The Direct Integral Method for Confidence Intervals for the Ratio of Two Location Parameters

Summary

In a relative risk analysis of colorectal caner on nutrition intake scores across genders, we show that, surprisingly, when comparing the relative risks for men and women based on the index of a weighted sum of various nutrition scores, the problem reduces to forming a confidence interval for the ratio of two (asymptotically) normal random variables. The latter is an old problem, with a substantial literature. However, our simulation results suggest that existing methods often either give inaccurate coverage probabilities or have a positive probability to produce confidence intervals with infinite length. Motivated by such a problem, we develop a new methodology which we call the Direct Integral Method for Ratios (DIMER), which, unlike the other methods, is based directly on the distribution of the ratio. In simulations, we compare this method to many others. These simulations show that, generally, DIMER more closely achieves the nominal confidence level, and in those cases that the other methods achieve the nominal levels, DIMER has comparable confidence interval lengths. The methodology is then applied to a real data set, and with follow up simulations.

1. Introduction

We use data on the relationship between diet and colorectal cancer from a subset of the NIH-AARP Study of Diet and Health (Reedy et al., 2008), which itself is a large cohort study with approximately 250,000 men and 200,000 women. The data subset that we have access to includes 1,075 males that developed colorectal cancer during the course of the study, along with 479 females who also developed colorectal cancer. In addition, the data set includes 3,225 randomly selected men and 1,437 randomly selected women who did not develop colorectal cancer. Hence, there are 4,300 males and 1,916 females in the data set.

It is traditional in nutritional epidemiology to examine the risk of cancer from single foods or nutrients normalized by energy (caloric) intake, e.g., the percentage of calories coming from fat, the amount of whole grants per 1,000 calories, etc. However, nutritionists have increasingly turned to dietary indices, which account for the patterns of energy-adjusted intake for multiple foods and nutrients. There are many such indices, e.g., the Healthy Eating Index - 2005 (HEI-2005, see Guenther et al., 2008), the Alternative Healthy Eating Index, the Mediterranean Index, etc., and they have been shown to be related to many chronic diseases and cancers. We use here the HEI-2005, which is based on the intakes of 12 interrelated dietary components, adjusted for energy intake. These intakes are then scored individually, and their sum is the HEI-2005, which is then used to predict disease. The Supplementary Material Table S.3 describes the components and how they are scored.

In our analysis of colorectal cancer, we fit a model where the scores are weighted and summed, but the weights are common for men and women, as in any dietary index. We show in Section 4.1 that, surprisingly, when comparing the relative risks for men and women based on this common index, the problem reduces to forming a confidence interval for the ratio of two (asymptotically) normal random variables. The latter is an old problem, with a substantial literature, one that we revisit based on our example.

One popular method for computing a confidence interval for the ratio of two location parameters is due to Fieller (Fieller, 1932; Fieller, 1954). Details of this method are described in the Supplementary Material Appendix S.1.

Consequently, other methods have been developed, most of which are based on the distribution of the ratio of the estimates of two location parameters (see for example recent papers by Beyene and Moineddin, 2005; Pham-Gia et al., 2006; Sherman et al., 2011). Most often, a normal approximation to the distribution is used, with subsequent intervals formed by Wald’s method. Hayya et al. (1975) showed that, under certain conditions, the distribution for the ratio of two estimators can be treated as a normal distribution with a second order Taylor expansion. This method is also defined in the Supplementary Material Appendix S.3. In addition, parametric and nonparametric bootstrap methods are also used. However, our empirical investigations suggest that confidence intervals constructed by these existing methods for the ratio often give inaccurate and sub-nominal coverage probabilities.

Motivated by such a problem, in this work we construct a new methodology, which we call the Direct Integral Method for Ratios (DIMER). This methodology is also based on the distribution of the ratio of the estimates of the two location parameters, a distribution that is Cauchy-like and has heavy tails. We show that DIMER can be computed easily by numerical integration. In our simulation studies, we show that DIMER closely achieves nominal coverage, unlike the Wald method and the method of Hayya et al. (1975). DIMER is also much faster computationally than bootstrap methods, which is important in large cohort studies, where the model is a nonlinear logistic regression based on samples of sizes in the tens of thousands or more.

In Section 2 we describe the methodology, while Section 3 compares various methods via simulation studies. Section 4 describes the analysis of the NIH-AARP Study. Simulations based on the actual data reinforce the conclusions of the simulations in Section 3 and shed more light on the data analysis. Technical details, proofs, definitions and additional simulations are given in the Supplementary Material.

2. Methodology

2.1 Basic Definitions

Consider two random variables T₁ and T₂ which have density functions f₁{(t₁ − μ₁)/υ₁} and f₂{(t₂ − μ₂)/υ₂}, respectively, with means μ₁ and μ₂ and standard deviations υ₁ and υ₂. In other words, f₁(x) and f₂(x) are the density functions of the standardized versions of T₁ and T₂, respectively. Let F₁(·) and F₂(·) denote the corresponding distribution functions. We are interested in making inference for the ratio μ₁/μ₂. We will outline a series of cases where it is possible to compute easily the cumulative distribution function of r̂ = T₁/T₂. All proofs are given in the Supplementary Material Section S.2.

2.2 Independent Case

Suppose that T₁ and T₂ are independent.

Lemma 1

Define

$$g(z\mid x,{\mu }_1,{\mu }_2,{\upsilon }_1,{\upsilon }_2)=\{\begin{array}{ll}(1-F_1[\{x({\mu }_2+{\upsilon }_{2}z)-{\mu }_1\}/{\upsilon }_1])f_2(z)\exp (z^2)& \mathrm{if}z\le -{\mu }_2/{\upsilon }_2,\\ F_1[\{x({\mu }_2+{\upsilon }_{2}z)-{\mu }_1\}/{\upsilon }_1]f_2(z)\exp (z^2)& \mathrm{if}z>-{\mu }_2/{\upsilon }_2.\end{array}$$

Then the cumulative distribution function of r̂ = T₁/T₂ is given by

$$\mathrm{pr}(\widehat{r}\le x)={\mathit{\int }}_{-\infty }^{\infty }g(z\mid x,{\mu }_1,{\mu }_2,{\upsilon }_1,{\upsilon }_2)\exp (-z^2)\mathit{dz},$$

a quantity that is easily computed by Gauss-Hermite quadrature.

In Lemma 1 x denotes a value of r̂ and z denotes a value of T₂, and similarly in Sections 2.3-2.4.

