# Improved Small Sample Inference on the Ratio of Two Coefficients of Variation of Two Independent Lognormal Distributions

Authors: A. Wong; L. Jiang
Journal: Journal of Probability and Statistics
 (2019)

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## Abstract

Without the ability to use research tools and procedures that yield consistent measurements, researchers would be unable to draw conclusions, formulate theories, or make claims about generalizability of their results. In statistics, the coefficient of variation is commonly used as the index of reliability of measurements. Thus, comparing coefficients of variation is of special interest. Moreover, the lognormal distribution has been frequently used for modeling data from many fields such as health and medical research. In this paper, we proposed a simulated Bartlett corrected likelihood ratio approach to obtain inference concerning the ratio of two coefficients of variation for lognormal distribution. Simulation studies show that the proposed method is extremely accurate even when the sample size is small.


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## Body

## 1. Introduction

In health and medical research, it is common that the variable of interest,X, such as the survival time, takes only positive values and the underlying distribution of this variable is highly skewed to the right. In this case, the frequently assumed normal distribution for X is not suitable. A standard approach to first transform X such that the transformed variable Y=g(X) is normally distributed. Then the existing statistical theories developed for the normal distribution can be applied. For X>0 and the distribution of X is highly skewed to the right, the most common transformation is the logarithmic transformation. In other words, Y=log(X) is normally distributed. Hence, X is lognormally distributed. Detailed review of the theories of the lognormal distribution can be found in Aitchison and Brown [1], and Crow and Simizu [2]. In practice, Fears et al. [3] investigated the variability and reproducibility of hormone assays used by laboratories with the capability of performing large numbers of tests. They assumed the hormone samples used in laboratories are independent lognormally distributed. In this case, it is of special interest to know if each sample yields consistent measurements.The coefficient of variation (τ) is defined as the ratio of the standard deviation to the mean, where the mean is assumed to be non zero. It is an important index for assessment of the reliability of a measuring procedure. Hence, the problem considered in Fears et al. [3] can be viewed as testing if the coefficients of variation used in each laboratory are the same or not.Mathematically, if a random variableX is distributed as lognormal(μ,σ), then Y=log(X) is distributed as normal with mean μ and variance σ2. It is well-known that(1)EX=expμ+σ22,andvarX=exp2μ+σ2expσ2-1.Hence, the coefficient of variation, τ, is(2)τ=varXEX=expσ2-1>0.Nam and Kwon [4] compared various approximate interval estimations of the ratio of two coefficients of variation for independent lognormal distributions. And their simulation results showed that empirical coverage rates of these methods are satisfactorily close to the nominal coverage rate for medium sample size. The aim of this paper is to develop a more accurate method to obtain inference for the ratio of two coefficients of variation for independent lognormal distributions. Moreover, the proposed method can be generalized to test if the coefficients of variation from k independent lognormal distributions are heterogeneous.The rest of the paper is organized as follows. Section2 reviewed the existing methods for obtaining inference concerning the ratio of two coefficients of variation from independent lognormal distribution. The simulated Bartlett corrected likelihood method is proposed in Section 3. A real data example is presented in Section 4 to illustrate the application of the method discussed in this paper. Simulation studies are performed to compare the accuracy of the methods discussed in this paper in Section 5. Extension to testing for homogeneity of coefficients of variations from k independent lognormal distributions is discussed in Section 6. Some concluding remarks are recorded in Section 7.


