Question

Twenty-four subjects were used in a clinical trial in which three factors were applied in a completely randomized 2³ arrangement with three replicates for each treatment. The three factors were A, B, and C. The response variable was measured at a subsequent stage and the data were:

	B1		B2
	C1	C2	C1	C2
A1	29 33 30	40 40 44	63 59 56	60 68 58
A2	36 39 37	48 52 52	60 65 59	73 61 68

a) write the linear model for the experiment, explain terms, and computer the analysis of variance for the data

b) Prepare a table of cell and marginal means with their respective standard errors.

c) Test the null hypothesis for all main and interaction effects.

d) Comment in detail on the difference between treatment means.

Answer 1

a)

b)

		sample size	marginal means	varince	standard errors
A	A1	12	48.33	191.52	3.99
	A2	12	55.00	168.91	3.75
B	B1	12	40.00	60.36	2.24
	B2	12	63.33	27.33	1.51
C	C1	12	48.00	230.91	4.39
	C2	12	55.33	124.42	3.22

c)

Tests of Between-Subjects Effects
Dependent Variable:Y
Source	Type III Sum of Squares	df	Mean Square	F	Sig.
Corrected Model	4002.667a	7	571.810	40.010	.000
Intercept	64066.667	1	64066.667	4482.799	.000
A	266.667	1	266.667	18.659	.001
B	3266.667	1	3266.667	228.571	.000
C	322.667	1	322.667	22.577	.000
A * B	10.667	1	10.667	.746	.400
A * C	2.667	1	2.667	.187	.672
B * C	130.667	1	130.667	9.143	.008
A * B * C	2.667	1	2.667	.187	.672
Error	228.667	16	14.292
Total	68298.000	24
Corrected Total	4231.333	23

Comment: The p-value of the interaction A, B, and C is 0.672. Hence, this interaction does not a significant effect on the model. Among the interaction of the two treatments, the interaction of the treatments B and C is 0.008. Hence, the interaction of this has a significant effect on the model. We know that if the higher order interaction of the treatments is significant than the effect of the lower order or main effect do not the meaning. Hence, the effect of the main effect in this model does not the meaning.

d)

Here, the interaction has a significant effect. Hence, the difference between treatment means do not have meaningful.