The Permutation Test as an Ancillary Procedure for Comparing Zero-Inflated Continuous Distributions

doi:10.4236/ojs.2012.23033

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Open Journal of Statistics, 2012, 2, 274-280

http://dx.doi.org/10.4236/ojs.2012.23033 Published Online July 2012 (http://www.SciRP.org/journal/ojs)

The Permutation Test as an Ancillary Procedure for

Comparing Zero-Inflated Continuous Distributions

Jixiang Wu1, Lei Zhang2*, William D. Johnson3

1Departments of Plant Science and Mathematics & Statistics, South Dakota State University,

Brookings, USA

2Office of Health Data and Research, Mississippi State Department of Health, Jackson, USA

3Pennington Biomedical Research Center, Louisiana State University System, Baton Rouge, USA

Email: [email protected]

Received January 10, 2012; revised February 12, 2012; accepted February 28, 2012

ABSTRACT

Empirical estimates of power and Type I error can be misleading if a statistical test does not perform at the stated rejec-

tion level under the null hypothesis. We employed the permutation test to control the empirical type I errors for

zero-inflated exponential distributions. The simulation results indicated that the permutation test can be used effectively

to control the type I errors near the nominal level even when the sample sizes are small based on four statistical tests.

Our results attest to the permutation test being a valuable adjunct to the current statistical methods for comparing dis-

tributions with underlying zero-inflated data structures.

Keywords: Central Limit Theorem; Likelihood Ratio Test; Modified Central Limit Theorem; Permutation Test;

Wald Test; Zero-Inflated Distribution

1. Introduction

Statistical analysts sometimes encounter data that have

an excessive number of zeros and these data often pre-

sent analytical difficulties because traditional methods

rely on assumptions that may be unrealistic and plausible

transformations may not be found. Many studies have

reported on statistical methods for analyzing count data

with excessive zeros [1-6]. Some zero inflated data may

be viewed as having a mixed distribution where zeros

have a point distribution and the distribution of non-zero

observations is positive and continuous. This distribution

has not been investigated adequately and statistical meth-

ods with favorable Type I and Type II errors for com-

paring these non-traditional distributions are desired.

Testing equivalence of zero-inflated populations in the

context of underlying mixed distributions is equivalent to

testing equality of the probabilities of zeros and simulta-

neously equality of the parameters of the non-zero ob-

servations [7]. The likelihood ratio (LR) [8] and Wald [9]

tests are two widely used methods. These two methods

typically perform well if the probability density function

that applies under the null hypothesis is known. Recently,

Monte Carlo simulations were employed to compare

several approaches including the LR, Wald, central limit

theorem (CLT), modified central limit theorem (MCLT)

tests with respect to their empirical Type I errors and

testing powers for three zero-inflated continuous distri-

butions [7]. The LR, Wald, and MCLT tests were found

to be preferable to the tests based on central limit theory.

There are two important issues when several popula-

tions with zero-inflated data structure are compared. First,

the underlying distribution is usually unknown and, there-

fore, the assumptions of specific distributions can be eas-

ily violated by using assumption-constrained methods.

Second, empirical Type I errors and testing powers are

difficult to determine because the relevant parameters are

almost always unknown even if the assumed distribution

is correct. Moreover, a small sample size may contribute

to higher Type I and Type II errors. Thus, a test that con-

trols the empirical Type I errors and yields valid esti-

mates of testing powers is helpful.

Permutation tests are advocated for data analysis when

assumptions required to validate parametric procedures

are violated [10-13]. Unlike parametric tests, permutation

tests can generate probabilities by repeatedly “resam-

pling” the data and evaluating the obtained results with

reference to an empirically derived distribution [14,15].

Permutation tests have two major advantages: 1) they can

be used to adjust the empirical Type I errors and the

testing powers and, 2) they can be used when some as-

sumptions required to justify parametric tests are violated.

*Corresponding author.

J. X. WU ET AL. 275

Hence, their use may lead to more appropriate statistical

conclusions.

