Precise and Accurate Power of the Rank-Sum Test for a Continuous Outcome

Abstract

Accurate power calculations are essential in small studies containing expensive experimental units or high-stakes exposures. Herein, power of the Wilcoxon Mann-Whitney rank-sum test of a continuous outcome is formulated using a Monte Carlo approach and defining P(X < Y) ≡ p as a measure of effect size, where X and Y denote random observations from two distributions hypothesized to be equal under the null. Effect size p fosters productive communications because researchers understand p = 0.5 is analogous to a fair coin toss, and p near 0 or 1 represents a large effect. This approach is feasible even without background data. Simulations were conducted comparing the empirical power approach to existing approaches by Rosner & Glynn (2009), Shieh et al. (2006), Noether (1987), and O’Brien-Castelloe (2006). Approximations by Noether and O’Brien-Castelloe are shown to be inaccurate for small sample sizes. The Rosner & Glynn and Shieh et al. approaches performed well in many small sample scenarios, though both are restricted to location-shift alternatives and neither approach is theoretically justified for small samples. The empirical method is recommended and available in the R package wmwpow.

1. Introduction

Despite the current era of big data, there remains a practical need for power calculations of small preclinical, first-in-human, and basic science studies involving two independent samples. Accurate power calculations are critical when each experimental unit is expensive (e.g., macaques for preclinical HIV vaccine experiments) or the study is high stakes (e.g., novel HIV cure strategies where toxicity risks are unknown). Anti-conservative power approximations can result in an underpowered study, and conservative approximations can lead to using more experimental units than necessary. Owing to small sample sizes, determining power in this setting is challenging because asymptotic approximations may not be reliable. An additional challenge common in many studies, such as preclinical or first-in-human trials, is the absence of relevant background data to inform power calculations.

In small studies with a continuous outcome (e.g., Kulkarni et al. 2011; Archin et al. 2014; Denton et al. 2014), the Wilcoxon Mann-Whitney (WMW) rank-sum test is often utilized to test for differences between groups (Wilcoxon 1945; Mann and Whitney 1947). Thus it is of interest to compute power of the WMW test against different alternatives. Previous work on calculating power of the WMW test for a continuous outcome includes Haynam & Govindarajulu (1966), Noether (1987), Collings & Hamilton (1988), Lehmann (1998), Van de Wiel (2001), Chakraborti et al. (2006), Shieh et al. (2006), Zhao et al. (2008), Rosner & Glynn (2009), and Divine et al. (2010). Power of the WMW test for ordered categorical outcomes was considered previously by Hilton & Mehta (1993), Kolassa (1995), and Tang (2011, 2016).

In this paper, an approach for determining the power of the WMW test is formulated using Monte Carlo simulation. The approach is accurate and precise in that no asymptotic approximation is employed, and the amount of Monte Carlo error can be controlled by the user. In addition, an appealing aspect of this approach is that it can be implemented with or without background data. Effect size is defined by p = P(X < Y), where X and Y denote random observations from the two distributions being compared. Equivalently, the effect size can be expressed by the odds p/(1−p) (O’Brien & Castelloe 2006; Divine et al. 2013, 2017). Under a location-shift alternative, the WMW test null hypothesis is p = 0.5, analogous to a fair coin toss. This effect size can be easily understood by collaborative investigators. Moreover, when background data are lacking, it can be more productive to discuss plausible values for p with collaborators than to elicit parameterizations for each distribution or to quantify effect size using standard deviation units. Further, as shown here and by Rosner & Glynn (2009), in many design scenarios the underlying distributions have minimal impact on power for a fixed effect size p.

The outline of the remainder of this paper is as follows. Section 2 presents several approaches to calculating power of the WMW test (with details in the Appendices). Section 3 presents simulation results comparing WMW test power calculations. Section 4 provides a motivating example, and Section 5 concludes with a discussion.