If the parameters υ₁ and υ₂ are unknown, we can apply Lemma 1 using their estimated values. However, we have found that a more numerically efficient approximation can be developed in the case of normally distributed T₁ and T₂. We present this result in the following setting. Suppose the estimated variances are ${\widehat{\upsilon }}_1^2$ and ${\widehat{\upsilon }}_2^2$ which are independent of each other, and independent of T₁ and T₂, and have degrees of freedom d₁ and d₂, respectively. Thus, both (T₁ − μ₁)/υ̂₁ and (T₂ − μ₂)/υ̂₂ follow the t-distribution with d₁ and d₂ degrees of freedom, respectively. In addition, assume that d = min(d₁, d₂) increases to infinity, which is implied when the sample sizes increase to infinity. Suppose that ${\widehat{\upsilon }}_1^2={\upsilon }_1^2+O_p(d_1^{-1/2})$ and ${\widehat{\upsilon }}_2^2={\upsilon }_2^2+O_p(d_2^{-1/2})$. Then we have the following lemma.

Lemma 2

With an error of order O_p(d^−1/2), g(z∣x, μ₁, μ₂, ${\widehat{\upsilon }}_1^2$, ${\widehat{\upsilon }}_2^2$) defined in Lemma 1 can be approximated by

$$h(z\mid x,{\mu }_1,{\mu }_2,{\widehat{\upsilon }}_1^2,{\widehat{\upsilon }}_2^2)=\{\begin{array}{ll}(1-F_{t,d_1}[\{x({\mu }_2+{\widehat{\upsilon }}_{2}z)-{\mu }_1\}/{\widehat{\upsilon }}_1])f_{t,d_2}(z)\exp (z^2)& \mathrm{if}z\le -{\mu }_2/{\widehat{\upsilon }}_2,\\ F_{t,d_1}[\{x({\mu }_2+{\widehat{\upsilon }}_{2}z)-{\mu }_1\}/{\widehat{\upsilon }}_1]f_{t,d_2}(z)\exp (z^2)& \mathrm{if}z>-{\mu }_2/{\widehat{\upsilon }}_2,\end{array}$$

where f_t,d(·) and F_t,d(·) are the t-density with d degrees of freedom and the corresponding cumulative distribution function, respectively.

2.3 Dependent Case of Two Normally Distributed Variables with Known Covariance Matrix

Suppose now that (T₁, T₂) are jointly normally distributed with means (μ₁, μ₂), variances ( ${\upsilon }_1^2$, ${\upsilon }_2^2$), covariance υ₁₂ and suppose that ( ${\upsilon }_1^2$, ${\upsilon }_2^2$, υ₁₂) are known. Let ϕ(·) and Φ(·) denote the standard normal density and distribution function.

Lemma 3

Define g(z∣x, μ₁, μ₂, ${\upsilon }_1^2$, ${\upsilon }_2^2$, υ₁₂) as follows. If z ≤ −μ₂/υ₂, then

$$g(z\mid x,{\mu }_1,{\mu }_2,{\upsilon }_1^2,{\upsilon }_2^2,{\upsilon }_{12})={(2\pi )}^{-1/2}(1-\mathrm{\Phi }[\{x({\mu }_2+{\upsilon }_{2}z)-({\mu }_1+z{\upsilon }_{12}/{\upsilon }_2)\}{\upsilon }_2/\sqrt{{\upsilon }_1^2{\upsilon }_2^2-{\upsilon }_{12}^2])}\exp (z^2/2).$$

If z > −μ₂/υ₂, then

$$g(z\mid x,{\mu }_1,{\mu }_2,{\upsilon }_1^2,{\upsilon }_2^2,{\upsilon }_{12})={(2\pi )}^{-1/2}\mathrm{\Phi }[\{x({\mu }_2+{\upsilon }_{2}z)-({\mu }_1+z{\upsilon }_{12}/{\upsilon }_2)\}{\upsilon }_2/\sqrt{{\upsilon }_1^2{\upsilon }_2^2-{\upsilon }_{12}^2]}\exp (z^2/2).$$

Then the distribution function of r̂ is

$$\mathrm{pr}(\widehat{r}\le x)={\mathit{\int }}_{-\infty }^{\infty }g(z\mid x,{\mu }_1,{\mu }_2,{\upsilon }_1^2,{\upsilon }_2^2,{\upsilon }_{12})\exp (-z^2)\mathit{dz},$$

which again can be computed by Gauss-Hermite quadrature.

Of course, when υ₁₂ = 0, Lemma 3 is a special case of Lemma 1.

2.4 Dependent Case of Two Normally Distributed Variables with Estimated Covariance Matrix

Here we discuss the cumulative distribution of the ratio r̂ = T₁/T₂ when T₁ and T₂ are jointly normally distributed with jointly estimated variance and covariance which have the same number of degrees of freedom d, and these estimates are independent of T₁ and T₂. These are the same assumptions noted in Fieller (1954). Define the estimates of the variances and covariance of T₁ and T₂ as ${\widehat{\upsilon }}_1^2$, ${\widehat{\upsilon }}_2^2$ and υ̂₁₂. Let $\eta ={\upsilon }_{12}/{\upsilon }_2^2$. For fixed η, write W = T₁ − ηT₂, Then W and T₂ are independent. In addition, if ${\widehat{\upsilon }}_1^2$, ${\widehat{\upsilon }}_2^2$ and υ̂₁₂ are computed from the sample covariance matrix of normal random variables from a sample of size d + 1, then we also have that T₁ − ηT₂ and T₂ are independent of their estimated variances ${\widehat{\upsilon }}_1^2-2\eta {\widehat{\upsilon }}_{12}+{\eta }^2{\widehat{\upsilon }}_2^2$ and υ̂₂, which are independent of each other and also have d degrees of freedom.

We use the following algorithm, based on the approximation used in Section 2.2. Under our assumptions, the variables $Z_1=\{(T_1-\eta T_2)-({\mu }_1-\eta {\mu }_2)\}/\sqrt{{\widehat{\upsilon }}_1^2-2\eta {\widehat{\upsilon }}_{12}+{\eta }^2{\widehat{\upsilon }}_2^2}$ and Z₂ = (T₂ − μ₂)/υ̂₂ are independent and both have t-distributions with d degrees of freedom. As in Lemma 2, we then make the approximation that the density of (T₁, T₂), having fixed the estimated covariance matrix, is approximately

$${\widehat{\upsilon }}_2^{-1}{({\widehat{\upsilon }}_1^2-2\eta {\widehat{\upsilon }}_{12}+{\eta }^2{\widehat{\upsilon }}_2^2)}^{-1/2}{f}_{t,d}[\{(t_1-\eta t_2)-({\mu }_1-\eta {\mu }_2)\}/\sqrt{{\widehat{\upsilon }}_1^2-2\eta {\widehat{\upsilon }}_{12}+{\eta }^2{\widehat{\upsilon }}_2^2]}{f}_{t,d}\{(t_2-{\mu }_2)/{\widehat{\upsilon }}_2\}.$$

