ON MODIFIED SKEW LOGISTIC REGRESSION MODEL AND ITS APPLICATIONS

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1 STATISTICA, anno LXXV, n. 4, 015 ON MODIFIED SKEW LOGISTIC REGRESSION MODEL AND ITS APPLICATIONS C. Satheesh Kumar 1 Department of Statistics, University of Kerala, Trivandrum , India L. Manu Department of Community Medicine, Sree Goulam Medical College, Trivandrum , India 1. Introduction Regression methods are usually used for studying the relationship between a response variable and one or more explanatory variables. Over the last decade the logistic regression model, also nown as logit model, has become the standard method of analysis when the outcome variable is dichotomous in nature and it has been found applications in several areas of scientific studies such as bioassay problems (Finney, 195, study of income distributions (Fis, 1961, analysis of survival data(placett, 1959 and modelling of the spread of an innovation(oliver, The main drawbac of a logit model is that it consider variables of only symmetric and unimodal nature. But asymmetry may arise in several practical situations where the logit model is not appropriate. So through this paper we develop certain regression models based on a modified version of the sew-logistic distribution and compare it with the existing logit model as well as a regression model based on the sew-logistic distribution of (Nadaraah, 009. The paper is organized as follows. In section, we describe some important aspects of the sew-logistic regression model(slrm and propose a modified version of the SLRM, which we termed as the Modified Sew Logistic Regression Model (MSLRM. In section 3, we obtain some structural properties of the modified sew logistic distribution. In section 4, we consider the estimation of the parameters of the MSLRM and in section 5, two real life medical datasets are considered for illustrating the usefulness of the model compared to both the logit and sew logit models. In section 6, a generalized lielihood ratio test procedure is suggested for testing the significance of the parameters and a simulation study is also conducted to test the efficiency of the maximum lielihood estimators(mles of MSLRM. 1 Corresponding Author. drcsumar0@gmail.com

2 36 S. Kumar and L. Manu. Sew Logistic and Modified Sew Logitic regression models A random variable X is said to follow the logistic distribution(ld if its probability density function (p.d.f is of the following form. e x f(x = (1+e x, (1 where x R = (, +. The logistic regression model(lrm is given by in which p = z = a+ 1 1+e z, ( s b r X r. (3 r=1 The importance of the logistic regression model is due to its mathematical flexibility and in several medical applications it provides clinically meaningful interpretations (Hosmer and Lemeshow, 000. Nadaraah (009 developed a modified version of the logistic distribution, similar to the sew-normal distribution of Azzalini (1985, namely sew-logistic distribution(sld, which he defined through the following p.d.f, in which x R,β > 0 and λ R. f(x = e ( β x β 1+e ( x β 1+e ( λx β (4 When λ = 0, (4 reduces to p.d.f of the standard logistic distribution given in (1. The main feature of sew-logistic distribution is that a new parameter λ is included here for controlling the sewness and urtosis. Now the sew-logistic regression model (SLRM can be obtained through the following double series representation. =0 =0 p = 1 =0 =0 1 e((1+λ+λ+z 1+λ+λ+ 1 e ((1+λ+z 1+λ+ β if z < 0 β if z > 0 wherezisasdefinedin(3. Hereweconsideramodifiedversionofthesew-logistic distribution namely the modified sew-logistic distribution(msld through the following p.d.f, in which x R, α 1 and β > 0. f(x;α,β = e x 1+ αe βx (1+e x 1+e βx (6 Clearly when α = 0 and/or β = 0, the p.d.f reduces to the p.d.f of the logistic distribution and when α = -1 the p.d.f reduces to the p.d.f of the sew-logistic (5

3 On Modified sew logistic regression model 363 density given in (4. The p.d.f f(x;α,β of the MSLD(α,β can also be expressed interms of the following single series as well as double series representations, f(x;α,β = e x (1+e x + e x (1+e x + α =0 α =0 e ( 1+βx (1+e x e (1+β+βx (1+e x, if x < 0, if x > 0 (7 f(x;α,β = ( e (1+x +α =0 ( e (1+x +α =0 e (1+β+x e (1+β+β+x, if x < 0, if x > 0 The modified sew-logistic regression model(mlrm is given by the following double series representation based on (8. ( e (1+z (1+ +α e (1+β+z (1+β+, if z < 0 =0 p = (9 ( 1 e (1+z (1+ +α e (1+β+β+z (1+β+β+, if z 0 =0 where z is as given in (3. For the derivation of (9, see Appendix I. A graphical representation of MSLRM for particular values of α and β = is given in Figure 1. (8 3. Some structural properties of modified sew logistic model Proposition 1. If X follows MSLD(α,β, then Y = X follows a convex mixture of standard logistic and sew-logistic distributions. Proof. The p.d.f f (y of Y is the following, for y R, β R and α 1. f (y = f ( y;α,β dx dy = e y 1+α eβy (1+e y 1+e βy = e y (1+e y 1+ α 1+e βy = e y + α e y 1 (1+e y, (1+e y 1+e βy

