Causal Analytics for Applied Risk Analysis Louis Anthony Cox, Jr

Fig. 4.3. Plot of hydroquinone (HQ), catechol (CA), and muconic acid (MA) concentrations (µm) vs. air benzene (AB) (ppm) for all exposed workers in Factories 1 and 2. The plot does not clearly suggest a nonlinear or supra-linear relationship between benzene and these metabolites for AB between 0 and 3 ppm.


Table 4.3 gives the numerical values for the data points in Figures 4.2 and 4.3, showing the mean values of each of the benzene metabolites (and also creatinine (CT), air toluene (AT), and urinary toluene (UT), and the number of person-days of data, N) for each level of air benzene concentration, 0-5, among workers in Factory 1 and Factory 2. The “PH” column in Table 4.3 corresponds to the left plot in Figure 4.2, and the column “UB” corresponds to the right plot.
Table 4.3. Mean values of each benzene metabolite and creatinine (CT), air toluene (AT), and urinary toluene (UT), and number of person-days of data, N, for each level of air benzene concentration (AB), 0-5, for workers in Factories 1 and 2.

AB rounded to nearest ppm	N	UB Means		MA Means		SPMA Means		PH Means		CA Means			HQ Means		Creat Means		AT Means		UT Means
0			42		205.9		6.5		0.1		99.7	14.7		11.3		1.5		18.8		78.7
1			144		409.5		12.6		0.2		152.7	20.7		17.6		1.4		14.5		108.3
2			51		687.0		24.9		0.6		196.6	24.7		30.2		1.2		12.9		84.7
3			17		1353.2		33.2		1.3		269.7	26.0		39.6		1.3		16.1		113.2
4			17		1103.3		27.2		1.0		217.3	29.2		38.5		1.3		10.9		143.4
5			18		2033.6		35.0		0.9		271.7	35.4		44.1		1.2		8.5		62.8
All Grps			289		626.3		17.3		0.4		170.8	22.3		23.1		1.4		14.2		99.3

Since the modeling work of Kim et al. (2006 a and b) specifically addressed non-smoking women, it is natural to wonder whether Figures 4.2 and 4.3 may be obscuring a true nonlinearity in the metabolism of this sub-population by averaging over all exposed workers. Figure 4.4 shows analogous plots specifically for non-smoking women. In this sub-population, there is again no evidence of supra-linear metabolism at low doses.

Fig. 4.4. Plots of phenol (left) and hydroquinone (HQ), catechol (CA), and muconic acid (MA) concentrations (µm) (right) vs. air benzene (AB) (ppm) for non-smoking women in Factories 1 and 2. These plots do not show clear nonlinear or supra-linear relationship between benzene and these metabolites for AB between 0 and 3 ppm.


Since almost all observations at these low concentrations come from Factory 1, focusing on the population of non-smoking women specifically in Factory 1 leaves these plots almost unchanged.

Rappaport et al. (2009) hypothesized saturation of metabolism and supra-linearity of dose-response relationships specifically below 1 ppm of benzene in air as a conjectured mechanism whereby leukemia risks in the general population might have been underestimated. Accordingly, Figure 4.5 focuses on metabolites for air benzene exposures below 1 ppm, down to the lowest recorded levels. The horizontal axis now has increased resolution, with concentrations rounded to the nearest 10^th of a ppm. Even with this sharpened focus, there is no evidence for the conjectured saturation and nonlinear (supra-linear) metabolism in this range.

Fig. 4.5. Plots of phenol (left) and hydroquinone (HQ), catechol (CA), and muconic acid (MA) concentrations (µm) (right) vs. air benzene (AB) for workers in Factories 1 and 2 exposed to < 1 ppm. These plots do not show a clear nonlinear or supra-linear relationship between benzene and these metabolites for AB between 0 and 1 ppm.

