotheses are:

• H₀: ?₁ = ?₂ = ?₃ = ?₄
• H₁: The means are not all equal.

Test Statistic for ANOVA

The test statistic for testing H₀: ?₁ = ?₂ = ... = ?_k is:

$equation_image35.gif$

and the critical value is found in a table of probability values for the F distribution with (degrees of freedom) df₁ = k-1, df₂=N-k.

In the test statistic,

n_j = the sample size in the j^th group (e.g., j =1, 2, 3, and 4 when there are 4 comparison groups)

$equation_image36.gif$ is the sample mean in the j^th group, and  is the overall mean.

k represents the number of independent groups (in this example, k=4),

and N represents the total number of observations in the analysis. Note that N does not refer to a population size, but instead to the total sample size in the analysis (the sum of the sample sizes in the comparison groups, e.g., N=n₁+n₂+n₃+n₄). The test statistic is complicated because it incorporates all of the sample data. While it is not easy to see the extension, the F statistic shown above is a generalization of the test statistic used for testing the equality of exactly two means.  

The ANOVA Procedure:

Source of Variation	Sums of Squares (SS)	Degrees of Freedom (df)	Mean Squares (MS)	F
Between Treatments	$equation_image38.gif$	k-1	$equation_image39.gif$	F_sB MSE 2
Error (or Residual)	SSE-x-x,)'	N-k	MSE = MSE y-t
Total	SST= ΣΣ(X-X)	N-1

where

• X = individual observation,

• $equation_image44.gif$ = sample mean of the j^th treatment (or group),
• $equation_image45.gif$ = overall sample mean,
• k = the number of treatments or independent comparison groups, and
• N = total number of observations or total sample size.

The ANOVA table above is organized as follows.

• The first column is entitled "Source of Variation" and delineates the between treatment and error or residual variation. The total variation is the sum of the between treatment and error variation.
• The second column is entitled "Sums of Squares (SS)". The between treatment sums of squares is

$equation_image46.gif$  

and is computed by summing the squared differences between each treatment (or group) mean and the overall mean. The squared differences are weighted by the sample sizes per group (n_j). The error sums of squares is:

  

SSE-x-x,)'

and is computed by summing the squared differences between each observation and its group mean (i.e., the squared differences between each observation in group 1 and the group 1 mean, the squared differences between each observation in group 2 and the group 2 mean, and so on). The double summation ( SS ) indicates summation of the squared differences within each treatment and then summation of these totals across treatments to produce a single value. (This will be illustrated in the following examples). The total sums of squares is:

$equation_image49.gif$  

and is computed by summing the squared differences between each observation and the overall sample mean. In an ANOVA, data are organized by comparison or treatment groups. If all of the data were pooled into a single sample, SST would reflect the numerator of the sample variance computed on the pooled or total sample. SST does not figure into the F statistic directly. However, SST = SSB + SSE, thus if two sums of squares are known, the third can be computed from the other two.

• The third column contains degrees of freedom. The between treatment degrees of freedom is df₁ = k-1. The error degrees of freedom is df₂ = N - k. The total degrees of freedom is N-1 (and it is also true that (k-1) + (N-k) = N-1).

• The fourth column contains "Mean Squares (MS)" which are computed by dividing sums of squares (SS) by degrees of freedom (df), row by row. Specifically, MSB=SSB/(k-1) and MSE=SSE/(N-k). Dividing SST/(N-1) produces the variance of the total sample. The F statistic is in the rightmost column of the ANOVA table and is computed by taking the ratio of MSB/MSE.

Follow the above procedure to find out the problem.The steps are described in detail manner.