Comparing the Means of. Two Log-Normal Distributions: A Likelihood Approach

Transcription

1 Journal of Statistical and Econometric Methods, vol.3, no.1, 014, ISSN: (print), (online) Scienpress Ltd, 014 Comparing the Means of Two Log-Normal Distributions: A Likelihood Approach L. Jiang 1, M. Rekkas and A. Wong 3 Abstract The log-normal distribution is one of the most common distributions used for modeling skewed and positive data. In recent years, various methods for comparing the means of two independent log-normal distributions have been developed. In this paper a higher-order likelihoodbased method is proposed. The method is applied to two real-life examples and simulation studies are used compare the accuracy of the proposed method to some existing methods. Mathematics Subject Classification: 6F03 Keywords: Confidence interval; Coverage probability; Modified signed loglikelihood ratio statistic; r -formula 1 Beijing Education Examination Authority, No. 9 Zhixin Road, Haidian District, Beijing, P.R. China, , jiangl@bjeea.cn Department of Economics, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada, V5A 1S6, mrekkas@sfu.ca 3 Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada, M3J 1P3, august@yorku.ca Article Info: Received : December 18, 013. Revised : January 3, 014. Published online : February 7, 014

2 138 Comparing the Means of Two Log-Normal Distributions 1 Introduction In real life applications, cases arise when random variables take on strictly positive values where the use of the normal distribution is not appropriate for statistical inference. The use of the log-normal distribution features prominently in these settings. For example, in medical research, the log-normal distribution is commonly used to model the incubation periods of certain diseases, and the survival times of cancer patients. In economics and business studies, the log-normal distribution is used to model income, firm size and stock prices. In this paper, we use higher-order likelihood-based analysis to derive a test statistic for comparing the means of two independent log-normal distributions. We compare the performance of this test statistic with three existing approaches in the literature. These approaches are due to Zhou et al. (1997), Abdollahnezhad et al. (01) and the signed log-likelihood ratio statistic method. Simulation studies show that our proposed method outperforms the three existing methods. Let X and Y be independently distributed with log X N(µ x, σx) and log Y N(µ y, σy). The random variables X and Y are said to have a log-normal distribution with means M x = exp{µ x + σx/} and M y = exp{µ y + σy/}, respectively. Then testing: H 0 : M x = M y vs H a : M x < M y is equivalent to testing: H 0 : ψ = 0 vs H a : ψ < 0, where ψ = log M x log M y = log(m x /M y ) = µ x µ y + (σx σy)/. In this paper, we consider a slightly more general hypothesis: H 0 : ψ = ψ 0 vs H a : ψ < ψ 0. (1) Consider (x 1,..., x n ) and (y 1,..., y m ) to be independent samples from N(µ x, σx) and N(µ y, σy), respectively. Zhou et al. (1997) propose a likelihoodbased test and show that the p-value for testing hypothesis (1) can be approximated by p Z (ψ 0 ) = Φ(z), ()

3 L. Jiang, M. Rekkas and A. Wong 139 where Φ() is the cumulative distribution function of N(0, 1) and z = ˆµ x ˆµ y + (s x s y)/ ψ 0 ( ), (3) s x n + s y + 1 s 4 x + s4 y m n 1 m 1 where ˆµ x = log x i /n and ˆµ y = log y i /m are the maximum likelihood estimators of µ x and µ y, respectively. The terms s x = (log x i ˆµ x ) /(n 1) and s y = (log y i ˆµ y ) /(m 1) are the unbiased estimators of σ x and σ y, respectively. Moreover, a (1 α)100% confidence interval for ψ is given by ( ) ˆψ z α/ var( ˆψ), ˆψ + zα/ var( ˆψ), (4) where ˆψ = ˆµ x ˆµ y + (s x s y)/ var( ˆψ) = s x n + s y m + 1 ( ) s 4 x n 1 + s4 y m 1 and z α/ is the (1 α/)100 th percentile of N(0, 1). Abdollahnezhad et al. (01) apply a generalized p-value approach to obtain the p-value for testing hypothesis (1). This generalized p-value is obtained using the following algorithm: Step 1: For k = 1 to K, where K is reasonably large: (a) generate u x from χ n 1 and u y from χ m 1 (b) calculate p k = Φ ˆµ x ˆµ y + nˆσ x u x mˆσ y u y ψ 0, (5) ˆσ x u x + ˆσ y u y where ˆσ x = (log x i ˆµ x ) /n, and ˆσ y = (log y i ˆµ y ) /m Step : p A (ψ 0 ) = K k=1 p k/k. The corresponding (1 α)100% confidence interval for ψ using the method in Abdollahnezhad et al. (01) is (ψ L, ψ U ) such that p(ψ L ) = 1 α/ and p(ψ U ) = α/. Note that this method requires extra computing time because for each calculation, we need to generate K values of u x and K values of u y respectively. Thus, obtaining confidence intervals will require substantial computing time, especially when K is large.

