Reducing Bias and Mean Squared Error Associated With Regression-Based Odds Ratio Estimators

Abstract

Ratio estimators of effect are ordinarily obtained by exponentiating maximum-likelihood estimators (MLEs) of log-linear or logistic regression coefficients. These estimators can display marked positive finite-sample bias, however. We propose a simple correction that removes a substantial portion of the bias due to exponentiation. By combining this correction with bias correction on the log scale, we demonstrate that one achieves complete removal of second-order bias in odds ratio estimators in important special cases. We show how this approach extends to address bias in odds or risk ratio estimators in many common regression settings. We also propose a class of estimators that provide reduced mean bias and squared error, while allowing the investigator to control the risk of underestimating the true ratio parameter. We present simulation studies in which the proposed estimators are shown to exhibit considerable reduction in bias, variance, and mean squared error compared to MLEs. Bootstrapping provides further improvement, including narrower confidence intervals without sacrificing coverage.

1. INTRODUCTION

Generalized linear models yield fitted coefficients that are commonly used to estimate odds ratios or other measures of association. Standard fitting techniques such as maximum-likelihood and estimating equation methods yield consistent estimators with first-order asymptotically normal sampling distributions (Cox and Oakes 1984; McCullagh and Nelder 1989; Davidian and Giltinan 1995; Diggle et al. 2002; Agresti 2002). Outside of linear models, however, these estimators can suffer from considerable higher-order asymptotic bias.

Most research on bias reduction has targeted the fitted regression coefficients, e.g., Byth and McLachlan (1978), Anderson and Richardson (1979), McLachlan (1980), Schaeffer (1983), Cook, Tsai, and Wei (1986), Cordeiro and McCullagh (1991), and Firth (1993), although King and Zeng (2001) studied bias-reduced estimators for probabilities based on logistic regression with rare events. While some statisticians see arithmetic bias and mean squared error (MSE) as less relevant on the scale of skewed estimators of effect (e.g., odds ratios), this is not a universal view. Several authors (e.g., Jewell 1984, 1986; Greenland 2000) have targeted small-sample bias on the odds ratio scale in tabular-based analyses, noting that researchers generally report and interpret point estimates on that scale. Such reporting has added practical rationale in that for uncommon diseases, the odds ratio approximates the risk ratio and thus is proportional to the change in caseload associated with a unit increase in exposure (Greenland, 2000).

We propose an approach to bias reduction for estimating measures of effect in general regression settings. Exponentiation of estimated regression coefficients to transform to an absolute-effect scale induces positive bias, a well-known consequence of Jensen’s inequality (Jensen 1906). We derive a form for this positive second-order bias that leads to a reduced-bias estimator. We then explore further benefits of an initial first-order bias correction on the log scale. Our main estimator targets second-order unbiasedness at the expense of increasing median bias (Read, 1985) relative to standard estimators. We propose alternative bias-reduced estimators to control the increased risk of underestimation due to mean bias adjustment.

Throughout, we use “asymptotic” and “approximate” to indicate standard first-order results (which apply on a scale proportional to n^−½ where n is the sample size) and will contrast those to second-order (scale n⁻¹) properties, comparing finite-sample behavior via simulations.

2. METHODS AND EXAMPLES

2.1 Bias-Corrected Odds Ratio Estimation

Odds ratio (OR) and rate or risk ratio (RR) estimators typically arise from exponentiating coefficient estimators from logistic, Poisson, or Cox models (Cox and Oakes 1984; McCullagh and Nelder 1989; Agresti 2002). Consider a generalized linear model of the form

$$\text{g}[\text{E}(\text{Y}\mid \mathbf{X}=\mathbf{x})]={\beta }_0+\sum _{\text{j}=1}^{\text{k}}{\beta }_{\text{j}}{\text{x}}_{\text{j}},$$ (1)

where g(.) is a strictly increasing link function. Inferences based on (1) commonly make use of the asymptotic normality of maximum-likelihood (ML) or other standard estimators:

$${\widehat{\beta }}_{\text{j}}\stackrel{•}{\sim }\text{N}({\beta }_{\text{j}},{\sigma }_{\text{j}}^2),$$ (2)

where ${\sigma }_{\text{j}}^2$ is the asymptotic variance of β̂_j. Thus, β̂_j is asymptotically median unbiased.