If z ≤ −μ₂/υ̂₂, define

$$\begin{array}{l}g(z\mid x,{\mu }_1,{\mu }_2,{\widehat{\upsilon }}_1^2,{\widehat{\upsilon }}_2^2,{\widehat{\upsilon }}_{12},\eta )\\ =\left(1-F_{t,d}\left[\{(x-\eta )({\mu }_2+{\upsilon }_{2}z)-({\mu }_1-\eta {\mu }_2)\}/\sqrt{{\widehat{\upsilon }}_1^2-2\eta {\widehat{\upsilon }}_{12}+{\eta }^2{\widehat{\upsilon }}_2^2}\right]\right)f_{t,d}(z)\exp (z^2),\end{array}$$

while if z > −μ₂/υ̂₂, define

$$\begin{array}{l}g(z\mid x,{\mu }_1,{\mu }_2,{\widehat{\upsilon }}_1^2,{\widehat{\upsilon }}_2^2,{\widehat{\upsilon }}_{12},\eta )\\ =F_{t,d}\left[\{(x-\eta )({\mu }_2+{\upsilon }_{2}z)-({\mu }_1-\eta {\mu }_2)\}/\sqrt{{\widehat{\upsilon }}_1^2-2\eta {\widehat{\upsilon }}_{12}+{\eta }^2{\widehat{\upsilon }}_2^2}\right]f_{t,d}(z)\exp (z^2).\end{array}$$

Then, using the same device as in Lemma 2, we have that

$$\mathrm{pr}(\widehat{r}\le x)={\mathit{\int }}_{-\infty }^{\infty }g(z\mid x,{\mu }_1,{\mu }_2,{\widehat{\upsilon }}_1^2,{\widehat{\upsilon }}_2^2,{\widehat{\upsilon }}_{12},\eta )\exp (-z^2)\mathit{dz}+O_p(d^{-1/2}).$$ (1)

In practice, η is unknown, so we use $\widehat{\eta }={\widehat{\upsilon }}_{12}/{\widehat{\upsilon }}_2^2$ to estimate it.

2.5 Algorithm for Computing the Confidence Interval of Ratios

In Sections 2.2-2.4, we express the distribution function of r̂ as F(x; r) = pr(r̂ ≤ x; r = μ₁/μ₂) when μ₂ ≠ 0. The ratio μ̂₁/μ̂₂ is an estimate of r = μ₁/μ₂, so that we can view F(x; μ̂₁/μ̂₂) as an estimate of the population distribution function F(x; r). Efron (1981) and Benton and Krishnamoorthy (2002) pointed out that if we generate values r̂_i, i = 1,…, m, from F(x; μ̂₁/μ̂₂), we can make inference about r using the distribution of the generated r̂_i’s.

The main difference between our approach and that of Benton and Krishnamoorthy is that instead of generating a larger number of r̂_i’s and then obtaining its percentiles, we compute the percentile of r̂_i directly. Consequently, our method is much faster computationally. Specifically, our simulation results indicate that DIMER usually needs less than 30 iteration steps to obtain the quantile of a distribution, but in Benton and Krishnamoorthy (2002), they used m = 100, 000 r̂_i’s to get the quantiles.

Define the α/2 quantile for F(x; μ̂₁/μ̂₂) as r̂_{α/2∣μ̂1/μ̂2}. Then an approximate 100(1 − α)% confidence interval for r is (r̂_{α/2|μ̂1/μ̂2}, r̂_{1−α/2| μ̂1/μ̂2}). Here we give the steps of our iterative, bisection-based algorithm to obtain the quantiles.

• Step 1. Give two initial values of r̂_{α/2∣μ̂1/μ̂2} as r̂_α1 < 0 < r̂_α2 and both have sufficiently large absolute values to make sure that r̂_{α/2∣μ̂1/μ̂2} is inside the interval (r̂_α1, r̂_α2). How we did this is described in the Supplementary Material Appendix S.4. Our method, being based on bisection, is not sensitive to these starting values.

• Step 2. Apply the Gauss-Hermit quadrature to the cumulative distribution function of r̂ to obtain c_α/2 = pr{r̂ ≤ (r̂_α1 + r̂_α2)/2}. If c_α/2 < α/2, let r̂_α1 = (r̂_α1 + r̂_α2)/2; if c_α/2 > α/2, let r̂_α2 = (r̂_α1 + r̂_α2)/2; if c_α/2 = α/2, stop the iteration and let r̂_{α/2∣μ̂1/μ̂2} = (r̂_α1 + r̂_α2)/2.

• Step 3. Repeat Step 2 until c_α/2 is close to α/2 and/or the difference |r̂_α2 − r̂_α1| is negligible. Then we have r̂_{α/2∣μ̂1/μ̂2} = (r̂_α1 + r̂_α2)/2, the lower limit of interval.

• Step 4. Repeat Steps 1–3 to obtain r̂_{1−α/2∣μ̂1/μ̂2}, the upper limit of the interval.

3. Simulations

3.1 Overview

We performed simulations on two simple linear regression models. The first (Section 3.4) is to illustrate an application of the formulas in Section 2.2 where the two variables are independent. The second (in Supplementary Material Appendix Section S.6) is an example to demonstrate the performance of our method developed in Section 2.4 when the two variables are dependent. In both simulations, some other possible methods are outlined and compared with DIMER. Since the dependent case is developed with the normality assumption, it is important to evaluate how sensitive DIMER is to the violation of this assumption. Therefore, we also considered such a case in the second part of our simulations.

3.2 Comments Upon and Applications of Fieller’s Intervals

Fieller’s interval, defined in Supplementary Material Section S.1, is sometimes of infinite length, being either the entire real line or the union of two disconnected infinite length intervals, e.g., when the denominator of the ratio is not significantly different from zero.

Fieller’s intervals have been used in a variety of contexts. Here are three cases, the first two of which are illustrated in our simulations. The simulation of the first case appears here, while the second in the Supplementary Material, Section S.6.

• In a slope ratio assay (Finney, 1978; Hubert, 1984; Redmond, 2005c), data are fit to a standard and treatment, observing Y_S = α_S + X_Sβ_S + ε_S for the standard, while the treatment is fit to the model Y_T = α_T + X_Tβ_T + ε_T. The relative potency ρ is a function of β_T/β_S, where the estimates of β_T and β_S are independent. In a common setting, it is assumed that α_S = α_T but the doses X_S = X_T = X, and by centering X the described model holds with different intercepts. This is an example of two independent slope estimates.

