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Issues with ratios in statistical analysis in kinesiology

This is an excerpt from Statistics in Kinesiology 6th Edition by Joseph P. Weir,Anthony B. Ciccone,Jacob A. Siedlik,William J. Vincent.

In kinesiology, we routinely calculate ratios in our data analyses. Sometimes we wish to create a proportion where the ratio has the same units in the numerator and the denominator. For example, we could make a ratio of quadriceps muscle cross-sectional area (derived from ultrasound measures) before versus after a resistance training program. When we have a proportion, the units in the numerator and denominator cancel, and we can convert the values to a percentage by multiplying the ratio by 100. Alternatively, we can calculate a ratio to try to standardize or normalize the variable in the numerator to the variable in the denominator. For example, we could make a ratio of ultrasound-derived thigh muscle cross-sectional (units = cm²) area to body mass (units = kg). (Here, the units are not the same, so converting to a percentage is nonsensical.)

Recall in chapter 1 that of the levels of measurement (nominal, ordinal, interval, ratio), the highest level of measurement was the ratio scale. Data measured on a ratio scale have a zero value that indicates the absence of whatever it is being measured. When you have a meaningful zero, you can make interpretable ratios. For example, if my 1RM squat = 100 kilograms and your 1RM squat = 200 kilograms, we can correctly state that your 1RM squat is two times my 1RM squat. (In contrast, 100 °C is not two times as hot as 50 °C since 0 °C is not the absence of thermal energy.)

Certain measurements, such as velocity and force, are, by definition, ratios. Velocity is the change in position per unit of time and can be reported as meters per second (m/sec, or m ∙ sec⁻¹). If we are testing isometric strength, we can record the force output in Newtons. One Newton is equal to 1 kilogram ∙ meter ∙ second squared, such that 1 Newton equals the force required to accelerate 1 kilogram of mass by 1 meter per second squared. Further, in chapter 8, we covered bivariate regression; recall that the slope of the regression line is a ratio defined as the change in Y divided by the change in X.

Therefore, ratios can be central to our research projects. However, just because we can calculate a ratio does not mean that we should create a ratio. Ratios can create issues that may complicate our statistical analyses. Take, for example, a test of maximal oxygen consumption (V̇O₂max). When a participant performs a V̇O₂max test, we quantify the oxygen consumption in units of liters of oxygen consumed per minute (L/min). However, we know that body size affects how much oxygen one can consume per unit of time. Larger people, on average, have bigger lungs, more muscle mass, larger left ventricles, higher blood volumes and hemoglobin mass, and so on. It is common to divide oxygen consumption by body mass and then convert from liters to milliliters, resulting in units of milliliters of oxygen per kilogram of body mass per minute (mL O₂ ∙ kg⁻¹ ∙ min⁻¹). The division by body mass is an attempt to “normalize” or standardize the V̇O2 values to adjust for differences in body size and make “fair” comparisons between different people and groups. Attempting to normalize data via dividing a score by a body size variable is very common in kinesiological studies.

Caution is warranted, however. A classic paper by Tanner in 1949 entitled “Fallacy of per-weight and per-surface area standards, and their relation to spurious correlation” (Tanner, 1949) warned our field about the dangers of ratio scaling. Figure 19.1 shows the relationship between body mass (kg) and peak V̇O₂ (L/min) for a sample of 35 males of varying ages. The mean of the peak V̇O₂ values = 3.13 liters per minute, and the mean body mass = 82.2 kilograms. The ratio of the means = 3.13 liters per minute ÷ 82.2 kilograms ~0.038 liters per minute per kilogram. The regression line (solid line) has a Y-intercept = 1.64 liters per minute and the slope = 0.018 liter per minute per kilogram (95% CI = 0.006 to 0.03 L ∙ min⁻¹ ∙ kg⁻¹), r² = .22, p = .004. In contrast, the dashed line represents the fit to the data assuming a ratio standard. With ratio scaling, we assume the best fit line goes through the origin (intercept = 0) and is of the relationship: peak V̇O₂ (L/min) = 0.038 (L ∙ min⁻¹ ∙ kg⁻¹) × body mass (kg), where 0.038 is the ratio of the means of peak V̇O₂ and body mass. Notice that the 95% CI for the slope from the regression line excludes the ratio value of .038 (L ∙ min⁻¹ ∙ kg⁻¹). Researchers should keep in mind that ratio scaling will not necessarily “remove” the effect of the body size variable. Ratio scaling may result in a spurious correlation, where we report a statistical relationship between the variables where none truly exists, or it may obscure a relationship where one exists (Curran-Everett, 2013). Further, the magnitude of the true effect may be misestimated. In our example in figure 19.1, we noted that the ratio standard estimated the relationship between peak VO₂ as ~0.038 liter per minute per kilogram, while the more statistically appropriate regression-based estimate was = .018 liter per minute per kilogram. As a general rule, ratio scaling is appropriate only when the true relationship is linear, and the line of best fit goes through the origin (Curran-Everett, 2013). An alternative way of noting this is that if you were to logarithmically transform the X and Y data and conduct a regression analysis, and if the slope of the line of the log transformed data = 1.0, then ratio scaling would be appropriate (Atkinson and Batterham, 2012).

As a final comment, note that the data in figure 19.1 look somewhat curvilinear, and perhaps both ratio scaling and standard bivariate linear regression are not ideal approaches to model this relationship. An alternative approach to scaling for body size is called allometric scaling. Allometric scaling is beyond the scope of this book, but interested readers are referred to Nevill et al. (1992), Thompson et al. (2010), Vanderburgh et al. (1995), and Weir et al. (1999) for examples of kinesiology studies employing allometric scaling to human performance measures.

More Excerpts From Statistics in Kinesiology 6th Edition

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