An Email Experiment to Identify the Effect of Racial Discrimination on Access to Lawyers: A Statistical Approach

Brian Libgober; Tirthankar Dasgupta

REVIEW 2 major objections 6 minor 21 references

An Email Experiment to Identify the Effect of Racial Discrimination on Access to Lawyers: A Statistical Approach

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read An email audit with six graded name levels can estimate the full response curve of lawyer bias, not just a black-white gap.

desk verdict A genuine statistical template for graded audit studies, with a known-but-unaddressed measurement error problem in the race levels and a Section 5 inconsistency that should be fixed before publication. read the letter →

arxiv 1908.08116 v2 pith:H24D2NMO submitted 2019-08-21 stat.AP

Brian Libgober , Tirthankar Dasgupta This is my paper · ORCID

classification stat.AP MSC 62K0562F1262P25

keywords accesstojusticedesignofexperimentsmaximumlikelihoodbinaryresponseemailauditstudiesracialdiscriminationracelevelinverted-Sfunction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Standard email-audit experiments compare only very-black-sounding and very-white-sounding names, so they can detect discrimination but cannot say whether the effect is confined to extreme names or grows steadily with every hint of race. This paper establishes a statistical framework in which each sender name carries a graded "race level" $\xi$ in $(0,1)$, and the goal is inference on the shape of the response probability $\pi(\xi)$ as a function of $\xi$. The response function is parameterized by boundary rates $\alpha$ and $\beta$ and a single shape parameter $\gamma$, with $\gamma=1$ giving a linear relationship, $\gamma>1$ a logistic curve, and $\gamma<1$ an inverted-S curve; the paper derives maximum-likelihood estimation, asymptotic standard errors, and prediction variances for this model. On the design side, simulations of the Fisher information determinant indicate that six equally spaced race levels are near-optimal across a wide range of parameter values, and the paper demonstrates a complete workflow of name selection, blocking, randomization, and analysis on a pilot sample of lawyers. The pilot's estimate of the treatment gap is not statistically significant, but the paper's contribution is the inference-and-design machinery, which it argues is ready for markets where discrimination is already known to exist.

What carries the argument

The load-bearing object is the probability-weighting function $g(\gamma,\xi)=\xi^{\gamma}/(\xi^{\gamma}+(1-\xi)^{\gamma})$, taken from the psychology literature on how people weight probabilities. It converts the shape of the race-response curve into a single scalar: $\gamma=1$ recovers a straight line between the boundary rates, $\gamma>1$ produces the steep-midrange curve the paper calls logistic, and $\gamma<1$ produces the extreme-sensitive inverted-S curve. The function enters the likelihood through the reparameterization $\pi(\xi)=(a+g)/(a+b)$ or $(b-g)/(a+b)$, where $a$ and $b$ are deterministic functions of the boundary rates $\alpha$ and $\beta$, and it is the derivative $g'(\gamma,\xi)$ that appears in the score equations and the Fisher information matrix. All of the paper's inference results, including the MLE algorithm, Proposition 1, the delta-method corollaries, and the D-optimality simulations, flow from this single parametric family.

What would settle it

Run the six-level design with a total sample of at least 2000 units in a market where a prior binary-name audit already shows a large black-white response gap, and estimate $\alpha$, $\beta$, and $\gamma$; if the fitted 95% confidence interval for $\alpha-\beta$ still contains zero, the design fails to recover a known effect. Separately, ask the lawyers themselves to rate the six sender names on the same black-white scale used to fix $\xi$; if their average ratings deviate from the assigned levels by more than sampling error, the fixed-$\xi$ assumption is violated.

