# The Dose-Response Curve

An Important Feature of Linear Dose-Response Curves

By Bill Masters

*Wallace, Klor & Mann, P.C.*

Toxic tort litigation will invariably involve expert testimony about the relationship between the dose or level of exposure of the plaintiff or those similarly situated to a toxin and the physiological response to that toxic exposure. This relationship is schematized in what is called a “dose-response curve.” ^{1} When there is only one variable about dose, the schema is constructed in a two-dimensional rectangular Cartesian co-ordinate system. The “x” axis (abscissa) is the horizontal axis and on it is scaled the independent variable of the dose or exposure to the toxin; the “y” axis (ordinate) is the vertical axis and on it is scaled the dependent variable or physiological response to the dose.

The nature of relationship between the dose and the response is empirical.^{2} A series of doses or exposures to the toxin are identified and plotted on the coordinate system, and then the corresponding responses are identified and plotted. The relationship between the domain of doses and range of their correlated responses is established through “simple linear regression,” a statistical technique used to assess the relationship between a single explanatory variable and a single response variable that varies over a range of values. Regression seeks to use one variable (the independent variable) “to **predict”** another variable (the dependent variable). ^{3}

The relationship between the independent and dependent variables is “linear.” That is, the line graphed from one set of independent and dependent variables (coordinates) to another set is straight. The equation for this linear relationship is y = ax + b. ^{4} “Y” is the dependent variable; “x” is the independent variable. “a” is the “slope of the line” or the ratio of the amount of change in the dependent variable to the amount of change in one unit of the independent variable. When the line is linear, the slope is constant; that is, the increase in y is proportionate to an increase in x. And “b” is the y-intercept when x is 0.

The location of the y-intercept is important in dose-response curves. When there is no y-intercept, that is, when the line or curve bisects the origin (or vertex), there is no dose-response threshold. But if the y-intercept intersects the y axis above the origin when x is 0, then there is a dose-response threshold. In that event, some small doses of the toxin will not result in a measurable physical response. The defense prefers a dose-response curve with a threshold. Then, the defense can eliminate potential claims of injury where the dose was below the threshold. But the plaintiffs’ bar angles to eliminate thresholds. It seeks experts who will testify that there is no threshold, that there is no y-intercept and the curve bisects the origin.

Plaintiffs’ experts also seek to describe the dose-response curve vaguely. The more vague the description of the curve, the more difficult it is for the defense to pin down exactly what is being described other than that any exposure to the putative toxin results in a harmful response.

A case in point was an asbestos case involving one of the plaintiffs’ more persuasive experts in occupational and environmental medicine. He sought to establish, through simple linear regression, a predictive relationship between exposure to asbestos and the risk of lung cancer. He also sought to establish that as to this predictive relationship, there was no threshold. He testified, that is, that there was no y-intercept of the dose response curve, that the line bisected the origin of the co-ordinate system.

He next testified about how he *scaled* the x and y axes. He said that the x axis was scaled on the basis of three categories of exposure to asbestos: the first category represented “low exposure,” the second category “medium exposure” and the third category “high exposure.” The y axis was scaled on an index of the risk of lung cancer from exposure to asbestos (risk ratios or odds ratios). So, in this expert’s dose-response analysis, an exposure falling in the low category would result in a risk of lung cancer. Any such increased risk, of course, would be sufficient to persuade a jury to find for a minimally exposed plaintiff.

This all seems unassailable. But it is not! The reason, the expert has failed to attend carefully to the matter of how the x and y axes are scaled. He acknowledged that he had scaled the y axis on a “ratio scale” on the “real number line” so that the y axis represented a continuous range of values (no gaps between the numbers in the scale) for the dependent variable. He also acknowledged that he had scaled the x axis on the basis of rank or “ordinal” metric so that the independent variable could take only one of three values—low, medium or high. In short, plaintiff’s expert wished to create a predictive relationship, through simple linear regression, between an “ordinal” variable and a “continuous” variable.

