Introduction: In 1976 Allison and Cicchetti studied the sleep patterns of mammalian species, looking for correlations with a variety of constitutional and ecological factors to better understand “the significance of sleep in the life of mammals.” The ecological variables examined included among others: 1) predation index (pred), 2) sleep exposure index (exp, and 3) danger (danger). These were assigned based on professional opinions, and are explained in Variables on page 2. The constitutional variables included: 1) brain weight in grams (brain), 2) maximum life span in years (span), and 3) gestation time in days (gest). The data were collected from a number of cited sources published previously.
For this paper, which is based on the Allison and Cicchetti data, each of the 42 mammalian species was assigned to one of two groups according to their sleep habits. Species were classed as light sleepers if they slept less than 10 hours per day (n=22), while those sleeping more than 10 hours per day (n=20) were classified as sleepers. The two groups were compared according to the variables described above as well as a seventh variable: percent of sleep time spent dreaming (dream), a ratio derived from the Allison and Cicchetti data on a specie’s hours of paradoxical (dream) sleep to total hours of sleep. This analysis determined if heavy and light sleepers differ in terms of these variables, and quantified and described those differences when found.
Exploratory Analysis: Distributions for the three constitutional variables (brain weight, lifespan, and gestation) were positively skewed (Figure 1), and summary statistics showed great differences between light and heavy sleepers (Table 1). Brain weight, in particular, showed a severe positive skew in light sleepers as well as vast differences in summary measures (med1 = 142 IQR1 = 276.45, Median2 = 3.95 IQR2 = 9.275). There were also two extreme outliers in the light sleeper brain weight data. In each case, log transformations not only brought the distributions closer to normal and to each other, but also brought intra-group variations closer together and removed most outliers so that t-tests could be applied.
The distributions of the categorical ecological data were also generally not normal, with at least one group in each variable lacking at least one tail.(Figure 1) Variance was similar in groups for predation and danger (Table 1), but the heavy and light sleepers differed in their exposure measures (mean1 = 3.0, SD1 = 1.662, mean2 = 1.65, SD2 = .9880) Permutation tests were used here to explore differences between groups primarily because the original data were categorical and subjective and therefore not very descriptive, and because the distributions deviated so grossly from normal and from each other. The great number of “ties” in rank transformations made the rank sum test useless in these cases.
The distributions of the final variable, dream (mean1=.1736 SD1=.0957, mean2=.1815 SD2=.0876), were slightly skewed, and in one case contained an outlier (Figure 1). As logit transformation did not greatly improve the distributions, and as it would also have delivered results of limited inferential utility, a t-test was performed on the data in natural scale.
Findings: (Table 2) There is sufficient evidence to reject the null hypotheses in tests of each of the constitutional variables. The log transformations made in these cases permit comparison of the population medians of light and heavy sleepers. Median brain weight in light sleepers was estimated to be 15.75 times that of heavy sleepers (one sided p=.0001). The lower bound of a one-sided 95% CI for this estimate was 5.253. Median lifespan for light sleepers was estimated to be 2.36 times that of heavy sleepers (one-sided p=.0024), with a lower one-sided 95% CI for the ration of 1.46. Median gestation periods for light sleepers were estimated to be 2.871 times those of heavy sleepers (one-sided p=.0003) with a lower 95% CI of 1.771.
The analyses failed to detect differences between light and heavy sleepers with respect to the population probability distributions of the ecological variables, predation (one-sided p = .1597), exposure (one-sided p=.6926), and danger (one-sided p =.4353). The tests also failed to detect differences in the dream variable (two-sided p =.7833).
Scope of Inference: These findings suggest that light sleepers tend to have larger brains, to live longer, and to have longer gestation periods than heavy sleepers. No differences were detected between light and heavy sleepers with respect to any of the other variables. As the data were observational, the results only indicate associations between sleep and brain weight, life span, or gestation period. It is possible that the three are closely correlated with one another, and sorting out the effects of one variable from those of another is beyond the scope of this study. Although a wide variety of mammalian species are represented in this study, it would be inappropriate to assume that they were chosen at random. Many of the animals were probably chosen for reasons such as ease of study. In addition between 10 and 12 (24-29%) of the species listed are domesticated animals, and domestication could affect sleep in predictable ways. Inferences to mammals in general based on a non-random sample so heavily biased toward domestication would have very limited value. Future investigations might render more generalizable results by using a randomly chosen sample not so heavily biased by domesticated animals.
Mark Mobius
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Variables:
Ecological: Values for these variables were assigned by the Allison and Cicchetti.
1) Predation index (pred) - rates a species from 1(low) to 5(high) based on the extent to which it was preyed upon.