The purpose of this study was to investigate the issues

raised above pertaining to the use of ancillary permuta-

tion tests to compare several populations when the ran-

dom variable of interest has either a known or unknown

zero-inflated continuous distribution. Four statistical tests

were compared with respect to both their empirical type I

errors and testing powers. First we assumed the data fol-

lowed a zero-inflated exponential distribution as reported

by Zhang et al. [7]. Empirical Type I errors and testing

powers for these tests were compared with and without

adjunct permutation tests by empirical estimates obtained

using Monte Carlo simulations. Section 2 describes a

general permutation test that generates an empirical prob-

ability for each test. Simulated results for four carefully

selected parameter configurations are presented in Sec-

tion 3. Finally, Section 4 demonstrates the results with

the permutation test for a data set reported by Koopmans

[16].

2. Statistical Methods

2.1. Four Testing Methods

Performances of four tests including the likelihood ratio

(LR) [8], Wald [9], central limit theorem (CLT), and

modified central limit theorem (MCLT) tests [7] were

evaluated. The CLT test considers only the population

means calculated over all zero and non-zero observations

while the MCLT test considers both the probability of

zeros and simultaneously the mean of non-zero observa-

tions. The first two tests are distribution-based while the

other two are distribution-free based. Maximum likeli-

hood (ML) estimators [17] are required for both the LR

and Wald tests. For the CLT and MCLT tests, the Wald

test was incorporated to derive the probability for each

test [7]. These methods were detailed in one of our pre-

vious papers [7] and were not repeated in this study.

2.2. Permutation Test

The procedures of using the permutation test in zero-

inflated data are:

Step 1: Calculate the p-value using each of the above

mentioned four tests (e.g. LR) to analyze the original

data;

Step 2: Reshuffle the original data and randomly as-

sign the data to different populations without replace-

ment;

Step 3: Calculate the p-values by the same method

used in Step 1 for the reshuffled data obtained in Step 2;

Step 4: Repeat Steps 2 and 3 “N times”;

Step 5: Construct the sampling distribution of p-values

obtained in Steps 2 through 4;

Step 6: Locate the p-value in this distribution that cor-

responds to each p-value calculated in Step 1. If the p-

value from the original data is in the main body of the

distribution (α/2 to (1 − α/2)), then there is no significant

difference at probability level α among populations. Oth-

erwise, there is evidence that the difference between

(among) populations is significant.

The above procedures from Steps 1 to 6 were applied

to all four tests in this study.

3. Simulation Study

3.1. Simulation Procedure

In our empirical investigation we assumed interest was in

testing the hypothesis that three zero-inflated distribu-

tions had identical means. We simulated data from three

zero-inflated distributions with sample sizes ranging from

25 to 300 and performed each of four tests repeatedly

using the replicate samples to test the null hypothesis.

We tabulated the number of rejections of the hypothesis

under each known scenario to estimate Type I errors and

powers. Twelve sample sizes (n = 25 × s, where s = 1, 2,

···, 12) were considered and the nominal probability level

was set at 0.05 throughout. Although different configura-

tions were considered only one was listed for the null

distributions and three for alternative distributions as

described in Table 1. The first configuration in Table 1

was designed to estimate the empirical Type I errors and

the remaining three configurations were designed to es-

timate the empirical testing powers. Each set of simu-

lated data was analyzed by the four tests with and with-

out employment of the permutation test. Repetitions of

1000 simulated samples were used for each case. All

simulations were conducted by a C++ program written

by the authors of this paper.

3.2. Simulation Results

First the number of permutations sufficient for statistical

tests at a given probability level is determined. The Type

I errors and testing powers from 100 to 2000 different

permutations for configurations 1 and 2 with sample size

200 are summarized in Figures 1 and 2, respectively.

These figures clearly demonstrate that both empirical

Type I errors and testing powers became reasonably sta-

ble after the sample size surpassed 100 permutations.

Results from additional simulations for various different

sample sizes and configurations showed similar trends.

Thus samples of 500 permutations were chosen for all

the remaining simulations.

The empirical Type I errors of the four tests with and

without permutation tests are summarized in Table 1 for

the case of a zero-inflated exponential distribution. The

J. X. WU ET AL.

276

Table 1. Four parameter configurations for the simulation

study under zero-inflated expone ntial distr i bution.

Design 11





† 22









1 0.35, 1.00 0.35, 1.00 0.35, 1.00

2 0.25, 0.75 0.25, 1.00 0.25, 1.25

3 0.15, 0.20 0.35, 0.25 0.55, 0.30

4 0.15, 0.75 0.25, 1.00 0.35, 1.25

Design 1 is for null hypothesis and 2 to 4 are alternative hypotheses. †: δj

and βj are zero probability level and mean of exponential distribution for jth

population.