4 364 S. Kumar and L. Manu Figure 1 Plots of regression function of MSLD(α, for different values of α. which shows that the p.d.f of Y can be considered as a convex mixture of the p.d.f of the standard logistic and and sew-logistic distributions. Proposition. If X follows MSLD(α,β, then Y = X follows a half logistic distribution. Proof. For y > 0, the p.d.f f (y of Y is f (y = f (y;α,β +f ( y;α,β = = e y (1+e y, dx dy dx dy e y 1+α e βy + (1+e y 1+e βy which is the p.d.f of the half logistic distribution. e y (1+e y 1+ α 1+e βy Proposition 3. If X follows MSLD(α,β, then Y = X 1/c,c (0,1 follows a distribution with p.d.f f(y = c y c 1 e 1+α yc e βyc (1+e yc 1+e βyc Proof. For any y > 0, the p.d.f f (y of Y = X 1/c is given by f(y = y c 1 e yc 1+α e βyc dx 1+e βyc dy (1+e yc = c y c 1 e yc 1+α e βyc (1+e yc 1+e βyc

5 On Modified sew logistic regression model 365 Proposition 4. The first four raw moments of MSLD(α,β are given by µ 1 = α ( ( 1 1 (1+β +β + + ( 1 (1+β +, (10 =0 µ = 4 ( (1+ 3 +α =0 1 (1+β +β (1+β + 3, (11 µ 3 = 1α =0 1 (1+β +β ( 1 (1+β + 4 (1 and µ 4 = 48 ( (1+ 5 +α =0 1 (1+β +β (1+β + 5.(13 Proof. By using the double series representation of the p.d.f of the M SLD(α, β as given in (8 we obtain its first raw moment as µ 1 = = xf(xdx ( 0 xe (1+x dx+α =0 xe (1+x dx+α ( 0 =0 0 xe (1+β+x dx + xe (1+β+β+ dx, which gives (10, by using the standard results of integration. In a similar way one can obtain (11,(1 and (13. Now by using proposition 4 we can evaluate the mean, variance, sewness and urtosis of the MSLD(α,β with the help of R softwares as obtained in Table 1. 0

6 366 S. Kumar and L. Manu TABLE 1 Mean, Variance, Sewness and Kurtosis of the MSLD for particular choice of α and β. (α, β Mean Variance Sewness Kurtosis (-0.5, (, (5, (10, (0, (50, (-00, Proposition 5. The median of MSLD(α,β is given by the following equations ( e (1+xm (1+ +α e (1+β+xm (1+β+ = 1 if x m < 0 α =0 =0 ( ( e (1+xm (1+ + ( 1 (1+β+ + (1 e (1+β+β+xm (1+β+β+ = 1 if x m > 0 Proof. The median of a probability density function f(x is a point x m on x m the real line which satisfies the equation f (xdx = 1. Using the double series expansion of the p.d.f we get, Case i: If x m < 0 x m ( e (1+x m +α =0 e (1+β+x m dx = 1 ( e (1+x m (1+ +α =0 e (1+β+x m = 1 (1+β + (14 Case ii: If x m > 0 0 x m 0 ( e (1+xm (1+ +α ( e (1+xm (1+ =0 +α =0 e (1+β+xm (1+β+ e (1+β+β+xm (1+β+β+ dx+ dx = 1