• In a radioimmunoassay (Finney, 1978; Redmond, 2005b) with dose denoted by X and response Y, if one is in the linear part of the calibration cure a reasonable model is Y = α + Xβ + ε. The logarithm of ID₅₀, the dose required for 50% of binding inhibition, is given by log(ID₅₀) = α/β. The parameter estimates (α̂, β̂) are generally correlated, and this is an example of estimating the ratio of the intercept to the slope when the parameter estimates are correlated.

• In a parallel line assay (Finney, 1978; Redmond, 2005a), a standard is fit to the linear model Y_S = α_S + X_Sβ + ε_S while the treatment is fit to the model Y_T = α_T + X_Tβ + ε_T : the slope is the same in both, hence parallel line. The log-relative potency in this assay is log(ρ) = (α_T − α_S)/β. In the homoscedastic case, unless X_S = X_T, the estimates of the numerator and denominator are not independent.

In radioimmunoassays, it is often the case that the variance of the responses is proportional to a power θ of the mean, but with 1 < θ < 2. Generalized least squares can then be used to estimate θ (Davidian, et al., 1988), but once the estimates in these examples are obtained, we still have a problem of forming a confidence interval for a ratio of two parameters.

3.3 Comments on Sample Sizes and Parameter Choices

Fieller intervals for a ratio θ₁/θ₂ are of infinite length if the null hypothesis H₀ : θ₂ = 0 cannot be rejected. If the power for rejecting this hypothesis is low, Fieller intervals will have terrible properties. In simulations not reported here, the behavior of the alternative methods is also very poor. If the power for rejecting the hypothesis is essentially 100%, then all the methods will be essentially the same, with minor fluctuations depending on the sample size. The interesting cases lie on the boundary between low and perfect power, e.g., 80%-90% power with Type I error 0.05. Our simulations include settings with low power, perfect power and in between.

In our simulations, which are based on linear regression with error standard deviation υ_ε, we have set the covariates to be Normal(0, 1), and we set υ_ε = 1, so that the standard error of the slope is roughly n^−1/2υ_ε/s_x, where s_x is the sample standard deviation of the covariates. On average, $s_x^{-1}\approx 1.0$, so the standard error of the slope estimate ≈ n^−1/2. Consequently, the sample sizes we have chosen, n = 18, 25 and 50, result in reasonable standard errors that illustrate a range of powers for the test that the slope = 0.0. In Table 2, had we changed υ_ε = 2, 3 and 4, the sample sizes needed to get roughly the same percentage of infinite length Fieller intervals are roughly 60, 130 and 225, respectively. In the Supplementary Material, Table S.4, we show what happens to Table 1 when we set (n, υ_ε) = (55, 2) and (115, 3), showing that roughly the same results apply in this setting.

Table 2.

Confidence intervals for β₂₁ in a simulation study with 2000 replications and true parameter values (β₁₀, β₂₀, β₂₁, ω) = (0.00, 0.00, 1.00, 0.75) for the linear regression model

	Mean of Coverage			Mean of Length			Median of Length			90% Quantile of Length
Method	90% CI	95% CI	99% CI	90% CI	95% CI	99% CI	90% CI	95% CI	99% CI	90% CI	95% CI	99% CI
n1 = n2 = 18, cv(ω̂) = 0.35, cv(λ̂) = 0.35
mean(β̂10, β̂20, β̂21, ω̂) = (0.01, 0.01, 2.85, 0.75), median(β̂10, β̂20, β̂21, ω̂) = (0.01, 0.01, 1.00, 0.75)
IF	83.60	88.60	94.15	4.51	5.38	7.07	1.46	1.74	2.29	4.59	5.47	7.19
HM	86.45	91.65	95.55	2975	3544	4658	1.54	1.83	2.41	4.06	4.84	6.36
NB	93.35	95.10	97.75	74.05	88.24	116.0	4.54	5.41	7.11	105.1	125.2	164.6
PB	93.05	94.55	97.35	1634	1948	2559	3.55	4.23	5.56	94.25	112.3	147.6
FI	91.50	96.10	99.40	∞	∞	∞	2.13	2.97	7.75	∞	∞	∞
DIMER	92.80	96.55	99.55	7.57	15.87	56.16	2.15	3.05	8.28	10.03	25.88	105.1
LR Test	86.55	92.00	97.85	∞	∞	∞	1.92	2.70	7.84	∞	∞	∞
INL–LR	11.8%	18.8%	44.5%
INL–FI	11.8%	18.2%	39.2%
n1 = n2 = 25, cv(ω̂) = 0.28, cv(λ̂) = 0.28
mean(β̂10, β̂20, β̂21, ω̂) = (−0.00, 0.00, 1.17, 0.75), median(β̂10, β̂20, β̂21, ω̂) = (0.00, 0.00, 1.00, 0.75)
IF	85.65	91.40	96.30	2.02	2.40	3.16	1.27	1.51	1.99	3.20	3.81	5.01
HM	89.45	94.15	97.75	5.06	6.03	7.92	1.30	1.55	2.03	2.69	3.20	4.21
NB	93.40	95.55	98.25	42.36	50.47	66.33	2.07	2.47	3.24	45.18	53.84	70.75
PB	93.10	95.50	98.30	53.39	63.62	83.61	1.91	2.28	2.99	35.70	42.54	55.90
FI	91.05	96.50	99.65	∞	∞	∞	1.59	2.11	3.90	5.72	15.03	∞
DIMER	92.40	96.95	99.75	4.53	9.82	35.54	1.62	2.16	4.15	4.64	7.96	61.76
LR Test	88.25	93.90	98.75	∞	∞	∞	1.49	1.96	3.65	5.14	11.50	∞
INL–LR	4.5%	8.0%	24.6%
INL–FI	4.2%	7.1%	20.2%
n1 = n2 = 50, cv(ω̂) = 0.19, cv(λ̂) = 0.20
mean(β̂10, β̂20, β̂21, ω̂) = (0.00, 0.00, 1.04, 0.75), median(β̂10, β̂20, β̂21, ω̂) = (−0.00, 0.01, 1.00, 0.75)
IF	89.20	93.00	97.25	1.19	1.41	1.86	0.89	1.07	1.40	2.00	2.38	3.13
HM	90.80	95.30	98.45	0.99	1.17	1.54	0.89	1.06	1.39	1.43	1.70	2.23
NB	93.00	95.60	98.35	2.57	3.06	4.02	1.01	1.20	1.58	2.33	2.77	3.64
PB	92.75	96.10	98.65	3.79	4.52	5.93	1.00	1.19	1.57	2.19	2.61	3.43
FI	91.30	95.80	99.10	∞	∞	∞	0.97	1.21	1.77	1.73	2.28	4.13
DIMER	91.55	96.05	99.10	1.16	1.52	3.70	0.98	1.22	1.81	1.75	2.31	4.23
LR Test	90.20	95.00	98.95	∞	∞	∞	0.95	1.18	1.73	1.68	2.20	3.98
INL–LR	0.1%	0.4%	1.9%
INL–FI	0.1%	0.2%	1.3%

Y_1i = β₁₀ + X_1iω + ε_1i; Y_2j = β₂₀ + β₂₁X_2jω + ε_2j. “INL–LR” depicts the % of times that the interval by the likelihood ratio test was of infinite length, and “INL–FI” depicts the % of times that Fieller’s interval was infinite length, either the entire real line or two infinite length disconnected intervals.