Figure 1. Empirical type I errors obtained by 20 different

numbers of permutations. (LR = likelihood ratio, CLT =

central limit theorem, and MCLT = modified central limit

theorem).

Figure 2. Empirical testing powers obtained by 20 different

numbers of permutations. (LR = likelihood ratio, CLT =

central limit theorem, and MCLT = modified central limit

theorem).

differences between observed Type I errors and the

nominal 0.05 level tend to be smaller as the sample size

increases for all four tests without permutation tests, in-

dicating that all these tests tend to perform better as the

sample size increases. However, with the permutation

tests, the empirical Type I errors are close to the nominal

0.05 level for different methods and various sample sizes

including small sample sizes (Table 2). The results indi-

cate that the permutation tests can reduce the high Type I

errors that are prevalent with small sample sizes. When

the sample sizes are large, i.e., at least 100, the empirical

Type I errors for the four statistical methods are almost

identical irrespective of using the permutation tests.

Tables 3-5 present the empirical powers of the four

tests for three parameter configurations as defined in

Table 1. As expected, the testing power increased for all

four tests as the sample size increased. The testing pow-

ers obtained without permutation tests were typically

lower than those obtained with permutation tests for all

methods when the sample size is small (100 and below).

However, as the sample size increases, the testing powers

were similar irrespective of using permutation tests. As

for parameter configuration 2 described in Table 1, the

CLT test and the other three tests have similar testing

powers because only means for the non-zero observa-

tions contributed the differences (Table 3). As for de-

signs 3 and 4, the CLT test has an extremely low testing

power compared with other three tests (Tables 4 and 5).

The increase or decrease of both zero probability level

and the non-zero mean made the differences among popu-

lations hard to detect with the CLT method, while the

other three tests are sensitive and maintain desirable test-

ing powers. This indicates that the LR, Wald, and the

MCLT tests are better than the CLT test in general. When

the zero probability levels among populations are similar,

the CLT test is still a good option.

In many situations, the distribution for a given zero-

inflated data set is unknown. It will be interesting to re-

veal the empirical Type I errors and testing powers ob-

tained using these methods by assuming the following

distributions. In this study, we generated 1000 simulated

data sets based on different parameter configurations as

described in Table 1 with the zero-inflated exponential

distribution. Then the LR and Wald methods were ap-

plied to test the differences among three populations by

assuming the data follow zero-inflated gamma and log-

normal distributions. Although simulations for various

sample sizes were conducted only the results for con-

figurations 1 and 2 with sample size of 200 were reported

(Table 1) because the similar patterns were observed for

different configurations with different sample sizes (data

not shown). Given zero-inflated exponential data, both

the LR and Wald tests resulted in unfavorably high Type

I errors if no permutation tests were applied; however,

these type I errors were adjusted substantially to be close

to the nominal level on using the permutation test. On the

other hand, the testing powers obtained by the LR and

Wald tests were lower when the lognormal distribution

was assumed. For the gamma distribution, both the LR

and Wald tests have similar and desirable testing powers

when the permutation tests are applied (Table 6). The

results suggested that the tests could have caused either

higher Type I errors or lower testing powers when an

J. X. WU ET AL.

277

Table 2. Empirical Type I errors for zero-inflated exponential distribution based on 1000 simulations.

Without permutation With permutation‡

Size

LR† Wald CLT MCLT LR Wald CLT MCLT

25 0.060 0.060 0.082 0.101 0.050 0.044 0.061 0.056

50 0.062 0.056 0.060 0.073 0.056 0.051 0.050 0.056

75 0.047 0.048 0.059 0.059 0.047 0.047 0.047 0.046

100 0.046 0.046 0.064 0.062 0.051 0.049 0.061 0.050

125 0.051 0.048 0.055 0.055 0.054 0.054 0.054 0.053

150 0.039 0.038 0.048 0.047 0.045 0.043 0.053 0.041

175 0.042 0.040 0.057 0.043 0.041 0.039 0.054 0.045

200 0.042 0.044 0.052 0.045 0.043 0.044 0.052 0.043

225 0.048 0.054 0.053 0.053 0.043 0.051 0.050 0.050

250 0.049 0.047 0.042 0.051 0.049 0.045 0.038 0.049

275 0.050 0.051 0.055 0.054 0.048 0.048 0.054 0.050

300 0.052 0.053 0.053 0.053 0.055 0.052 0.052 0.051

†: LR = likelihood ratio, CLT = central limit theorem, and MCLT = modified central limit theorem; ‡: 500 permutations were used.

