Question

A study is done to examine the retention of two forms of protein (P1, P2) given to rabbits at each of two concentrations (10.0 and 2.0 milimoles). Twelve rabbits were randomly divided into four groups, one group for treatment combination. There were three rabbits assigned to each combination. The following sums of squares are computed.

source	df	SS	MS	F
Protein		18.8
Concentration	1
interaction		2.1
error		133.3
total	11	406.7

a. State the hypothesis to be tested first. Fill in the values missing from the ANOVA table.

b. Test for significant interaction. a = 0.05

c. Test for significance differences among .two concentrations. a = 0.05

Answer 1

(a) The Hypothesis:

FACTOR A (Protein)

H0: There is no difference in the means of Factor A

Ha: At least one of the means differs from the others.

FACTOR B (Concentration)

H0: There is no difference in the means of Factor B.

Ha: At least one of the means differs from the others.

INTERACTION

H0: There is no interaction between the two factors. They are independent.

Ha: There is an interaction between the two factors. They are not independent.

____________

The ANOVA table

	df	SS	MS	Fobt
Protein	1	18.8	18.8	1.13
Concentration	1	252.5	252.5	15.15
Interaction	1	2.1	2.1	0.13
Error	8	133.3	16.7
Total	11	406.7

Calculations

n = 3, Number of rows (protein) = 2, Number of columns (concentration = 2), N = 12

df protein = n - 1 = 2 - 1 = 1

df concentration = n - 1 = 2 - 1 = 1

df Interaction = df protein x df concentration = 1 * 1 = 1

df error = N - (protein x concentration) = 12 - (2 * 2) = 8

df Total = 1 + 1 + 1 + 8 = 11

_____________

SS concentration = SStotal - SS protein - SS interaction - SS error = 406.7 - 18.8 - 2.1 - 133.3 = 252.5

___________

MS Protein = SS protein / df protein = 18.8 / 1 = 18.8

MS concentration = SS concentration / df Concentration = 252.5 / 1 = 252.5

MS interaction = SS interaction / df Interaction = 2.1 / 1 = 2.1

MS error = SS error / df error = 133.1 / 8 = 16.6

_____________

F protein = MS protein / MS error = 18.8 / 16.6 = 1.13

F concentration = MS concentration / MS error = 252.5 / 16.6 = 15.15

F interaction = MS interaction / MS error = 2.1 / 16.6 = 0.13

________________

(b) The p value for Interaction is 0.7278, which is greater than (0.05). Therefore we fail to reject H0.

There is no significant interaction between the factors protein and concentration.

___________________

(c) The p value for the concentrations is 0.0046, which is less than (0.05). Therefore we reject H0.

There is a significant difference between the means of the concentration.

_____________________