What is the largest possible positive sum that could be achieved, given the observed absolute values of the differences?

Biological changes that result from psychological manipulations, although typically not well understood, have captured attention in many areas such as health psychology. One early study examined the effects of the social environment on the anatomy of the brain in an effort to find evidence for the kinds of changes in the brain as a result of experience demanded by learning theories. The experiments are described in Bennett, E. L., Diamond, M. C., Krech, D., & Rosenzweig, M. R. (1964). “Chemical and anatomical plasticity of the brain” Science 146, 610-619, and some of the raw data are presented in Freedman et al. (1998, p. 501). Pairs of male rats from a litter were used as subjects, with one member of each litter being chosen at random to be reared with other rats in an enriched environment, complete with playthings and novel areas to explore on a regular basis, whereas another member of the litter was randomly selected to be reared in isolation in a relatively deprived environment. Both groups were permitted to consume as much as they wanted of the same kinds of food and drink. After a month, the deprived environment animals were heavier and had heavier brains overall. Of critical interest though was the size of the cortex, or gray matter portion, of the brain in the two groups. The experiment was replicated a number of times. However, in the current exercise, we are considering the data from only one of the replications (labeled Experiment 3 in Freedman et al., 1998, p. 501). The weights of the cortex (in milligrams) for the pairs of experimental (enriched) and control (deprived) subjects are shown in the following table: Test for the effect of the treatment in this experiment by doing a randomization test. That is, perform a test of the hypothesis that the sum of the difference scores is no different than you would expect if the + and — signs had been assigned with probability .5 to the absolute values of the obtained difference scores. Although a large number of rerandomizations are possible with 12 pairs of subjects, the randomization test can be carried out here with even less computation than a t test by thinking a bit about the possibilities. To carry out the test, you should answer the following questions: a. What is the observed sum of differences here? b. How many assignments of signs to differences are possible? c. What proportion of these would result in a sum at least as large in absolute value as that observed? To answer this question, use the following approach: (i) What is the largest possible positive sum that could be achieved, given the observed absolute values of the differences? (ii) By considering how much this largest sum would be reduced by changing one or two of the signs of the absolute differences from positive to negative, determine which assignments of signs to differences would result in sums between (or equal to) the maximal sum and the observed sum. (iii) Considering the symmetry of the distribution of sums resulting from rerandomizations, what is the total number of sums as extreme or more extreme, either positive or negative, as the observed sum?


 

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