Question

Twelve dogs from three different breeds (basenji, Shetland sheepdog, beagle) were either indulged or disciplined between the third and eight weeks of their lives. The indulged animals were encouraged in play, aggression, and climbing on their caretaker. In contrast, the disciplined dogs were retrained to their handler’s lap, taught to sit, stay, come, and so on. The indulged-disciplined treatment was inspired by reports that overindulged children cannot often inhibit their impulses in structured situations. Each dog was taken into a room with a bowl of meat. The dog was hungry but the handler prevented it from eating for 3 minutes by hitting on the rump with a newspaper and shouting ‘no’. The handler left the room and the length of time it took the dog to begin eating the meat was recorded. Presumably, the indulged dog should go to the food more quickly than the disciplined dogs.

Basenjis	Shetlands	Beagles
Indulged	1	7	9
	4	10	7
	3	10	10
	1	9	10
	2	6	8
	2	8	9
Disciplined	5	9	2
	1	9	6
	4	8	3
	1	10	4
	2	5	5
	3	8	3

a. State the hypotheses for each of the three separate tests included in the two-factor ANOVA.

b. Use SPSS or manual calculation to test the significance of main and the interaction effects.

c. Present the effect size for each of the three tests

Answer 1

a)

Null and Alternative Hypothesis:

We will have three hypotheses:

H₀: µ_Indulged = µ_Disciplined

H₁: Not all Means are equal

H₀: µ_Basenjis = µ_Shetlands= µ_Beagles

H₁: Not all Means are equal

H₀: An interaction is absent

H₁: Interaction is present

Alpha = 0.05

b)

Degress of Freedom:

Df_Treatment(A) = a-1 = 2-1 = 1

Df_Dog (B) = b-1 = 3-1 = 2

df _{Treatment * Dog} (A*B) = (a-1) * (b-1) = 1*2 = 2

df _error = N – ab = 36 – 2*3 = 30

df _total= N – 1 = 36 – 1 = 35

Decision Rule (3):

We have three hypotheses, so we have three decision rules:

Critical Values:

Time (df_Treatment(A), df _error) : (1,30) = 4.17

Intensity (df_Dog (B),df _error) : (2,30) = 3.32

Time*Intensity (df _{Treatment * Dog} (A*B), df _error) : (2,30) = 3.32

[Treatment] If F is greater than 4.17, reject the null hypothesis

[Dog] If F is greater than 3.32, reject the null hypothesis

[Interaction] If F is greater than 3.32, reject the null hypothesis

Test Statistics:

SS_Treatment = ∑(∑a_i)²/b*n - T²/N = 21.78

SS_Dog = ∑(∑b_i)²/a*n - T²/N = 212.17

SS_{Treatment*Dog} = ∑(∑a_i * b_i)²/n - ∑(∑a_i)²/b*n - ∑(∑b_i)²/a*n + T²/N = 54.06

SS_Total = ∑(Y)² - T²/N = 354

SS_Error = SS_Total - SS_Treatment - SS_Dog - SS_{Treatment*Dog} = 66

MS = SS/df

               

F = MS_effect / MS_error

Hence,

F_Treatment = 21.78/2.2 = 9.90

F_Dog = 106.08/2.2 = 48.22

F_Interaction = 1056.57/2.2 = 12.29

Results:

[Treatment] If F is greater than 4.17, reject the null hypothesis

Our F = 9.90, we reject the null hypothesis.

[Dog] If F is greater than 3.32, reject the null hypothesis

Our F = 48.22, we reject the null hypothesis.

[Interaction] If F is greater than 3.32, reject the null hypothesis

Our F = 12.29, we reject the null hypothesis.

c)

Effect Size using Eta Squared is calculated as:

ȵ² = SS _between / SS _total

Treatment

ȵ² = 21.78/354 = 0.062 or 6.2%

Dog Type

ȵ² = 212.17/354 = 0.599 or 59.9%

Interaction

ȵ² = 54.06/354 = 0.153 or 15.3%