This is a solution to the first STA 250 assignment for Fall 2000 (PS, PDF) on the "s99testg" 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 that one mouse (examination of the data show's it's the one with ID 749) supposedly ate an average of 15.61 grams of food per day. Since this mouse was only allowed 8 grams of food per day, and since even the mice allowed 10 grams per day didn't eat more than 8 grams on average, this must be an error. I changed this value to "*" to indicate that its correct value is unknown.
The stemplot for weight200 showed one unusually low point - a value of 10.0 (for the mouse with ID 703). This point also appears very strange in a scatterplot of weight200 versus weight100 (PS, PDF). Either the weight at age 200 recorded for this mouse is an error, or this is a very unusual mouse. It seems more likely that the value is an error however, since it seems quite unlikely that a mouse would drop in weight from 36.1 grams to 10.0 grams, when it is fed 5 grams of food per day (more than the lowest food group, who didn't lose that much weight). It also seems unlikely that this mouse lost weight due to extreme sickness, since it lived for 572 days, which is not an unusually short lifespan. If the value for weight at age 200 is correct, it seems that the weight loss must have been due to some strange illness or other strange circumstance, which does not relate to the purpose of this experiment. If the mouse really was this strange, it might be best to ignore it completely, but since it seems more likely that this was just a recording error, I replaced the value of 10.0 for weight200 for this mouse by "*" to indicate that this mouse should be ignored in plots or regressions that involve the weight200 variable, but still looked at when the weight200 variable isn't relevant.
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 weight 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). From these plots, it appears that eating more food is associated with longer lifespan (except for one mouse), and that exercising more is also associated with longer lifespan. It also seems that female mice live longer than male mice. Female mice also tend to eat more, which isn't surprising given that we saw earlier that the females generally weigh more at the start of the experiment.
Interpreting these relationships requires care, because food-amount and ex-amount are themselves correlated (r=+0.583). To investigate these relationships further, I did a regression of lifespan on exercise amount (PS, PDF). The relationship of lifespan to exercise amount is fairly strong (R-squared of 38%). I plotted the residual of this regression versus amount of food eaten (PS, PDF). The lack of any relationship in this residual plot shows that once ex-amount is known, food-amount provides littel or no additional information about lifespan. In contrast, when I did a regression of lifespan on food amount (PS, PDF), the relationship was weaker (R-squared of 13%), and a plot of the residuals of this regression against ex-amount (PS, PDF) shows an apparent upward trend, which if real would indicate that knowing how much exercise the mouse did helps predict lifespan even if you already know how much food they ate.
These results are consistent with the idea that exercise increases lifespan, but consuming more food does not in itself increase lifespan. The mice who exercised more did tend to eat more, however, which is not surprising.
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 lifespan is increased when exercise is allowed. The plot of lifespan versus food-allowed (PS, PDF) shows a lower lifespan in the group allowed only 4 grams of food per day, but there is no clear pattern beyond that.
To get a clearer idea of how food-allowed influences lifespan, 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) shows an increase in lifespan as the amount of food allowed goes from 4 grams per day to 5 grams per day, but a decrease in lifespan as the amount of food allowed is increased beyond 5 grams per day. For mice allowed to exercise, lifespan increases with the amount of food allowed (PS, PDF), though the increase levels off past about 6 grams per day. Comparing these two sets of boxplots, it seems that the greatest benefit of exercise is for the mice allowed 6 or more grams of food per day. There seems to be a lot of variability in lifespan for mice allowed only 4 grams of food per day, as seen from the large inter-quartile ranges in both the plot for mice allowed exercise and the plot for those not allowed exercise.
Smaller mice presumably need less food than larger mice. I therefore computed the ratio of the amount of food allowed to the weight at age 100 days (I called this variable fall/w100), and plotted lifespan versus this variable, for the mice allowed exercise (PS, PDF), and those those not allowed exercise (PS, PDF), with sex marked. From these plots it is clear that with or without exercise allowed, mice that are allowed less than 0.1 grams of food per day per gram of weight at age 100 days tend to die quite early. This plot also shows a tendency for the female mice to live longer (provided the amount of food is adequate).
One possible explanation for these relationships is that 4 grams of food per day is not really adequate (at least for the bigger mice), leading to a short lifespan due to malnutrition, whereas 6 or more grams of food per day is enough to shorten lifespan due to obesity, if the mice are not allowed to work off the extra food intake by exercise.
To see if this is the real explanation, I first looked at side-by-side boxplots of weight at age 200 days versus the amount of food allowed for the mice not allowed exercise (PS, PDF) and for the mice that were allowed exercise (PS, PDF). (Note that mice who died before age 200 are not included in these plots.) The plots confirm that weight tends to be reduced by both restricting the amount of food allowed and by allowing exercise.
Next, I looked at whether weight actually has an effect on lifespan, by plotting lifespan versus weight at age 200 days (PS, PDF). In this plot, mice allowed exercise are marked with "+", those not allowed exercise with "o". It seems from this plot that mice who weigh between about 30 and 40 grams live longer than those who weigh more or less than this. The mice who weigh less may die from malnutrition, which explains the early deaths when famt/w100 is small, and also at least partly explains the bimodal distribution of lifespan and the large variability in lifespan when food is restricted to only 4 grams per day (since for some mice, this is adequate, but for others it is not). On the other hand, the mice who weighed more than 40 grams may have had a shorter lifespan due to obesity.
However, it also appears from this plot that exercise may have a benefit beyond preventing obesity, since the mice allowed exercise seem to live longer than mice of the same weight who were not allowed exercise. To investigate this further, I plotted lifespan versus exercise amount, for those mice allowed exercise, with the amount of food allowed indicted by the plotting symbol (PS, PDF). A strong relationship is apparent, except for the mice allowed only 4 grams of food. Lifespan generally goes up as exercise increases, though there were a few mice who died fairly early even though they exercised a fair amount. On its own, this plot wouldn't be strong evidence that exercise causes an increases in lifespan, since it could instead be that naturally healthier mice both live longer and exercise more. However, in combination with the plot discussed in the previous paragraph, in which whether the mice were allowed to exercise or not was determined randomly by the experimenter, there is fairly good reason to think that exercise does increase lifespan, when the amount of food is adequate.
Looking at the same plot of lifespan versus exercise amount, but this time with sex marked (PS, PDF), we can see that long-lived female mice tend to live longer than long-lived male mice, even when they exercise the same amount. However, for mice who die young, either due to malnutrition or other reasons, there seems to be no advantage to being female.
The observational data showed an increase in lifespan with exercise amount. The experimental data provides some evidence that this increase does represent an actual benefit of exercise.
The observational data also showed a positive correlation between amount of food eaten and lifespan, but this appeared to be just a consequence of mice eating more when they exercise more, and not a direct effect of the amount of food eaten on lifespan. The full experiment included mice who where not allowed to exercise, and for these mice, the amount of food they are allowed to eat has a strong effect on lifespan, with mice who are allowed to each a lot living less long. Restricting the amount of food to only 4 grams per day also leads to some mice dying of malnutrition. These effects weren't seen in the observational data because they are not seen when the mice are allowed to eat and to exercise as much as they want.