Watch

Extended reading notes

Core claim

On the paper's own terms, the core discovery is that a graded-treatment email audit can be designed and analyzed so that the qualitative question "is lawyer discrimination linear or concentrated at extreme names?" becomes a parameter-estimation problem. Using the odds-ratio function $g(\gamma,\xi)=\xi^{\gamma}/(\xi^{\gamma}+(1-\xi)^{\gamma})$ inside a likelihood for binary responses, the paper shows that the boundary response probabilities $\alpha$ and $\beta$ and the shape parameter $\gamma$ are identifiable, gives their asymptotic normal approximation, and derives a delta-method covariance for $\alpha$ and $\beta$ and a prediction variance for $\pi(\xi)$ at any new level. Design simulations then show that six equally spaced levels of $\xi$ keep the determinant of the Fisher information high over a wide grid of $(\alpha,\beta,\gamma)$, making the design a near-optimal default. The pilot application, using 899 lawyers, six names per gender, and blocking on lawyer race, gender, and practice area, yields point estimates $\hat{\alpha}=0.3687$, $\hat{\beta}=0.2118$, and $\hat{\gamma}=0.688$, with a 95% confidence interval for $\beta-\alpha$ that contains zero, so the pilot cannot reject a null treatment effect but does demonstrate the intended analysis.

Load-bearing premise

The load-bearing premise, flagged by the authors themselves, is that the race level $\xi$ assigned to each sender name, meaning the proportion of a survey panel who read the name as black, is an exact and noise-free constant describing how lawyers perceive that name; if lawyers' perceptions differ systematically, every estimated response curve, standard error, and design choice is miscalibrated.

Editorial extensions

If this is right

The distinction between linear and nonlinear discrimination becomes a Wald test of $\gamma=1$, so an adequately powered audit can say whether bias is a smooth gradient or switches on only for very distinctive names.
With six equally spaced levels and equal allocation, the design is near-D-optimal across a wide grid of $\alpha$, $\beta$, and $\gamma$, meaning future audits can adopt it without running bespoke optimal-design calculations for each new market.
The iterative MLE and asymptotic standard errors apply to any binary-outcome correspondence audit with a continuous treatment, so the method generalizes beyond legal services to rental, employment, and other service-provision contexts.
Power simulations show that resolving shape requires large samples, with $\gamma$ far from 1 and large $|\beta-\alpha|$ most favorable; the 899-lawyer pilot is underpowered for small effects, and its non-significant result should not be read as evidence against discrimination.
The design is presented as ready for use where discrimination has already been established; in undiagnosed markets, the sample sizes needed for shape inference are prohibitive.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the same shape parameter $\gamma$ could be exported to other continuous signals, such as accent strength, neighborhood cues, or credit-score proxies, so that a single number characterizes whether any service market discriminates in a threshold or gradient fashion; this is a direct generalization the paper does not spell out.
Editorial inference: the fixed-$\xi$ assumption could be tested by collecting race ratings of the actual sender names from a sample of the same profession being audited; if lawyer ratings differ systematically from the survey panel's proportions, the model would need a measurement-error layer for $\xi$, which the authors flag as future work.
Editorial inference: an adaptive design, with a first wave to localize the response curve and a second wave concentrating treatment levels there, could recover shape information at lower total sample cost than the fixed equally spaced design, an extension suggested by the paper's own power constraints.
Editorial inference: the pilot's wide confidence interval, combined with the paper's power analysis, implies that conclusions of "no discrimination" from small correspondence audits should be treated cautiously; failure to reject could simply reflect sample size rather than the absence of bias.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear [Colleague],

Here's the quick take on Libgober and Dasgupta: it's a real methods contribution to audit studies, giving a way to estimate a dose-response curve over racial signaling rather than the usual two-name black/white comparison. The statistical core is sound; the soft spots are known but not resolved.

What's new: The paper adapts a probability weighting function to model response as a function of ξ, the probability a receiver reads a name as black, and works out the MLE, Fisher information, and delta-method standard errors. The design section goes beyond platitudes: simulations to choose the number of levels (k=6), a naive Bayes pipeline for name selection from the Florida voter file, and MTurk validation. The pilot is honestly reported as underpowered, and the power analysis in Appendix B is unusually frank about sample size demands. The math in Section 3 checks out as far as I can see.

Where it's soft. First and most important: the likelihood and the design treat each name's ξ as a known constant, but in the actual experiment ξ is a point estimate from MTurk. Measurement error in a nonlinear covariate doesn't just inflate standard errors; it biases the MLE of α, β, and γ, and it means the k=6 design simulations are run on levels that may not be the ones actually administered. The authors flag this in Section 6 but don't quantify or fix it. This isn't a reason to kill the paper, but a serious revision needs to confront it, even if only with a sensitivity analysis or a measurement-error model.

