3. Two-factor analysis of variance Emphasis on calculations Aa Aa w. Thomas Boyc

Question

3. Two-factor analysis of variance Emphasis on calculations Aa Aa w. Thomas Boyce, a professor and pediatrician at the University of British Columbia, Vancouver, has studied interactions between individual differences in physiology and differences in experience in determining health and well-being. Dr. Boyce foun that some children are more sensitive to their environments. They do exceptionally well when the environment is supportive bu are much more likely to have mental and physical health problems when the environment has challenges. You decide to do a similar study, conducting a factorial experiment to test the effectiveness of one environmental factor and one physiological factor on a physical health outcome. As the environmental factor, you choose two levels of stressful life events. As the physiological factor, you choose three levels of immune reactivity. The outcome is number of respiratory illnesses in the previous 12 months, and the research participants are kindergartners. You conduct a two-factor ANOvA on the data. The two-factor ANOvA involves several hypothesis tests. Which of the following are null hypotheses that you could use this ANOVA to test? Check all that apply. There is no interaction between stressful life events and immune reactivity. O Immune reactivity has no effect on number of respiratory illnesses Stressful life events have no effect on number of respiratory illnesses. The effect of stressful life events on number of respiratory illnesses is no different from the effect of immune reactivity. The results of your study are summarized by the corresponding sample means below. Each cell reports the average number of respiratory illnesses for 11 kindergartners. Factor B: Immune Reactivity High Medium Low M 1.82 M 1.55 M 1.36 TRowi 52 T 20 17 Low T m 15 SS a 2.5455 ss 2.7273 ss m 1.6364 EX2 237 Factor A

Explanation / Answer

There is no interaction between stressful events and immune reactivity
Immune reactivity has no effect on number of respiratory illnesses
Stressful life events have no effect on number of respiratory illnesses

Number of replicates, r = 11
Number of levels Factor A,a = 2
Number of levels Factor B,b = 3
Corrected mean, CM = 1192/66 = 214.5606061

SS(Factor A) = ?(row wise sum2)/rb - CM = (522 + 672)/(11*3) - 214.5606061 = 3.4091
SS(Factor B) = ?(column wise sum2)/ra - CM = (342 + 402 + 452)/(11*2) - 214.5606061 = 2.7576

SS(Within) = SS(Total) - SS(Treatment) = 22.4394 - 6.2575 = 16.1819

df(Factor A) = 2 - 1 = 1
df(Factor B) = 3 -1 = 2
df(Interaction AB) = 5 - (1+2) = 2
df(Within) = df(Total) - df(Treatment) = 65 - 5 = 60

MS(AxB) = SS(AxB)/df(AxB) = 0.0908/2 = 0.0454
MS(Within) = SS(within)/df(Within) = 16.1819/60 = 0.2697

F(AxB) = MS(AxB)/MS(Within) = 0.0454/0.2697 = 0.1683

p(Factor A) = .0007
p(Factor B) = .0089
p(AxB) = .8454

At the significance level 0.01, the main effect due to factor A is significant, the main effect due to factor B is significant and the interaction effect between the two factors is not significant

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