The Use of Bayes Coefficients to Assess the Racial BiasHair Analysis Conjecture for Detection of Cocaine in Hair Samples
Tom Mieczkowski
Professor/Chair
Department of Criminology
University of South Florida
Tampa, Florida
Chris Sullivan
Assistant Professor
Department of Criminology
University of South Florida
Tampa, Florida
Michael Kruger
Statistical Analyst
C. S. Mott Center for Human Growth and Development
Wayne State University
Detroit, Michigan
Introduction and Background
 Data
 Findings
 Discussion and Conclusions
 References
Introduction and Background
For more than a decade, the allegation that hair analysis is “racially biased” in the identification of cocaine and its metabolites has persisted in the face of a lack of substantial empirical support. It has been argued, contrary to the bias claim, that assessment of cocaine hair assays based on large data sets has failed to show any significant racial bias when also accounting for selfreported use and when appropriate statistical analysis is done. Yet, in spite of these criticisms and a general rejection by the courts of the claim of race bias in hair analysis, the belief in the racial bias of hair analysis continues to surface.
It is not our intention to revisit in detail the totality of the literature and the series of publications bearing on this topic, which are quite extensive. This has been done elsewhere, and those interested in the history of this controversy are referred to that literature (see Cone and Joseph 1996; Henderson et al. 1993, 1996, 1998; Hoffman 1999; Joseph et al. 1996; Kelly et al. 2000; Kidwell 1996, 1999; Kidwell and Blank 1990, 1991; Mieczkowski 2000, 2001, 2003; Mieczkowski and Kruger 2001, 2007; Mieczkowski and Newel 1993, 1995, 1999; Schwartz 2001; Stepan 1982; Zuckerman 1990).
The racebias argument has taken two forms: one is based on largely uncontrolled clinical assay outcomes in which African Americans have been reported to test positive for cocaine more frequently than Caucasians in several small N studies. An analysis of those studies, we believe, calls into question any conclusions relating hair analysis to racial bias. The second basis for this argument is derived from laboratory animal studies in which blackpigmented animal fur has been found to retain cocaine at higher concentrations than nonpigmented fur. These studies, using mainly guinea pigs and rats, have suggested, by inference, that “darkhaired persons” may retain more cocaine in their hair than “lighthaired persons,” and therefore by further inference, that African Americans, being largely darkhaired, would be more likely to test positive for cocaine. Hence the conclusion deduced from this chain of premises is that the test is “biased.”
We have taken issue with this conclusion on several different levels. Our response to the reported bias findings has included the following arguments:
 There is no scientific consensus on the biological meaning of the terms used to describe race as applied to humans or even what aspects of race, if any, have a biological component. There are important sociological definitions, certainly, but the systematic biological dimensions attached to race, if any, are largely unknown.
 If such a biological dimension existed, it is unknown how these putative “racial characteristics” such as hair and dermal pigmentation are distributed. Not having knowledge of the differential distribution of these traits precludes any generalizations about racial difference. In spite of this, the concept of racial identity has been treated as though it were a welldefined set of biological attributes whose distribution is well documented. Also, within the analytic framework of these studies, race has been treated as an exclusive and exhaustive property—an assumption that is clearly erroneous (e.g., see Henderson et al. 1998).
 On the preanalytic side of this issue, across these studies, there is no uniformity in the protocols for preparing hair samples, making the results largely noncomparable because they use very different preparatory methods.
 The subjects constituting the samples in these studies are, at best, convenience samples. No attempt has been made to evaluate the race hypothesis with a view to the problem of generalization, although these studies have readily suggested generalized conclusions—in some cases, rather extreme ones. We note as well that the statistical procedures used, when they have been used, are often inappropriate given the data.
 Largescale epidemiological studies using statistical regression techniques, which account for the critical factor of reported drug consumption, have found that the hypothesized bias cannot be demonstrated as either statistically significant or as demonstrating an effect size of any clinical consequence. Furthermore, these studies have demonstrated a consistent equality of outcome by selfreported race when controlling for level and type of cocaine use.