Second, Section 5 has an internal inconsistency: the text says the fitted model assumed α<β, but the reported estimates (Table 2) all have α>β. That has to be a typo or a leftover from an earlier draft, and it obscures what the pilot actually showed.

Third, and more of a design concern: there is one name per level. Any idiosyncratic effect of "Latasha" versus "Nicole" is completely confounded with the race level. The method assumes names differ only through ξ. Given the whole point is to vary ξ continuously, this is a real limitation, though common in early dose-response work.

Bottom line: this deserves a serious referee. It's a useful template for statisticians designing audit studies, and the honest power analysis is a plus. I wouldn't cite the empirical null, but I'd consider citing the model and design approach, with a caveat about ξ. Send it to review, with instructions to push on measurement error and name-specific effects.

Referee Report

2 major / 6 minor

Summary. The paper develops a design-of-experiments and maximum-likelihood framework for email audit studies that measure racial discrimination in access to lawyers using more than two race-signal levels. It defines a race level ξ∈(0,1) as the probability that a receiver identifies a sender's name as black, models the response probability π(ξ) with a parametric family (2)-(5) indexed by boundary probabilities α, β and shape parameter γ, and derives an iterative MLE, the asymptotic normal distribution of the estimator (Proposition 1), and delta-method covariance results for α, β and predictions (Corollaries 1-2). The design section uses D-optimality simulations over k levels to recommend six equally spaced treatment levels, describes name selection using Florida voter-file data and MTurk perception surveys, and presents a blocked randomization of 899 Florida lawyers. The empirical analysis finds no statistically significant treatment effect, and the authors are explicit that resource constraints prevented a definitive test. The paper closes by acknowledging that the treatment level is assumed known without noise.

Significance. If the statistical claims hold, the paper makes a useful methodological contribution to correspondence audit design: it replaces the common binary black/white name manipulation with a multi-level perceived-race treatment and provides a coherent likelihood-based estimation and power-analysis workflow. The derivations are standard but carefully executed, the design simulations are honest about the large sample sizes required, and the paper explicitly flags the fixed-ξ assumption and the pilot nature of the empirical application. The main value is as a template for planning multi-level audit experiments rather than as a source of substantive estimates, since the Florida pilot lacks power.

major comments (2)

[Section 3 (Eqs. (6), (10)), Section 4.2, Section 6] The inference and design results treat each ξ_i as a fixed, exactly known treatment level, but in the implemented experiment ξ_i is not controlled; it is a point estimate (an MTurk proportion or an objective Bayes probability) of the perception probability for a selected name. Measurement error in a nonlinear covariate generally biases the MLE of (α, β, γ) and makes the plug-in covariance matrix from Proposition 1 inconsistent, so the nominal 95% intervals and the power curves in Appendix B are not calibrated to the actual randomized treatment. The acknowledgment in Section 6 ("we have assumed that the input level ξ is a constant and can be chosen without noise") states the problem but does not quantify its effect on the central claims. I ask the authors to add a sensitivity analysis (e.g., simulation under plausible true ξ values and MTurk sample sizes) or an errors-in-variables formulation, and to state explicitly which ξ values (objective, subjective, or a combination) were used in the Section 5 analysis.
[Section 5 and Table 2] The text says the preliminary proportions showed a lower response at the lowest race level (very likely black) than at the highest (very likely white), and that the model was therefore fit under α<β; however, all three rows of Table 2 report α̂>β̂, which is the decreasing-branch case, and the reported intervals are centered at α̂−β̂ rather than β̂−α̂ (e.g., overall midpoint 0.1569 equals 0.3687−0.2118). The heading "95% CI for ˆβ−ˆα" is therefore inconsistent with the entries, and the description of which model branch was fitted is internally contradictory. This must be corrected and reconciled because it determines the sign convention for the estimated response curve and the interpretation of the confidence interval for the treatment effect.

minor comments (6)