## 2. Existing Methods for Inference on the Ratio of Two Coefficients of Variation of Two Independent Lognormal Distributions

Let(xi1,…,xini) be the ith sample from the lognormal(μi,σi) distribution, where i=1,…,k. Then (yi1,…,yini)=(log⁡xi1,…,log⁡xini) is the ith sample from the normal distribution with mean μi and variance σi2. From (2), the ith coefficient of variation is τi=expσi2-1. Nam and Kwon [4] compared four methods in obtaining confidence intervals for ψ=τ1/τ2. The following is the summary of the methods discussed in Nam and Kwon [4]:(1)
Wald type methodLet the observed test statistic be(3)zWψ=ψ^-ψvar^ψ^whereψ^=τ^1/τ^2, τ^i=expσ^i2-1, σ^i2=1/ni∑j=1ni‍log⁡xij-∑h=1ni‍log⁡xih/ni2, and var^(ψ^)=n1ψ4(1+τ^22)σ^222+n2(1+τ^12)σ^122/2n1n2(τ^1τ^2)2. Then ZW(ψ) is asymptotically distributed as standard normal distribution. The significance function of ψ is p(ψ)=ΦzW(ψ), where Φ() is the cumulative distribution function of the standard normal distribution.(2)
Fieller type methodLet the observed test statistic be(4)zFψ=τ^1-ψτ^2var~τ^1+ψ2var~τ^2where(5)var~τ^i=σ^i21+τ^i22niτ^i2i=1,2.ThenZF(ψ) is also asymptotically distributed as standard normal distribution. The significance function of ψ is p(ψ)=ΦzF(ψ).(3)
Log methodLet the observed test statistic be(6)zLψ=log⁡τ^1-log⁡τ^2-log⁡ψvar~log⁡τ^1+var~log⁡τ^2where(7)var~log⁡τ^i=var~τ^iτ^i2i=1,2.ThenZL(ψ) is also asymptotically distributed as standard normal distribution. The significance function of ψ is p(ψ)=ΦzL(ψ).(4)
Method of variance estimates recovery (MOVER)This is a method that will directly obtain an approximate(1-α)100% confidence interval for ψ only. Let(8)li=log⁡τ^i-zα/2var~log⁡τ^i,andui=log⁡τ^i+zα/2var~log⁡τ^i.Then an approximate(1-α)100% confidence interval for log⁡ψ is (L,U) where (9)L=log⁡τ^1-log⁡τ^2-log⁡τ^1-l12+u2-log⁡τ^22,U=log⁡τ^1-log⁡τ^2+log⁡τ^1-l12+u2-log⁡τ^22.Thus, an approximate(1-α)100% confidence interval for ψ is exp{L},exp{U}. If var~(log⁡τ^i), for i=1,2, to be the same as that obtained in the Log method, the MOVER method is identical to the Log method. Note that Hasan and Krishamoorthy [5] proposed an improved version of the MOVER method.


## 3. Proposed Method

In this section, we will first review the likelihood based methods and the Bartlett corrected likelihood ratio method. Since the required Bartlett adjustment for the Bartlett corrected likelihood ratio method is very difficult to obtain, a numerical algorithm is proposed to approximate the Bartlett adjustment. Then the methods are applied to obtain inference for the ratio of two coefficients of variaation of two independent lognormal distribution.