7 On Modified sew logistic regression model 367 which on simplification yields α =0 ( ( e (1+xm (1+ + ( 1 (1+β+ + (1 e (1+β+β+xm (1+β+β+ = 1 (15 Closed form for x m is not obtainable, on solving equations (14 or (15 using mathematical softwares MATHEMATICA OR MATHCAD one can obtain the median. Proposition 6. The mode of MSLD(α,β is given by the following equation e x 1 e x +(+α+αβe βx (+α αβe (1+βx + (1+αe βx (1+αe (1+βx (+α(1+e x (1+e βx = 0 (16 Proof. The mode of a probability density function is obtained by equating the derivative of the density function to zero and solving for the variable. Thus, differentiating f(x;α,β with respect to x yields (16. Using the mathematical softwares MATHCAD OR MATHEMATICA one can solve (16 and obtain the mode. Proposition 7. The mean deviation about the average A denoted by δ 1 (X is given by the following { δ11 (A, ifa < 0 δ 1 (X = δ 1 (A, ifa 0 where, δ 11 (A = δ 1 (A = α =0 ( e (1+A +α (1+ ( e (1+A + (1+ =0 e (1+β+A (1+β+ ( (e (1+β+β+A 1 (1+β+β+ + 1 Proof. The mean deviation about A is defined by δ 1 (X on simplification yields δ 1 (X = AF (A A A δ 1 (X = x A f (xdx xf (xdx+ xf (xdx A (1+β+ A, +A

8 368 S. Kumar and L. Manu Case (i. A < 0 δ 1 (X = AF (A A A xf (xdx+ 0 A xf (xdx+ 0 xf (x dx (17 Using the double series expansion of p.d.f we get, { A xf (xdx = ( e (1+A A( (1+ } α (18 A(1+β + 1 (1+β+ =0 0 A { xf (xdx = ( (1+ e (1+A (1 A( α (1+β+ e (1+β+A (1 A(1+β =0 xf (xdx = ( (1+ +α =0 Substituting (18-(0 in (17 we get the equation of δ 11 (A. Case (ii. A 0 } (1+β +β + (19 (0 A 0 δ 1 (X = AF (A A 0 xf (xdx = 0 xf (xdx A 0 ( (1+ +α xf (xdx+ =0 A xf (x dx (1 (1+β + { xf (xdx = ( (1+ e (1+A (1+A(1+ 1 α (1+β+β+ e (1+β+β+A (1+A(1+β + 1 =0 ( } (3 A { xf (xdx = ( e (1+A 1+A(1+ + (1+ } α e (1+β+β+A 1+A(1+β + (1+β+β+ =0 (4 Substituting ( - (4 in (1 we get the equation of δ 1 (A.

9 On Modified sew logistic regression model Estimation This section deals with the maximum lielihood estimation of the parameters of the MSLRM. Suppose we have a sample of n independent observations of the pair (x i,y i, i=1,,...,n, where y i denotes the value of the dichotomous outcome variable and x i is the value of the independent variable for the i th subect. Let p i = P(Y i = 1 X i, so that P(Y i = 0 X i = 1 p i. The probability of observing the outcome Y i whether it is 0 or 1 is given by P(Y i X i = p yi i (1 p i 1 yi. If there are n sets of values of X i, say X, the probability of observing a particular sample of n values of Y, say Y is given by the product of n probabilities, since the observations are independent. That is, P(Y X = n i=1 p yi i (1 p i 1 yi (5 Let z = a+ s b r X r and Θ = (α,β,a,b 1,b,...,b s be the vector of parameters r=1 of the MSLD regression model and let Θ = ( α, β,â, b 1, b,..., b s be the maximum lielihood estimator(mle of Θ. The log-lielihood function of MSLRM is given by n n l = log L(y z,θ = y i logp i + (1 y i log(1 p i (6 i=1 The MLE of the parameters are obtained by solving the following set of lielihood equations, in which, δ(p,y = n y i p i n (1 y i (1 p, with p i i is as defined in (9. i=1 i=1 Case 1 : For z 0, l α = 0 or equivalently i=1 δ(p,y ( ( 1 e (1+z or equivalently δ(p,y α ( =0 l β = 0 =0 ( 1 + (1+e (1+β+β+z (1+β +β + (7 = 0, ( 1 + (1+(1+e (1+β+β+z (1+β +β + (1+β +β +z +1 = 0, l a = 0 (8

10 370 S. Kumar and L. Manu or equivalently δ(p,y ( 1 (1+e (1+z +α ( 1 + (1+e (1+β+β+z = 0 ( and for r = 1,,..., s, or equivalently δ(p,y x r ( Case : For z <0 l b r = 0 =0 =0 (9 ( 1 (1+e (1+z +α ( 1 + (1+e (1+β+β+z = 0. =0 l α = 0 or equivalently δ(p,y ( ( 1 + (1+ e(1+β+z (1+β + ( 1 e (1+z = 0, or equivalently δ(p,y α ( =0 l β = 0 ( 1 + (1+ e(1+β+z (1+β +z 1 (1+β + =0 (30 (31 = 0, (3 l a = 0 or equivalently δ(p,y ( 1 (1+e (1+z +α ( 1 + (1+e (1+β+z = 0, ( and for r = 1,,..., s, or equivalently δ(p,y x r ( l b r = 0 =0 (33 ( 1 (1+e (1+z +α ( 1 + (1+e (1+β+z = 0. (34

