Dose-response statistics

In 1996, we received a grant from the Texas Institute for Advanced Chemical Technology (TIACT) at Texas A M to assess whether the hormesis hypothesis was toxicologically credible. We set forth to make initial judgments on the existence of hormesis based on the conformity of published dose responses to the hormetic /3-curve (Figure 1). In order to assess this in an objective manner, we developed a priori criteria based on study designfeatures, quantitative characteristics of the dose response, statistical power, and reproducibility of experimental findings. These... 

Wong WK, Lachenbruch PA (1996) Tutorial in biostatistics. Designing studies for dose response. Statistics in Medicine 15 343-359. 

Lowcst-Obsciwcd-Advcrsc-El fcct-Lcvel (LOAEL) In dose-response experiments, the lowest exposure level at which there are statistically or biologically significant increases in frequency or severity of adverse effects between the exposed population and its appropriate control group. 

If the exposure level (E) exceeds tliis tlireshold (i.e., E/RfD exceeds unity), tliere may be concern for potential noncancer effects. As a rule, tlie greater tlie value of E/RfD above unity, tlie greater tlie level of concern. However, one should not interpret ratios of E/RfD as statistical probabilities a ratio of 0.001 does not mean tliat tliere is a one in one tliousand cliance of the effect occurring. Furtlier, it is important to empliasize tliat tlie level of concern does not increase linearly as tlie RfD is approached or exceeded because RfDs do not have equal accuracy or precision and are not based on tlie same severity of toxic effects. Thus, tlie slopes of the dose-response curv e in excess of the RfD can range widely depending on tlie substance. 

Uncertainty on tlie other hand, represents lack of knowledge about factors such as adverse effects or contaminant levels which may be reduced with additional study. Generally, risk assessments carry several categories of uncertainly, and each merits consideration. Measurement micertainty refers to tlie usual eiTor tliat accompanies scientific measurements—standard statistical teclmiques can often be used to express measurement micertainty. A substantial aniomit of uncertainty is often inlierent in enviromiiental sampling, and assessments should address tliese micertainties. There are likewise uncertainties associated with tlie use of scientific models, e.g., dose-response models, and models of environmental fate and transport. Evaluation of model uncertainty would consider tlie scientific basis for the model and available empirical validation. 

There are statistical procedures available to determine whether the data can be fit to a model of dose-response curves that are parallel with respect to slope and all share a common maximal response (see Chapter 11). In general, dose-response data can be fit to a three-parameter logistic equation of the form... 

This value is identified in F tables for the corresponding dfc and dfs. For example, for the data in Figure 11.13, F = 7.26 for df=6, 10. To be significant at the 95% level of confidence (5% chance that this F actually is not significant), the value of F for df = 6, 10 needs to be > 4.06. In this case, since F is greater than this value there is statistical validation for usage of the most complex model. The data should then be fit to a four-parameter logistic function to yield a dose-response curve. 

FIGURE 11.14 Data set consisting of a control dose-response curve and curves obtained in the presence of three concentrations of antagonist. Panel a curves fit to individual logistic functions (Equation 11.29) each to its own maximum, K value, and slope. Panel b curves fit to the average maximum of the individual curves (common maximum) and average slope of the curves (common n) with only K fit individually. The F value for the comparison of the two models is 2.4, df = 12,18. This value is not significant at the 95% level. Therefore, there is no statistical support for the hypothesis that the more complex model of individual maxima and slopes is required to fit the data. In this case, a set of curves with common maximum and slope can be used to fit these data. 

FIGURE 11.16 Control dose-response curve and curve obtained in the presence of a low concentration of antagonist. Panel a data points. Panel b data fit to a single dose-response curve. SSqs = 0.0377. Panel c data fit to two parallel dose-response curves of common maximum. SSqc = 0.0172. Calculation of F indicates that a statistically significant improvement in the fit was obtained by using the complex model (two curves F = 4.17, df=7, 9). Therefore, the data indicate that the antagonist had an effect at this concentration. 

There are statistical methods to determine the verisimilitude of experimental data to models. One major procedure to do this is nonlinear curve fitting to dose-response curves predicted by receptor models. 

The squared deviations between the calculated and actual responses are shown in Table 12.6c (see column labeled SSq). The AIC values are calculated according to Equation 11.30. The values are shown in Table 12.6c. It can be seen that the fit to the curves with a mean Emax and slope gives a lower AIC value. Therefore, this model is statistically preferable. It is also the most unambiguous model for simple competitive antagonism since it fulfills the criteria of parallel dextral displacement of dose-response curves with no diminution of maxima. The calculated curves are shown in Figure 12.7b. 

Log normal distribution, the distribution of a sample that is normal only when plotted on a logarithmic scale. The most prevalent cases in pharmacology refer to drug potencies (agonist and/or antagonist) that are estimated from semilogarithmic dose-response curves. All parametric statistical tests on these must be performed on their logarithmic counterparts, specifically their expression as a value on the p scale (-log values) see Chapter 1.11.2. 

Benchmark Dose Model—A statistical dose-response model applied to either experimental toxicological or epidemiological data to calculate a BMD. 

Experimental exposure studies have attempted to associate various neurological effects in humans with specific trichloroethylene exposure levels. Voluntary exposures of 1 hours resulted in complaints of drowsiness at 27 ppm and headache at 81 ppm (Nomiyama and Nomiyama 1977). These are very low exposure levels, but the results are questionable because of the use of only three test subjects per dose, lack of statistical analysis, sporadic occurrence of the effects, lack of clear dose-response relationships, and discrepancies between the text and summary table in the report. Therefore, this study is not presented in Table 2-1. No effects on visual perception, two-point discrimination, blood pressure, pulse rate, or respiration rate were observed at any vapor concentration in this study. Other neurobehavioral tests were not performed, and the subjects were not evaluated following exposure. 

In 1981, Lawson et al.,87 for example, compared a group of 210 women hospitalized for fibrocystic disease with 241 women who had breast cancer and were drawn from two ongoing studies in different countries. They matched each case to three female control patients on age, current smoking habits, country, and study. Recent coffee and tea consumption in cases and controls were compared and were shown to have a modest positive association with hot beverage consumption for both fibrocystic disease and breast cancer, but there was no dose-response relationship. The risk of fibrocystic disease associated with heavy consumption of hot beverages (7+ cups per day) vs. none was elevated but not statistically significant. 

The following example is based on a risk assessment of di(2-ethylhexyl) phthalate (DEHP) performed by Arthur D. Little. The experimental dose-response data upon which the extrapolation is based are presented in Table II. DEHP was shown to produce a statistically significant increase in hepatocellular carcinoma when added to the diet of laboratory mice (14). Equivalent human doses were calculated using the methods described earlier, and the response was then extrapolated downward using each of the three models selected. The results of this extrapolation are shown in Table III for a range of human exposure levels from ten micrograms to one hundred milligrams per day. The risk is expressed as the number of excess lifetime cancers expected per million exposed population. 

Basketter, D.A., et al., A comparison of statistical approaches to the derivation of EC3 values from local lymph node assay dose responses, J. Appl. Toxicol., 19, 261, 1999. 

---