2) Sleep exposure index (exp) - measures from 1 to 5 how exposed to danger or predation a specie’s place of sleep is.
3) Danger (danger) - a general measure of specie’s overall risk when sleeping. It incorporates factors such as low predation rate despite highly exposed sleep location.
Constitutional:
1) Brain weight in grams (brain)
2) Maximum life span in years (span)
3) Gestation time in days (gest)
Table 1 - Summary Statistics:
Variable / Group / Important statistics / Number of OutliersBrain / Light Sleepers (group1) N=22 / Median = 142 IQR = 276.45 / 2
Heavy Sleepers (group2) N=20 / Median = 3.95 IQR = 9.275 / 1
Span / Light (group1) / Median = 23 IQR = 30.825 / 1
Heavy (group2) / Median = 6.75 IQR = 10.15 / 0
Gest / Light (group1) / Median = 172 IQR = 201.75 / 1
Heavy (group2) / Median = 44 IQR = 93.5 / 0
Pred / Light (group1) / Mean = 3.183 SD = 1.593
Median = 3 / 0
Heavy (group2) / Mean = 2.70 SD = 1.261
Median = 2 / 0
Exp / Light (group1) / Mean = 3.0 SD = 1.662
Median = 3 / 0
Heavy (group2) / Mean = 1.65 SD = .9880
Median = 1.0 / 1
Danger / Light (group1) / Mean = 3.136 SD = 1.5211
Median = 3.5 / 0
Heavy (group2) / Mean = 2.20 SD = 1.0563
Median = 2 / 0
Dream / Light (group1) / Mean = .1736 SD = .0957 / 1
Heavy (group2) / Mean = .1815 SD = .0876 / 0
Table 2 - Tests and Results
VariableName / Hypothesis Tested (HA)
0 = light sleepers, 1 = heavy sleepers / Test Used / Test Stat. / p-value / 95% Confidence
Interval **
Brain / mean(log(brain0))-mean(log(brain1)) > 0 / 1 sided t-test / t = 4.228 / .0001 / (1.658891, NA)1
Span / Mean(log(span0))-mean(log(span1)) > 0 / 1 sided t-test / t = 2.993 / .0024 / (.3762095, NA)2
Gest / Mean(log(gest0))-mean(log(gest1))> 0 / 1 sided t-test / t = 3.674 / .0003 / (.5712874, NA)3
Pred / mean pred1 – mean pred2 0 / Permutation / .-.35 / .1597 / NA
Exp / mean exp1 – mean exp2 0 / Permutation / .11 / .6926 / NA
Danger / mean danger1 – mean danger2 0 / Permutation / -.15 / .4353 / NA
Dream / mean(dream0) – mean(dream1) 0 / 2 sided t-test / t = -.2768 / .7833 / (-.0652, .04954)
** - Where one sided t-tests were done, only single sided confidence intervals are reported.
1. 95% CI for ratio of population median brain0 to population median brain1 (5.253482, NA)
2. 95% CI for ratio of population median span0 to population median span1 (1.456752, NA)
3. 95% CI for ratio of population median gest0 to population median gest1 (1.770545, NA)
Mark Mobius
Page 3 of 4
Figure 1.
Box Plots of Data in Natural Scale (0 = light sleepers 1 = heavy sleepers)
Mark Mobius
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Supporting Material:
Estimates and Confidence Intervals for log transformations:
Brian:
Mean(log(brain1))-mean(log(brain2)) = difference in mean(log(brain))
4.09711 - 1.340336 = 2.756774, and to relate to natural scale, e2.756774 = 15.74895 = ratio med0:med1
Lower CI for difference in log(brain) = 1.658891 and relating to natural scale, e1.658891 = 5.253482
Span:
Estimate: Mean(log(span1))-mean(log(span2)) = difference in mean(log(span))
2.892611 – 2.032333 = .860278, and to relate to natural scale, e.860278 = 2.36382 = ratio med0:med1
see Table 2 for CI.
Gest:
Estimate: Mean(log(gest1))-mean(log(gest2)) = difference in mean(log(gest))
4.876722 – 3.822066 = 1.054656, and to relate to natural scale, e1.054656 = 2.87099 = ratio med0:med1
see Table 2 for CI.
In support of decision to use non-parametric tests for Ecological variable:
Examples of QQ normal plots for Ecological variables
Note in Exp – Heavy Sleeper that the median = 1st quartile = minimum value. Other distributions in this group showed similar (though to lesser degrees) deviations from normal and differences in center and variation. Common transformations, like log, would not correct these distributions due to the great number of duplicate values. This duplication would also lead to unacceptably high numbers of ties in a rank sum test.
Mark Mobius
Page 3 of 4