Table 3. Empirical testing power for zero-inflated exponential distribution based on 1000 simulations for configuration 2.

Without permutation With permutation‡

Size

LR† Wald CLT MCLT LR Wald CLT MCLT

25 0.213 0.170 0.242 0.252 0.177 0.123 0.183 0.157

50 0.383 0.347 0.435 0.406 0.356 0.322 0.383 0.340

75 0.557 0.534 0.589 0.579 0.531 0.519 0.562 0.532

100 0.680 0.658 0.671 0.678 0.660 0.644 0.659 0.656

125 0.799 0.791 0.785 0.796 0.792 0.782 0.779 0.786

150 0.893 0.878 0.885 0.884 0.881 0.874 0.870 0.882

175 0.931 0.932 0.933 0.929 0.931 0.924 0.928 0.924

200 0.960 0.964 0.955 0.963 0.957 0.955 0.950 0.958

225 0.980 0.982 0.975 0.984 0.983 0.979 0.975 0.982

250 0.987 0.988 0.982 0.992 0.988 0.987 0.982 0.990

275 0.993 0.996 0.993 0.994 0.991 0.994 0.994 0.994

300 0.997 0.997 0.994 0.995 0.996 0.997 0.993 0.995

†: LR = likelihood ratio, CLT = central limit theorem, and MCLT = modified central limit theorem; ‡: 500 permutations were used.

inappropriate distribution was assumed. However, with

the permutation tests, the chance to make Type I errors

can be greatly decreased, yet the testing powers can be

desirable in many cases.

4. Application

Koopmans [16] reported results of a study of seasonal

activity patterns of field mice. Data consisted of the av-

erage distances traveled between captures by field mice

at least twice in a given month. The distances were

rounded to the nearest meter. A large number of zero

distances were observed in addition to non-zero distances

resulting in data with a zero-inflated distribution. The

exact distribution of the non-zero observations is unknown.

J. X. WU ET AL.

278

Table 4. Empirical testing power for zero-inflated exponential distribution based on 1000 simulations for configuration 3.

Without permutation With permutation‡

Size

LR† Wald CLT MCLT LR Wald CLT MCLT

25 0.766 0.810 0.161 0.828 0.771 0.795 0.121 0.746

50 0.984 0.988 0.143 0.988 0.982 0.987 0.125 0.981

75 1.000 1.000 0.151 1.000 0.999 0.999 0.142 0.999

100 1.000 1.000 0.206 1.000 1.000 1.000 0.192 1.000

125 1.000 1.000 0.215 1.000 1.000 1.000 0.203 1.000

150 1.000 1.000 0.270 1.000 1.000 1.000 0.251 1.000

175 1.000 1.000 0.264 1.000 1.000 1.000 0.260 1.000

200 1.000 1.000 0.307 1.000 1.000 1.000 0.299 1.000

225 1.000 1.000 0.330 1.000 1.000 1.000 0.326 1.000

250 1.000 1.000 0.325 1.000 1.000 1.000 0.314 1.000

275 1.000 1.000 0.367 1.000 1.000 1.000 0.351 1.000

300 1.000 1.000 0.398 1.000 1.000 1.000 0.388 1.000

†: LR = likelihood ratio, CLT = central limit theorem, and MCLT = modified central limit theorem; ‡: 500 permutations were used.

Table 5. Empirical testing power for zero-inflated exponential distribution based on 1000 simulations for configuration 4.

