Math 11, Lab 3

Lab 3: Metabolic rates in mammals (Regression and transformations)

In this lab, you will investigate the relationship between body mass and basal metabolic rate in mammals using linear regression.

The Data
First, open the data set MAMMALS, which is available in TritonEd.

The data were assembled from a variety of sources and used in the study [C. R. White and R. S. Seymour (2003). Mammalian basal metabolic rate is proportional to body mass^2/3. Proceedings of the National Academy of Sciences. 100, 4046-4049]. (Here is a link to the article. The data appear in the Supporting Table, but this table contains some errors and a corrected data set was provided by Craig White.) Only mammals with a body mass of less than 18,000 grams are included in the data set MAMMALS. The data set contains 599 rows, one for each mammal, and the following four columns:

Variable Name	Description
Species	The name of the species
Order	The order to which the species belongs (for example, humans belong to the order Primates)
Mass	Body mass in grams
BMR	Basal metabolic rate (a measure of the rate at which an organism uses energy while at rest, determined by measuring oxygen intake), in milliliters of oxygen per hour

Regression of basal metabolic rate against body mass
Your goal for this lab is to determine a good way of predicting the basal metabolic rate of mammals, given the body mass. A natural place to start is with linear regression. Make a scatterplot with the body mass as the X-variable and the basal metabolic rate as the Y-variable. Recall that you can do this by going to Graph --> Scatterplot, and that Minitab will draw the regression line on the plot if you click on "With Regression". Find the equation of the regression line. Recall that you can perform linear regression by going to Stat --> Regression --> Regression --> Fit Regression Model. To make sure you get a residual plot with your output, click on "Graphs" before clicking "OK", then click in the box under "Residuals versus the variables" and select your X-variable, which for now is Mass. You may also find it useful to store the fitted values and the residuals in separate columns. You can do this by clicking on "Storage" and checking "Residuals" and "Fits". Now answer the following questions, providing plots when necessary to support your answers.

In Minitab Express, recall that you can find the equation of the regression line, and get a scatterplot at the same time, by going to Statistics --> Regression --> Simple Regression. To get the residual plot as well, click on "Graphs" before clicking "OK", then check the box "Residuals versus the variables" and select your X-variable, which for now is Mass. Minitab Express does not provide the option of storing the residuals and fitted values in separate columns. However, once you have done a regression, you can go to Statistics --> Regression --> Predict for Regression. Then enter the value for the body mass for which you want to predict the basal metabolic rate. To find the prediction, look in the output at the "Prediction" chart under "Fit". Of course, you can then compute the residual by hand from the fitted value and the actual value.

What is the equation of the regression line? Provide a scatterplot with the line included.
What does the regression line predict for the basal metabolic rate of the Tasmanian Devil (Sarcophilius harrisii, row 548 in the data set)? What about for the Eastern Chipmunk (Tamias striatus, row 509)? There are two ways to go about this. One is to use the equation of the regression line. The other is to get the fitted value directly from Minitab, as explained above. You should verify that you get the same answer both ways. How do your predictions compare with the observed values?
Judging from the scatterplot and the residual plot, do you think that your linear regression model is appropriate for predicting basal metabolic rate from body size? Explain your answer, and show the necessary plots.

Transforming the variables
When working with highly skewed data, or scatterplots that show curvature or an uneven spread around the regression line, it is often useful to transform the variables. For this data set, try taking logarithms of both variables. This will require defining two new variables, one for the log of body mass and one for the log of basal metabolic rate. To define a variable for the log of body mass, first go to Calc --> Calculator. Under "Functions", scroll down to "Natural log", click on it, and then click on the "Select" button in the lower right. Then click on "Mass" in the variables box and click on the "Select" button in the lower left. Type in a name for your new variable in the box at the top of the screen labeled "Store result in variable", and click "OK". Then do the same for basal metabolic rate. Now you can work with the new variables.