11 On Modified sew logistic regression model 371 Second order partial derivatives of equation(6 with respect to the parameters are observed with the help of MATHEMATICA software and find that the equation gives negative values for all α 1, β > 0, a,b R. Now we can obtain the Θ by solving the lielihood equations (7 to (34 with the help of some mathematical softwares such as MATHCAD, MATHEMATICA, R Softwares etc. 5. Applications Here we illustrate the procedures discussed in section 4 with the help of the following two data sets: Data Set 1. Shoc data set obtained from (Afifi and Azen, (see These data were collected at the Shoc Research Unit at the University of Southern California, Los Angeles, California. Data were collected on 113 critically ill patients. Here we consider the explanatory variable as the urine output (ml/hr at the time of admission and the dependent variable y as whether the person survived or not. Data Set.Prostate cancer data set: (see data is on 380 subects of which 153 had tumor that penetrated the prostatic capsule. The variable capsule denotes the status of the tumor, whether it is penetrated or not, which we consider as the dichotomous dependent variable(y and Prostatic Specimen Antigen Value(PSAin mg/ml as the explanatory (X variable. These data set is also studied in (Hosmer and Lemeshow, 000. These data are copyrighted by John Wiley and Sons Inc. Here we consider the simplest model, Z = a+bx. We obtained the MLE of the parameters α,β,a,b with the help of R software by using the nlm pacage. The estimated parameters of logit and sew logit model are also obtained by the same procedure. The estimated values of the parameters of the LRM, SLRM and MSLRM along with the computed values of the Aaie Information Criterion (AIC, Bayes Information Criterion(BIC and pseudo R values (Mcfadden, 1973; Cameron and Windmeier, 1996 are given in Table. Also, we have plotted the Empirical cummulative distribution function (ECDF of the data sets and the three fitted regression models in Figure and Figure 3. The values of AIC, BIC are relatively less in case of the MSLRM compared to other existing models while the values of the of pseudo R (such as McFadden s R, McFadden s Ad R, Cox Snell R, Cragg-Uhler(NagelereR are more in case of MSLRM. This shows that the MSLRM gives a better fit to the given data sets compared to other existing models. Also Figure and Figure 3 support this conclusion.

12 37 S. Kumar and L. Manu TABLE Estimated values of the parameters with the corresponding Pseudo R values and Information criteria values. Distribution Data set LRM(a, b SLRM(λ, β, a, b MSLRM(α, β, a, b ˆα ˆλ ˆβ Data set 1 â ˆb AIC BIC McFadden s R McFadden s Ad R Cox Snell R Cragg-Uhler(NagelereR ˆα ˆλ ˆβ Dataset â ˆb AIC BIC McFadden s R McFadden s Ad R Cox Snell R Cragg-Uhler(NagelereR

13 On Modified sew logistic regression model 373 Figure ECDF of the data set1 and fitted regression models - LRM, SLRM and MSLRM Figure 3 ECDF of the data set and fitted regression models - LRM, SLRM and MSLRM

14 374 S. Kumar and L. Manu Data set TABLE 3 Calculated values of the test statistic ( ( Ω;y x Hypothesis log L logl Ω;y x Test statistic Data set1 H 0 : α = H 0 : α = Data set H 0 : α = H 0 : α = Testing of hypothesis and Simulation In this section we discuss certain test procedures for testing the significance of the additional parameter α of the MSLRM and carry out a brief simulation study for examining the performance of the maximum lielihood estimators. First we discuss the generalized lielihood ratio test procedure for testing the following hypothesis: Test 1. H 0 : α = 0 against the alternative hypothesis H 1 : α 0 Test. H 0 : α = 1 against the alternative hypothesis H 1 : α 1. Here the test statistic is, ( ( Ω;y x logλ = logl logl Ω;y x (35 where Ω is the maximum lielihood estimator of Ω = (α,β,a,b with no restriction, and Ω is the maximum lielihood estimator of Ω when α = 0 in case of Test 1 and α = 1 in case of Test. The test statistic logλ given in (35 is asymptotically distributed ( as χ with( one degree of freedom (Rao, The Ω;y x computed values of logl,logl Ω;y x and test statistic in case of both data sets are listed in Table 3. Since the critical value at the significance level 0.05 and degree of freedom one is 3.84, the null hypothesis is reected in both the case, which shows the appropriatness of the MSLRM to the data sets. Next we conduct a simulation study for assessing the performance of the MLEs of the parameters of the MSLRM. We consider the following two sets of parameters. 1. α=1.134, β= 1.793, a= 0.006, b= α=11.086, β= 9.170, a= -0.00, b=0.000