Typically, the link function g is the logit or logarithm, and the target parameter is ψ_j = e^β_j, with estimator Ψ̂_j = e^β̂_j (j=1,…, k). Although the first-order limiting distribution of Ψ̂_j is also normal with mean ψ_j, the log-scale normal approximation is far more accurate for typical sample sizes. It follows from (2) that the distribution of Ψ̂_j will be more closely lognormal with

$$\text{E}({\widehat{\mathrm{\Psi }}}_{\text{j}})\stackrel{•}{=}{\text{e}}^{{\beta }_{\text{j}}+{\sigma }_{\text{j}}^2/2}={\text{e}}^{{\sigma }_{\text{j}}^2/2}{\psi }_{\text{j}}.$$ (3)

Expression (3) may also be derived via Taylor-series arguments given that E(β̂_j) ≈ β_j, without assuming normality for β̂_j. The bias factor ${\text{e}}^{{\sigma }_{\text{j}}^2/2}$ disappears asymptotically.

Approximate median unbiasedness of β̂_j for β_j ensures approximate median unbiasedness of Ψ̂_j for ψ_j. However, Ψ̂_j can be subject to large overestimation errors unless ${\sigma }_{\text{j}}^2$ is small. Such errors might be especially detrimental if the estimate is used for resource-allocation decisions, e.g., if it is taken as representing the best prediction of the excess caseload to be expected from a harmful (ψ_j>1) exposure (e.g., Greenland, 2000). The bias factor ( ${\text{e}}^{{\sigma }_{\text{j}}^2/2}$) is negligible for small but increases rapidly with σ_j: The approximate expectation of Ψ̂_j overestimates ψ_j by more than 50% for σ_j values ≥ 0.9 (see Supplementary Figure 1).

To reduce the bias of Ψ̂_j, consider the following corrected estimator based on eqn. (3):

$${\widehat{\mathrm{\Psi }}}_{\text{j},\text{corr}}={\text{e}}^{-{\widehat{\sigma }}_{\text{j}}^2/2}{\widehat{\mathrm{\Psi }}}_{\text{j}},$$ (4)

where ${\widehat{\sigma }}_{\text{j}}^2$ is the variance estimate for β̂_j. This “plug-in” estimator is necessarily smaller than Ψ̂_j. A criticism of (4) is that it only reduces bias due to exponentiation, without addressing higher-order bias in the log-scale estimator β̂_j. We will investigate the extent to which initial bias correction to β̂_j followed by (4) improves performance.

2.2 Bias-Reduced OR Estimators Controlling the Risk of Underestimation

The estimator in (4) designed to reduce mean bias is first-order equivalent to the MLE, but has lower second-order variance due to multiplying by a correction factor between 0 and 1. As it converges more slowly to median unbiasedness, we also consider estimators that compromise between Ψ̂_j and Ψ̂_j,corr. To this end, note that (2) implies that

$$Pr({\widehat{\mathrm{\Psi }}}_{\text{j},\text{corr}}<{\text{e}}^{{\beta }_{\text{j}}})\stackrel{•}{=}\mathrm{\Phi }({\sigma }_{\text{j}}/2),$$ (5)

where Φ(.) is the standard normal CDF. Thus Φ(σ̂_j/2), which always exceeds 0.5, estimates the probability that Ψ̂_j,corr would underestimate ψ_j. To reduce median bias and target a risk of underestimation no larger than p (≥ 0.5), we consider estimators of the form Ψ̂_j,p = e^cΨ̂_j for some constant c such that Pr(Ψ̂_j,p < Ψ_j) ≈ p. From eqn. (2), the latter relation holds when c = −σ_jz_p, where z_p is the 100×p-th percentile of the standard normal distribution. This leads to a class of estimators that are consistent for all p, i.e.,

$${\widehat{\mathrm{\Psi }}}_{\text{j},\text{p}}={\text{e}}^{-{\widehat{\sigma }}_{\text{j}}{\text{z}}_{\text{p}}}{\widehat{\mathrm{\Psi }}}_{\text{j}}.$$ (6)