Here the acronyms are IF–Inverse Fisher score method, HM–Hayya’s Method, NB–Nonparametric Bootstrap, PB–Parametric Bootstrap, FI–Fieller’s Interval, DIMER–Direct Integral Method for Ratios and LR Test—Likelihood ratio test.

Table 1.

Confidence intervals for β₂₁ in a simulation study with 2000 replications and true parameter values (β₁₀, β₂₀, β₂₁, ω) = (0.00, 0.00, 1.00, 1.00) for the linear regression model

	Mean of Coverage			Mean of Length			Median of Length			90% Quantile of Length
Method	90% CI	95% CI	99% CI	90% CI	95% CI	99% CI	90% CI	95% CI	99% CI	90% CI	95% CI	99% CI
n1 = n2 = 18, cv(ω̂) = 0.26, cv(λ̂) = 0.26
mean(β̂10, β̂20, β̂21, ω̂) = (0.01, 0.01, 1.10, 1.00), median(β̂10, β̂20, β̂21, ω̂) = (0.01, 0.01, 1.00, 1.00)
IF	84.05	89.40	94.60	1.63	1.95	2.56	1.09	1.30	1.71	2.83	3.38	4.44
HM	88.50	92.90	96.70	1.74	2.08	2.73	1.15	1.37	1.80	2.31	2.75	3.61
NB	92.15	94.50	97.75	20.66	24.62	32.35	1.67	1.98	2.61	31.39	37.40	49.15
PB	92.00	94.20	97.35	38.84	46.28	60.83	1.49	1.78	2.34	22.75	27.10	35.62
FI	89.85	95.05	99.35	∞	∞	∞	1.39	1.80	3.08	4.28	8.25	∞
DIMER	91.45	95.90	99.50	2.69	4.92	63.53	1.43	1.88	3.35	3.74	6.12	37.32
LR Test	86.50	92.00	97.85	∞	∞	∞	1.27	1.65	2.78	3.72	6.50	∞
INL–LR	3.2%	6.6%	17.3%
INL–FI	2.9%	5.7%	14.8%
n1 = n2 = 25, cv(ω̂) = 0.21, cv(λ̂) = 0.21
mean(β̂10, β̂20, β̂21, ω̂) = (−0.00, 0.00, 1.05, 1.00), median(β̂10, β̂20, β̂21, ω̂) = (0.00, 0.00, 1.00, 1.00)
IF	86.15	92.15	96.50	1.35	1.60	2.11	0.95	1.13	1.49	2.17	2.59	3.41
HM	90.15	94.75	98.20	1.12	1.33	1.75	0.97	1.16	1.52	1.65	1.97	2.59
NB	92.55	95.30	98.45	10.25	12.21	16.05	1.17	1.40	1.83	7.55	8.99	11.82
PB	92.55	95.45	98.40	7.23	8.61	11.31	1.13	1.35	1.77	3.84	4.57	6.01
FI	90.15	95.90	99.60	∞	∞	∞	1.10	1.38	2.12	2.17	3.02	6.92
DIMER	91.15	96.30	99.70	1.78	2.75	10.96	1.12	1.42	2.23	2.20	3.06	6.74
LR Test	88.25	93.85	98.70	∞	∞	∞	1.04	1.31	1.99	2.02	2.79	6.15
INL–LR	0.6%	1.2%	5.9%
INL–FI	0.5%	1.0%	4.5%
n1 = n2 = 50, cv(ω̂) = 0.14, cv(λ̂) = 0.15
mean(β̂10, β̂20, β̂21, ω̂) = (0.00, 0.00, 1.02, 1.00), median(β̂10, β̂20, β̂21, ω̂) = (−0.00, 0.01, 1.00, 1.00)
IF	90.00	93.10	97.25	0.94	1.12	1.47	0.67	0.79	1.04	1.38	1.64	2.15
HM	90.65	95.50	98.65	0.71	0.84	1.11	0.67	0.79	1.04	0.95	1.13	1.48
NB	92.15	95.40	98.55	0.84	1.00	1.32	0.70	0.84	1.10	1.10	1.31	1.72
PB	92.15	96.10	98.65	0.80	0.95	1.25	0.71	0.84	1.11	1.09	1.29	1.70
FI	91.20	95.75	99.00	0.76	0.93	1.35	0.70	0.86	1.19	1.04	1.29	1.90
DIMER	91.50	95.80	99.10	0.77	0.94	1.36	0.71	0.87	1.21	1.05	1.31	1.94
LR Test	90.20	95.00	98.95	0.74	0.91	∞	0.69	0.83	1.16	1.01	1.26	1.85
INL–LR	0.0%	0.0%	0.1%
INL–FI	0.0%	0.0%	0.0%

Y_1i = β₁₀ + X_1iω + ε_1i; Y_2j = β₂₀ + β₂₁ X_2jω + ε_2j.”INL–LR” depicts the % of times that the interval by the likelihood ratio test was of infinite length, and “INL–FI” depicts the % of times that Fieller’s interval was infinite length, either the entire real line or two infinite length disconnected intervals.

Here the acronyms are IF–Inverse Fisher score method, HM–Hayya’s Method, NB–Nonparametric Bootstrap, PB–Parametric Bootstrap, FI–Fieller’s Interval, DIMER–Direct Integral Method for Ratios and LR Test—Likelihood ratio test.

3.4 Linear Model When the Two Estimates are Independent

3.4.1 Setup

Consider the 2-group linear regression model

$$\begin{array}{l}{Y}_{1i}={\beta }_{10}+X_{1i}{\beta }_{11}+{\epsilon }_{1i},i=1,\dots ,n_1;\\ Y_{2j}={\beta }_{20}+X_{2j}{\beta }_{21}+{\epsilon }_{2j},j=1,\dots ,n_2,\end{array}$$

where (Y_1i, X_1i) and (Y_2j, X_2j) are the same outcomes and predictors from different populations. See Section 3.2 for an example. Also ε_1i and ε_2j are independently normally distributed with mean zero and variances ${\upsilon }_{{\epsilon }_1}^2$ and ${\upsilon }_{{\epsilon }_2}^2$, respectively. Our interest is in the ratio of the two slopes β₂₁/β₁₁.