 The term “bias,” we have argued, is a misnomer and, as applied to race in this context, is unnecessarily inflammatory. A term more accurately describing the alleged phenomenon is “effect.” Consequently, we point out that all kinds of biological drug tests (such as urine, blood and blood products, oral fluid, and sweat and hair matrices) have “effects”; that is, their outcomes are in some measure influenced by the base state or condition of the individual. Factors such as age, general health, gender, body weight, etc., have effects on many types of bioassays, but these are not labeled as “bias.” Such factors are either ignored because they are clinically unimportant or because their effect sizes are very small. If they are important, the interpretation of the assay values is typically adjusted to correct for this condition. Therefore, in any case where a bioassay needs to be modified in light of a controlling condition (e.g., age, body mass, etc.), a scalar adjustment is made.
In this paper, we extend the examination of the issue of putative race bias as applied to hair analysis. We compare the outcomes of urine assays, hair analysis, and selfreported cocaine use by selfreported racial identity. The basis of our study is a conjecture that the relationship between the various measures of drug use, both bioassays, and selfreported use will be logically consistent. Our hypothesis is that selfreported cocaine use will parallel the outcomes of both urine and hair assays for cocaine using current conventional workplacerelated drugtesting cutoffs (not limits of detection), regardless of race. We posit that departures from this consistency in a positive direction for hair assays—that is, that the hair assays are positive for cocaine at a higher rate than either selfreport or urinalysis—would be consistent with, though not conclusive of, a “racial bias” as that term has been used.
We employ a Bayesian approach in this analysis for several reasons. Earlier analyses have used regression techniques, so we are attempting to assess whether an alternative approach produces findings consistent with those published previously. Also, a Bayes approach does not require assumptions about parametric characteristics of the data as regression does. By calculating and comparing Bayes coefficients for four categorical criteria related to cocaine (hair assay; urine assay; shortterm, selfreported use; and longterm, selfreported use), we can compare the relative likelihood of a positive report on each of these items for categories commonly applied and defined as race. We seek to know whether these four measures are different for the racial groups we analyzed.
The hair and urine specimens and survey data used in this analysis were collected during the operation of the Pinellas County Drug Study, a drugmonitoring program conducted at the Pinellas County Jail from 1991 through 1993, as part of a National Institute of Justice grant examining the rate of drug use by persons arrested in Pinellas County, Florida. Approximately twice a year the Pinellas County Drug Study staff interviewed approximately 300 male and female arrestees at the booking stage of arrest. All interviews were anonymous and confidential, and no results were used as evidence for any charges or other action by legal officials.
The sample of arrestees for any one collection period was stratified into the following categories:
nondrug felonies (200 cases, approximately 64 percent); drug felonies (60 cases, approximately 20 percent); and driving under the influence of intoxicants (40 cases, approximately 16 percent). All prostitutionrelated arrestees who consented to be included were interviewed. The stratification plan reflected a desire to address local concerns regarding the use of drugs by persons arrested for prostitution and driving under the influence. The data analyzed here comprise a base of 1857 cases. This is a sample of adults who had recently been arrested and were held in detention while waiting for official processing. The arrestees were a minimum of 18 years of age (mean age of 30.3 years; standard deviation [SD] = 9.6). The detained persons were asked during their booking to participate as volunteers in the study. If they agreed (greater than 96 percent did agree), they were interviewed and asked to provide a urine and hair specimen. Of those asked, 92.9 percent provided a urine sample, and 84.9 percent provided a hair sample. There were no notable differences in rates of selfreported responses. For the 48hour time period, the African American nonresponse rate was 0.2 percent greater than the Caucasian nonresponse rate. For the 60day retrospective period, there was no difference in response rates between African Americans and Caucasians. All respondents were guaranteed confidentiality and immunity by the authorities for anything they revealed in the interview. No personal identification was recorded for any participants, and the specimens were coded so no retroactive linkage to a specific subject could be made.