At the “ordinal” level of measurement, things are assigned numbers such that the order of the numbers reflects an order relation defined on the attribute. ^{5} Two things x and y with attribute values a(x) and a(y) are assigned numbers m(x) and m(y) such that if m(x) > m(y), then a(x) > a(y). (An example is grades—A, B, C, D, F–for academic performance.) The permissible “transformation” of this scale is any “monotone increasing” transformation. A “transformation” generally is the process of changing all the variables in an equation by using some authorized mathematical operation. ^{6} A “monotone increasing transformation” is a function with domain (x) and co-domain (y) that are sets of real numbers such that the dependent variable increases or stays the same as the independent variable increases.

In simple linear regression, where the curve has a y intercept, both the dependent and independent variables must be on an “interval” or stronger scale (viz., ratio and absolute scales).^{7} At the interval level of measurement, things are assigned numbers such that differences between the numbers reflect differences of the attribute. ^{8} If m(x) – m(y) > m(u) – m(v), then a(x) – a(y) > a(u) – a(v). The permissible transformation is any “affine transformation” t(m) = c* m + d, where c and d are constants. (The origin and unit of measurement are arbitrary.) An “affine transformation” is a composition of a “translation” and a “linear transformation.” ^{9} A “translation,” in the plane, maps the point with Cartesian coordinates (x, y) onto the point (x + a, y + b) where a and b are constants. ^{10} A “linear transformation” is the process of changing a number, group of numbers or an equation by adding, subtracting, multiplying or dividing by a constant. ^{11}

When the y intercept is through the origin, both variables must be on a “ratio” or stronger scale (viz., the absolute scale). ^{12} At the ratio level of measurement, things are assigned numbers such that differences and ratios between the numbers reflect differences and ratios of the attribute. ^{13} A ratio scale has an absolute 0 point reflecting the complete absence of the attribute. The permissible transformations are any “linear” transformation t(m) = c* m, where c is a constant (the unit of measurement is arbitrary). [At the “absolute” level of measurement, things are assigned numbers such that all properties of the numbers reflect analogous properties of the attribute. The only permissible transformation is the identity transformation. ^{14}]

In this case, because the expert sought to have no y-intercept, that the curve bisect the origin, he needed both the dependent and independent variables to be on a ratio or stronger scale. He had the dependent variable on a ratio scale, but the independent variable was merely on an ordinal scale. So the expert failed to meet the requirements for simple linear regression curve with no y-intercept.

In that situation, no linear transformation of the curve is possible. “If we are estimating a parameter that lacks invariance or equivariance under permissible transformations, we are estimating a chimera.” ^{15}

**ENDNOTES**

1. M.O. Amdur et al. *Casarett and Doul’s Toxicology.* pps. 18-25 (4^{th} ed. 1991).

2. M.O. Amdur et al. *Casarett and Doul’s Toxicology.* pps. 18-25 (4^{th} ed. 1991).

3. T. Lang & M. Secic. *How to Report Statistics in Medicine*. p. 108 (1997).

4. T. Lang & M. Secic. *How to Report Statistics in Medicine*. p. 109 (1997).

5. W.S. Sarle. *Measurement Theory: Frequently Asked Questions in Disseminations of the International Statistical Applications Institute*. p. 3 (4^{th} ed 1996).

6. W. Paul Vogt. *Dictionary of Statistics and Methodology.* p. 234 (1993).

7. W.S. Sarle. *Measurement Theory: Frequently Asked Questions in Disseminations of the International Statistical Applications Institute*. p. 9 (4^{th} ed 1996).

8. W.S. Sarle. *Measurement Theory: Frequently Asked Questions in Disseminations of the International Statistical Applications Institute*. p. 4 (4^{th} ed 1996).

9. D. Nelson (ed.) *Dictionary of Mathematics.* p. 5 (Penguin 1998).

10. D. Nelson (ed.) *Dictionary of Mathematics.* p. 421 (Penguin 1998).

11. W. Paul Vogt. *Dictionary of Statistics and Methodology.* p. 130 (1993).

12. W.S. Sarle. ^{th} ed 1996); A.R. Feinstein, *Multivariable Analysis*. p. 7 (1996).

13. W.S. Sarle. ^{th} ed 1996).

14. W.S. Sarle. ^{th} ed 1996).

15. W.S. Sarle. ^{th} ed 1996).