To improve mean bias and squared error while limiting the increased chance of underestimating Ψ_j in repeated samples, we propose the following bias-reduced estimator:

$${\widehat{\mathrm{\Psi }}}_{\text{j},\text{corr}}^{\ast }=max({\widehat{\mathrm{\Psi }}}_{\text{j},\text{corr}},{\widehat{\mathrm{\Psi }}}_{\text{j},\text{p}}),$$ (7)

with p selected by the investigator. This encompasses two extremes: 1) An investigator who demands median unbiasedness takes p = 0.5, so ${\widehat{\mathrm{\Psi }}}_{\text{j},\text{corr}}^{\ast }={\widehat{\mathrm{\Psi }}}_{\text{j},.50}={\widehat{\mathrm{\Psi }}}_{\text{j}}$, the usual MLE; 2) An investigator unconcerned with median unbiasedness uses Ψ̂_j,corr. Note that (7) is equivalent to

$${\widehat{\mathrm{\Psi }}}_{\text{j},\text{corr}}^{\ast }=\{\begin{array}{ll}{\widehat{\mathrm{\Psi }}}_{\text{j},\text{corr}}\hfill & \text{if}{\widehat{\sigma }}_{\text{j}}\le 2{\text{z}}_{\text{p}}\hfill \\ {\widehat{\mathrm{\Psi }}}_{\text{j},\text{p}}\hfill & \text{otherwise}\hfill \end{array}$$

Thus, the bias-reduced estimator retains the full mean-bias correction as in (4) when σ̂_j ≤ 2z_p ; otherwise, it reduces the correction factor to a degree determined by p, the acceptable probability of underestimation. In Section 3, we summarize a simulation study to evaluate the performance of a version of (7) in which a bias correction is initially applied to β̂_j.

2.3 Case Studies 1 and 2: 2×2 Tables for Unpaired and Paired Data

For brevity, we confine attention to odds ratios. To investigate the bias correction (4), let X and Y respectively denote exposure and disease status and consider 2×2 tables for unadjusted odds-ratio estimation in unpaired and pair-matched case-control settings as in Table 1.

Table 1.

Cell Counts and Cell Probabilities for Unpaired and Paired Data

Unpaired Data			Paired Data
				Controls (Y=0)
	X=1	X=0	Cases (Y=1)	X=1	X=0
Y=1	A (p11)	B (p10)	X=1	A (p11)	B (p10)
Y=0	C (p01)	D (p00)	X=0	C (p01)	D (p00)

The number of subjects in the unpaired setting is n = A+B+C+D, whereas for paired data n is the number of pairs. Under Poisson, product-binomial and multinomial models for the unpaired data cell counts, the MLE for the odds ratio ψ = p₁₁p₀₀/(p₁₀p₀₁) in Table 1 is Ψ̂_ML = AD/BC = p̂₁₁p̂₀₀/(p̂₁₀p̂₀₁), and σ̂² = 1/A+1/B+1/C+1/D is the usual first-order variance estimator for β̂_ML = ln(Ψ̂_ML). Under Poisson and multinomial models for the pair counts, the MLE for the paired-data odds ratio ψ = p₁₀/p₀₁ in Table is Ψ̂_ML = B/C = p̂₁₀/p̂₀₁, and σ̂² = 1/B+1/C is the usual first-order variance estimator for β̂_ML.

The O(n⁻¹) bias in Ψ̂_ML and β̂_ML = ln(Ψ̂_ML) follows from the Taylor-series expansion

$$\text{E}[\text{f}(\widehat{\mathbf{p}})]=\text{f}(\mathbf{p})+\text{g}(\mathbf{p})+\text{O}({\text{n}}^{-2}),$$ (8)

where g(p) = E[(p̂ − p)′ D₂ (p)(p̂ − p)/2] and D₂ (p) is the Hessian of f evaluated at p (Jewell 1984). Along with the previous estimator Ψ̂_corr1 = e^−σ̂2/2 Ψ̂_ML (we now add the subscript 1), we consider Ψ̂_{corr 2} = e^−σ̂2/2 exp(β̂^*_ML) where β̂^*_ML is the bias-corrected estimator for β = ln(OR) derived from (8). For unpaired data, β̂^*_ML = β̂_ML − (1/2)(1/B+1/C−1/A−1/D), while for paired data β̂^*_ML = β̂_ML −(1/2)(1/C−1/B). Table 2 summarizes the resulting O(n⁻¹) bias terms.