Urinalysis Data and Hair Analysis Data
Urine samples were tested at a local National Institute on Drug Abuse (NIDA)certified urinalysis laboratory using enzymemultiplied immunoassay (Syva EMIT) technology (cocaine threshold, 300 ng/mL) with gas chromatographymass spectrometry (GCMS) confirmation testing (benzoylecognine threshold, 300 ng/mL). The hair was analyzed for cocaine by the Psychemedics Corporation, of Culver City, California, using radioimmunoassay (RIA) following the preparatory method and analytic protocol described by Baumgartner and Hill (1992). All positive hair samples were confirmed with GCMS. The hair samples ranged from 1 to 2 cm in length and consisted of 20 to 40 strands of hair, cut at the scalp by surgical scissors. The hair was subjected to an initial anhydrous isopropanol wash and three subsequent phosphate buffer washes. After the third washing in buffer, the hair was digested by a proteinase enzyme at a neutral pH followed by centrifugation and pellet removal. Each wash and final hair digest was assayed by RIA. A complete technical description of the Psychemedics sample assessment, preparation, and confirmation procedure has been published elsewhere (Baumgartner et al. 1995). The criteria required to consider a specimen cocainepositive are discussed in detail in Baumgartner and Hill (1992). We report the laboratory values in measured concentration units of ng/mg for hair and ng/ml for washes, with the threshold for a positive at 0.5 ng/mg of analyte.
The analysis presented in this paper compares the ratio of the probability of having a positive outcome for all three measures: selfreport, urine assay, and hair assay. In order to evaluate the hypotheses, we used an analysis based on the application of Bayes’ theorem (Hays and Winkler 1970). Bayes’ theorem is a calculation algorithm that produces a probability value and derives this value by a combination of conditional and marginal probabilities in a tabular array. The theorem can relate the outcome of combining information from a measurement A with the distribution of a variable B such that if B has an effect on A, it will be reflected in a change in the probability of an outcome of A given B. This outcome of A given B—expressed as P(AB)—is called either the “conditional probability” or “posterior probability” of A under B. One interpretation of this is a reductionoferror process that provides an infrastructure for informationbased decision making. The probability of A alone constitutes a prior probability. Added information from B may modify the probability of A. This effect may be described in the form of a hypothetical statement that B has an effect on the probability of A. The Bayes value, or coefficient, is a measure of that effect. Bayes’ theorem is expressed in its definitional form as:
P(AB) = [P(BA)P(A) ÷ P(BA)P(A)] + P(BA^{c})P(A^{c}),
which is generally read as “the probability of A given a probability of B,” and where each term is a probability of occurrence ranging from 0 to 1 for any particular event, and if designated with superscript “c,” it is the nonoccurrence of the event such that P(A^{c}) = 1 – P(A). The calculation formula used here is:
P(AB) = P(BA)P(A) ÷ P(B)
where
P(A) is the prior or marginal probability of A (the probability of a cocainepositive outcome).
P(B) is the prior or marginal probability of B (the probability of a particular ethnicity).
P(AB) is the conditional or posterior probability of A given B. It is the value of A specified by a particular condition of B (the probability of a positive cocaine outcome given that an individual is a member of a specific ethnic or racial grouping).
P(BA) is the conditional or posterior probability of B given A. It is the value of B specified by a particular condition of A (the probability that an individual is a particular race/ethnicity given a positive cocaine outcome).
The comparison values used here are the ratios of the Bayescalculated posterior probabilities of a positive hair or urine assay when race is added as information to the prior probability. The prior probabilities are the overall odds of having a positive hair or urine assay. The posterior probability is the change in odds when selfreported race is added as information. The calculated Bayes values are then expressed as ratios of African American to Caucasian subject values, and the ratios are compared to see if they are consistent or inconsistent across race. Further, the ratios of Bayes factors are then compared across measurement type (i.e., selfreport, urine test, hair assay) to determine whether the relative likelihood of a positive test is differentially apparent given the respondent’s race. This responds to the essential question of whether there is a race bias in positive cocaine tests given the method of assessment.
The measured variables used in the study are as follows:
Urinalysis for Cocaine—Dichotomous categorical variable coded as “positive” or “negative.” All urine samples at or above 300 ng/mL of crossreacting substances responding to the EMIT assay for cocaine and metabolites were labeled “positive.”
Hair Analysis for Cocaine—Dichotomous categorical variable coded as “positive” or “negative.” All hair samples with an assay value at or above 0.5 ng/ml of cocaine were labeled “positive.”
SelfReported Cocaine Use Within 48 Hours of Arrest—Dichotomous categorical variable coded as “yes” or “no.” Subjects were asked to answer yes or no to the question of any cocaine consumption within the past 48 hours.
