Most sports injuries are immediate and obvious, like a broken leg. However, some

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.

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