Evolutionary biologists are often interested in “form-function relationships” to help explain the evolutionary history of say an animal species. Harris and Steudel (2002) investigated factors that are related to the jumping ability of domestic cats.
Because jump ability and height are largely dependent on takeoff velocity, several traits were recorded for 18 healthy adult cats such as relative limb length, relative extensor muscle mass, body mass, fat mass relative to lean body mass, and the percentage of fast-twitch muscle fibers to see which might best explain maximum takeoff velocity (based on high-speed videos).
JMP output for velocity: boxplot and histogram of takeoff velocity from 280 to 420 cm/sec, with Quantiles (median 339.55, quartiles 316.175 and 369.625, maximum 410.8, minimum 286.3) and Summary Statistics (mean 343.28, standard deviation 33.08, n = 18).
R iscamsummary output for cats velocity: n = 18, minimum 286.3, Q1 317.25, median 339.55, Q3 368.4, maximum 410.8, mean 343.28, SD 33.077, skewness 0.222.
R histogram of takeoff velocity with bins from 280 to 420 cm/sec; tallest bar between 300 and 320, roughly mound-shaped with single observations in the 280, 380, and 400 bins.
The distribution is pretty symmetric or slightly skewed to the right (though depending on bin choices might see a visual outlier in the 420 bin). Most cats had a velocity around 320-380 cm/sec but one was as small as 286.30 cm/sec and the high outlier was at 410.80 cm/sec.
Do you think there will be a relationship between a cat’s takeoff velocity and its body mass? If so, do you think heavier cats will tend to have larger or smaller takeoff velocities than lighter cats?
Describe the relationship between a cat’s takeoff velocity and its body mass, as displayed in this scatterplot. Does this pattern confirm your expectation in Question 4?
Do any of these cats appear to be outliers in the sense that its pair of values (body mass, takeoff velocity) does not fit the pattern of the majority of cats? If so, identify the ID for that cat and describe what’s different about this cat (in context).
Scatterplots are useful for displaying the relationship between two quantitative variables. If one variable has been defined as the response variable and the other as the explanatory variable, we will put the response variable on the vertical axis and the explanatory variable along the horizontal axis.
The above scatterplot reveals a fairly strong, negative association between body mass and takeoff velocity, meaning that heavier cats tend to have a smaller takeoff velocity than lighter cats. The relationship is somewhat linear but has a bit of a curved pattern. There is one outlier cat (cat C) with a very high takeoff velocity despite having a very large body mass.
Now produce a scatterplot of takeoff velocity vs. percentage of body fat. Describe the association. Would you say that the association with velocity is stronger than with body mass? More or less linear?
For the other two variables (hind limb length and muscle mass), would you expect to see a positive or negative association with takeoff velocity? Explain. Then look at scatterplots, and comment on whether the association is as you expected.
There are only 5 males but they tend to be heavier than the females. When the body mass is similar the velocities are similar though a little higher for males. The larger outlier is a male and there is also a very large male in terms of body mass.
These researchers reported that variation in cat maximum takeoff velocity was significantly explained by both hind limb length (cats with longer limbs tended to have higher takeoff velocities) and fat mass relative to lean body mass (cats with lower fat mass tended to have higher takeoff velocities), but not to extensor muscle mass relative to lean mass or fast-twitch fiber content. They explained the “pervasive effect” of body mass by the increase in muscle work invested in increasing the center of mass potential energy as compared with kinetic energy during takeoff.
Later in this chapter you will learn how they determined the statistical significance of these relationships. First, we will examine a numerical measure of the strength of the association between two variables.
The dataset KYDerby26.txt contains information on each running of the Kentucky Derby since 1875. The speeds of the winning horses have been calculated (taking into account the change in track length in 1896).