SelfReported Cocaine Use Within 60 Days of Arrest—Dichotomous categorical variable coded as “yes” or “no.” Subjects were asked to answer yes or no to the question of any cocaine consumption within the past 60 days.
Race—Dichotomous variable coded as “African American” or “Caucasian.” Subjects were asked to select one of five racial categories to describe themselves: Caucasian, African American, Asian, Hispanic, or “other.” This analysis used only subjects who reported either Caucasian or African American.
Null Hypotheses
Null hypothesis 1 states that the rates of selfreported cocaine use across the two time measures (48 hours and 60 days) will not differ when compared by race.
Null hypothesis 2 states that the ratios of the measurement of the Bayes probabilities of a cocainepositive outcome will show no significant difference between racial groups when accounting for selfreported use of cocaine.
Null hypothesis 3 states that hair analysis will not have a higher rate of frequency for cocainepositive outcomes than urinalysis when comparing ethnicities and accounting for selfreported cocaine use.
Alternate Hypotheses
Alternative hypothesis 1 states that the rates of selfreported cocaine use across the two time measures will vary significantly by race.
Alternative hypothesis 2 states that the ratios of the measurement of the odds of a cocainepositive outcome will vary significantly across racial groups when accounting for selfreported cocaine use.
Alternate hypothesis 3 states that, in hair analysis, African Americans will have a higher rate of frequency than Caucasians for cocainepositive outcomes than in urinalysis when accounting for selfreported cocaine use. This hypothesis and its associated null represent the essential test of differential racial bias across testing techniques.
The generation of the Bayes values is based on the use of simple 2 x 2 tabular arrays. Applied to a tabular array of two categorical dichotomous variables in this analysis, the Bayes values can be calculated as shown in Table 1.
Table 1: Bayes Component Identification
Prior Probability Function

Race 
Negative Cocaine Assay, Urine Assay, or SelfReport 
Positive Cocaine Assay, Urine Assay, or SelfReport 
Total 
African American 
Cell 1 
Cell 2 
Total_{r1} 
Caucasian 
Cell 3 
Cell 4 
Total_{r2} 
Total 
Total_{c1} 
Total_{c2} 
Total_{all} 
Given the definitions in Table 1, the following cell and marginal frequency values are used to calculate the Bayes functions:
For selfreported African American race
P(A) = Total_{c2}/Total_{all}
P(B) = Total_{r1}/Total_{all}
P(BA) = Cell 2/Total_{c2}
For selfreported Caucasian race
P(A) = Total_{c2}/Total_{all}
P(B) = Total_{r2}/Total_{all}
P(BA) = Cell 4/Total_{c2}
The subscript designations indicate the particular values used to make the probability calculations. So, for example, “Total_{c2}/Total_{all}” indicates that the total of cells in column 2 of the table should be divided by the grand total for the table in order to calculate P(A).
We calculated for each racial group a Bayes factor for the following four variables: dichotomous (±)
hairassay outcome, dichotomous (±) urinalysis outcome, yes/no response to query on cocaine use within the last 48 hours, yes/no response to query on cocaine use within the last 60 days. The data used in each of these calculations and the crosstabulated data for all four variables are presented in Table 2.
Table 2: CrossTabulated Data for Hair Assay, Urinalysis, and SelfReported Cocaine Use

Assay Outcomes 
Hair Assay 
(–) 
(+) 
Total 
African American

76 
126 
202 
Caucasian 
745 
320 
1065 
Total 
821 
446 
1267 
Urinalysis 
(–) 
(+) 
Total 
African American

304 
188 
492 
Caucasian 
1076 
192 
1268 
Total 
1380 
380 
1760 

SelfReport Outcomes 
48Hour SelfReport 
(–) 
(+) 
Total 
African American 
450 
63 
513 
Caucasian 
1274 
70 
1344 
Total 
1724 
133 
1857 
60Day SelfReport 
(–) 
(+) 
Total 
African American 
421 
92 
513 
Caucasian 
1211 
130 
1341 
Total 
1632 
222 
1854 
The Bayes factors and the ratios of the Bayes factors are presented in
Table 3 along with each value in calculating the prior and posterior distributions. Table 3 presents the various probabilities for the
hairassay outcomes; the urinalysis outcomes; the 48hour, selfreported cocaine use; and the 60day, selfreported cocaine use, as well as the Bayes factor [P(AB)] for each of the four measures for each racial/ethnic group. The first row is for Caucasians; the second, for African Americans. The bottom row of Table 3 reports the AfricanAmericantoCaucasian ratio (the relative odds ratio of the two Bayes factors) for each measure. The Bayes factor ratio can be read as relative odds of occurrence. For example, the ratio of African Americans to Caucasians for a cocainepositive is 2.52 to 1. Thus, considering all cases of cocainepositive urine samples, the odds that such a person is African American are approximately 2.5 times greater than the odds of the person’s being Caucasian.
