Take Home Midterm Exam 2 Martin Kaltenpoth
Sleeping habits in mammals
Although every mammalian species studied in the laboratory spends at least some time asleep, the biological function of sleep is still not completely understood. This study tries to shed some light on the significance of sleep in the life of mammals. Therefore, 42 species of mammals were divided into two groups according to their sleeping habits: light sleepers spend usually less than 10 hours per day sleeping, heavy sleepers sleep more than 10 hours per day. For every species, three constitutional variables (brain weight, life span, and gestation time) and the percent of time dreaming were measured. Three ecological variables (predation index, exposure index, and danger index), were rated according to a five-point scale on the basis of field observations in the literature. However, it is important to notice that the danger index is neither independent of the other two indices, nor is it simply a linear combination of them.
The study addresses the question whether these seven characteristics differ significantly for the two groups with different sleeping habits. Since there is no apparent reason to assume that constitutional variables and percent of time dreaming take on higher values for one of the two groups, two-sided hypothesis-tests were performed for these variables. However, species that are subject to heavy predation, sleep in more exposed places or seem to face a higher overall danger should tend to be light sleepers in order to reduce their risk of being preyed upon. Therefore, one-sided tests were performed for the ecological variables to address the question whether – on average – these indices take on higher values for light sleepers.
Because the sample distributions for all three constitutional variables were skewed and showed larger variance in the group with the larger median, log-transformations were tried to achieve approximately normal distributions. Box plots show that the log-transformed data for life span (Fig.2) and gestation time (Fig.3) are approximately normally distributed and the two groups have almost equal sample variances, whereas the variances of the groups in the log-transformed brain data (Fig.1) seem to be different. But since the distributions for both groups look approximately normal and the sample sizes are nearly equal and sufficiently large, the two-sample t-tools are assumed to be applicable despite unequal standard deviations. Summary statistics for the original and the transformed data are given in Table 1.
To test the claim that median brain weight, median life span and median gestation time differ in light and heavy sleepers, two-sided two-sample t-tests were performed on the log-transformed data. The results of the t-tests indicate highly significant differences in group medians for all three variables (brain: p=0.0001, life span: p=0.0047, gestation time: p=0.0007). The median brain weight, life span and gestation time of light sleepers were estimated to be 15.8, 2.4 and 2.9 times as high as those of heavy sleepers, respectively (for confidence limits see Table 2, Table 3 App.).
The ecological variables consisted of indices on a five-point scale and produced ordinal data with distributions that differed clearly from normal distributions and showed many ties (Fig.4-6). Therefore, one-sided permutation tests were performed on the differences in group means (summary statistics see Table 1). The results indicated no significant difference in predation indices between the two groups (p=0.147, see Fig.8 App.) but significantly higher exposure and danger indices for light sleepers (exposure: p=0.003, danger: p=0.022, see Table 2, Fig.9-10 App.).
The data for the relative time spent dreaming (summary statistics given in Table 1) were arcsin-square root transformed for variance stabilization and because the distributions on the original scale were skewed for both groups (Fig.11 App.). After transformation, the data of the light sleepers still showed a mildly skewed distribution with one outlier (Echidna: 0% dreaming) but seemed to be sufficiently close to normal distributions for applying the two-sample t-tools (Fig.7). A two-sided two-sample t-test revealed no significant difference in the median relative time spent dreaming in light and heavy sleepers, regardless whether the test was performed with or without the outlier in the light sleepers group (p=0.648 with outlier, p=0.973 without outlier, see Table 2).
The statistical tests performed on the data of this study lead to the conclusion that median brain weight, life span, gestation time, exposure index and danger index are significantly higher in light than in heavy sleepers for the investigated animal species whereas predation index and relative time spent dreaming do not differ significantly. However, since the study is observational no causal inferences can be drawn from this conclusion. But concerning the ecological variables, there is obviously an evolutionary advantage in being a light sleeper for animals that sleep at exposed places or face a high overall danger of being preyed upon, so the conclusions are consistent with the theory suggesting that different selective pressures because of predation risk lead to different sleeping habits. In this context, it is surprising that no significant difference in predation indices of light and heavy sleepers could be found in this study.
The conclusions can be generalized to other mammalian species, if one assumes that the species studied represent a simple random sample of all mammals, which need not necessarily be the case. A problem with this study is that it does not consider phylogenetic relationships between the animal species. Some species are closely related to each other, so the claim that these data are independent has to be considered with caution.
Table 1: Summary statistics for the seven variables before and after transformation.
Table 2: Hypotheses and statistical tests for the seven variables.
Take Home Midterm Exam 2 Martin Kaltenpoth
Fig.1: Brain weights in light (n=22) and heavy (n=20) sleepers (log-transf.). / Fig.2: Life span in light (n=22) and heavy (n=20) sleepers (log-transf.).Fig.3: Gestation time in light (n=22) and heavy (n=20) sleepers (log-transf.). / Fig.4: Relative time spent dreaming in light (n=22) and heavy (n=20) sleepers (arcsin-square root transf.).
Fig.5: Predation indices for light (n=22) and heavy (n=20) sleepers. / Fig.6: Exposure indices for light (n=22) and heavy (n=20) sleepers.
Fig.7: Overall danger indices for light (n=22) and heavy (n=20) sleepers.
Take Home Midterm Exam 2 Martin Kaltenpoth
Appendix
Table 3: Difference in means and confidence limits of the log-transformed data, and back-transformed values for brain weight, life span, and gestation time.
Fig.8-10: Permutation distributions of differences in means for the ecological variables.
Fig.8: Permutation distribution of differences in mean predation indices (light-heavy) based on 1000 resamplings. The black point indicates the observed difference of 0.48. / Fig.9: Permutation distribution of differences in mean exposure indices (light-heavy) based on 1000 resamplings. The black point indicates the observed difference of 1.35.Fig.10: Permutation distribution of differences in mean danger indices (light-heavy) based on 1000 resamplings. The black point indicates the observed difference of 0.94.