### 3.1. Likelihood Based Methods and Bartlett Corrected Likelihood Ratio Method

Let(x1,…,xn) be a sample from a known distribution with probability density function f(·,θ), where θ is a p-dimensional vector of parameters. Let ψ=ψ(θ), which has dimension d<p be the parameter of interest. The log-likelihood function is(10)lθ=lθ;x1,…,xn=∑i=1nlog⁡fxi;θ.Under the regularity conditions stated in Barndorff-Nielsen and Cox [6], we have the standardized maximum likelihood estimate (MLE) statistic θ^-θ′var(θ^)-1(θ^-θ) and the likelihood ratio statistic 2l(θ^)-l(θ) that are asymptotically chi-square distributed with p degrees of afreedom, χp2, where θ^ is the overall MLE, which is the value of θ that maximized l(θ), and var(θ^) is approximately the inverse of the Fisher’s expected information. When the parameter of interest is ψ=ψ(θ), Barndorff-Nielsen and Cox [6] showed that similar statistics can be obtained. The standardized MLE statistic becomes(11)Qψ=ψ^-ψ′varψ^-1ψ^-ψwhere ψ^=ψ(θ^), and var(ψ^) can be approximated by the delta method, which takes the form(12)varψ^≈∂ψθ∂θ′varθ^∂ψθ∂θ.The likelihood ratio statistic is(13)Wψ=2lθ^-lθ~,where θ~ is the constrained MLE, which is obtained by maximizing l(θ) for the given ψ value. Both Q(ψ) and W(ψ) are asymptotically χd2. As defined in Fraser [7], the significance function for ψ is defined as p(ψ)=P(χd2≤q(ψ)) or p(ψ)=P(χd2≤w(ψ)) can be used to obtain inference concerning ψ where q(ψ) and w(ψ) are the observed values of Q(ψ) and W(ψ), respectively. In particular, the (1-α)100% confidence region of ψ is(14)ψ:qψ≤χd,1-α2andψ:wψ≤χd,1-α2,respectively, where χd,1-α2 is the (1-α)100th percentile of χq2.It is well-known that these two asymptotic methods have rate of convergenceO(n-1/2), and they are referred to as the first-order methods. In statistics literature, there exists various adjustments to improve the accuracy of the above methods. In particular, Barndorff-Nielsen [8, 9] introduced the modified signed log-likelihood ratio statistics, a third-order method. However, this method is restricted to scalar parameter of interest only. On the other hand, Bartlett [10] proposed a transformation of the likelihood ratio statistic such that the mean of the transformed statistic matched the mean of the asymptotic distribution. More specifically,(15)W∗ψ=WψBwhere B is the Bartlett adjustment such that EW∗(ψ)=d. And W∗(·) is known as the Bartlett corrected likelihood ratio statistic. An obvious choice of B is(16)B=EW∗ψd.Bartlett [10] showed that the Bartlett corrected likelihood ratio statistic is also asymptotically χd2 distributed and it has rate of convergence O(n-2). Therefore, it is an extremely accurate method. Nevertheless, except in a few well-defined problem, EW∗(ψ) is very difficult to obtain which hinders the use of this method in applied statistics. A review of the Bartlett corrected likelihood ratio method can be found in Barndorff-Nielsen and Cox [6].Although, mathematically, the explicit closed form ofB, or even an asypmptotic expansion of B, is difficult to obtain, we propose the following algorithmic way to obtain EW(ψ) numerically, and hence, an estimated B.Given: (x1,…,xn) is a sample of size n from a distribution with known probability density function f(·;θ).Interest: Inference concerning ψ=ψ(θ).Have: Overall maximum likelihood estimate θ^, the constrained maximum likelihood estimate θ~, and the observed likelihood ratio statistic w(ψ).Step 1: Simulate M samples of data of size n from f(·;θ~).Step 2: For each set of simulated data, obtain the simulated observed likelihood ratio statistic. As a result, we have w1(ψ),…,wM(ψ).Step 3: Calculate(17)w¯ψ=∑i=1MwiψM,which is an estimate of the mean of the likelihood ratio statistic. Hence, we have B^=w¯(ψ)/d.Step 4: The observed simulated Bartlett corrected likelihood ratio statistic is(18)w∗ψ=wψB^,is asymptotically distributed as χd2 with fourth order rate of convergence. Thus, the significance function is p(ψ)=Pχd2≤w∗(ψ), and the (1-α)100% confidence region of ψ is(19)ψ:w∗ψ≤χd,1-α2.As a final note on the proposed algorithm, theoretically, the choice ofM should be as large as possible. However, the larger M is, the more calculations are required to obtain w¯(ψ). Moreover, the more nuisance parameters exist in the model, the larger M has to be. We recommend to use trial-by-error of M until w¯(ψ) is stablized.