Table 2.

O(n⁻¹) (second-order) Bias Terms for MLEs and Bias-Corrected Estimators in Unpaired and Paired Data Settings

	Unpaired Data	Paired Data
β̂ML	(1/2n)(1/p01 + 1/p10 − 1/p00 − 1/p11)	(1/2n)(1/p01 − 1/p10)
Ψ̂ML	(ψ/n)(1/p01 + 1/p10)	(ψ/n)(1/p01)
Ψ̂corr 1	(ψ/2n)(1/p01 + 1/p10 − 1/p11 − 1/p00)	(ψ/2n)(1/p01 − 1/p10)
Ψ̂corr 2	0	0

In each case the arithmetic bias term for Ψ̂_ML is positive, and could be quite large. The O(n⁻¹) bias of Ψ̂_corr1 is not null, but tends to be markedly smaller than that of Ψ̂_ML when ψ>1. Most importantly, implementing an O(n⁻¹) bias correction to the log-scale estimator β̂_ML before applying the correction factor e^−σ2/2 to obtain Ψ̂_{corr 2}, completely eliminates the O(n⁻¹) bias.

The estimator Ψ̂_{corr 2} eliminates O(n⁻¹) bias even though it uses the first-order variance estimator (σ̂²) for β̂_ML in the correction factor e^−σ̂2/2. One might expect small-sample performance improvements by using an alternative estimator for σ² that reflects the (slightly reduced) variance of the bias-corrected estimator β̂^*_ML. We explore this idea below.

2.4 Bias-Corrected OR Estimation in Logistic Regression

Our general strategy to obtain bias-corrected adjusted OR estimates begins with model (1), assuming a logit link. While direct application of eqn. (4) reduces bias, further benefits follow from initial application of an O(n⁻¹) coefficient-scale bias correction as available for all generalized linear models (e.g., McCullagh and Nelder 1989; Cordeiro and McCullagh 1991; Firth 1993). Because we encountered no separation problems in our simulations (see Discussion), we focus here on the direct method of Cordeiro and McCullagh (1991) which yields the following O(n⁻¹) bias corrected estimator for the vector of logistic regression coefficients β:

$${\widehat{\mathbf{\beta }}}^{\ast }={\widehat{\mathbf{\beta }}}_{\text{ML}}-{\widehat{\mathbf{b}}}_1/\text{n}={\widehat{\mathbf{\beta }}}_{\text{ML}}-{({\mathbf{X}}^{\prime }\widehat{\mathbf{W}}\mathbf{X})}^{-1}{\mathbf{X}}^{\prime }\widehat{\mathbf{W}}\mathbf{\xi }.$$ (9)

In (9), i indexes the observation and π_i the estimated binomial probability for observation i, W is the diagonal matrix of binomial variances n_iπ_i(1−π_i), and Wξ is a vector with ξ_i = h_i(π_i−½) and h_i the i-th diagonal element of W^1/2X(X′WX)⁻¹X′W^1/2. We apply the bias-corrected estimator in (9) and then the exponentiation correction (4). For the variance estimate ( ${\widehat{\sigma }}_{\text{j}}^2$) to accompany each ${\widehat{\beta }}_{\text{j}}^{\ast }$, there are several options. Using the variance associated with the MLE is justifiable asymptotically (Firth, 1993). However, the bias-corrected ${\widehat{\beta }}_{\text{j}}^{\ast }$‘s tend to shrink toward their means, resulting in finite-sample variance smaller than that of the MLE. Thus, in Section 3 we also evaluate the use of bootstrap variance estimates in (4).

2.5 Interval Estimation and Invariance

Our primary focus is point estimation on the OR scale, whereas Wald-type confidence intervals (CIs) are best set on the log scale. Thus, for CIs to accompany the bias-corrected OR estimates, we favor use of the exponentiated limits of Wald-type intervals based on bootstrap estimates of standard errors for bias-corrected coefficient estimates. Potential benefits of such intervals relative to standard ML-based alternatives are illustrated via simulations below.

