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:

Data Set

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?


Boxplot Diagram


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


Assuming random assignments to treatment levels and observations reasonably close to normal, these are the procedures.

(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.

Notes: No small dietary study of this kind, standing alone, can be a guide to health. (i) It is likely that in such a study several measurements other than antioxidant capacity would have been measured. If there were others, we are not hearing about them in this report. That would mean this one measurement was singled out for attention because it showed a remarkable effect. (ii) Moreover, with only 36 subjects altogether, even a high level of significance may not be a reliable indication of benefits of dark chocolate. (iii) I do not know enough about the practical importance of a 16 point difference in antioxidant levels to say whether the findings are important to health. (iv) It is typical for dietary studies over time to contradict one another.

Meanwhile, because dark chocolate is a great favorite of mine, I will enjoy fantasizing that it is a magnificent health food--until a study comes along that says dark chocolate clogs arteries, or has some other horrible health consequence. At that point I will briefly re-assess my overindulgence in dark chocolate, and probably make no changes.

  • $\begingroup$ that was insightful. If you realised, I just added the Analysis of Variance Table with F-Test distribution. How does the calculations relate to question (a)? $\endgroup$ – J.R. Sep 3 '15 at 8:21
  • $\begingroup$ 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.) $\endgroup$ – BruceET Sep 3 '15 at 21:25

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