What Is the Numerical Nature of Pain Relief?

Abstract

Pain relief, or a decrease in self-reported pain intensity, is frequently the primary outcome of pain clinical trials. Investigators commonly report pain relief in one of two ways: using raw units (additive) or using percentage units (multiplicative). However, additive and multiplicative scales have different assumptions and are incompatible with one another. In this work, we describe the assumptions and corollaries of additive and multiplicative models of pain relief to illuminate the issue from statistical and clinical perspectives. First, we explain the math underlying each model and illustrate these points using simulations, for which readers are assumed to have an understanding of linear regression. Next, we connect this math to clinical interpretations, stressing the importance of statistical models that accurately represent the underlying data; for example, how using percent pain relief can mislead clinicians if the data are actually additive. These theoretical discussions are supported by empirical data from four longitudinal studies of patients with subacute and chronic pain. Finally, we discuss self-reported pain intensity as a measurement construct, including its philosophical limitations and how clinical pain differs from acute pain measured during psychophysics experiments. This work has broad implications for clinical pain research, ranging from statistical modeling of trial data to the use of minimal clinically important differences and patient-clinician communication.

Keywords: ANCOVA; clinical trials; pain; statistical models; treatment effects.

Conflict of interest statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2021 Vigotsky, Tiwari, Griffith and Apkarian.

Figures

Figure 1

Graphical illustration of the hierarchical model from which patients' pain scores are sampled. The broad yellow (light gray) distribution is the between-patient distribution (level 2), from which each patient's mean pain score is sampled. Each red (dark gray) distribution is a within-patient distribution (level 1), from which single measurements are sampled.

Figure 2

Properties of additive and multiplicative data. We simulated data with additive (left) and multiplicative (right) assumptions. (A) Relationships between pre- and post-intervention pain scores when improvements are additive (left) and multiplicative (right). Note the additive post-intervention scores are relatively homoscedastic, while the variance of multiplicative post-intervention scores increases with increasing pre-intervention scores. (B) Negative relationships between change scores and pre-intervention scores. Gray areas in (B) represent regions where points are not possible due to measurement constraints; that is, because a change score cannot be >|100|.

Figure 3

Simulations of additive and multiplicative changes reveal the effect of different intraclass correlation coefficients on the slope between change scores and pre-intervention scores. Additive effects have slopes that trend toward zero with increasing ICC's, while multiplicative effects always have a negative slope no matter their ICC.

Figure 4

Simulations of additive and multiplicative changes reveal differential residual behavior for raw and log-transformed ANCOVA models. (Left) data generated with have an additive structure have homoscedastic residuals when fit with a standard ANCOVA (top) but heteroscedastic residuals when fit with a log-transformed ANCOVA (bottom). (Right) data generated with a multiplicative structure have heteroscedastic residuals when fit on their raw scale (top) but homoscedastic residuals when log-transformed (bottom).

Figure 5

Relationships between pre-intervention scores and change scores (top) and post-intervention scores (bottom). (Top) Relationship between pre-intervention scores and change scores. Note that most of the studies have a negative relationship. This could be explained by regression toward the mean or multiplicative effects, in addition to ceiling/floor effects. (Bottom) Relationship between pre-intervention and post-intervention pain scores across all studies. Each study shows a positive relationship between pre- and post-intervention scores; however, the Levodopa study appears to have greater variance in post-intervention scores with greater pre-intervention scores.

Figure 6

Increasing the number of points used for each patient's pre- and post-intervention scores increases the slope between change scores and pre-intervention scores. Each patient's pre- and post-intervention scores were calculated using the mean of x points. By averaging over more points, we should increase the intraclass correlation coefficient. Negative slopes between change scores and pre-intervention scores are indicative of one of two things: (1) regression toward the mean or (2) multiplicative effects. In the datasets that show evidence of being additive, we see marked increases in slopes, indicating that we are decreasing regression toward the mean by including more points. However, because the Levodopa Trial displays multiplicative properties, it is only minimally affected by adding more points.

Figure 7

Absolute values of residuals from additive ANCOVA models. We fit an ANCOVA to each dataset using pre-intervention score and group membership as covariates. From these models, we plotted the absolute values of the residuals as a function of the fitted value. Additive models should be homoscedastic, meaning the magnitudes of the residuals do not change as a function of the response variable. However, multiplicative models have compounding error, such that if you fit them using an additive model, greater predicted values will be associated with larger magnitudes of residual error. Placebo I, Placebo II, and the Prospective Cohort study all exhibit features of additive data. However, the Levodopa Trial exhibits multiplicative properties, as evidenced by the increasing error residual magnitude with increasing fitted values.