### 3.2. Applying Likelihood Based Method to Obtain Inference on the Ratio of Two Coefficients of Variation of Two Independent Log Normal Distribution

LetYi¯=∑j=1ni‍Yij/ni and (ni-1)Si2=∑j=1ni‍(Yij-Yi¯)2. Then Yi¯ is normally distributed with mean μi and variance σi2/n, and (ni-1)Si2/σi2 is χn-12. Moreover, Yi¯ and (ni-1)Si2/σi2 are independent. Hence, inference concerning σi2 will be based on χni-12. Since τi is a function of σi2 only, inference concerning τi will be based on χni-12. Let θ=(σ12,σ22). Then the likelihood function for θ can be written as(20)lθ=lσ12,σ22;s12,s22=-n1-12log⁡σ12-n1-1s122σ12-n2-12log⁡σ22-n2-1s222σ22.It is easy to show that the overall MLE(21)θ^=σ^12,σ^22=s12,s22.Since our parameter of interest is ψ=ψ(θ)=τ1/τ2, where τi=exp{σi2}-1, we have(22)σ12=log⁡ψ2exp⁡σ22-ψ2+1.For a given ψ value, the log-likelihood function in (20) can be expressed as a function of σ22 only, and is(23)lσ22=lσ22;s12,s22-n1-12log⁡log⁡ψ2exp⁡σ22-ψ2+1-n1-1s122log⁡ψ2exp⁡σ22-ψ2+1-n2-12log⁡σ22-n2-1s222σ22.Hence, to solve for the constrained MLE θ~=(σ~12,σ~22), we have to find σ~22 that maximized (23), and then σ~12=logψ2exp{σ~22}-ψ2+1. Once we have both the overall and constrained MLEs, we can obtain the observed likelihood ratio statistic w(ψ) as given in (13). Therefore, the significance function is p(ψ)=P(χ12≤w(ψ)). Moreover, by applying the algorithm given in the previous section, we can also obtain the observed simulated modified likelihood ratio statistic w∗(ψ) and the corresponding significance function is p(ψ)=P(χ12≤w∗(ψ)).