The interaction plots in Figures 4.2-4.5 do not provide reason to reject the null hypothesis of a linear relationship between benzene metabolite concentrations in urine and benzene concentrations in air of 5 ppm or less, but neither do they provide a quantitative test of the hypothesis of linearity. To help close this gap, Table 4.4 quantifies the Pearson product-moment linear correlations between air benzene and its metabolites over the whole range of observed values. (The occasional missing values were pair-wise deleted; sensitivity analyses showed that case-wise deletion or imputation made little difference.)

Table 4.4. Linear correlations between benzene and its biomarkers

Variable
		AB		UB		MA		SPMA		PH		CA		HQ
AB		1.000		0.56		0.77		0.66		0.71		0.68		0.70
UB		0.557		1.00		0.54		0.69		0.53		0.51		0.50
MA		0.766		0.54		1.00		0.72		0.94		0.90		0.95
SPMA		0.659		0.69		0.72		1.00		0.76		0.78		0.68
PH		0.711		0.53		0.94		0.76		1.00		0.96		0.89
CA		0.684		0.51		0.90		0.78		0.96		1.00		0.86
HQ		0.700		0.50		0.95		0.68		0.89		0.86		1.00
Creat		-0.043		0.08		0.08		0.06		0.11		0.11		0.10
AT		0.049		-0.02		-0.02		0.01		0.02		0.01		-0.01
UT		-0.089		0.03		0.04		0.05		0.05		0.08		0.04

All correlations greater than 0.08 in Table 4.4 are statistically significantly greater than zero (p < 0.05). The high correlation coefficients suggest that linear relationships between air benzene and metabolite concentrations and between MA and other metabolites fit the data well. Spearman rank correlation coefficients between AB and PH, CA, and HQ are 0.61, 0.60, and 0.68, respectively, indicating that non-linear increasing relationships between AB and these metabolites do not provide a better description of the data by this criterion than straight-line relationships. Benzene metabolite concentrations are even more strongly correlated with each other than with air benzene (e.g., the correlation is 89% between phenol and the phenolic metabolite hydroquinone, HQ; 94% between phenol and CA; and 96% between phenol and MA), suggesting that these metabolites increase roughly linearly with phenol (PH) and with each other, as predicted by the hypothesis of low-dose linear metabolism.

Inter-individual variability and Declining DSM ratios
We have arrived at an apparent puzzle: although recent articless have repeatedly claimed disproportionately large dangers from benzene metabolites at low concentrations of benzene, stating that the Tianjin data “provide extremely strong statistical evidence” of a putative high-affinity enzyme active primarily below 1 ppm (Rappaport et al., 2010) and show “highly nonlinear” benzene metabolism with increasing benzene exposure above 0.03 ppm (Kim et al., 2006b), and, more specifically, disproportionately large production of benzene metabolites at these relatively low exposure concentrations (Rappaport et al., 2009), the raw data in Figures 4.2-4.5 show no such phenomenon. We see no highly nonlinear metabolism or saturation below 3 ppm (Kim et al., 2006b) or below 1 ppm (Rappaport et al., 2009) in these data. This section attempts to resolve the paradox.

A key point is that, unlike the direct plots of metabolite concentrations vs. air benzene in Figures 4.2-4.5, the analyses by the Berkeley team focused on the ratios of metabolites to air benzene at different levels of air benzene. This amounts to studying how a ratio varies as its denominator increases. Thus, for example, Rappaport et al. (2009) note that “Intriguingly, the exposure-specific production of major metabolites (phenol, muconic acid, hydroquinone, and catechol, in micromolar per parts per million benzene) decreased continuously with estimated exposure levels over the range of 0.03–88.9 ppm, with the most pronounced decreases occurring at benzene concentrations < 1 ppm” and Kim et al. (2006b) wrote that “Mean trends of dose-specific levels (micromol/L/ppm benzene) of E,E-muconic acid, phenol, hydroquinone, and catechol all decreased with increasing benzene exposure, with an overall 9-fold reduction of total metabolites. Surprisingly, about 90% of the reductions in dose-specific levels occurred below about 3 ppm for each major metabolite.” Both findings are commenting on the fact that a ratio decreases as its denominator increases. But this is not surprising: it is a consequence of the algebra of random variables that, even if Y is directly proportional to X, the ratio Y/X may still be a decreasing function of X (rather than a constant), with the steepest decline occurring for the smallest values of X, if there is substantial variance in the Y values for any X value, as is the case in the Tianjin data for benzene metabolites. It is therefore a mistake to interpret a declining ratio of Y/X as X increases as a sign of an intriguing or surprising low-dose nonlinear underlying toxicological mechanism if this is just what should be expected from linear metabolism with substantial variance in measured metabolite concentrations for each X level.