2. Methods

Suppose X₁,…,X_m and Y₁,…,Y_n are independent identically distributed (iid) random variables with continuous cumulative distribution functions F and G, respectively. It is of interest to test the null hypothesis H₀ : F = G versus the two-sided alternative hypothesis H_A : F ≠ G. The WMW test statistic is $W={\sum }_{i=1}^m{\sum }_{j=1}^n\phi (Y_j-X_i)$ where φ(Y_j − X_i) =1 when Y_j > X_i, and 0 otherwise; i.e., the WMW statistic counts the number of times a Y_j is larger than a X_i. Under H₀, the WMW statistic has mean μ₀ = mn/2 and variance ${\sigma }_0^2=mn(N+1)/12$ where N = n + m; as m and n tend to infinity,(W−μ₀)/σ₀ has a limiting standard normal distribution under H₀ (Mann & Whitney 1947).

Shieh et al. (2006) derived a large-sample approximation for power of the WMW test using an exact variance of W under the alternative hypothesis H_A, and demonstrated that their approach was more accurate than the Noether (1987) and Lehmann (1998) approximations. Effect size in Shieh et al. was defined in terms of G(x) = F(x−θ), where θ is the location shift in the cumulative distribution function (CDF) and H₀ : θ = 0. The Shieh et al. method is reformulated here using effect size p (Appendix A) to facilitate interpretation and comparison to other approaches to estimating power of the WMW test. For large m and n, power for the two-sided WMW test against a specific alternative hypothesis can be approximated by:

$$P\left\{\left|\frac{W-{\mu }_0}{{\sigma }_0}\right|>z_{\alpha /2}|H_A\right\}\approx \Phi \left(\frac{\mu -{\mu }_0-z_{\alpha /2}{\sigma }_0}{\sigma }\right)+\Phi \left(\frac{{\mu }_0-\mu -z_{\alpha /2}{\sigma }_0}{\sigma }\right)$$ (1)

where α is the significance level, Φ(·) is the CDF of a standard normal distribution, z_α/2 = Φ⁻¹(1−α/2), and μ and σ are the mean and standard deviation of the WMW statistic under H_A, respectively. The mean under H_A, μ = mn/p, depends upon effect size p, and the variance under H_A can be expressed as:

$${\sigma }^2=mn\{p(1-p)+(n-1)(p_2-p^2)+(m-1)(p_3-p^2)\}$$ (2)

where σ² depends upon effect size p and underlying distributions F and G through p₂ and p₃ (Lehmann 1998; Shieh et al. 2006).

The approach from Noether (1987) provides an approximation to the power of the WMW test which also relies on the normal approximation in Equation 1, but does not require selecting parametric models for F and G. Instead, two additional assumptions are supposed: (i) ${\sigma }^2={\sigma }_0^2$, i.e., the variance of W under H_A is equal to the variance of W under H₀; and (ii) N/(N + 1) ≈ 1 (Appendix B). Assumptions (i) and (ii) may be dubious for small sample sizes. Clearly the approximation N/(N + 1) ≈ 1 only holds for large N. In addition, a study with small m and n will have adequate power only for large effect sizes, in which case σ will not, in general, equal σ₀ (Shieh et al. 2006).

Rosner and Glynn (2009) also provide a method for estimating the power of the WMW test which relies on the normal approximation in Equation 1 but does not require selecting parametric models for F and G. Rosner and Glynn derive a closed-form estimate of power for location-shift alternatives defined after first applying a probit transformation to F and G.

With modern computing, empirical (Monte Carlo) power calculation for the WMW test is feasible and accurate, particularly for small studies. As described below, empirical power computation entails repeated sampling from F and G. Options for selecting F and G include: (i) specifying parametric distributions for both F and G; (ii) specifying a parametric distribution for F and choosing a value for p, which in turn imply a distribution for G; or (iii) resampling from a sufficient amount of background data (Collings & Hamilton 1988; Hamilton & Collings 1991). For studies where background data are unavailable or sparse, the resampling approach (iii) is not feasible. While approach (i) is feasible for small studies, it can be harder to interpret (e.g., presenting a mean difference in standard deviation units) compared to (ii) where one selects effect size p or odds. Options (i) and (ii) are available in the R package described below.