## 3.1. Likelihood Based Methods and Bartlett Corrected Likelihood Ratio Method

Let(x1,…,xn) be a sample from a known distribution with probability density function f(·,θ), where θ is a p-dimensional vector of parameters. Let ψ=ψ(θ), which has dimension d<p be the parameter of interest. The log-likelihood function is(10)lθ=lθ;x1,…,xn=∑i=1nlog⁡fxi;θ.Under the regularity conditions stated in Barndorff-Nielsen and Cox [6], we have the standardized maximum likelihood estimate (MLE) statistic θ^-θ′var(θ^)-1(θ^-θ) and the likelihood ratio statistic 2l(θ^)-l(θ) that are asymptotically chi-square distributed with p degrees of afreedom, χp2, where θ^ is the overall MLE, which is the value of θ that maximized l(θ), and var(θ^) is approximately the inverse of the Fisher’s expected information. When the parameter of interest is ψ=ψ(θ), Barndorff-Nielsen and Cox [6] showed that similar statistics can be obtained. The standardized MLE statistic becomes(11)Qψ=ψ^-ψ′varψ^-1ψ^-ψwhere ψ^=ψ(θ^), and var(ψ^) can be approximated by the delta method, which takes the form(12)varψ^≈∂ψθ∂θ′varθ^∂ψθ∂θ.The likelihood ratio statistic is(13)Wψ=2lθ^-lθ~,where θ~ is the constrained MLE, which is obtained by maximizing l(θ) for the given ψ value. Both Q(ψ) and W(ψ) are asymptotically χd2. As defined in Fraser [7], the significance function for ψ is defined as p(ψ)=P(χd2≤q(ψ)) or p(ψ)=P(χd2≤w(ψ)) can be used to obtain inference concerning ψ where q(ψ) and w(ψ) are the observed values of Q(ψ) and W(ψ), respectively. In particular, the (1-α)100% confidence region of ψ is(14)ψ:qψ≤χd,1-α2andψ:wψ≤χd,1-α2,respectively, where χd,1-α2 is the (1-α)100th percentile of χq2.It is well-known that these two asymptotic methods have rate of convergenceO(n-1/2), and they are referred to as the first-order methods. In statistics literature, there exists various adjustments to improve the accuracy of the above methods. In particular, Barndorff-Nielsen [8, 9] introduced the modified signed log-likelihood ratio statistics, a third-order method. However, this method is restricted to scalar parameter of interest only. On the other hand, Bartlett [10] proposed a transformation of the likelihood ratio statistic such that the mean of the transformed statistic matched the mean of the asymptotic distribution. More specifically,(15)W∗ψ=WψBwhere B is the Bartlett adjustment such that EW∗(ψ)=d. And W∗(·) is known as the Bartlett corrected likelihood ratio statistic. An obvious choice of B is(16)B=EW∗ψd.Bartlett [10] showed that the Bartlett corrected likelihood ratio statistic is also asymptotically χd2 distributed and it has rate of convergence O(n-2). Therefore, it is an extremely accurate method. Nevertheless, except in a few well-defined problem, EW∗(ψ) is very difficult to obtain which hinders the use of this method in applied statistics. A review of the Bartlett corrected likelihood ratio method can be found in Barndorff-Nielsen and Cox [6].Although, mathematically, the explicit closed form ofB, or even an asypmptotic expansion of B, is difficult to obtain, we propose the following algorithmic way to obtain EW(ψ) numerically, and hence, an estimated B.Given: (x1,…,xn) is a sample of size n from a distribution with known probability density function f(·;θ).Interest: Inference concerning ψ=ψ(θ).Have: Overall maximum likelihood estimate θ^, the constrained maximum likelihood estimate θ~, and the observed likelihood ratio statistic w(ψ).Step 1: Simulate M samples of data of size n from f(·;θ~).Step 2: For each set of simulated data, obtain the simulated observed likelihood ratio statistic. As a result, we have w1(ψ),…,wM(ψ).Step 3: Calculate(17)w¯ψ=∑i=1MwiψM,which is an estimate of the mean of the likelihood ratio statistic. Hence, we have B^=w¯(ψ)/d.Step 4: The observed simulated Bartlett corrected likelihood ratio statistic is(18)w∗ψ=wψB^,is asymptotically distributed as χd2 with fourth order rate of convergence. Thus, the significance function is p(ψ)=Pχd2≤w∗(ψ), and the (1-α)100% confidence region of ψ is(19)ψ:w∗ψ≤χd,1-α2.As a final note on the proposed algorithm, theoretically, the choice ofM should be as large as possible. However, the larger M is, the more calculations are required to obtain w¯(ψ). Moreover, the more nuisance parameters exist in the model, the larger M has to be. We recommend to use trial-by-error of M until w¯(ψ) is stablized.


## 3.2. Applying Likelihood Based Method to Obtain Inference on the Ratio of Two Coefficients of Variation of Two Independent Log Normal Distribution

LetYi¯=∑j=1ni‍Yij/ni and (ni-1)Si2=∑j=1ni‍(Yij-Yi¯)2. Then Yi¯ is normally distributed with mean μi and variance σi2/n, and (ni-1)Si2/σi2 is χn-12. Moreover, Yi¯ and (ni-1)Si2/σi2 are independent. Hence, inference concerning σi2 will be based on χni-12. Since τi is a function of σi2 only, inference concerning τi will be based on χni-12. Let θ=(σ12,σ22). Then the likelihood function for θ can be written as(20)lθ=lσ12,σ22;s12,s22=-n1-12log⁡σ12-n1-1s122σ12-n2-12log⁡σ22-n2-1s222σ22.It is easy to show that the overall MLE(21)θ^=σ^12,σ^22=s12,s22.Since our parameter of interest is ψ=ψ(θ)=τ1/τ2, where τi=exp{σi2}-1, we have(22)σ12=log⁡ψ2exp⁡σ22-ψ2+1.For a given ψ value, the log-likelihood function in (20) can be expressed as a function of σ22 only, and is(23)lσ22=lσ22;s12,s22-n1-12log⁡log⁡ψ2exp⁡σ22-ψ2+1-n1-1s122log⁡ψ2exp⁡σ22-ψ2+1-n2-12log⁡σ22-n2-1s222σ22.Hence, to solve for the constrained MLE θ~=(σ~12,σ~22), we have to find σ~22 that maximized (23), and then σ~12=logψ2exp{σ~22}-ψ2+1. Once we have both the overall and constrained MLEs, we can obtain the observed likelihood ratio statistic w(ψ) as given in (13). Therefore, the significance function is p(ψ)=P(χ12≤w(ψ)). Moreover, by applying the algorithm given in the previous section, we can also obtain the observed simulated modified likelihood ratio statistic w∗(ψ) and the corresponding significance function is p(ψ)=P(χ12≤w∗(ψ)).


