# Experimental design

An experimental is interested in studying the effects of consuming chocolates on cardiovascular health. She decides to use three different types of chocolates: 100g of dark chocolates, 100g of dark chocolates with 200ml of milk, and 200g of milk chocolate. She randomly selects 12 subjects, with an average age of [31.2,33.2] years, and an average weight of [137.8,151.43], and an average body mass index of [21.5,22.30]kg/m. On different days, a subject consumes one of the three types (factor levels), and one hour later the total anti-oxidants capacity of the blood plasma is measured. The data set is in the table below:

The experimenter would like to answer questions such as

(a) Do the data indicate any differences in mean antioxidants capacity? (the effects of the three levels of the factor(chocolate) are different, or at least one of them is different from the others)

(b) Which of the three levels has the most significant effect on antioxidant capacity?

(c) Can we perform pairwise comparisons?

(d) What methods of data analysis should we use in each case?

My worry now is how do I approach each of these questions. What do i need to do?

(a) F-test in a standard one-factor ANOVA. Null hypothesis that population means $\mu_D = \mu_B = \mu_M$ rejected against alternative of some difference(s) among means.
(b-c) Because of the result in (a), some sort of paired comparison procedure is appropriate. Perhaps Tukey HSD. Just looking at standard errors it is pretty clear that $\mu_D$ is significantly greater than either of the other population means, and that there is no significant difference between the other two. So, as the boxplots indicate, Dark Chocolate has the greatest effect; notice that ALL 12 D measurements are greater than ANY of the other 24, signaling a very clear separation, even with no formal test.
• Tiny P-value, so strong rejection of null hypothesis that all 3 pop means are equal. Now you know there is a 'pattern of differences' to be investigated. Possibilities: $\mu_D > \mu_B > \mu_M$? $\mu_D > \mu_B \approx \mu_M$? Etc, etc. There are ${3 \choose 2} = 3$ possible paired comparisons. You could do 95% CI for each difference, but then 5% 'error probabilities' might accumulate to give 'family' confidence rate below 95%. Tukey HSD is a procedure that allows control of family rate for assessing patterns at 95%. (If ANOVA F didn't reject, then no point in doing HSD.) – BruceET Sep 3 '15 at 21:25