If a covariate X_j is binary, then the MLE (Ψ̂_j) for the OR has inversion symmetry. For example, if the coding of X_j is switched from (0,1) to (1,0), then Ψ̂_j is correspondingly inverted. In contrast, bias corrections on the OR scale are asymmetric. As a result, the proposed bias-corrected OR estimators should not be inverted to estimate Ψ_j upon recoding of a covariate or outcome. Doing so yields an estimator no longer bias-corrected on the inverted scale, thus negating the purpose of the proposed estimation method. The proper approach is to compute the corrected estimate directly, after first selecting the coding and scale for reporting.

3. SIMULATION STUDIES

3.1 Bias-Corrected OR Estimator Based on Discriminant Function Approach

We first replicate a simulation study originally conducted by Lyles, Guo and Hill (2009) to evaluate a discriminant function-based estimator for β, the ln(OR) relating a continuous exposure X to a binary outcome Y and covariates C. Data were simulated under the following linear model with independent normal(0,0.04) errors:

$$\text{E}(\text{X})=4.52+0.083\text{y}+0.11{\text{c}}_1-0.07{\text{c}}_2-0.04{\text{c}}_3+0.26{\text{c}}_4+0.01{\text{c}}_5.$$

The parameters and covariate distributions were chosen to mimic a birthweight data example in Hosmer and Lemeshow (2000). This special case allows a unique application of the proposed bias correction approach because a strictly unbiased estimator (β̂_disc) for the log OR (β), as well as an unbiased estimator Vâr(β̂_disc) for its exact sampling variance, are available (Lyles et al. 2009). The estimator β̂_disc showed lower variance than the MLE β̂_ML from logistic regression of Y on (X, C). However, the OR estimator Ψ̂_disc = exp(β̂_disc) retained substantial bias due to exponentiation despite improved bias and variance relative to the MLE.

Supplementary Table 1 summarizes a repeat of the prior simulation study adding bias-corrected OR estimators

$${\widehat{\mathrm{\Psi }}}_{\text{corr}1}={\text{e}}^{-\text{V}\widehat{\text{a}}\text{r}({\widehat{\beta }}_{\text{ML}})/2}{\widehat{\mathrm{\Psi }}}_{\text{ML}}\text{and}{\widehat{\mathrm{\Psi }}}_{\text{corr}2}={\text{e}}^{-\text{V}\widehat{\text{a}}\text{r}({\widehat{\beta }}_{\text{disc}})/2}{\widehat{\mathrm{\Psi }}}_{\text{disc}}.$$ (10)

Note that Ψ̂_corr1 is the estimator in (4), while Ψ̂_{corr 2} refines it by substituting β̂_disc and its unbiased variance estimator. Based on 10,000 simulations with a true adjusted OR of exp(0.083/0.04) = 7.96, the estimators Ψ̂_ML (mean =13.51) and Ψ̂_disc (mean =11.53) display extreme positive bias. In contrast, Ψ̂_corr1 is largely corrected (mean=8.96) and Ψ̂_{corr 2} in (10) is virtually unbiased (mean = 7.91), with both showing striking variance and MSE reduction.

3.2 Bias-Corrected OR Estimators in Logistic Regression

Tables 3 and 4 summarize simulations based on 10,000 independent trials from a logistic model with three covariates distributed as follows: X₁ ~ N(0, 0.25²), X₂ ~ Bernoulli(0.30), and X₃ ~ Uniform(0, 0.5), and sample size n=200. The parameters β₁, β₂, and β₃ were set to 1.75, 1, and −0.5, corresponding to ORs of Ψ₁=5.75, Ψ₂=2.72, and Ψ₃=0.61.

Table 3.