In practice, the empirical method can provide power estimates that are effectively exact. The general approach entails simulating multiple datasets from F and G, and computing the proportion of simulated datasets where the WMW test rejects the null. As the number of simulated datasets approaches ∞, empirical power converges in probability to the true power of the WMW test. For a finite number of simulated data sets, the Monte Carlo error can be quantified, such that the number of simulations may be chosen to ensure this error is within an acceptable tolerance. In particular, let Q be the number of rejections of H₀ among S simulated datasets and let p_q be the probability of rejecting H₀ with Q ~ Binomial(S, p_q). For simulations under H₀, p_q equals the type I error rate, and for simulations under a particular alternative hypothesis H_A, p_q equals power. Power (or type I error) is estimated empirically by ${\widehat{p}}_q=Q/S$. By the central limit theorem, for large S, ${\widehat{p}}_q$ will be approximately normal with mean p_q and the standard error of ${\widehat{p}}_q$ will be no larger than $1/\sqrt{4S}$, which is ≈ 0.0016 for S = 100,000. This implies that S = 100,000 simulated datasets will provide a precise power estimate to two decimal places. For example, suppose S = 100,000 and Q = 80,000; then ${\widehat{p}}_q=0.8$ and the corresponding 99% Wald confidence interval (CI) for p_q rounded to two decimal places is (0.80, 0.80). With S = 10,000 the standard error of ${\widehat{p}}_q$ is no larger than ≈ 0.005, and for Q/S = 8,000/10,000 the 99% CI for p_q is (0.79, 0.81).

The wmwpow R package provides three functions for estimating power: wmwpowp, wmwpowd, and shiehpow. For all three functions, the user inputs the sample sizes (m,n) and the significance level (α). The function wmwpowp also takes inputs of the distribution for F and the effect size p, and returns empirical power. For example, suppose the user inputs an exponential distribution with rate parameter μ for F and a particular value for p; then wmwpowp solves for G. Available choices in wmwpowp for F are the exponential, normal, and double exponential (Laplace) distributions, corresponding to the derivations in Appendix C. These distributions were selected following from the work by Shieh et al. In each case, F and G are assumed to be in the same family or class of distributions; e.g., if F is specified to be normal with mean μ_x and variance ${\sigma }_x^2$, then G is assumed to be normal as well. If F is exponential with rate μ and p is fixed, then G is completely specified. On the other hand, if F is normal or double exponential and p is fixed, then G is not completely specified without additional assumptions. Therefore, for the normal and double exponential distributions, the function wmwpowp also takes as an input the scalar k which specifies the ratio of standard deviations for F and G. For k = 1, choosing p ≠ 0.5 corresponds to a location-shift alternative. Choosing k ≠ 1 allows for unequal variances and thus a wider class of alternative hypotheses.

When specifying parametric distributions for both F and G is preferred, the function wmwpowd can be used to compute empirical power; wmwpowd allows the user to select from many standard continuous parametric distributions, including beta, exponential, normal, and Weibull. The function wmwpowd outputs the empirical power as well as the effect size p and the equivalent odds corresponding to the F and G specified by the user.

The wmwpow package also includes the function shiehpow, which implements the Shieh et al. method for location-shift alternatives assuming normal, shifted exponential, or double exponential distributions. The function shiehpow uses a shifted exponential distribution, whereas the exponential distribution in wmwpowp uses one rate parameter that defines both shape and location such that a common support [0,∞) is maintained for F and G.

3. Empirical Comparisons

The performance of methods by Noether (1987), O’Brien-Castelloe (2006), Shieh et al. (2006), and Rosner & Glynn (2009) were compared to empirical power results. Each method was formulated such that α, m, n, and p were the inputs, as well as an assumed probability distribution, when required. Power was estimated for effect size p ranging from 0.50 to 0.95 by 0.05 (odds ranging from 1 to 19).

