Question

A researcher studied the effects of three experimental diets with varying fat contents on

the total lipid (fat) level in plasma. Total lipid level is a widely used predictor of coronary heart disease.

Fifteen male subjects who were within 20 percent of their ideal body weight were grouped into five

blocks according to age. Within each block, the three experimental diets were randomly assigned to the

three subjects. Data on reduction in lipid level (in grams per liter) after the subjects were on the diet for

a fixed period of time follow.

Fat Content of Diet

Block		j=1	j=2	j=3
i		Extremely Low	Fairly Low	Moderately Low
1	Ages 15-24	.73	.67	.15
2	Ages 25-34	.86	.75	.21
3	Ages 35-44	.94	.81	.26
4	Ages 45-54	1.40	1.32	.75
5	Ages 55-64	1.62	1.41	.78

How would I obtain an analysis of variance table and test whether or not the mean reductions in lipid levels differ for the three diets using alpha(a = .05)?

Answer 1

Since the problem is to determine whether or not the mean reductions in lipid levels differ for the three diets, we should use TWO-WAY ANOVA to solve this. I have used R to solve this problem.

R CODE:

#TWO WAY ANOVA

Two_way=aov(Lipid~Diet+Age,data=data)
summary(Two_way)

OUTPUT:

HYPOTHESIS:

The hypothesis is given by:

(a) : The mean reduction in lipid levels for three different diets are same.

​    : The mean reduction in lipid levels for atleast two different diets differs.

LEVEL OF SIGNIFICANCE:

INFERENCE:

From the ANOVA table, since the p value is 4.33e-08 (with respect to three diets) which is less than 0.05, hence we reject at 5% level and conclude that the mean reduction in lipid levels for atleast two different diets differs.

We have found out that the mean reduction in lipid levels for atleast two different diets differs.​ Now to find that between which of the two diets there is different mean reduction in lipid levels, we use tukey test for pairwise comparison.

PAIRWISE COMPARISON:

Since the p-value is less than 0.05 for all three diet levels, there is a statistically significant difference in the mean reduction in lipid levels between diet levels Fairly low and extrememly low (p-value=0.013), Moderately low and Extremely low (p-value=0.000) and between Moderately low and Fairly low (p-value=0.000) at 5% level of significance.

The difference in mean reduction in lipid levels between diet levels Fairly low and extremely low is -0.1 (i.e.) the mean reduction in lipid levels for diet levels extremely low is 0.1 units higher than diet levels Fairly low.

The difference in mean reduction in lipid levels between diet levels moderately low and extremely low is -0.68 (i.e.) the mean reduction in lipid levels for diet levels extremely low is 0.68 units higher than diet level moderately low.

The difference in mean reduction in lipid levels between diet levels moderately low and fairly low is -0.56 (i.e.) the mean reduction in lipid levels for diet levels fairly low is 0.56 units higher than diet level moderately low.