Simulation Results Comparing Estimators for Regression Coefficients and Corresponding Standard Errors

	Regression Coeff. Estimates		Mean of Standard Error Estimates		Mean Width of 95% CI for Ψ (Coverage)
Variable	β̂ML Mean (SD)	β̂corr Mean (SD)	ML	Parametric Bootstrap	ML	β̂corr w/Parametric Bootstrap
X1 (β1=1.75, Ψ1=5.75)	1.803 (0.646)	1.748 (0.627)	0.636	0.626	25.55 (95.3%)	22.13 (94.8%)
X2 (β2=1, Ψ2=2.72)	1.030 (0.349)	1.001 (0.338)	0.345	0.340	4.44 (95.2%)	4.13 (94.9%)
X3 (β3=−.5, Ψ3=0.61)	−0.516 (1.067)	−0.503 (1.040)	1.051	1.037	8.29 (94.8%)	7.94 (94.6%)

^*10,000 replications with n=200 in each case; Covariate distributions described in text

Table 4.

Simulation Results Comparing Performance of Proposed OR Estimators in Logistic Regression^*

	Mean Bias [Median Bias]
	(SD)
	{Empirical MSE}
	Percentage of estimates < true value
Variable	Ψ̂ML†	Ψ̂corr1‡	Ψ̂corr 2¶	${\widehat{\mathrm{\Psi }}}_{\text{corr}2}^{\ast }$ ||
X1 (β1=1.75, Ψ1=5.75)	1.81 [0.22]	0.33 [−0.86]	0.05 [−1.11]	0.77 [−0.52]
	(6.11)	(4.64)	(4.48)	(5.04)
	{40.58}	{21.59}	{20.06}	{26.01}
	48%	60%	63%	57%
X2 (β2=1, Ψ2=2.72)	0.26 [0.05]	0.09 [−0.11]	0.00 [−0.18]	0.05 [−0.15]
	(1.13)	(1.04)	(0.99)	(1.01)
	{1.34}	{1.08}	{0.98}	{1.01}
	48%	55%	58%	56%
X3 (β3=−.5, Ψ3=0.61)	0.46 [−0.01]	0.00 [−0.26]	0.00 [−0.25]	0.31 [−0.07]
	(1.67)	(0.94)	(0.91)	(1.37)
	{2.99}	{0.88}	{0.83}	{1.98}
	50%	70%	70%	55%

^*Based on 10,000 replications with n=200 in each case; Covariate distributions described in text

^† Usual MLE for adjusted OR

^‡ Bias-corrected estimate computed using β̂_ML and its standard error [eqn. (4)]

^¶ Bias-corrected estimate computed using β̂_corr and parametric bootstrap-based standard error

^||Bias-reduced estimate [eqn. (7)] using β̂_corr and parametric bootstrap-based standard error; using p=0.55 to limit risk of underestimating true OR to approximately 55% or less

Table 3 compares the coefficient MLEs (β̂_ML) versus the bias-corrected alternatives β̂_corr) in (9). The latter display smaller bias and variance in each case. The ML-based standard errors match the empirical standard deviation of (β̂_corr) relatively well on average, but noticeably better matches are obtained from a parametric bootstrap (Efron and Tibshirani 1993). Wald-type confidence intervals are evaluated based on ML and on β̂_corr with its bootstrap standard error; the latter CIs exhibit reduced mean width, yet retain near-nominal coverage..

Table 4 summarizes the comparison of four adjusted OR estimators: Ψ̂_ML, Ψ̂_corr1, Ψ̂_{corr 2}, and ${\widehat{\mathrm{\Psi }}}_{\text{corr}2}^{\ast }$. The corrected estimators are defined as follows:

$${\widehat{\mathrm{\Psi }}}_{\text{corr}1}={\text{e}}^{-\text{V}\widehat{\text{a}}\text{r}({\widehat{\beta }}_{\text{ML}})/2}{\widehat{\mathrm{\Psi }}}_{\text{ML}},{\widehat{\mathrm{\Psi }}}_{\text{corr}2}={\text{e}}^{{\widehat{\beta }}_{\text{corr}}-\text{V}\widehat{\text{a}}\text{r}({\widehat{\beta }}_{\text{corr}})/2},$$

and

$${\widehat{\mathrm{\Psi }}}_{\text{corr}2}^{\ast }=max({\widehat{\mathrm{\Psi }}}_{\text{corr}2},{\widehat{\mathrm{\Psi }}}_{\text{p},\text{corr}}),$$ (11)