The approach of Shieh et al. was implemented using the R package wmwpow, function shiehpow with the formulae shown in Appendix A. The Noether approach (Appendix B) was also implemented in R. The O’Brien-Castelloe approach was applied using the SAS Power procedure (twosamplewilcoxon, SAS/STAT v14.2); default settings were used and distributional assumptions were X ~ N(0,1) and Y ~ N(μ_y,1), solving for μ_y by inputting values of p into the equation shown in Appendix C.2. Rosner & Glynn (2009) provided a SAS macro (%WilcxPowerContinuousNties) for their approach. Empirical power was computed as the proportion of rejections of H₀ under a specific alternative hypothesis over S simulated datasets; S = 100,000 simulated datasets were used for n,m < 20, and S = 10,000 simulated datasets for n,m ≥ 20. Computations were conducted in R version 3.4.3 and SAS version 9.4 (Cary, NC).

Empirical power for large effect sizes (p ≥ 0.8) over a range of small samples sizes (n = m = 6 to 15) is shown in Figure 1a as calculated using wmwpowp. Comparisons between the empirical power calculations and results from Shieh et al., Rosner-Glynn, Noether, and the O’Brien-Castelloe methods are shown in Table 1 and Figure 1b–1d. For a given p, varying the assumed distributions for F and G had only a small effect on power. For n = m = 6 per group, the Shieh et al. and Rosner-Glynn methods provided very similar results (Figure 1b). The O’Brien-Castelloe approximation was typically anti-conservative for small m and n (e.g., n = m = 6). The Noether approximation was both anti-conservative or over-conservative depending upon effect size p and sample sizes (Figures 1b and 1c). For example, when n = m = 6 and effect size p = 0.95 the Noether approximate power was 77% and normal empirical power was 92%, a 15% power under-estimation from the Noether formula (Figure 1b). For n = m = 6 and p = 0.85 the O’Brien-Castelloe power was 69% and normal empirical power was 56%, resulting in a 13% over-estimation of power from the O’Brien-Castelloe approximation (Table 1). As m and n increase, power results from the methods evaluated here became increasingly similar, as expected. For m,n ≥ 50, all of the methods yielded similar results (Figure 1d).

Figure 1:

Power and type I error for the 2-sided Wilcoxon Mann-Whitney rank-sum test, α = 0.05. Sim = empirical simulation. Panel (a) presents power from the empirical Monte Carlo method with a normal distribution and equal standard deviations (k = 1), and panels (b-d) compare power results for a given sample size per group.

Table 1:

Power and type I error for the 2-sided Wilcoxon Mann-Whitney rank-sum test, α = 0.05

Sample Size	Method	Effect Size
		p=0.5	p=0.7	p=0.75	p=0.8	p=0.85	p=0.9
n=6, m=6	Noether	3	22	32	44	56	67
	O’Brien-Castelloe	n/a	26	39	54	69	82
	Rosner-Glynn	5	19	27	38	53	74
	Empirical - Normal	4	18	28	40	56	75
	Empirical - Exp	4	18	28	40	56	74
	Empirical - Laplace	4	18	28	39	55	72
	Shieh - Normal	5	18	27	38	53	74
	Shieh - Shifted Exp	5	19	28	39	53	72
	Shieh - Laplace	5	19	27	38	53	72
n=15, m=15	Noether	3	48	66	81	91	97
	O’Brien-Castelloe	n/a	52	72	87	96	99
	Rosner-Glynn	5	46	67	87	98	>99
	Empirical - Normal	5	47	67	85	96	>99
	Empirical - Exp	5	46	68	86	96	>99
	Empirical - Laplace	5	46	68	85	95	99
	Shieh - Normal	5	46	67	86	98	>99
	Shieh - Shifted Exp	5	46	67	85	97	>99
	Shieh - Laplace	5	46	67	86	97	>99
n=6, m=12	Rosner-Glynn	5	25	37	53	73	92
	Empirical - Exp	4	24	37	54	73	90
	Shieh - Shifted Exp	5	23	36	54	74	93
n=12, m=6	Rosner-Glynn	5	25	37	53	73	92
	Empirical - Exp	4	26	39	55	72	86
	Shieh - Shifted Exp	5	27	39	53	69	86