Table 3: The Outcomes for Bayes Coefficients: Urinalysis, Hair Assays, and SelfReports
Race 
Hair Assay 
Urinalysis 
SelfReport—
48 Days 
SelfReport—
60 Days 
Ethnicity:
African American 
P(A) 
0.352 
P(A) 
0.216 
P(A) 
0.071 
P(A) 
0.119 
P(B) 
0.159 
P(B) 
0.279 
P(B) 
0.276 
P(B) 
0.276 
P(BA) 
0.282 
P(BA) 
0.494 
P(BA) 
0.473 
P(BA) 
0.414 
P(AB) 
0.623 
P(AB) 
0.382 
P(AB) 
0.122 
P(AB) 
0.179 
Ethnicity:
Caucasian 
P(A) 
0.352 
P(A) 
0.216 
P(A) 
0.071 
P(A) 
0.119 
P(B) 
0.840 
P(B) 
0.720 
P(B) 
0.723 
P(B) 
0.723 
P(BA) 
0.717 
P(BA) 
0.505 
P(BA) 
0.526 
P(BA) 
0.585 
P(AB) 
0.300 
P(AB) 
0.151 
P(AB) 
0.052 
P(AB) 
0.096 
Ratio of Bayes Factors:
African American to Caucasian 
2.07 
2.52 
2.55 
1.85 
Table 4 summarizes the Bayes factors for each variable by racial group and summarizes the relative ratios.
Table 4: Summary of the Relative Ratios of the Bayes Factors by Racial Group
Drug Measure 
Bayes Factors x Race 
Ratio of Bayes Factors: 

African American 
Caucasian 
African American to Caucasian 
Urinalysis 
0.382 
0.151 
2.52 
Hair Assay 
0.623 
0.300 
2.07 
SelfReport— 48 Hours 
0.122 
0.052 
2.35 
Self Report— 60 Days 
0.179 
0.096 
1.84 
Findings
Null Hypotheses
In regard to the null hypotheses, null hypothesis 1 stated that the rates of selfreported cocaine use across the two time measures will have no significant difference when compared by race. We reject the null hypothesis based on a Pearson’s chisquare indicating that African Americans have a significantly higher rate of selfreported cocaine use for the proximal 48hour time period (chisquare, 27.93; p < 0.001) and the more distal 60day time period (chisquare, 23.89; p < 0.001).
Null hypothesis 2 stated that the ratios of the measurement of the Bayes probabilities of cocainepositive outcomes will demonstrate no significant difference between racial groups when accounting for selfreported cocaine use. We cannot reject the null in this case because the ratio of Bayes factors for African Americans regardless of bioassay type remains consistent with selfreported levels of cocaine use. Urinalysis positiveoutcome probabilities for African Americans are about 2.52 times greater than for Caucasians, and African American selfreported cocaine use within 48 hours of interview is about 2.55 times that of Caucasians. We note the same outcome when comparing the outcomes of hair assays for cocaine with reported cocaine use over the previous 60 days. African Americans report 60day past use of cocaine at rates 1.85 times that of Caucasians, and hairassay values for African American subjects are approximately 2 times that of Caucasians. Comparing the four ratios, there is no significant difference among them. All four ratios fall well within the 95 percent confidence interval (2.201 ± 0.4251). The calculation of the confidence interval is described below.