## 4. Real Data Example

To illustrate the application of the methods discussed in this paper, we revisit the example discussed in Nam and Kwon [4]. Faupel-Badger et al. [11] compare concentrations of estrogen metabolites by RIA with the concentrations obtained using a novel and high-performance liquid chromatography-tandem mass spectrometry (LC-MS/MS). The 10% blinded quality control samples were used for assessment of quality control of the laboratory assay. Partial summary of data were presented in Nam and Kwon [4] and we have(24)n1=48,σ^12=0.16872,n2=53,σ^22=0.05622where the first sample is taken from RIA, and the second sample is taken from LC-MS/MS. Table 1 records the 95% confidence interval for the ratio of the two coefficients of variation assuming that the data are obtained from independent lognormal distributions obtained by the methods discussed in this paper. Note that the MOVER method is identical to the Log method and Hasan and Krishnamoorthy [5] showed that results from the improved version of the MOVER method are still similar to those obtained by the Log method. Hence, both the MOVER method and its improved version are not included in the calculations. Except for the Wald type, the intervals obtained in Nam and Kwon [4] seem to be close to each other. Notice that the results from the Fieller type are different from that reported in Nam and Kwon [4]. Moreover, we observed that the likelihood ratio method and the proposed Bartlett correction method seem to be different from the other methods by having a larger upper confidence limit.Table 1
95% confidence interval forψ for the experiment by Faupel-Badger et al. [11].
Method 95% confidence interval forψ  Wald type (2.1798, 3.8620) Fieller type (2.2705, 3.9992) Log method (2.2867, 3.9907) Likelihood ratio (2.2776, 4.0126) Bartlett correction (2.2770, 4.0138)With the above observation, it is of interest to compare the accuracy of the methods discussed in this paper, especially when the sample size is small.