Above, n and m are the two sample sizes. p is a measure of effect size, i.e., the probability that the first random variable is less than the second variable, P(Y > X). Statistical power (%) is displayed rounded to the nearest whole percentage. Effect size H₀ : p = 0.5 presents type I error. Exp=Exponential

Generally, the Shieh et al. and Rosner-Glynn approaches tended to well approximate the empirical power. However, for small unequal sample sizes (e.g., m = 6,n = 12), the Shieh et al. and Rosner-Glynn power estimates can differ, as demonstrated in the bottom of Table 1. Note the Rosner-Glynn approach gives the same power estimate when m = 6,n = 12 compared to m = 12,n = 6, for a fixed effect size p. In contrast, Shieh et al. power estimates need not be the same when the values of m and n are interchanged, as can be seen from Equation 2 and Appendix A (p₂ and p₃ are unequal for non-symmetric distributions).

Empirical power for alternative hypotheses where F and G are normal with unequal variances is shown in Figure 2a. For m = n = 6, power decreases as the degree of variance heterogeneity increases (i.e., as k increases). Varying k had less impact for m = n = 15. Note that if k ≠ 1, then the null hypothesis H₀ : F = G does not hold even if p = 0.5. Hence, in Figure 2b the empirical power is above α = 0.05 for p = 0.5 and k = 3,4.

Figure 2:

Empirical power for F and G normal with standard deviation ratio k = σ_y/σ_x = 1,2,3,4, α = 0.05.

4. Motivating Example

Consider a proposed study of m = n = 15 per group with the sample size limited by ethical (e.g., safety), recruitment, or budgetary constraints. Given the limited feasible sample size, an accurate assessment of power is crucial for deciding whether the study should proceed as planned. In this study, background data on the outcome are dearth, and yet power calculations are still needed if null hypothesis significance testing is planned. In some cases, the study team should change the study design to focus on estimation and collection of pilot data without hypothesis testing. Here we proceed assuming that group comparisons are essential to the study objectives.

For example, in early phase clinical trials evaluating potential cures for HIV, sample sizes are typically limited to mitigate potential risks to participants. An outcome of interest, HIV replication index, is a relatively new measure used in HIV cure research with limited background data. Replication index is defined as the mean number of live daughter cells created by each parent cell over a specific length of time; this provides proliferative capacity on a per-cell basis, independent of the number of cells that originally started to proliferate (Clutton et al. 2016), and is closely related to proliferation index (Migueles et al. 2009).

Suppose the investigators choose a 0.05 significance level and decide p = 0.8 or larger is a meaningful effect size, i.e., an 80% or larger true probability that the HIV replication index for any given individual in the placebo group is higher than for any given individual in the treatment group. Assuming p = 0.8 (or equivalently, a true odds of 4 or larger) and using an empirical power approach, thirty individuals (m = n = 15 per group) will provide 85% power to detect a difference between two independent groups (placebo versus treatment). Rosner-Glynn, and O’Brien-Castelloe power estimates were both 87% and the Shieh et al. estimate was 86%, whereas the Noether approximation was conservative in this example (81% power). Empirical power for the exact 2-sided WMW test was conducted assuming a normal distribution for log₁₀ replication index and 100,000 datasets of size m = n = 15 were generated.

5. Discussion

Empirical power calculation is accurate and feasible for many power scenarios, including small sample settings, unequal variances, and unequal group sample sizes. The power approximations of Noether and O’Brien-Castelloe are not reliably accurate for small sample sizes. The Rosner & Glynn and Shieh et al. approaches performed well in many small sample scenarios, though both are restricted to location-shift alternatives and neither approach is theoretically justified to provide accurate power estimates for small samples. In contrast, the empirical power approach can evaluate a wider class of alternative hypotheses and is valid for any sample size.