15 On Modified sew logistic regression model 375 TABLE 4. Bias and Mean Square Error(M SE within bracets of the simulated data sets Data Set sample size: α β a b E E E E-03 (7.30E-01 (1.3E-0 (8.9E-07 (4.11E-06 Data Set E E E E-04 (3.98E-01 (4.4E-03 (.56E-07 (9.9E E E E E-04 (.E-01 (1.44E-03 (5.6E-08 (7.18E E E E E-04 (7.E-0 (.54E-04 (1.64E-08 (.9E E E E E-04 (1.01E+00 (8.4E-01 (1.84E-06 (6.89E-07 Data Set E E E E-04 (5.9E-01 (.E-0 (4.46E-07 (1.10E E E E E-05 (4.54E-01 (4.55E-03 (6.46E-08 (.94E E-0.60E E-05.07E-05 (7.76E-0 (5.65E-05 (.45E-08 (1.87E-09 The computed values of the bias and mean square error(mse corresponding to sample sizes 100, 00, 300 and 500 respectively are given in Table 4. From the table it can be seen that both the absolute bias and MSEs in respect of each parameters of the MSLRM are in decreasing order as the sample size increases. Acnowledgements The authors would lie to express their sincere thans to the Editor and the anonymous Referee for for their valuable comments on an early version of the paper which greatly improved the quality and presentation of the manuscript.

16 376 S. Kumar and L. Manu Appendix A. Proofs of equation (9 Bydefinition,usingthedoubleseriesexpansionforp.d.f, thec.d.fofthemsld(α,β taes the following form, for x < 0. ( x F(x = ( ( x 1 e (1+x dx+α e (1+β+x dx =0 ( = e (1+x (1+ +α ( ( 1 e (1+β+x (36 (1+β + =0 In a similar way, the c.d.f of the MSLD(α,β can be written as given below, for x 0. F(x = 1 x = 1 = 1 f (xdx Thus (36 and (37 gives (9. ( x e (1+x dx+α ( e (1+x (1+ +α =0 =0 x e (1+β+β+x dx e (1+β+β+x (37 (1+β +β + References A. A. Afifi, S. P. Azen (1979. Statistical Analysis: A Computer Oriented Approach. Academic Pres, New Yor. A. Azzalini (1985. A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 1, pp A. C. Cameron, F. A. Windmeier (1996. R-squared measures for count data regression models with applications to health-care utilization. Journal of Business and Economic Statistics, 14, no., pp D. J. Finney (195. Statistical methods in Biological Assay. Hafner Publications, NewYor. P. R. Fis (1961. The graduation of income distributions. Econometrica, 9, pp D. W. Hosmer, S. Lemeshow (000. Applied Logistic Regression. JohnWiley and Sons, NewYor.

17 On Modified sew logistic regression model 377 D. Mcfadden (1973. Conditional logit analysis of qualitative choice behavior in zaremba,p. (ed.. In Frontiers in Econometrics, Academic Press, NewYor, pp S. Nadaraah (009. The sew logistic distribution. AStA. Adv. Stat. Anal, 93, pp F. R. Oliver (1969. Another generalisation of the logistic growth function. Econometrica, 37, pp R. L. Placett (1959. The analysis of life test data. Technometrics, 1, pp C. R. Rao (1973. Linear statistical inference and its applications. NewYor, Wiley. Summary Here we consider a modified form of the logistic regression model useful for situations where the dependent variable is dichotomous in nature and the explanatory variables exhibit asymmetric behaviour. Certain structrual properties of the modified sew logistic model is discussed and the proposed regression model has been fitted to some real life data sets by using the method of maximum lielihood estimation. Further, two data illustrations are given for highlighting the usefulness of the model in certain medical applications and a simulation study is conducted for assessing the performance of the estimators. Keywords: Logistic regression; Maximum lielihood estimation; Asymmetric distributions; Simulation.