The model can be rewritten as follows in order to use a simple expression for the ratio:

$$\begin{array}{r}{Y}_{1i}={\beta }_{10}+X_{1i}\omega +{\epsilon }_{1i},i=1,\dots ,n_1;\\ Y_{2j}={\beta }_{20}+{\beta }_{21}{X}_{2j}\omega +{\epsilon }_{2j},j=1,\dots ,n_2.\end{array}$$ (2)

Then the ratio of the slopes now is β₂₁ and ω is the slope for the first group.

Our interest is to construct a confidence interval for β₂₁. Let (β̂₂₁, ω̂) denote the maximum likelihood estimate (mle) of (β₂₁, ω), and define λ = β₂₁ω and its estimate λ̂ = β̂₂₁ω̂. Both (λ̂ − λ)/υ̂_λ and (ω̂ − ω)/υ̂_ω follow independent t-distributions with degrees of freedom n₂ − 2 and n₁ − 2, respectively, where υ̂_λ and υ̂_ω are corresponding estimated standard deviations.

The estimated cumulative distribution function of β̂₂₁ is obtained as in Section 2.2. We can then apply the DIMER algorithm in Section 2.5 to obtain confidence intervals. To compare with other methods, in Section 3.4.2 we outline an application of Fieller’s interval. In addition, we apply the Wald interval by inverting the Fisher score matrix, Hayya’s method, the nonparametric bootstrap, the parametric bootstrap, and the likelihood ratio test; see the details in the Supplementary Material, Appendices S.1, S.3 and S.5.

3.4.2 Comparison with the Fieller’s Interval

To form a confidence interval for β₂₁, one common method in practice is Fieller’s interval. However, in this linear regression setting, it cannot be applied directly since ${\widehat{\upsilon }}_{\widehat{\omega }}^2$ and ${\widehat{\upsilon }}_{\widehat{\lambda }}^2$ are obtained independently. In this case, by the Welch-Satterthwaite equation (Satterthwaite, 1946; Welch, 1947), the degrees of freedom of ( ${\widehat{\upsilon }}_{\widehat{\lambda }}^2+{\beta }_{21}^2{\widehat{\upsilon }}_{\widehat{\omega }}^2$) are approximately given by $d_F={({\widehat{\upsilon }}_{\widehat{\lambda }}^2+{\beta }_{21}^2{\widehat{\upsilon }}_{\widehat{\omega }}^2)}^2/\{({{\widehat{\upsilon }}_{\widehat{\lambda }}^2)}^2/(n_2-2)+{({\beta }_{21}^2{\widehat{\upsilon }}_{\widehat{\omega }}^2)}^2/(n_1-2)\}.$ We use β̂₂₁ instead of β₂₁ in the expression to obtain the estimated degrees of freedom $d_F^{\ast }={({\widehat{\upsilon }}_{\widehat{\lambda }}^2+{\widehat{\beta }}_{21}^2{\widehat{\upsilon }}_{\widehat{\omega }}^2)}^2/\{{({\widehat{\upsilon }}_{\widehat{\lambda }}^2)}^2/(n_2-2)+{({\widehat{\beta }}_{21}^2{\widehat{\upsilon }}_{\widehat{\omega }}^2)}^2/(n_1-2)\}$. Then we have $a={\widehat{\omega }}^2-t_{d_F^{\ast },\alpha /2}^2{\widehat{\upsilon }}_{\widehat{\omega }}^2$, b = −2ω̂λ̂ and $c={\widehat{\lambda }}^2-t_{d_F^{\ast },\alpha /2}^2{\widehat{\upsilon }}_{\widehat{\lambda }}^2$ used in the Supplementary Material. Here ρ = 0 since ω̂ and λ̂ are independent.

3.4.3 Simulation Results

Our simulations for model (2) compare the seven methods mentioned in Section 3.4.1. For simplicity, in all settings, we first fixed ${\upsilon }_{{\epsilon }_1}^2={\upsilon }_{{\epsilon }_2}^2=1$, and without loss of generality, let the intercepts β₁₀ and β₂₀ be 0. Supplementary Material Table S.4 gives some additional results when ${\upsilon }_{{\epsilon }_1}^2={\upsilon }_{{\epsilon }_2}^2=2$ and 3. We generated X_1i and X_2j independently from the standard normal distribution.

We considered two parameter configurations: (β₁₀, β₂₀, β₂₁, ω) = (0, 0, 1, 1), (0, 0, 1, 0.75). For each parameter setting, we report simulation results for (n₁, n₂) = (18, 18), (25, 25), (50, 50) with 2000 runs. In our experience in linear regression cases, and with these effect sizes, sample sizes higher than that typically lead to good numerical performances for all methods. Following Efron and Tibshirani (1994, p. 52), we used B = 400 bootstrap replications for all the bootstrap results reported in this article.

The results for the first parameter configuration (β₁₀, β₂₀, β₂₁, ω) = (0, 0, 1, 1) are given in Table 1 while Table 2 presents the results for setting (β₁₀, β₂₀, β₂₁, ω) = (0, 0, 1, 0.75). QQ plots (not shown here) comparing the quantiles of β̂₂₁ with the quantiles of the standard normal distribution in the two parameter configurations with n₁ = n₂ = 18 clearly show that for small to moderate sample sizes, normal approximations are not appropriate.

Table 2 shows that when n₁ = n₂ = 18 the empirical mean of β̂₂₁ is 2.85 when the true value is 1.00. The reason for this difference is that β̂₂₁ follows a Cauchy like distribution, and one of characteristics for this distribution is that it has severely heavy tails. For example, the maximum of the absolute values of β̂₂₁ reached 3, 138 over the 2,000 runs in this case. Therefore, some severe outliers dramatically affected the empirical mean. In sharp contrast, the empirical median of β̂₂₁ is 1.00.

Table 2 also displays the percentage of times the Fieller and likelihood ratio confidence intervals have infinite length.

The inverse Fisher information matrix algorithm has the lowest coverage probabilities. Hayya’s method and the likelihood ratio test also have sub-nominal coverage probabilities when the sample sizes are small. Moreover, the latter has a positive probability to get infinite length. The performance of the two bootstrap methods is acceptable when the sample sizes are relatively large. When the sample sizes are small to moderate, the coverage rate of the bootstrap methods for the 90% confidence intervals are higher than 90%, while the coverage rate of the 99% confidence intervals is lower than 99%.

Fieller’s interval has good performance overall in coverage. Here we focus on cases that the sample sizes are small and moderate (n₁ = n₂ = 18 and n₁ = n₂ = 25), where Fieller’s interval can be the real line or otherwise of infinite length. The inverse Fisher information method produced the shortest confidence interval lengths, not surprising, since its coverage rates are below the nominal values. Hayya’s method remains stable but has a low coverage when the sample sizes are small. Compared with the two bootstrap methods, our method obviously has markedly shorter lengths in the 90% and 95% confidence intervals when the sample sizes are small and moderate, especially when (n₁, n₂) = (18, 18). When the sample sizes are small, DIMER and Fieller’s interval have similar median and interquartile ranges of lengths, but our method is much shorter in terms of mean and 90^th percentile of length.

