Lord's Paradox
In this question, you will consider a statistical "paradox" (described in Frederic Lord's 1967 paper, "A paradox in the interpretation of group comparisons") and relate it to the potential outcomes framework.
A large university is interested in investigating the effects on the students of the diet provided in the university dining halls and any sex difference in these effects. Various types of data are gathered. In particular, the weight of each student at the time of his arrival in September and his weight the following June are recorded. (Lord, 1967, p. 304)
a.) Under the potential outcomes framework, what are the possible treatments/causes considered in the study? Choose all that apply.
i) Students
ii) Diet
iii) Sex
iv) Initial weight
v) Final weight
vi) Weight change (gain score)
b.) Which of the following could represent potential outcomes of interest in this study? Choose all that apply.
i) Students, with and without treatment
ii) Diet, with and without treatment
iii) Gain score, with and without treatment
iv) Sex, with and without treatment
v) Initial weight, with and without treatment
vi) Final weight, with and without treatment
Part 2
Two different statisticians are asked to comment on the data. The first looks at the differences in average weight for each group, males and females.
a.) What is the average difference in weight (final minus initial) for females (Gender = 1)?
i) +10
ii) +5
iii) 0
iv) 5
v) -10
b.) What is the average difference in weight for males (Gender = 0)? i) +10
ii) +5
iii) 0
iv) 5
v) -10
c.) Based on the observations of the first statistician alone, what might you conclude about the effect of the university diet based on sex?
i) The diet has a differential positive effect on females compared to males.
ii) The diet has a differential negative effect on females compared to males.
iii) There is no evidence of any differential effect based on sex.
Part 3
The second statistician looks at the data in a more "sophisticated" way, and compares the average difference in weight for subgroups of males and females of the same initial weight.
a.) Consider the subgroup of individuals with a starting weight of 140. Compute the average gain score for females in the subgroup minus the average gain score for males in this subgroup.
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