Figure 4.6 shows scatter plots of the concentrations of major metabolites of benzene against air benzene concentrations of 10 ppm or less on the left side, and corresponding DSM ratios of metabolite concentrations per ppm of air benzene vs. ppm of air benzene on the right, with phenol in the top panels and CA, HQ, and MA in the bottom panels. To aid visualization, a non-parametric regression (lowess) curve is shown for each scatter plot.
Fig. 4.6. Plots of phenol (PH) vs. air benzene (AB) (upper left) and PH/AB DSM ratio vs. AB (upper right) for workers in Factories 1 and 2 exposed to < 10 ppm. The declining DSM ratios on the right are compatible with approximately linear metabolism on the left; they reflect the arithmetic fact that small denominators are associated with large ratios and large denominators are associated with small ratios.




In agreement with Kim et al. (2006b) and Rappaport et al. (2009, 2010), we observe that the DSM ratios on the right side of Figure 4.6 decline as air benzene increases, with the majority of the decline taking place at air benzene concentrations below 1 ppm. However, the left side of Figure 4.6 shows that this decline does not correspond to any strongly nonlinear metabolism at low concentrations of air benzene. Rather, the downward slopes of the right-side DSM plots are consequences of the fact that metabolite concentrations are distributed with substantial variance and skew around their air benzene exposure concentration-dependent means (or medians, geometric means, etc.), so that the ratios of these random variables are negatively correlated with their denominators. As cautioned by Liermann et al. (2004), “Ratio data, observations in which one random value is divided by another random value, present unique analytical challenges. …[S]everal authors have pointed out that interpreting results of analyses based on ratio data can be non-intuitive, potentially leading to unintended inference and incorrect conclusions… They advocate reformulating the hypothesis in terms of the numerator, using the denominator as an explanatory variable.” The left-side plots in Figure 4.6 accomplish this reformulation.

To further clarify the logic of ratios of random variables, Figure 4.7 presents results from an artificial data set with 1000 simulated data points. Here, the true relationship between exposure concentration X and metabolite concentration Y is known exactly: it is specified to be purely linear, Y = 10X, as X ranges from 0 to 10 ppm and Y ranges from 0 to 100. However, there is a relatively modest error variance in the measured values of Y, much less than in the realistic data in Figure 4.6 but sufficient to illustrate methodological points. It is described by addition of an error term uniformly distributed between -10 and 10, but truncated to prevent negative concentration values. Negative values of Y are assumed to be impossible, so any negative value due to random error is rounded up to 0; this imparts an upward bias to errors, as is probably realistic with real-world measurements that allow large positive values but no negative values. The left side of Figure 4.7 shows this simulated data set; a nonlinear regression (lowess) curve fit to the data provides an excellent approximation to the true linear relationship, E(Y | X) = 10X. The right side of Figure 4.7 shows how the ratio Y/X varies with X. Even though this simulated data set has been constructed so that metabolism is exactly linear, the DSM ratio on the right is declining, with most of the decline occurring below 1 ppm. It is not necessary to postulate a hidden high-affinity enzyme or other toxicological mechanism to explain this pattern: it is a consequence of the algebra of random variables and of the assumed error model, rather than of toxicological mechanisms. Thus, contrary to the inferences drawn by the Berkeley group, a declining DSM ratio does not necessarily provide evidence against the null hypothesis of linear metabolism at low doses: the two are entirely compatible both in principle (Figure 4.7) and in practice (Figure 4.6).