Future related work could consider evaluating power of the rank-sum test when the outcome data are anticipated to include ties. Ties can arise when the underlying variable is continuous and the variable is measured or recorded with limited granularity such that two or more individuals may have the same recorded value. Estimating power of the WMW test when ties may or may not be present was not considered here because it is often not practical to ascertain the a priori probability of a tie occurring. Zhao et al. (2008) generalized the Noether (1987) method to handle ties, making the assumption that the variance of the test statistic W under the alternative H_A is the same as under H₀; this assumption may be dubious when group sample sizes are small. If adequate background information is available regarding ties, one can simulate data accordingly (e.g., resample from the background data), and proceed with empirical power calculation. Ordered categorical data can be thought of as an extreme case of ties, and can be simulated directly using category probabilities (e.g., the tabled distribution within the SAS function RAND). WMW test power calculation for ordered categorical data is also available in StatXact software (Hilton & Mehta 1993).

The empirical Monte Carlo approach has a few limitations. One limitation is the use of a finite number of samples, though in practice it is often computationally feasible to generate 100,000 simulated datasets, which provides a precise power estimate to two decimal places at a 99% confidence level. Still, to get fully reproducible power results a random seed will need to be set. For large study sizes, computation time may become an issue. With modern computing we did not experience a computational burden for small sample sizes, which is the design setting where a Monte Carlo approach is most useful. At equal sample sizes of 50 or more per group, each power approach evaluated here provided a similar result. Another limitation of the Monte Carlo approach used here entails sampling from a user-specified parametric distribution. If this parametric assumption is incorrect, the resulting power estimates may not be accurate. Alternatively, if sufficient background data are available, one could sample from the empirical distribution of the background data, rather than from an assumed parametric distribution.

Accurate and precise power calculation via Monte Carlo simulation is recommended whenever computationally feasible. Empirical power calculation for the rank-sum test is available in the commercial software PASS by inputting parametric distributions for F and G. However, PASS version 16 does not yet provide p or odds as an input or output value. The R package wmwpow can be used to compute empirical power with either p or odds as an input (or alternatively F and G), and is free and publicly available on CRAN.

Appendices

Appendix A

As shown in Lehmann (1998) and Shieh et al. (2006), the variance σ² of the WMW statistic under H_A depends on F and G; this dependence can be formulated using p₂ and p₃ for a location-shift alternative, with p_{2 =} p₃ when distributions F and G are symmetric. When the underlying distributions of F and G are shifted exponential (a non-symmetric distribution),θ= −ln[2(1−p)] for p in (0.5,1), p₂ = 1−2/3e^−θ, and p₃ = 1−e^−θ+1/3e^−2θ. When the distributions of F and G are double exponential (Laplace), θ= −L(4(p−1)/e²) where L is the Lambert-W function used to solve for x when y = xe^x and p₂ = p₃ = 1−(7/12+θ/2)e^−θ−1/12e^−2θ. Lastly, for the normal case, F ~ N(0,1), $\theta =\sqrt{2}{\Phi }^{-1}(p)$ and p₂ = p₃ = E[{Φ(Z+θ)}²], where Z ~ N(0,1).

Appendix B

Noether (1987) provided an approximation to the power of the WMW test assuming σ = σ₀, and N/(N+1) ≈ 1, where N = m + n. Consider a one-sided WMW test, in which case the power equals:

$$1-\beta =P\left(Z>\frac{{\mu }_0-\mu }{\sigma }+\frac{z_{\alpha }{\sigma }_0}{\sigma }\right)\mathrm{.}$$

Let c = m/N and z_β = Φ⁻¹(1−β). Then under the assumption σ = σ₀, it follows that

$${\left(\frac{{\mu }_0-\mu }{{\sigma }_0}\right)}^2=\frac{12N^{2}c(1-c)(p-{0.5)}^2}{N+1}=(z_{\alpha }+z_{\beta }{)}^2$$

or equivalently

$$\frac{N^2}{N+1}=\frac{{(z_{\alpha }+z_{\beta })}^2}{12c(1-c)(p-{0.5)}^2}\mathrm{.}$$