[Section 3, Algorithm Step 7] The equation for updating b is said to be solved "for a"; it should be solved for b.
[Section 4.1] In "det(IN(θ) = ∑Ni=1 Ii(θ))" a closing parenthesis is missing, and the cross-reference to "Ii(θ) is given by (1)" should point to Eq. (10), not Eq. (1).
[Figure 5] The y-axis label "Estimated subjective probability" does not say whether the probability is of identifying the name as black or as white; since ξ is defined as black-identification probability, this needs to be stated.
[Appendix B, Figure 7 caption] The clause "when there is a large treatment effect (α−β = 0.6, the asymptotic intervals..." is missing a closing parenthesis.
[References] The Wald (1943) entry ("Transactions of the American Mathematical Society. Health Educ. Res. 54, 426–482") and the Pager et al. (2006) entry (whose title appears to belong to a different paper) need to be corrected.
[Sections 1 and 4.2] Minor typographical errors include "adu it studies" in Section 1 and "nave Bayesian" in Section 4.2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the likelihood inference and design simulations are self-contained under an explicitly assumed parametric model; the only self-citation is contextual and the fixed-ξ caveat is an acknowledged measurement-error limitation, not a circular reduction.

full rationale

The paper's derivation chain is not circular. Section 3 develops standard maximum-likelihood inference for independent Bernoulli outcomes under the explicitly assumed response family (2)-(5); the asymptotic results (Proposition 1, Corollaries 1 and 2) follow from textbook likelihood theory and the delta method, not from the experimental data. The functional form of g(γ, ξ) is imported from the probability-weighting literature (Goldstein and Einhorn 1987; Gonzalez and Wu 1997), and the parameters α, β, γ are free unknowns estimated from data, so the model is not defined in terms of the quantities it is used to infer. The design recommendation k = 6 in Section 4.1 is obtained by evaluating the D-optimality criterion across a grid of assumed true parameters and is explicitly framed as a robustness check under the model family, not as an out-of-sample prediction. Section 5's fitted response curve is a standard plug-in prediction from the estimated model, not a separate empirical claim. The only self-citation, Libgober (2020), is used as background evidence that market conditions may violate blocking assumptions; it does not carry the statistical derivation. The acknowledged fixed-ξ assumption in Section 6 is a measurement-error and identification limitation, not a case of a result reducing to its own input. Accordingly, no circular step is present.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central method does not require a specific fitted value; the free parameters listed are the targets of inference in the demonstration and the design choice from simulations. The axioms are the modeling assumptions that must hold for the proposed inference to be valid.

free parameters (4)

α (boundary response probability at ξ→0+) = 0.3687 (overall fit, Table 2)
Estimated by MLE from the Florida experiment demonstration; the method itself does not require a specific value.
β (boundary response probability at ξ→1−) = 0.2118 (overall fit, Table 2)
Estimated by MLE from the Florida experiment demonstration.
γ (shape parameter) = 0.688 (overall fit, Table 2)
Estimated by MLE from the Florida experiment demonstration; γ=1 linear, >1 logistic, <1 inverted-S.
Number of treatment levels k = 6
Chosen from D-optimality simulations over a grid of assumed α,β,γ values; not fitted to the experimental data but a design choice that affects the method's operating characteristics.

assumptions (5)

domain assumption The response function π(ξ) is monotone in ξ and belongs to the parametric family defined in Eqs. (2)-(5).
This is the modeling assumption on which all inference on α,β,γ rests. If the true response curve is non-monotone or outside this family, the shape parameter γ does not have its intended interpretation. Introduced in Section 2.
domain assumption The race level ξ of each sender name is a known constant, equal to the measured subjective probability of being identified as black.
Used throughout the likelihood and design simulations; the paper acknowledges in Section 6 that in reality the applied level is only a point estimate of a true level, so this is an idealization.
domain assumption First and last names combine independently in the naive Bayes calculation of race level (Eq. 12).
Borrowed from Fryer and Levitt (2004); if first and last names are correlated markers of race, the computed race level is biased. Used in Section 4.2 to choose names.
domain assumption Responses Y_i are independent Bernoulli given ξ_i.
Standard model for binary email responses; used to construct the likelihood in Section 3.
standard math Standard asymptotic MLE regularity conditions hold so that Proposition 1 applies.
The paper cites Boos and Stefanski (2013) for the asymptotic theory; regularity conditions are not verified for the empirical sample.

