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

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 |

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

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

- 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.

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

In Minitab Express, you can make these transformations by going to

- 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*. Consequently, the number^{a}*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?

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

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

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

- 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.