Null hypothesis 3 stated that hair analysis will not have a higher rate of frequency for cocainepositive outcomes compared to urinalysis when comparing ethnicities and controlling for selfreported cocaine use. We are not able to reject the third null hypothesis because the ratio of the relative odds for a positive hair assay appears to have no significant difference from the probabilities of urinalysis. In fact, the relative odds for a cocainepositive hair assay are somewhat less than for a positive urinalysis, although this difference is not statistically significant. This was assessed using the four observed ratios of Bayes factors associated with each of the techniques displayed in Table 4. In an effort to further examine the apparent proximity of these ratios (suggesting no racial bias), a bootstrap resampling of the ratio values was done (mean, 1000 iterations) in order to estimate a standard error and construct a confidence interval.
Bootstrapping is a technique that creates a sampling distribution for a statistic by resampling from the data at hand. The method allows an estimate of how closely the sample data resemble the general population data. It does this by creating a synthetic estimate of the population (the “bootstrap”) and modeling the relationship between the sample and the population by considering it analogous to the relationship between the sample and the bootstrapped data. Several things can be estimated using this relationship, including confidence intervals. If the confidence interval is very large, it indicates that the two comparison populations are very likely distant from each other and, therefore, potentially different from each other. However, if the variation between them is modest—that is, within the range of confidence intervals expected in a normally distributed population—then the data do not support an argument of difference but, rather, an argument that the data have come from the sample population.
The results of the bootstrap were as follows: observed mean, 2.201; estimated mean, 2.201; bias, 0.000176; standard error, 0.1336. A 95 percent confidence interval calculated using the standard error value is 2.201 ± 0.4521 (confidence interval = 1.7759 ↔ 2.6261). The range of ratio values falls well within this confidence interval. Figure 1 is the histogram plot of the bootstrap, and Figure 2 is the normal quantile plot for the bootstrapped value estimations.
Figure 1: Histogram of the Bootstrap Distribution of 1000 Resampled Means
Figure 2: Quantile Plot for the Bootstrapped Value Estimations
Both of these graphs support the normality assumption for the bootstrap distribution. The normal quantile plot shows that replicates from the bootstrap largely fall along the diagonal and that the histogram of the bootstrap means is approximately normal. This further affirms that these ratio values are not significantly different from one another—supporting the assumption that they are values pulled from a random set of ratios of Bayes factors. In fact, it would take a far greater discrepancy between the assessment types to produce a significant, differential race effect across these techniques.
Alternate Hypotheses
Alternative hypothesis 1 stated that the rates of selfreported cocaine use across the two time measures will have a significant difference by race. We confirm this hypothesis. African Americans have a significantly higher rate of selfreported cocaine use for the proximal 48hour time period (chisquare, 27.93; p = 0.000) and the more distal 60day time period (chisquare, 23.89; p = 0.000).
Alternative hypothesis 2 stated that the ratios of the measurement of the odds of a cocainepositive outcome will have a significant difference between ethnic groups when controlling for selfreported cocaine use. We have stated that the null hypothesis for this relationship cannot be rejected. And, likewise, for alternate hypothesis 3—that hair analysis will have a higher rate of frequency for cocainepositive outcomes than urinalysis when comparing ethnicities and controlling for selfreported cocaine use—we have already noted that we cannot reject the null hypothesis in this situation.
Discussion and Conclusions
While a simple inspection of the data may seem to be indicative of a race bias for hair analysis, the findings do not support such a conclusion. Given that the ratios of all of the Bayes posterior probabilities are in the same direction—namely, that African American racial identification is associated with a relatively higher probability of being assaypositive—one can understand how, on first glance, such a conjecture might arise. But the data reveal two important phenomena. Regardless of the biospecimen, African Americans are more likely to test positive than Caucasians. Indeed, for this group, the likelihood of a positive bioassay is actually marginally greater for urinalysis than hair assay. We also note that African American racial identification is also associated with higher probability of selfreported cocaine use in both proximal and distal time frames.
The findings, we conclude, are not supportive of a racialbias hypothesis for hair analysis. The ratio for cocainepositive hairassay outcomes by race is lower for hair analysis than for urinalysis. So, in fact, if one were to assign a label of “most racially biased” to any of these toxicological/drugingestion assessment methods based on empirical analysis, it would be urinalysis. In that sense, the empirical work here demonstrates the folly of previous arguments of racial bias in hairassay methods, given acrosstheboard differences in the identified levels of cocaine use for African Americans and Caucasians. The ratios for all assays versus selfreports are consistent across their respective time frames, and none of the ratios are significantly different from one another.
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