4 140 Comparing the Means of Two Log-Normal Distributions This paper proceeds as follows. In Section, we present the likelihoodbased inference methods for a scalar parameter from an exponential family model. In Section 3 we apply these methods for inference on the difference of the means of two independent log-normal distributions. In Section 4 we present our numerical results and some concluding remarks are given in Section 5. Likelihood-Based Inference for a Scalar Parameter from an Exponential Family Model In this section we provide an overview of the third-order likelihood-based inference method we will use in this paper for testing the difference of means of two independent log-normal distributions. We begin by reviewing the standard first-order likelihood method. Consider a random sample (y 1,... y n ) obtained from a distribution with density f(y; θ), where θ = (ψ, λ ) with ψ being the scalar interest parameter and λ the vector of nuisance parameters. The loglikelihood function based on the given sample is then l(θ) = l(ψ, λ) = log f(y i ; θ). From this function, the signed log-likelihood ratio statistic (r) can be obtained: r r(ψ) = sgn( ˆψ ψ){[l(ˆθ) l( θ]} 1/, (6) where ˆθ represents the maximum likelihood estimator of θ satisfying the firstorder conditions, l(θ)/ θ = 0, and θ represents the constrained maximum likelihood estimator of θ for a given ψ. The constrained maximum likelihood estimator can be solved by maximizing the log-likelihood function subject to the constraint ψ(θ) = ψ. The method of Lagrange multipliers can be used to solve this optimization problem. The function H(θ, α) = l(θ) + α[ψ(θ) ψ] (7) is known as the Lagrangean, where α is termed the Lagrange multiplier. Then ( θ, ˆα) is the solution from solving the equations H(θ, α) θ = 0 and H(θ, α) α = 0

5 L. Jiang, M. Rekkas and A. Wong 141 simultaneously. The tilted log-likelihood is defined as l(θ) = l(θ) + ˆα[ψ(θ) ψ]. (8) Note that the tilted log-likelihood function has the property that l( θ) = l( θ). The signed log-likelihood ratio statistic in (6) is asymptotically distributed as N(0, 1) with rate of convergence O(n 1/ ) and is referred to as a first-order method. Hence, the p-value for testing H 0 : ψ = ψ 0 vs H a : ψ < ψ 0 using the signed log-likelihood ratio statistic method is The (1 α)100% confidence interval for ψ is p r (ψ 0 ) = Φ(r(ψ 0 )). (9) {ψ : r(ψ) < z α/ }. (10) It is well-known that the signed log-likelihood ratio statistic method does not perform satisfactorily well in small sample situations. Many improvements have been suggested in the literature. In particular, Barndorff-Nielsen (1986, 1991) proposes a modified signed log-likelihood ratio statistic, given by { } q(ψ) r r (ψ) = r(ψ) + r(ψ) 1 log, (11) r(ψ) where r(ψ) is the signed log-likelihood ratio statistic given in (6) and q(ψ) is a statistic that can be derived in various ways depending on the information at hand. It is shown in Barndorff-Nielsen (1986, 1991) that the modified signed log-likelihood ratio statistic is asymptotically distributed as N(0, 1) with rate of convergence O(n 3/ ) and is referred to as a third-order method. Hence the corresponding p-value is p BN (ψ 0 ) = Φ(r (ψ 0 )) (1) and the corresponding (1 α)100% confidence interval for ψ is {ψ : r (ψ) < z α/ }. (13) Note that the most difficult aspect of the methodology is obtaining q(ψ).