In Minitab Express, you can make these transformations by going to Data --> Formula. Under "Functions", scroll down to "Natural log" and double click on it. Then double click on the variable "Mass", so that you should see LN('Mass') in the box labeled "Expression". Then click "OK". Do the same for the basal metabolic rate. You will probably want to give the new variables better names than "Expression" and "Expression_1", which you can do by clicking on these column titles and typing in a new variable name.

Compare histograms of the two original variables with the histograms of the two new transformed variables. Does taking logarithms of the variables reduce or eliminate the skewness?
Find the equation of the regression line using the transformed variables.
Your regression line should be of the form

ln(BMR) = a + b*ln(Mass),
where a and b are constants that you estimated using regression. By taking exponentials of both sides, we get that this model is the same as

BMR = c*Mass^b,
where c = e^a. Consequently, the number b is sometimes called the allometric exponent. Interest in estimating b goes back more than a century. Rubner's paper [M. Rubner (1883). Üden einfluss der körpergrösse auf stoff- und kraftwechsel. Zeitschrift fur Biologie, 19, 536-562] suggested b = 2/3, while Kleiber's 1932 paper [M. Kleiber (1932). Body size and metabolism. Hilgardia. 6, 315-353] argued for b = 3/4. Do your results lend support to either of these two values?
Based on the scatterplot and residual plot, do you think that this regression model is appropriate? Which of the two models (this one, or the one that you considered in question 1) do you think is better? Explain your answers. (Hint: when making your residual plot, make sure to plot the residuals against the explanatory variable for this regression, which is the log of body mass.)
What does this model predict for the basal metabolic rates of the Tasmanian Devil and the Eastern Chipmunk? (Hint: make sure you are reporting a prediction for the basal metabolic rate and not for the logarithm of the basal metabolic rate). How do these predictions compare with the observed values? Are these predictions more accurate than those obtained from the first model?
The largest mammal is the Blue Whale, which has a typical body mass of around 100 million grams. What does your model predict for the basal metabolic rate of a Blue Whale? How much confidence do you have in this prediction?

Comparing different types of mammals
Your next task is to examine specific groups of mammals to determine whether the relationship between body mass and basal metabolic rate that you have found holds for these groups. The mammals in the data set are classified according to their order. Begin by examining the rodents (Order Rodentia). You will need to create a new worksheet that includes only the rodents. To do this, go to Data --> Subset Worksheet. Give your new worksheet a name, then click on the button that says "Condition" and type

Order = "Rodentia"
in the box, then click "OK" twice. You should get a new worksheet with 289 rows because there are 289 rodents in the data set. You can then make scatterplots and carry our linear regression using this smaller data set. Of course, the same procedure also works for other groups of mammals.
[Note: if you downloaded Minitab onto your personal Windows computer, this works slightly differently than it does in the AP&M labs. First go to Data --> Subset Worksheet as above. Under "How do you want to create a subset?", select "Use rows that match a condition". In the Column box, select the Order variable, and then check the appropriate option. Then click "OK".]

In Minitab Express, you will have to do this by highlighting the relevant rows in the data set and copying and pasting them into a new Minitab worksheet. Recall that you can make a new Minitab worksheet by going to File --> New.

Using the transformed variables, find an equation for predicting basal metabolic rate from body mass for rodents. As always, examine a scatterplot and residual plot to make sure linear regression is appropriate. Include in your report these plots and a brief discussion of whether or not linear regression is appropriate.
How does the allometric exponent that you estimate for rodents compare to the exponent that you estimated for all mammals combined?
Next, consider the primates (Order Primates). Answer the same questions that you answered for the rodents.
Finally, consider the insectivores (Order Insectivora). Answer the same questions that you answered for the rodents and the primates.
Write a paragraph consisting of several sentences summarizing your conclusions about the relationship between body mass and basal metabolic rate for mammals in general and for the three groups of mammals that you studied in more detail.