## 5. Simulation Studies

To compare the accuracy of the methods discussed in this paper, simulations studies are performed. The parameters settings are given in Table2. Other settings have also been calculated but not reported because the results are very similar to those presented. However, they are available upon request. Since we are interested in developing a method that is accurate even for small sample sizes, hence the chosen sample sizes in the simulations studies are relatively small.Table 2
Parameters settings for simulation studies.
Study μ 1 σ 1 μ 2 σ 2 n 1 n 2  1 0.6 0.1 3.0 0.5 5 5 2     10 20 3     15 25 4     20 10  5 1.1 0.2 0.8 0.4 5 5 6     10 20 7     15 25 8     20 10  9 2.5 1.2 3.0 0.7 5 5 10     10 20 11     15 25 12     20 10  13 5.0 0.7 6.0 1.4 5 5 14     10 20 15     15 25 16     20 10For each study, we obtain 10,000 simulated samples. Theoretically,M should be as large as possible because we want to use w¯(ψ) to be the estimate of E[W(ψ)]. However, numerically, we have N simulated samples, and for each simulated sample, we have to do M simulations to obtain w¯(ψ). For these simulation studies, we use M=500. For each simulated sample, we compute the 95% confidence interval obtained by the methods discussed in this paper. Table 3 reported the percentage of samples where the true ψ is less than the lower 95% confidence limit (le), is within the 95% confidence interval (cc), and is greater than the upper 95% confidence limit (ue). The nominal values are 2.5%, 95% and 2.5%, respectively.Table 3  (a)
Empirical coverage rate for the simulation studies 1 to 8
Study Method le cc ue  1 Wald type 0.08 87.78 12.14 Fieller type 3.48 95.50 1.02 Log method 6.28 88.51 5.21 Likelihood ratio 3.12 93.39 3.49 Bartlett correction 2.40 94.90 2.70  2 Wald type 0.20 89.31 10.49 Fieller type 0.82 94.01 5.17 Log method 1.94 92.71 5.35 Likelihood ratio 2.22 94.78 3.00 Bartlett correction 1.92 95.29 2.79  3 Wald type 0.68 92.92 6.40 Fieller type 3.48 95.50 1.02 Log method 3.93 93.25 2.82 Likelihood ratio 2.77 94.57 2.66 Bartlett correction 2.53 94.85 2.27  4 Wald type 1.44 94.14 4.42 Fieller type 7.81 91.99 0.20 Log method 7.04 91.46 1.50 Likelihood ratio 3.15 94.23 2.62 Bartlett correction 2.88 94.85 2.27  5 Wald type 0.05 87.47 12.48 Fieller type 2.99 95.28 1.73 Log method 5.88 88.45 5.67 Likelihood ratio 3.13 93.39 3.48 Bartlett correction 2.42 94.90 2.68  6 Wald type 0.13 89.18 10.69 Fieller type 0.67 93.59 5.74 Log method 1.84 92.58 5.58 Likelihood ratio 2.25 94.75 3.01 Bartlett correction 2.00 95.23 2.77  7 Wald type 0.57 92.73 6.70 Fieller type 2.87 95.26 1.87 Log method 3.68 93.20 3.12 Likelihood ratio 2.70 94.66 2.64 Bartlett correction 2.52 94.90 2.58  8 Wald type 1.34 94.14 4.52 Fieller type 7.36 92.28 0.36 Log method 6.68 91.63 1.69 Likelihood ratio 3.15 94.34 2.49 Bartlett correction 2.88 94.83 2.29  (b)
Empirical coverage rate for the simulation studies 9 to 16
Study Method le cc ue 9 Wald type 0.00 82.10 17.90 Fieller type 0.01 98.88 1.11 Log method 1.27 90.26 8.47 Likelihood ratio 2.99 93.61 3.40 Bartlett correction 2.40 94.94 2.66  10 Wald type 0.00 83.91 16.09 Fieller type 0.00 90.18 9.82 Log method 0.07 90.86 9.07 Likelihood ratio 2.04 94.82 3.14 Bartlett correction 1.92 95.24 2.84  11 Wald type 0.00 89.36 10.64 Fieller type 0.00 96.70 3.30 Log method 0.96 93.93 5.11 Likelihood ratio 2.75 94.59 2.66 Bartlett correction 2.48 94.97 2.55  12 Wald type 0.02 92.69 7.29 Fieller type 0.47 99.16 0.37 Log method 3.62 93.86 2.52 Likelihood ratio 3.17 94.34 2.49 Bartlett correction 2.96 94.71 2.33  13 Wald type 0.00 79.33 20.67 Fieller type 2.01 97.99 0.00 Log method 10.56 88.87 0.57 Likelihood ratio 3.52 94.54 2.94 Bartlett correction 2.68 95.03 2.29  14 Wald type 0.01 91.26 8.73 Fieller type 0.64 99.32 0.04 Log method 3.15 94.73 2.12 Likelihood ratio 2.44 94.69 2.87 Bartlett correction 2.10 95.19 2.71  15 Wald type 0.59 93.88 5.53 Fieller type 5.44 94.56 0.00 Log method 6.29 93.43 0.28 Likelihood ratio 3.01 94.48 2.51 Bartlett correction 2.84 94.83 2.33  16 Wald type 1.60 94.18 4.22 Fieller type 12.16 87.84 0.00 Log method 10.64 89.36 0.00 Likelihood ratio 3.01 94.48 2.51 Bartlett correction 2.95 94.71 2.34From Table3, the three methods discussed in Nam and Kwon [4] do not give satisfactory coverage, especially when the sample sizes are small. The coverage of the likelihood ratio method is improving when the sample sizes increase and, in general, it has asymmetric errors. Nevertheless the proposed simulated Bartlett corrected likelihood ratio method is extremely accurate even when the sample sizes are as small as 5.


