Most sports injuries are immediate and obvious, like a broken leg. However, some
ID: 3172785 • Letter: M
Question
Most sports injuries are immediate and obvious, like a
broken leg. However, some can be more subtle, like
the neurological damage that may occur when soccer
players repeatedly head a soccer ball. To examine
long-term effects of repeated heading, Downs and
Abwender (2002) examined two different age groups
of soccer players and swimmers. The dependent
variable was performance on a conceptual thinking
task. Following are hypothetical data, similar to the
research results.
a. Use a two-factor ANOVA with = .05 to evaluate
the main effects and interaction.
b. Calculate the effects size (n2) for the main effects
and the interaction.
c. Briefly describe the outcome of the study
Factor B: Age
College Older
Soccer n= 20 n=20
Factor A: M=9 M= 4
Sport T= 180 T = 80
SS= 380 SS= 390
-----------------------------------------------------------------------------------------
swimming n= 20 n = 20
M = 9 M = 8
T = 180 T =160
SS = 350 SS= 400
----------------------------------------------------------------------------------------
EX2 = 6360
Explanation / Answer
Solution:
As this is a balanced design, we may easily determine the sport SS, the age SS, and the age sport interaction SS.
For soccer, the mean of the 40 subjects is (9+4)/2 = 13/2
For swimming, the mean of the 40 subjects is (9+8)/2 = 17/2
The overall mean is (13/2 + 17/2)/2 = 15/2
Thus, the Sport SS = 40(13/2-15/2)^2 + 40(17/2-15/2)^2 = 40 + 40 = 80
For age, the college mean is (9+9)/2=9
The older mean is (4+8)/2 = 6
The age SS is 40(9-15/2)^2 + 40(6-15/2)^2 = 40 * 9/4 + 40(9/4) = 180
Then, to get the interaction SS, we get the SS between for the four cells and subtract the age and sport SS
20(9-15/2)^2 + 20(9-15/2)^2 + 20(8-15/2)^2 + 20(4-15/2)^2 - 80 - 180 =
20*(3/2)^2 + 20*(3/2)^2 + 20*(1/2)^2 + 20*(7/2)^2 - 80 - 180 = 45 + 45 + 5 + 245 - 80 - 180 =
80
SSE = 380 + 390 + 350 + 400 = 1520
We have 1 df for sport, age, and sport-age interaction, and 4*(20-1) = 4*19 = 76 df for the error
Thus, MS sport = 80/1 = 80
MS age = 180/1 = 180
MS age-sport interaction = 80/1 = 80
MSE = 1520/76 = 20
Thus, F age-sport interaction = 80/20 = 4
F sport = 80/20 = 4
F age = 180/20 = 9
The critical value for F1,76 = finv(.05,1,76) = 3.96675978400878
F age-sport-interaction = 4 > 3.96675978400878
reject null hypothesis of no age-sport-interaction
F age = 9 > 3.96675978400878
reject null hypothesis of no age effect
F sport = 4 > 3.96675978400878
reject null hypothesis of no sport effect
There are different measures of effect size.
One is
Effective size for age = (6 - 9)/sqrt(20) = -0.670820393249937
Effect size for sport = (13/2-17/2)/sqrt(20) = -0.447213595499958
Another measure is the fraction of the SS explained. This makes more sense for interactions.
If we use this, then the TSS = 180 + 80 + 80 + 1520 = 1860
Then, the effect size for interaction is 80/1860 = 0.043010752688172,
the effect size for sport is 80/1860 = 0.043010752688172
the effect size for age is 180/1860 = 0.0967741935483871
The outcome of the study is that there is an interaction term, which suggests that playing soccer may have a negative on conceptual thinking.
As you can see reflecting on the results, once there is an interaction effect, it becomes difficult to assess the true main effect impact.