Psychology 318 Winter 2009

Example for one-way/ single factor ANOVA for a within subject design

Background: Alice, an undergrad honor student in the psychology department, decided to investigate factors that might influence performance in a game of Scrabble. In a “normal” game of Scrabble, players face different sides of a square game board and attempt to construct words on the board with letters they have drawn from a large pool of letters. After doing quite poorly in a game with friends, Alice suspected that she was at a disadvantage facing the board from the back and being forced to read the letters and words on the board upside down. Alice decided to conduct a simple experiment to test her question: “Does the seating of the game affect the performance on the game?”

Although she could use a between-subjects design, she decided to implement a within-subject design to minimize individual differences in participants’ scrabble-playing capability. Also, this design permitted her to have more power in the experiment with a relative scarcity of participants.

Alice recruited four friends of hers and let them play four games of Scrabble, each time from a different orientation with respect to the board (meaning each friend played one game from each of the four orientations). She counterbalanced the seating order so that subjects were not sitting next to the same people during each game. Each game ended when either the letters had all been played or no more moves were possible. They played four games on four successive days. The number of points each friend amassed during the course of the game was recorded. Table 1 summarizes the scores from each game in which each friend sat at a different seat orientation.

Table 1.

Seating
Subj / Front / Back / Left / Right / Mean
1 / 38 / 22 / 30 / 30 / 30
2 / 30 / 21 / 27 / 22 / 25
3 / 17 / 3 / 9 / 11 / 10
4 / 19 / 10 / 18 / 13 / 15
Mean / 26 / 14 / 21 / 19 / 20

Calculations for One-way ANOVA F test with within-subject design (repeated measures)

Please note: the grand mean is denoted by an X with a double bar.

  1. Compute the sums of squares (k is the # of treatment conditions):

SStotal = = SScond + SSsubj + SSerror

=

SScond = =

SSsubj = k =

SSerror =

  1. Calculate degrees of freedom:

dftotal = dfcond + dfsubj + dferror = (nk) – 1 =(4)(4)-1=15

dfcond = k – 1 = 4-1=3

dfsubj = n – 1 = 4-1=3

dferror = (n – 1)(k - 1) = (4-1)(4-1)=9

  1. F ratio:
  1. Make a decision (let α = .01) :

F critical = F α (df num, df den) = F α=.01 (3,9)=6.99

F obt > F crit  reject H0

  1. ANOVA summary table:

SOURCE SS dfs2=MS F

Between subjects10003333.33

Condition / treatment296398.6722.22

Error4094.44

Total133615

9. Draw a conclusion:

There is an effect of seating on Scrabble performance.