4. Empirical Example and Further Simulations

4.1 Method and Data Analysis

The HEI-2005 and the NIH-AARP data available to us were described in Section 1. The sample sizes were 4,300 males and 1,916 females. Let H (x) = exp(x)/{1 + exp(x)} be the logistic distribution function. Let ℓ = 1, 2 denote men and women, respectively. Let Y_iℓ denote the binary outcome of colorectal cancer for person i = 1, …, n_ℓ in sample ℓ and let X_ijℓ for j = 1, …, J = 12 denote the HEI-2005 score for the j^th dietary component. The traditional HEI-2005 analysis then posits a model $\mathrm{pr}(Y_{i\ell }=1\mid X_{i1\ell },\dots ,X_{\mathit{iJ\ell }})=H({\alpha }_{\ell }+{\beta }_{\ell }{\sum }_{j=1}^J{X}_{\mathit{ij\ell }})$, in other words, the HEI-scores are equally weighted. Notice here that the same predictor, ${\sum }_{j=1}^J{X}_{\mathit{ij\ell }}$, is to be used both for men and for women. In our case, we allow for the possibility that the predictor is the same in both populations, but the scores are weighted to be based on the data, so that our model is

$$\mathrm{pr}(Y_{i\ell }=1\mid X_{i1\ell },\dots ,X_{\mathit{iJ\ell }})=H({\alpha }_{\ell }+{\beta }_{\ell }{\sum }_{j=1}^J{\omega }_j{X}_{\mathit{ij\ell }}),$$ (3)

where the weights (ω₁, …, ω_J) are estimated through the data. The model as such is not identified, but if we make the restriction that β₁ = −1, then it is identified: the negative value is because higher HEI-2005 scores, i.e., better diets, lead to lower rates of colon cancer. Thus, with β₁ = −1, (3) becomes

$$\begin{array}{lll}\mathrm{pr}(Y_{i1}=1\mid X_{i11},\dots ,X_{\mathit{iJ}1})& =& H({\alpha }_1-{\sum }_{j=1}^J{\omega }_j{X}_{\mathit{ij}1});\\ \mathrm{pr}(Y_{i2}=1\mid X_{i12},\dots ,X_{\mathit{iJ}2})& =& H({\alpha }_2+{\beta }_2{\sum }_{j=1}^J{\omega }_j{X}_{\mathit{ij}2}).\end{array}$$

If we write $T_{\mathit{i\ell }}={\sum }_{j=1}^J{\omega }_j{X}_{\mathit{ij\ell }}$, then we see that if the relative risk in men for changing T_iℓ is R, the same change in women has a relative risk R^−β₂. Hence we wish to form a confidence interval for β₂. We fit model (3) by maximum likelihood, and the asymptotic covariance matrix Σ of (β₂, ω₁, …, ω_J)^T was estimated using the Fisher information matrix.

To see how this relates to the Fieller problem, let ω = (ω₁, …, ω_J)^T, λ = β₂ω, and e be the J × 1 vector of ones. From Σ and the delta method, the asymptotic covariance matrix for (ω̂, λ̂) can be constructed, and the covariance matrix of (e^T λ, e^T ω) is easily computed. Also, β₂ = e^T λ/e^T ω and β̂₂ = e^T λ̂/e^T ω̂. Thus, we see that β̂₂ is the ratio of two asymptotically normal random variables, and hence DIMER, Fieller’s method, etc. can be applied.

In the NIH-AARP study, the rate of colorectal cancer for men is 0.73%, while it is 0.48% for women. In the data analysis, we found that β̂₂ = −0.747, so that if relative risk of 0.60 for men who improve their diet by a fixed amount, it is 0.68 for women who improve their diet the same amount. Thus, for colorectal cancer, the indication is that men are more susceptible, a well-known fact, and that they will have greater benefit for the same change in diet.

In the top panel of Table 3, we present the confidence intervals for the various methods. We see there that the confidence intervals for the inverse Fisher score method and Hayya’s method are noticeably shorter than the others, the nonparametric bootstrap is quite a bit longer, and the parametric bootstrap, Fieller’s interval, DIMER, and the likelihood ratio test are intermediate. The nonparametric bootstrap does not suggest differences in risk between men and women even at 90% confidence. With the exception of the nonparametric bootstrap, whose intervals we believe are much too long, see Section 4.2, all indications are that the risk for men and women for the same change in diet is statistically significant, with a p-value of < 0.01 for DIMER.

Table 3.

Top Panel: confidence intervals for β₂ with the actual case-control data set of Section 4.1. The estimated values for parameter are (β̂₂, α̂₁, α̂₂) = (−0.747,−1.115,−1.024), ω̂ = (0.030, 0.018, 0.083, 0.033, −0.001, 0.081, 0.094, −0.043, 0.068, −0.020, 0.041, 0.098)^T. Bottom panel: average confidence intervals for β₂ in the simulation study of Section 4.2 for a logistic regression model with 2000 simulated case control data sets.

Data Analysis
Method	90% CI	95% CI	99% CI
IF	(−1.17, −0.33)	(−1.25, −0.25)	(−1.40, −0.09)
HM	(−1.18, −0.35)	(−1.26, −0.27)	(−1.41, −0.11)
NB	(−1.67, 0.18)	(−1.85, 0.35)	(−2.20, 0.70)
PB	(−1.26, −0.23)	(−1.36, −0.13)	(−1.56, 0.06)
FI	(−1.26, −0.33)	(−1.41, −0.24)	(−1.84, −0.02)
DIMER	(−1.26, −0.33)	(−1.41, −0.24)	(−1.84, −0.02)
LR Test	(−1.28, −0.30)	(−1.41, −0.22)	(−1.69, −0.05)
Simulation: Average Confidence Intervals
Method	90% CI	95% CI	99% CI
IF	(−1.17, −0.37)	(−1.24, −0.29)	(−1.39, −0.15)
HM	(−1.18, −0.38)	(−1.26, −0.31)	(−1.40, −0.16)
NB	(−1.46, −0.08)	(−1.59, 0.06)	(−1.85, 0.32)
PB	(−1.26, −0.27)	(−1.36, −0.18)	(−1.55, 0.01)
FI	∞	∞	∞
DIMER	(−1.28, −0.35)	(−1.47, −0.22)	(−2.52, 0.53)
LR Test	(−1.28, −0.36)	(−1.40, −0.29)	(−1.67, −0.14)

Here the acronyms are IF–Inverse Fisher score method, HM–Hayya’s Method, NB–Nonparametric Bootstrap, PB–Parametric Bootstrap, FI–Fieller’s Interval, DIMER–Direct Integral Method for Ratios and LR Test—Likelihood ratio test.