Assuming N/(N+1) ≈ 1, it follows that

$$N\approx \frac{{(z_{\alpha }+z_{\beta })}^2}{12c(1-c)(p-{0.5)}^2}$$

and therefore power of the WMW test is approximated by

$$1-\beta \approx \Phi [\sqrt{12Nc(1-c)(p-{0.5)}^2}-z_{\alpha }]\mathrm{.}$$

Appendix C

Consider the general form for $p=P(X<Y)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^y{f}_X(x)g_Y(y)\mathit{\text{dxdy}}={\int }_{-\infty }^{\infty }{g}_Y(y)F_X(y)dy$, where f_X(x) and g_Y(y) are probability density functions for X and Y, respectively. The following three distributions are implemented in the R package wmwpow, function wmwpowp.

C.1: Exponential

Let X ~ Exp(μ) and Y ~ Exp(λ), where μ and λ are exponential rate parameters. Then $p=P(X<Y)={\int }_0^{\infty }{\int }_x^{\infty }\mu \lambda e^{-\mu x}{e}^{-\lambda y}\mathit{\text{dydx}}=\mu /(\lambda +\mu )$, and therefore λ = μ(1−p)/p.

C.2: Normal

Let $X~N({\mu }_x\mathrm{,}{\sigma }_x^2)$ and $Y~N({\mu }_y\mathrm{,}{\sigma }_y^2)$ such that $X-Y~N({\mu }_x-{\mu }_y\mathrm{,}{\sigma }_x^2+{\sigma }_y^2)$. This implies $p=P(X-Y<0)=\Phi \left(\frac{{\mu }_y-{\mu }_x}{\sqrt{{\sigma }_x^2+{\sigma }_y^2}}\right)$, and therefore ${\mu }_y={\mu }_x+{\Phi }^{-1}(p)\sqrt{{\sigma }_x^2+{\sigma }_y^2}$.

C.3: Double Exponential

Let X ~ Laplace(μ_x,σ_x) and Y ~ Laplace(μ_y,σ_y). Then μ_y can be found as follows. Recall the cumulative distribution function of a Laplace random variable is

$$F_X(x)=\{\begin{array}{cc}\frac{1}{2}{e}^{\frac{x-{\mu }_x}{{\sigma }_x}}& \text{if }x\le {\mu }_x\\ 1-\frac{1}{2}{e}^{-\frac{x-{\mu }_x}{{\sigma }_x}}& \text{if }x>{\mu }_x\text{.}\end{array}$$

This implies p = P(X < Y)

$$={\int }_{-\infty }^{{\mu }_x}\left(\frac{1}{2}{e}^{\frac{y-{\mu }_x}{{\sigma }_x}}\right)\left(\frac{1}{2{\sigma }_y}{e}^{-\frac{|y-{\mu }_y|}{{\sigma }_y}}\right)dy+{\int }_{{\mu }_x}^{\infty }\left(1-\frac{1}{2}{e}^{-\frac{y-{\mu }_x}{{\sigma }_x}}\right)\left(\frac{1}{2{\sigma }_y}{e}^{-\frac{|y-{\mu }_y|}{{\sigma }_y}}\right)dy\mathrm{.}$$

Thus μ_y can be found by solving numerically

$$\left[{\int }_{\infty }^{{\mu }_x}\left(\frac{1}{2}{e}^{\frac{y-{\mu }_x}{{\sigma }_x}}\right)\left(\frac{1}{2{\sigma }_y}{e}^{-\frac{|y-{\mu }_y|}{{\sigma }_y}}\right)dy+{\int }_{{\mu }_x}^{\infty }\left(1-\frac{1}{2}{e}^{-\frac{y-{\mu }_x}{{\sigma }_x}}\right)\left(\frac{1}{2{\sigma }_y}{e}^{-\frac{|y-{\mu }_y|}{{\sigma }_y}}\right)dy\right]-p=0$$

using any standard one-dimensional root finding method given p, μ_x, σ_x, and σ_y.