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Cite this review

Pith. "Pith review of An Email Experiment to Identify the Effect of Racial Discrimination on Access to Lawyers: A Statistical Approach." pith.science (2026). https://pith.science/paper/H24D2NMO

@misc{pith2026190808116,
  author       = {Pith},
  title        = {Pith review of: An Email Experiment to Identify the Effect of Racial Discrimination on Access to Lawyers: A Statistical Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H24D2NMO}},
  note         = {Machine review of arXiv:1908.08116}
}

read the original abstract

We consider the problem of conducting an experiment to study the prevalence of racial bias against individuals seeking legal assistance, in particular whether lawyers use clues about a potential client's race in deciding whether to reply to e-mail requests for representations. The problem of discriminating between potential linear and non-linear effects of a racial signal is formulated as a statistical inference problem, whose objective is to infer a parameter determining the shape of a specific function. Various complexities associated with the design and analysis of this experiment are handled by applying a novel combination of rigorous, semi-rigorous and rudimentary statistical techniques. The actual experiment was attempted with a population of lawyers in Florida, but could not be performed with the desired sample size due to resource limitations. Nonetheless, it provides a nice demonstration of the proposed steps involved in conducting such a study.

Figures

Figures reproduced from arXiv: 1908.08116 by the authors.

**Figure 3.** D-optimality criteria versus levels of k when β − α ≤ 0.3 5 10 15 20 0.00 0.04 alpha= 0.2, beta= 0.3, gamma= 0.25 No. of levels (k) D−optimality 5 10 15 20 0.0 0.4 0.8 1.2 alpha= 0.2, beta= 0.3, gamma= 0.5 No. of levels (k) D−optimality 5 10 15 20 0.0 1.0 2.0 3.0 alpha= 0.2, beta= 0.3, gamma= 0.75 No. of levels (k) D−optimality 5 10 15 20 0.0 1.5 3.0 alpha= 0.2, beta= 0.3, gamma= 1 No. of levels (k) D−optimality 5 1… view at source ↗

**Figure 7.** Plots of δ1 and δ2 against N for α = .8, β = .2 1000 1200 1400 1600 1800 2000 0.0 0.2 0.4 0.6 0.8 1.0 δ1 versus N for α =.8, β =.2 N δ1 gamma=1 gamma=.5 gamma=1.5 1000 1200 1400 1600 1800 2000 0.0 0.2 0.4 0.6 0.8 1.0 δ2 versus N for α =.8, β =.2 N δ2 gamma=1 gamma=.5 gamma=1.5 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

Figures from the paper (1 more)

**Figure 8.** Figure 8: Plots of δ1 and δ2 against N for α = .3, β = .2 1000 1200 1400 1600 1800 2000 0.0 0.2 0.4 0.6 0.8 1.0 δ1 versus N for α =.3, β =.2 N δ1 gamma=1 gamma=.5 gamma=1.5 1000 1200 1400 1600 1800 2000 0.0 0.2 0.4 0.6 0.8 1.0 δ2 versus N for α =.3, β =.2 N δ2 gamma=1 gamma=.5 g…

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Donev, and R

Atkinson, A., A. Donev, and R. Tobias (2007). Optimum Experimental Designs, With SAS . Oxford: Oxford University Press

work page 2007

[2] [2]

Bertrand, M. and E. Duﬂo (2017). Field Experiments on Discrimination . North Holland

work page 2017

[3] [3]

Bertrand, M. and S. Mullainathan (2004). Are emily and greg more employable than lakisha and jamal? a ﬁeld experiment on labor market discrimination. The American Economic Review 94 , 991–1013

work page 2004

[4] [4]

Boos, D. and L. Stefanski (2013). Essential Statistical Inference . New York: Springer

work page 2013

[5] [5]

Butler, D. M. and D. Broockman (2011). Do politicians racially discriminate against constituents? a ﬁeld experiment on state legislators. American Journal of Political Science 55 , 463–477

work page 2011

[6] [6]

Carlsson, M. and D. O. Rooth (2007). Evidence of ethnic discrimination in the swedish labor market using experimental data. Labour Economics 14 , 716–729. 23

work page 2007

[7] [7]

Carpusor, A. G. and W. E. Loges (2006). Rental discrimination and ethnicity in names. Journal of Applied Social Psychology 36 , 934–952

work page 2006

[8] [8]

Chaloner, K. and I. Verdinelli (1995). Bayesian experimental design: A review. Statistical Sci- ence 10 , 273–304

work page 1995