## 6. Testing Homogeneity of Coefficients of Variation fromk Independent Lognormal Distributions

Fork samples from independent lognormal(μi,σi) distribution, the required log-likelihood function can be written as(25)lθ=lσ12,…,σk2;s12,…,sk2=∑i=1k-ni-12log⁡σi2-ni-1si22σi2where si2 is the unbiased sample variance estimate of the ith sample given in Section 3. It is well-known that the overall MLE is(26)θ^=σ^12,…,σ^k2=s12,…,sk2.The aim is to test(27)H0:τ1=⋯=τk=τvsHa:  not  all  coefficients  of  variation  are  the  same,which, in this case, is the same as testing(28)H0:σ12=⋯=σk2=σ2vsHa:  not  all  variances  are  the  same.Therefore, when H0 is true, the log-likelihood function can be re-written in terms of σ2 and is(29)lσ2=lσ2;s12,…,xk2=∑i=1k-ni-12log⁡σ2-ni-1si22σ2,and the constrained MLE is (30)σ~2=∑i=1kni-1si2∑i=1kni-1,which is the usual pooled variance estimate. The observed likelihood ratio statistic is(31)w=2lθ^-lσ~2,which is asymptotically distributed as χk-12. Hence, the observed simulated Bartlett corrected likelihood ratio statistic is(32)w∗=ww¯/k-1,where w¯ is obtained by the algorithm given in Section 2.Simulation studies are performed to compare the accuracy of the likelihood ratio method and the simulated Bartlett corrected likelihood ratio method. In particular, three samples of data from lognormal(μi,σ) distribution are generated. w is calculated and w∗ is also the calculation with M=1000. We repeat this process N=10,000. The proportion of samples that have p-values less than 5% is reported in Table 4 for various sample sizes. The choice of μi is not important because it does not involve in any of the calculations and, hence, we take it to be 0. Different choices of σ result in similar results and are not reported, but they are available upon request. Table 4 reported the cases μi=0 and σ=1. When sample sizes are small, the likelihood ratio method does not give satisfactory results, but it is improving when the sample sizes increase. The simulated Bartlett corrected likelihood ratio method consistently gives extremely accurate result even when the sample sizes are small.Table 4
The proportion of samples rejected at5% level of significance for testing H0:τ1=τ2=τ3=τ when data are generated from lognormal(0,1) distribution.
Proportion of samples rejected atα=0.05 (n1,n2,n3) Likelihood ratio Bartlett correction  (5,5,5) 0.0658 0.0512 (5,10,15) 0.0646 0.0524 (10,10,10) 0.0565 0.0495 (10,15,20) 0.0529 0.0487 (50,50,50) 0.0544 0.0523


## 7. Conclusion

The lognormal distribution has been frequently used for modeling positive valued right skewed data, which commonly arise in health and medical research. In this paper, we proposed a simulated Bartlett corrected likelihood ratio approach to obtain inference concerning the ratio of two coefficients of variation for lognormal distribution. Simulation studies show that the proposed Bartlett correction method is extremely accurate even when the sample size is small. Moreover, the proposed proposed Bartlett correction method is extended to test homogeneity ofk coefficients of variation from independent lognormal distributions.



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