As we have seen, the coefficient of an equation estimated using OLS regression analysis provides an estimate of the slope of a straight line that is assumed to be the relationship between the dependent variable and at least one independent variable. In Kinesiology, the slope tells us the magnitude of the impact of a one-unit change in the X variable (such as minutes of treadmill exercise) upon the value of the Y variable (such as milliliters of oxygen consumed), measured in the units of the Y variable. As we saw in the case of dummy variables, this can show up as a parallel shift in the estimated line—for example, a difference in baseline metabolism between biological sexes—or even a change in the slope of the line through an interactive variable. Here we wish to explore the concept of elasticity and how we can use regression analysis to estimate the various "sensitivities" in which Public Health researchers have an interest.
The concept of elasticity is borrowed from engineering and physics where it is used to measure a material’s responsiveness to a force, typically a physical force such as a stretching/pulling force. It is from here that we get the term an “elastic” band. In Public Health, the "force" in question is often a behavioral or environmental change, such as a change in program duration or community funding. Elasticity is measured as a percentage change/response. The value of measuring in percentage terms is that the units of measurement (e.g., kilograms vs. pounds, or steps vs. meters) do not play a role in the value, allowing direct comparison between different health interventions.
As an example, if the daily sodium intake of a patient increased say 500mg from an initial level of 3,000mg and generated an increase in systolic blood pressure from 120mmHg to 122mmHg, we calculate the elasticity. The "sodium elasticity" of blood pressure is the percentage change in pressure resulting from some percentage change in sodium. A 16.7% increase in sodium generated only a 1.6% increase in blood pressure: 16.7% stimulus change → 1.6% response change. This is an inelastic response, meaning a small relative response to the stimulus change. This might occur because the body has strong homeostatic mechanisms to regulate pressure. Conversely, some physiological markers are highly sensitive; for instance, a small percentage change in blood glucose can induce a large percentage change in insulin secretion. This would be considered an elastic response.
While this discussion has been about physiological changes, any of the independent variables in a public health model will have an associated elasticity. Thus, there is a participation elasticity that measures the sensitivity of health outcomes to changes in gym attendance: it might be low for general wellness but very sensitive for weight loss in controlled studies. If a model contains a term for competing behaviors, say "time spent swimming" in a model for "time spent running," the responsiveness of one to the other can be measured. This is called the cross-stimulus elasticity.
The mathematical formulae for various elasticities are:
\[\text {elasticity: } \eta_{\mathrm{p}}=\dfrac{(\% \Delta \mathrm{Q})}{(\% \Delta \mathrm{P})}\]
Where η is the Greek small case letter eta used to designate elasticity. Δ is read as "change". In our context, Q is the health outcome and P is the stimulus/intervention.
\[\text {elasticity:} \eta_{\mathrm{Y}}=\dfrac{(\% \Delta \mathrm{Q})}{(\% \Delta \mathrm{Y})}\]
Where Y is used as the symbol for the resource (like community health funding).
\[\text { Cross-behavior elasticity: } \eta_{\mathrm{p} 1}=\dfrac{\left(\% \Delta \mathrm{Q}_1\right)}{\left(\% \Delta \mathrm{P}_2\right)}\]
Where P2 is the "intensity" of a substitute or competing activity.
Examining closer the stimulus elasticity, we can write the formula as:
\[\eta_{\mathrm{p}}=\dfrac{(\% \Delta \mathrm{Q})}{(\% \Delta \mathrm{P})}=\dfrac{\mathrm{dQ}}{\mathrm{dP}}\left(\dfrac{\mathrm{P}}{\mathrm{Q}}\right)=\mathrm{b}\left(\dfrac{\mathrm{P}}{\mathrm{Q}}\right)\]
Where b is the estimated coefficient for the stimulus in the OLS regression.
The first form of the equation demonstrates the principle that elasticities are measured in percentage terms. Of course, the ordinary least squares coefficients provide an estimate of the impact of a unit change in the independent variable, X, on the dependent variable measured in units of Y. These coefficients are not elasticities, however; they are the derivative of the estimated function, which is simply the slope of the regression line. Multiplying the slope by QP provides an elasticity measured in percentage terms.
Along a straight-line regression curve, the percentage change, and thus elasticity, changes continuously as the scale changes, while the slope (the estimated regression coefficient) remains constant. In order to provide a meaningful estimate of the elasticity of a health intervention, the convention is to estimate the elasticity at the point of means. Remember that all OLS regression lines will go through the point of means. The formula to estimate an elasticity when an OLS model has been estimated becomes:
\[\eta_{\mathrm{p}}=\mathrm{b}\left(\dfrac{\overline{\mathrm{P}}}{\overline{\mathrm{Q}}}\right)\]
Where \(\overline{\mathrm{P}}\) and \(\overline{\mathrm{Q}}\) are the mean values of the data used to estimate b, the intervention coefficient. The same method can be used to estimate other elasticities, such as the sensitivity of patient recovery to mean age or mean protein intake.