In the next subsection, we study whether the different lengths of the confidence intervals are reproducible in simulations, and through these simulations, which methods attained nominal coverage.

4.2 Simulation

The sample sizes were the same as in the data set, namely 4,300 males and 1,916 females. We used a bootstrap resample of the HEI-2005 scores in the NIH-AARP data as the covariates, separately for men and women, and generated 2,000 data sets with binary outcome data according to the fit to the model (3), the parameter estimates of which are given in the caption to Table 3. The mean confidence intervals across the 2,000 simulations are given in the bottom panel of Table 3. The result reflects the same phenomenon that was observed in the actual data set, namely the inverse Fisher score method and Hayya’s method are noticeably shorter than the others, the nonparametric bootstrap is quite a bit longer, and the parametric bootstrap, DIMER, and the likelihood ratio test are intermediate. As seen in the previous simulations, the mean lengths of Fieller’s interval in this case are infinite for 90%, 95% and 99% intervals.

In Table 4, we show the confidence interval coverage performance of the various methods. The inverse Fisher score method and Hayya’s method both have short confidence intervals generally, but also much less than nominal coverage probability. The likelihood ratio test has longer intervals than the inverse Fisher score and Hayya’s method, but it is still under coverage. The nonparametric bootstrap had by far the longest intervals, and here we see great over coverage. The parametric bootstrap, Fieller’s method and DIMER have close to nominal coverage. For 95% confidence intervals, Fieller’s method was of infinite length for almost 4% of the simulations. In this simulation, the parametric bootstrap performed somewhat better than DIMER, with its confidence intervals being somewhat shorter, although computationally it is, on average, 35 times slower to compute for data sets of this size.

Table 4.

Analysis of the confidence intervals for β₂ in a simulation study of Section 4.2 for a logistic regression model with 2000 simulated data sets. There are 4300 males with 1075 individuals with colorectal cancer and 1916 females with 479 having the disease. “INL–LR” depicts the % of times that the interval by the likelihood ratio test was of infinite length, and “INL–FI” depicts the % of times that Fieller’s interval was infinite length, either the entire real line or two infinite length disconnected intervals.

Mean Coverage			Mean Length			Median Length			90% Quantile Length
Method	90% CI	95% CI	99% CI	90% CI	95% CI	99% CI	90% CI	95% CI	99% CI	90% CI	95% CI	99% CI
IF	84.10	91.65	97.20	0.79	0.95	1.24	0.76	0.91	1.19	0.99	1.18	1.55
HM	83.35	91.25	97.25	0.79	0.95	1.24	0.76	0.91	1.19	0.99	1.18	1.55
NB	96.55	98.45	99.60	1.38	1.65	2.17	1.19	1.41	1.86	2.08	2.48	3.26
PB	92.05	96.15	99.25	0.99	1.18	1.56	0.98	1.17	1.54	1.08	1.29	1.70
FI	87.60	94.30	99.00	∞	∞	∞	0.87	1.09	1.68	1.22	1.61	∞
DIMER	87.70	94.20	98.95	0.94	1.25	3.05	0.87	1.10	1.74	1.23	1.70	4.22
LR Test	86.85	92.80	98.25	0.92	1.11	1.53	0.87	1.05	1.43	1.18	1.44	2.01
INL–LR	0.0%	0.0%	0.0%
INL–FI	1.6%	3.5%	10.4%

Here the acronyms are IF–Inverse Fisher score method, HM–Hayya’s Method, NB–Nonparametric Bootstrap, PB–Parametric Bootstrap, FI–Fieller’s Interval, DIMER–Direct Integral Method for Ratios and LR Test—Likelihood ratio test.

For comparison purposes, the average computational time in these simulations for the Fisher Score, Hayya, nonparametric bootstrap, parametric bootstrap, Fieller’s interval, DIMER, and likelihood ratio test were 0.07, 0.15, 75.94, 55.25, 0.15, 1.02, and 54.29 seconds, respectively. To do a more severe time test, we also generated cohort data similar in size to the NIH-AARP Study data (293, 615 males and 198, 245 females). For one such data set, the computational time for the six former methods (without the likelihood ratio test) was 9.80, 19.25, 8709.00, 3223.44, 19.25, and 20.34 seconds, respectively, indicating that the time of the parametric bootstrap was 159 times larger than that of DIMER.

5. Discussion

We have developed DIMER for constructing confidence intervals for the ratio of two location parameters. The method, based on analytical results and further approximations to account for nuisance parameters, is computationally fast. Our simulations indicated that compared with other methods in the literature, DIMER achieves coverage probabilities close to the nominal levels in all the different scenarios under consideration while providing competitive confidence interval lengths.

While we have no definitive explanation, it is a reasonable conjecture that an important reason why the DIMER method works well is that the distribution of the estimated ratio is heavy tailed. Our DIMER method appeared to be less affected by this problem due to its direct probability computation, although it is not unaffected, see below.

However, there are obvious cases that any of the intervals, including DIMER, may have poor performance. In particular, in the cases that Fieller intervals are of infinite length, we found in our simulations that DIMER intervals also increase in length, sometimes dramatically, especially when the p-value for testing the denominator being 0 or not is large. We found the same thing to happen to the other methods we have discussed: the results were poor, although in some cases better than DIMER. In the case of normality, only Fieller’s interval is guaranteed to achieve its nominal coverage probability, at the potential cost of intervals of infinite length.

All the methods we have considered, other than Fieller’s interval, are first-order correct, i.e., their actual coverage probability is the nominal one +O(n^−1/2). There is a literature on second order correctness, i.e., nominal level +O(n⁻¹), such as Laplace approximations, second order bootstrap, etc. It would be interesting to see how and whether these methods can be applied to our problem of finding a confidence interval for the ratio of two parameters. The properties of such methods such as confidence interval lengths and actual coverage in the settings we have considered are not at all clear.

Supplementary Material

Supp Material. 6. Supplementary Material.

The NIH-AARP Study of Diet and Health references in Section 4.1 can be accessed from the National Cancer Institute, but they require a proposal and a Material Transfer Agreement. The Supplementary Material includes a representative simulated data set for men and for women as in Section 4.2. The Supplementary Material referenced in Sections 2.1-2.5 and Sections 3.1-3.4 also includes a simulation when the slope and the intercept are dependent, a series of additional simulations including two with skew-normal regression errors, and definition of the HEI-2005 scores. Matlab programs implementing our method are also available at the Biometrics web site on Wiley Online Library.