This is a solution to the first STA 250 assignment for Fall 2000 (PS, PDF) on the "s99testa" data set. Note: Links like the one in the previous sentence are to two version, one in Postscript format, the other in PDF format.
To start, I read in the original data, and gave names to all the variables. I then produced stem-and-leaf plots of ex-amount, food-amount, weight100, weight200, and lifespan.
The stemplot for food-amount showed an obvious recording error, in which one mouse (ID 876) supposedly ate 11.48 grams of food per day, even though they were allowed only 6 grams of food per day. I changed this value to "*" to indicate that its correct value is unknown. The stemplot for weight200 shows also shows an obvious error, with one mouse (ID 663) supposedly weighing only 7.4 grams at age 200 days, even though this mouse weighed 35.5 grams at age 100 days. Since 7.4 grams is also much less than any other mouse weighed, this must be a recording error. I replaced it by "*" as well.
The stemplot for lifespan shows that its distribution is bimodal, with some mice dying before age 400 days, but others surviving much longer.
I also looked at side-by-side boxplots of food-amount versus food-allowed (PS, PDF), after correcting the erroneous value mentioned above. There are no other points where the amount of food eaten was greater than the amount allowed. The amount of food eaten generally increases with the amount allowed, as one would expect, but the amount eaten is not much different when 8 or 10 grams are allowed than when 6 grams are allowed. Perhaps the group allowed 10 grams of food (and maybe the 8 gram group) could have been omitted from the experiment, saving money.
Side-by-side boxplots of weight at age 100 for each sex (PS, PDF) show that the female mice tend to weigh more than the male mice.
Using Manip > Subset worksheet , I created a worksheet with only the 20 mice that were allowed exercise and the maximum amount of food (10 grams).
I then looked for a relationship of lifespan to amount of food eaten by producing a scatterplot with sex marked (PS, PDF). In these and later plots, "o"=male and "+"=female. Similarly, I looked for a relationship of lifespan to amount of exercise by looking at a scatterplot for these variables, with sex marked (PS, PDF).
These plots show no clear relationship of lifespan to amount of food eaten, but they do show a fairly strong negative relationship of lifespan to amount of exercise (r=-0.590). Mice who exercised at a level above about 550 lived very short lives, less than 300 days, whereas most mice who exercised less often lived to twice that age, though a few died early.
I made plots of the residuals from this regression against sex, food-amount, weight100, weight200, and the ratio of weight200 to weight100, but I did not see any clear dependence of the residuals on any of these other variables. Exercise amount was positively correlated with the amount of food eaten, but it was not strongly correlated with any other of the variables.
From this observational data, it appears that exercising reduces lifespan. However, we cannot rule out the possibility that the observed association is due to some lurking variable, rather than an actual cause and effect relationship. If so, however, the lurking variable responsible is probably not one that has been observed.
Since the variables controlled by the experimenters were whether or not exercise was allowed and the amount of food allowed, I first looked at side-by-side boxplots of lifespan for groups differing in the values of one or the other of these variables. From the plot of lifespan versus ex-allowed (PS, PDF), it seems that exercise has little or no effect on lifespan. The plot of lifespan versus food-allowed (PS, PDF) also shows no clear relationship.
However, perhaps some relationships might be apparent looking at the mice who exercised and those who didn't exercise separately. To do this, I created two subset worksheets, one containing data on the 100 mice allowed exercise, the other containing data on the 100 mice not allowed exercise. I then produced side-by-side boxplots of lifespan versus food-allowed for the exercise and non-exercise groups.
For the mice not allowed exercise, the plot of lifespan versus food-allowed (PS, PDF) again shows no clear pattern. The median varies by a large amount, but not in any consistent way. For mice allowed to exercise, the plot of lifespan versus food-allowed (PS, PDF) is less erratic, and perhaps shows that lifespan is greater when the amount of food allowed is small, though it's hard to be sure this isn't just due to chance.
Since the stemplot of lifespan showed that its distribution for all mice is bimodal, perhaps boxplots and medians aren't the best way of looking at these distributions. Also, since mice who are bigger may need more food, looking at the ratio of the amount of food allowed to the weight at age 100 days (which I called fall/w100) may be better than looking at food-allowed itself. I produced a scatterplot of lifespan versus fall/w100 with ex-allowed marked (PS, PDF), and also a scatterplot of the same variables but with sex marked (PS, PDF).
Two groups of mice are clearly visible in these plots. Mice in one group lived more than 500 days; those in the other group lived less than 400 days. Very few mice fell in between, living from 400 to 500 days. Whether exercise is allowed seems to have little or no effect. Sex also seems to have little or no effect, except that the mice for which fall/w100 is large tend to be males, because males tend to be smaller. The amount of food allowed per gram of weight at age 100 days appeared to have a small effect on lifespan. There is some indication that mice in the longer-lived group lived longer when fed no more than 0.15 grams of food per gram of weight at age 100 days. The four longest-lived mice were in this category. There is also a slight indication that feeding more the 0.3 grams of food per gram of weight at age 100 reduces lifespan, since none of the mice fed this much lived longer than 700 days.
To investigate better how food affects the long-lived and short-lived mice, I created a worksheet with just the 104 mice who lived longer than 500 days, and another worksheet with just the 90 mice who lived less than 400 days. (The six mice who lived between 400 and 500 days were in neither of these worksheets.) For the short-lived mice, a regression of lifespan on fall/w100 shows no relationship at all (PS, PDF). For the long-lived mice, there is a definite, though rather weak relationship (PS, PDF), with more food resulting in shorter lifespan. Plots for the long-lived mice of the ratio of weight at age 200 to the weight at age 100 (w200/w100) versus fall/w100 (PS, PDF) and of lifespan versus w200/w100 (PS, PDF), both with ex-allowed marked, indicate that perhaps at least part of the reason for the relationship of lifespan to food allowed (for long-lived mice) is that when allowed more food, the mice tend to gain weight, and this leads them to live less long. However, a plot of the residuals of the regression of lifespan on w200/w100 (for long-lived mice) against fall/w100 (PS, PDF) indicates that perhaps allowing only a small amount of food (less than 0.15 grams per gram of weight at age 100) increases lifespan beyond what one would expect from just the effect of this restricted diet on weight. The effect is small, however, and might just be due to chance.
To see better how exercise is related to lifespan, I plotted lifespan versus ex-amount for the mice allowed exercise, marking the points by the amount of food allowed (PS, PDF). The two groups of mice are again apparent. Mice in the short-lived group tend to exercise more than mice in the long-lived group. Within the long-lived group, mice seem to exercise more if they are allowed to eat more food. The combined effect of this is that exercise amount is negatively correlated with lifespan, but this correlation seems to be due to lurking variables, not due to any actual effect of exercise on lifespan.
There appear to be two types of mice, one type that is long-lived (more than 500 days), another type that is short-lived (less than 400 days). Perhaps these mice differ genetically, or perhaps they differ in how stressful they find living in a cage. When given the opportunity, the short-lived mice tend to exercise a bit more than the long-lived mice, which leads to a negative correlation between exercise and lifespan, even though exercise does not appear to actually cause a shorter lifespan.
For the long-lived mice, restricting the amount of food they are allowed to eat increases lifespan a bit, probably in part because it reduces the weight of the mice, but perhaps partly for some other reason as well.