Logarithmic Transformation of the Data
Ordinary least squares estimates typically assume that the population relationship among the variables is linear, thus following the form presented in the standard regression equation. In this form, the interpretation of the coefficients is straightforward: the coefficient provides an estimate of the impact of a one-unit change in X on Y, measured in the units of Y. It does not matter where along the line you make the measurement because a straight line has a constant slope, meaning a constant estimated level of impact per unit change. It may be, however, that the kinesiologist or public health analyst wishes to estimate not the simple unit-measured impact, but the magnitude of t percentage impact on the Y variable of a one-unit change in the X variable.
To summarize, there are four cases:
1. Unit \(\Delta X \rightarrow\) Unit \(\Delta Y\) (Standard OLS case)
2. Unit \(\Delta X \rightarrow \% \Delta Y\)
3. \(\% \Delta \mathrm{X} \rightarrow\) Unit \(\Delta \mathrm{Y}\)
4. \(\% \Delta \mathrm{X} \rightarrow \% \Delta \mathrm{Y}\) (elasticity case)
Case 1: The ordinary least squares case begins with the linear model developed above:
\[Y=a+b X\]
where the coefficient of the independent variable \(b=\dfrac{\mathrm{d} Y}{\mathrm{~d} X}\) is the slope of a straight line and thus measures the impact of a unit change in \(X\) on \(Y\) measured in units of \(Y\).
Case 2: The underlying estimated equation is:
\[\log (\mathrm{Y})=a+b X\]
The equation is estimated by converting the \(Y\) values to logarithms and using OLS techniques to estimate the coefficient of the \(X\) variable, \(b\). This is called a semi-log estimation. Again, differentiating both sides of the equation allows us to develop the interpretation of the X coefficient b :
\[\mathrm{d}\left(\log _{\mathrm{Y}}\right)=b \mathrm{~d} X\]
\[\dfrac{\mathrm{d} Y}{Y}=b \mathrm{~d} X\]
Multiply by 100 to covert to percentages and rearranging terms gives:
\[100 b=\dfrac{\% \Delta \mathrm{Y}}{\text { Unit } \Delta \mathrm{X}}\]
\(100 b\) is thus the percentage change in \(Y\) resulting from a unit change in \(X\).
Case 3: In this case, the question is: "What is the unit change in Y resulting from a percentage change in X?" For example, a clinical manager might ask: "What is the total reduction in systolic blood pressure (measured in mmHg) if we increase the duration of a patient's weekly aerobic exercise by five percent?" or "What is the impact on mean hospital readmission rates (measured in number of patients) if a public health initiative receives a ten percent increase in funding?" The estimated equation for this case would be:
\[Y=a+B \log (X)\]
Here the calculus differential of the estimated equation is:
\[\begin{array}{c}
\mathrm{d} Y=b \mathrm{~d}(\log X) \\
\mathrm{d} Y=b \dfrac{\mathrm{~d} X}{X}
\end{array}\]
Divide by 100 to get percentage and rearranging terms gives:
\[\dfrac{b}{100}=\dfrac{\mathrm{d} Y}{100 \dfrac{\mathrm{~d} X}{X}}=\dfrac{\text { Unit } \Delta \mathrm{Y}}{\% \Delta \mathrm{X}}\]
Therefore, \(\dfrac{b}{100}\) is the increase in Y measured in units from a one percent increase in X .
Case 4: This is the elasticity case where both the dependent and independent variables are converted to logs before the OLS estimation. This is known as the log-log case or double log case, and provides us with direct estimates of the elasticities of the independent variables. The estimated equation is:
\[\log Y=a+b \log X\]
Differentiating we have:
\[\begin{aligned}
\mathrm{d}(\log Y) & =b \mathrm{~d}(\log X) \\
\mathrm{d}(\log Y) & =b \dfrac{1}{X} \mathrm{~d} X
\end{aligned}\]
thus:
\[\dfrac{1}{Y} \mathrm{~d} Y=b \dfrac{1}{X} \mathrm{~d} X \quad \text { OR } \quad \dfrac{\mathrm{d} Y}{Y}=b \dfrac{\mathrm{~d} X}{X} \quad \text { OR } \quad b=\dfrac{\mathrm{d} Y}{\mathrm{~d} X}\left(\dfrac{X}{Y}\right)\]
and \(b=\dfrac{\% \Delta \mathrm{Y}}{\% \Delta \mathrm{X}}\) our definition of elasticity. We conclude that we can directly estimate the elasticity of a variable through double log transformation of the data. The estimated coefficient is the elasticity. It is common to use double log transformation of all variables in the estimation of demand functions to get estimates of all the various elasticities of the demand curve.