6 14 Comparing the Means of Two Log-Normal Distributions Fraser and Reid (1995) provide a systematic approach to obtain q(ψ) for a full rank exponential model with density f(y; θ) = exp{ϕ (θ)t(y) c(θ) + h(t(y))}, where ϕ(θ) is the canonical parameter and t(y) is the canonical variable. For this model, q(ψ) takes the form q q(ψ) = sgn(ψ(ˆθ) ψ) χ(ˆθ) χ( θ) [ ( 1/ var χ(ˆθ) χ( θ))], (14) where χ(θ) is the standardized maximum likelihood estimator recalibrated in the ϕ(θ) scale: where ϕ θ (θ) = ϕ(θ). θ The variance term in (14) is computed as χ(θ) = ψ θ( θ)ϕ 1 θ ( θ)ϕ(θ) (15) ( ) var χ(ˆθ) χ( θ) ψ θ( θ) j 1 θθ ( θ)ψ θ ( θ) j θθ ( θ) ϕ θ ( θ) (16) j θθ (ˆθ) ϕ θ (ˆθ) where j θθ (ˆθ) and j θθ ( θ) are defined as and j θθ (ˆθ) = l(θ) θ θ j θθ ( θ) = l(θ) θ θ and are referred to as the observed information matrix obtained from l(θ) and l(θ) respectively, and ψ θ (θ) = ψ(θ). θ The idea of the method is to recalibrate (11) in the canonical parameter, ϕ(θ), scale. Since the signed log-likelihood ratio statistic is invariant to reparameterization, it remains the same as in (6). The only quantity that needs recalibration in the ϕ(θ) scale is therefore q(θ). For a detailed discussion of this derivation, see Fraser and Reid (1995). θ=ˆθ θ= θ

7 L. Jiang, M. Rekkas and A. Wong Inference for the Difference of Means of Two Independent Log-Normal Distributions We apply the modified signed log-likelihood ratio statistic method to our problem of interest. The log-likelihood function is: l(θ) = n log σ x nµ x + µ n x log x σx σx i 1 σx where m log σ y mµ y σ y = n log σ x nµ x σ x m log σ y mµ y σ y + µ y σ y i=1 m j=1 + µ x t σx 1 1 t σx 3 log y j 1 σ y n (log x i ) i=1 m (log y j ) j=1 + µ Y t σy 1 t σy 4, (17) ( (t 1, t, t 3, t 4 ) = log xi, log y i, (log x i ), ) (log y i ) is a minimal sufficient statistic, θ = (µ x, µ y, σ x, σ y) interest is and our parameter of ψ = ψ(θ) = (µ x µ y ) + (σ x σ y)/. (18) Moreover, the canonical parameter ϕ(θ) is ϕ ϕ(θ) = ( µx σ x, µ y, 1, 1 ). (19) σy σx σy The first and second derivatives of l(θ) are given in the Appendix. solving the first-order conditions, we have the maximum likelihood estimator of θ: where ˆµ x = t 1 n = t 1 ˆθ = (ˆµ x, ˆµ y, ˆσ x, ˆσ y), ˆµ y = t m = t ˆσ x = [ n n ( t 1 ) n( t 1 ) + 1 ] t 3 = 1 [ t3 n( t 1 ) ] n ˆσ y = [ m m ( t ) m( t ) + 1 ] t 4 = 1 [ t4 m( t ) ]. m By

8 144 Comparing the Means of Two Log-Normal Distributions The corresponding observed information matrix is given by: j θθ (ˆθ) = n ˆσ x m ˆσ y n ˆσ 4 x m ˆσ 4 y. (0) The constrained maximum likelihood estimator is derived using the Lagrangean defined in (7): H(θ, α) = l(θ) + α [(µ x µ y ) + 1 (σx σ y) ] ψ 0, (1) with derivatives given by: H(θ, α) = l(θ) + α µ x µ x H(θ, α) = l(θ) α µ y µ y H(θ, α) = l(θ) + 1 σx σx α H(θ, α) σ y H(θ, α) α = l(θ) σ y 1 α = (µ X µ Y ) + 1 (σ x σ y) ψ 0, from which ( θ, ˆα) can be obtained. The tilted log-likelihood from (8) is given as: l(θ) = l(θ) + ˆα [ (µ x µ y ) + 1 (σ x σ y) ψ 0 ]. () It is easy to show that l( θ) = l( θ) and j θθ ( θ) = j θθ ( θ). Since the canonical parameter ϕ(θ) is given in (19), we have: ϕ θ (θ) = 1 σ x 0 1 σ y 0 µx σ 4 x σ 4 x 0 0 µ y σy σ 4 y,