where ${\widehat{\mathrm{\Psi }}}_{\text{p},\text{corr}}={\text{e}}^{{\widehat{\beta }}_{\text{corr}}-{\text{z}}_{\text{p}}\sqrt{\text{V}\widehat{\text{a}}\text{r}({\widehat{\beta }}_{\text{corr}})}}$, Vâr(β̂_corr) is based on the parametric bootstrap, and Ψ̂_p,corr is computed taking p=0.55 to target a risk of underestimating the true OR no larger than 55%. Table 4 shows marked positive mean bias of the standard OR estimators, especially for those corresponding to larger sampling variances (i.e., for Ψ̂₁ and Ψ̂₃). In contrast, mean bias and MSE are dramatically reduced for the corrected estimators (Ψ̂_corr1 and especially Ψ̂_{corr 2}).

While the MLE for the OR approaches median unbiasedness, the bias–corrected estimators sacrifice that criterion. For example, Table 4 shows that the proportion of Ψ̂₁ values below the true OR was 48% for the MLE, vs. 60% and 63% for the ‘corr1’ and ‘corr2’ estimators. For comparison, the right-most column summarizes the performance of the estimator ${\widehat{\mathrm{\Psi }}}_{\text{corr}2}^{\ast }$, with p=0.55. This estimator achieves a mean/median bias compromise, with the risk of underestimating the true Ψ₁ approximately equal to the desired threshold of 55%.

Supplementary Figure 2 compares histograms representing the 10,000 standard and bias-corrected OR estimates (Ψ̂_ML and Ψ̂_{corr 2}) based on the simulations summarized in Table 4. The MLE histogram displays a longer and heavier tail, yielding an empirical mean of 7.56. In contrast, the mean bias-corrected estimate was 5.80, again nearly unbiased.

4. DISCUSSION

Ratio estimates of effect are the standard for reporting and interpreting epidemiologic results, reflecting their intuitive appeal as well as their proportionality to relative caseload or risk when the outcome is not common. While we focused primarily on estimates from logistic regression, the basic bias correction [Ψ̂_corr; eqn. (4)] is applicable in any case of exponentiation of an approximately unbiased estimate. Initial log-scale bias correction and bootstrap variance estimation yields further bias removal on the exponentiated scale. Nonetheless, as noted by a reviewer, an alternative coefficient estimator is necessary if the ML estimate is infinite. The simulations in Tables 3 and 4 yielded no separation problems and thus all could use Cordeiro and McCullagh’s (1991) correction. Firth’s (1993) approach is now commercially available for logistic and Cox regression (SAS Institute, Inc., 2008), making it attractive for the log-scale correction step when separation is encountered (e.g., Heinze, 1999; Heinze and Schemper, 2002).

Unlike the bias-corrected estimators Ψ̂_corr1 and Ψ̂_{corr 2}, Ψ̂_ML converges rapidly to median unbiasedness and is transformation invariant (e.g., 1/Ψ̂_ML is the MLE for 1/Ψ). But Ψ̂_ML is subject to potentially extreme positive mean bias when ${\sigma }_{\text{j}}^2$ is large, and always has higher variance and MSE than the proposed bias-corrected estimators. Thus the bias corrections discussed here will be valuable whenever loss is more proportional to Ψ rather than its log, as we think holds in most policy and planning settings. More generally, eqn. (3) reflects how the positive bias in Ψ̂_ML sacrifices traditional mean unbiasedness in favor of median unbiasedness. Arguably, the latter criterion ignores the magnitude of extreme estimates in the sampling distribution, producing an incomplete and perhaps misleading performance measure for an estimator whose realized value may be subject to interpretation with a view toward policy.

When standard errors are large, eqn. (5) indicates that the proposed bias-corrected estimators may entail substantial underestimation risk (median bias) in order to achieve approximate mean unbiasedness. The estimator Φ(σ̂_j/2) (Section 2.2) provides a convenient way to assess this risk and leads to the class of estimators ${\widehat{\mathrm{\Psi }}}_{\text{corr}}^{\ast }$ in (7), which can yield worthwhile reductions in mean bias, variance, and MSE without severe risk of underestimation.