9 L. Jiang, M. Rekkas and A. Wong 145 and ϕ 1 θ (θ) = σ x 0 µ x σ x 0 0 σy 0 µ y σy 0 0 σx σy 4 Moreover, the parameter of interest ψ = ψ(θ) is given in (18) and we have ( ) ψ θ (θ) = Hence, the recalibrated parameter χ(θ) can be obtained by (15). Given the computed quantities above, we are able to construct the modified signed loglikelihood ratio statistic given in (11), with q(ψ) defined as in (14). It is important to note that theoretically, ( θ, ˆα) is uniquely defined. However, special care must be taken when performing numerical calculations as standard optimization subroutines in standard statistical software may not converge to the true θ, which results in a negative definite j θθ ( θ). 4 Numerical Results We present two real-life examples followed by simulations to compare results obtained from the following methods: the Z-score method proposed by Zhou et al. (1997) (Zhou), the generalized test method proposed by Abdollahnezhad et al. (01) (Abdollahnezhad), the signed log-likelihood ratio statistic method (r), and the proposed modified signed log-likelihood ratio statistic method (r ). The first example is discussed in Abdollahnezhad et al. (01). They considered the amount of rainfall (in acre-feet) from 5 clouds, of which 6 were chosen at random and seeded with silver nitrate. The log-normal model is used and the summary statistics expressed in this paper s notation are: n = 6, m = 6, t 1 = , t = , t 3 = , t 4 = Let M x and M y be the mean rainfall for the seeded clouds and the mean rainfall for the unseeded clouds respectively. The p-values, obtained from the four methods discussed in this paper, for testing H 0 : M x = M y vs H a : M x > M y H 0 : ψ = 0 vs H a : ψ > 0,

10 146 Comparing the Means of Two Log-Normal Distributions where ψ is the logarithm of the ratio of the log-normal means, are recorded in Table 1. Table 1: p-values for testing H 0 : ψ = 0 vs H a : ψ > 0 for the rainfall example Method p-value Zhou Abdollahnezhad r r The 95% confidence interval for the ratio of the log-normal means are recorded in Table. Table : 95% confidence interval for M x M y for the rainfall example Method 95% Confidence interval Zhou (0.751, 11.34) Abdollahnezhad (0.600, ) r (0.681, 1.150) r (0.606, ) From these tables it is clear that the four methods produce quite different inferential results. The second example is a bioavailability study. A randomized, parallelgroup experiment was conducted with 0 subjects to compare a new test formulation (x), with a reference formulation (y), of a drug product with a long half life. The data from this study is given in Table 3. The Shapiro-Wilk tests for the normality on the log-transformed data give a p-value of for the test formulation group and for the reference formulation group. These tests suggest that the log-normal model is suitable for this data set. The p-values, obtained from the four methods discussed in

11 L. Jiang, M. Rekkas and A. Wong 147 Table 3: Data for the bioavailability study x y this paper, for testing H 0 : M x = M y vs H a : M x M y H 0 : ψ = 0 vs H a : ψ 0, where ψ is the logarithm of the ratio of the log-normal means, are recorded in Table 4. Table 4: p-values for testing H 0 : ψ = 0 vs H a : ψ 0 for the bioavailability example Method p-value Zhou 0.04 Abdollahnezhad 0.18 r r The 95% confidence interval for the ratio of the log-normal means are recorded in Table 5. Table 5: 95% confidence interval for Mx M y for the bioavailability example Method 95% Confidence interval Zhou (0.339, 1.59) Abdollahnezhad (0.6, 1.36) r (0.451, 1.181) r (0.4, 1.10)

12 148 Comparing the Means of Two Log-Normal Distributions Again the results obtained from the four methods discussed in this paper are quite different. To compare the accuracy of the four methods discussed in this paper, simulation studies are performed. 10,000 simulated samples from each combination of parameters are generated. For each generated sample, the 90% confidence interval for Mx M y is calculated for each of the four methods discussed in this paper. The performance of a method is judged using the following criteria: the coverage probability (CP): Proportion of the true M x M y the lower tail error rate (LE): falling below the lower limit of the 90% con- Proportion of the true M x M y fidence interval falling within the 90% confidence interval the upper tail error rate (LE): Proportion of the true Mx M y falling above the upper limit of the 905% confidence interval the average bias (AB): AB = LE UE The desired values are 0.9, 0.05, 0.05 and 0 respectively. These values reflect the desired properties of the accuracy and symmetry of the interval estimates of M x M y. Results are recorded in Table 6. It is clear that the method by Zhou et al. (1997) and the signed log-likelihood ratio method do not give satisfactory results. The method by Abdollahnezhad et al. (01) is an improvement of the other two methods but still it is not as good as the proposed modified signed log-likelihood ratio method. Simulation studies for other combinations of parameters have also been performed and the same pattern is observed. These results are available from the authors upon request.

13 L. Jiang, M. Rekkas and A. Wong 149 Table 6: Simulation results n µ x σ x m µ y σ y Method CP LE UE AB Zhou Abdollahnezhad r r Zhou Abdollahnezhad r r Zhou Abdollahnezhad r r Zhou Abdollahnezhad r r Discussion In this paper, four methods are studied for comparing the means of two independent log-normal distributions. In terms of computation, the method by Zhou et al. (1997) is the simplest. The method by Abdollahnezhad et al. (01) takes up the most computing time because for each calculation, we have to simulate K samples for u x and also K samples for u y. Both the signed log-likelihood ratio statistic method and the proposed modified signed log-likelihood statistic method require the constrained maximum likelihood estimator, θ. Blindly applying standard optimization subroutines may result in j θθ ( θ) being a negative definite matrix. The two real-life examples illustrate that the four methods can give very different inferential results. Simulation studies show the proposed modified signed log-likelihood ratio statistic method to be superior to the other three methods. From a theoretical perspective, the proposed modified signed log-likelihood ratio statistic method has the advan-

14 150 Comparing the Means of Two Log-Normal Distributions tage that it has a known rate of convergency, O(n 3/ ), whereas the signed log-likelihood ratio statistic method only has rate of convergency O(n 1/ ). The rate of convergency of the method by Zhou et al. (1997) and the method by Abdollahnezhad et al. (01) are unknown. Thus the proposed modified signed log-likelihood ratio statistic method is recommended for comparing the means of two independent log-normal distributions.

15 L. Jiang, M. Rekkas and A. Wong 151 Appendix The first and second derivatives of l(θ) given in (17): l(θ) = nµ x µ x σx + 1 t σx 1 l(θ) = mµ y µ y σy + 1 t σy l(θ) σ x l(θ) σy l(θ) µ x = n σ x + 1 σ 4 x = m σ y = n σ x l(θ) µ x µ y = 0 l(θ) µ x σx l(θ) µ x σy l(θ) µ y l(θ) µ y σx l(θ) µ y σy l(θ) σx σ x l(θ) σx σ y l(θ) σy σ y = nµ s σ 4 s = 0 = m σ y = 0 = mµ y σ 4 y + 1 σ 4 y 1 σ 4 x t 1 1 t σy 4 = n σ 4 x σ 6 x = 0 = m σ 4 y σ 6 y [ n µ x t 1 µ x + 1 ] t 3 [ m µ y t µ y + 1 ] t 4 [ n µ x t 1 µ x + 1 ] t 3 [ m µ y t µ y + 1 ] t 4.

16 15 Comparing the Means of Two Log-Normal Distributions References [1] K. Abdollahnezhad, M. Babanezhad and A. Jafari, Inference on Difference of Means of two Log-Normal Distributions: A Generalized Approach, Journal of Statistical and Econometric Methods, 1, (01), [] O.E. Barndorff-Nielsen, Inference on full or partial parameters based on the standardized signed log likelihood ratio, Biometrika, 73, (1986), [3] O.E. Barndorff-Nielsen, Modified signed log likelihood ratio, Biometrika, 78, (1991), [4] D.A.S. Fraser and N. Reid, Ancillaries and third order significance, Utilitas Mathemation, 7, (1995), [5] X. Zhou, S. Gao and S. Hui, Methods for Comparing the Means of Two Independent Log-Normal Samples, Biometrics, 53, (1997),