Without permutation† With permutation‡

Size

LR Wald CLT MCLT LR Wald CLT MCLT

25 0.398 0.366 0.088 0.427 0.366 0.326 0.066 0.312

50 0.731 0.725 0.126 0.749 0.724 0.707 0.108 0.708

75 0.924 0.918 0.164 0.920 0.916 0.900 0.146 0.898

100 0.980 0.976 0.211 0.978 0.975 0.972 0.187 0.971

125 0.990 0.989 0.257 0.991 0.989 0.988 0.248 0.984

150 0.997 0.997 0.288 0.997 0.997 0.996 0.273 0.996

175 1.000 1.000 0.328 1.000 1.000 1.000 0.330 1.000

200 1.000 1.000 0.345 1.000 1.000 1.000 0.340 1.000

225 1.000 1.000 0.435 1.000 1.000 1.000 0.435 1.000

250 1.000 1.000 0.457 1.000 1.000 1.000 0.446 1.000

275 1.000 1.000 0.485 1.000 1.000 1.000 0.486 1.000

300 1.000 1.000 0.554 1.000 1.000 1.000 0.555 1.000

†: LR = likelihood ratio, CLT = central limit theorem, and MCLT = modified central limit theorem; ‡: 500 permutations were used.

Various LR tests were used to identify which parameter(s)

were associated with the seasonal differences by assum-

ing the data followed a mixture of zero-inflated log-

snormal distributions [18]. To illustrate our approach, we

analyzed the data by four tests alone and by the permuta-

tion test with 1000 repetitions assuming the underlying

distribution was a zero-inflated exponential distribution

(Table 7). The results for all four tests, with and without

J. X. WU ET AL. 279

Table 6. Empirical type I errors and testing powers esti-

mated by the LR and Wald tests by assuming three differ-

ent distributions (exponential, Exp, Gamma, and log nor-

mal, LogN) as zero-inflated exponential distribution with

and without permutation tests based on 1000 simulations

for population size 200.

With no permutations

Type I error† Testing power‡

Exp 0.041 0.952

Gamma 0.423 1.000

LogN 0.167 0.806

Exp 0.038 0.952

Gamma 0.417 1.000 Wald

LogN 0.170 0.808

With permutation test∆

Type I error† Testing power‡

Exp 0.040 0.954

Gamma 0.045 0.978

LogN 0.051 0.543

Exp 0.035 0.947

Gamma 0.043 0.979

Wald

LogN 0.046 0.560

† and ‡: Based on design 1 and design 2 in Table 1, respectively; ∆: 500

permutations were used.

Table 7. P-values obtained by different methods by assum-

ing zero-inflated exponential distribution for the Koop-

mans’s data (1981) with and without permutation tests.

Method† No permutations With permutations

CLT 0.033 0.050

MCLT 0.031 0.041

LR 0.052 0.024

Wald 0.043 0.018

Note: 1000 permutation tests were used; †: LR = likelihood ratio, CLT =

central limit theorem, and MCLT = modified central limit theorem.

employment of the permutation tests, indicated that the

mice distances differed significantly among the three

seasons.

5. Discussion

It is desired that a statistical method sustains a preset

nominal Type I error and a high testing power. Many

methods are based on the appropriate statistical assump-

tions and require a large sample size. In some situations,

the sample size may be very small and test statistics may

yield unfavorable Type I errors and testing powers. In

addition, the real distribution is often unknown so desir-

able testing properties cannot be expected on employing

distribution-based tests. In this study, we investigated

statistical properties of the permutation tests integrated

with four distribution-based tests to compare populations

with zero-inflated data structures. Based on the results

from the simulated zero-inflated exponential data, several

conclusions can be made on use of the permutation test:

1) high Type I error caused by the appropriate statistical

tests without the permutation test for small sample sizes

can be adjusted to the preset nominal level when the

permutation test is used; 2) high Type I errors caused by

the inappropriate assumptions can be adjusted to the pre-

set nominal level; and 3) for a large data set, both the

type I errors and testing powers are similar regardless the

use of the permutation test for appropriate distribution

assumptions. The same conclusions applied for the other

two types of zero-inflated continuous distributions includ-

ing gamma and lognormal distributions (results not shown).

As reported by Zhang et al. [7] and in results of this

study, the LR and Wald tests hold similar type I errors

and testing powers but they are distribution dependent. If

an inappropriate distribution is assumed, both inflated

Type I errors and low testing powers can occur (Ta b le 6).

The CLT test is data structure dependent because it con-

siders only the population mean including zeros. When

the population means are similar (because the popula-

tions have similar probabilities of observations equal to

zero) and their non-zero observations have similar dis-

tributions, then the CLT test may have statistical proper-

ties similar to the other three tests. The MCLT test con-

siders two parameters: the zero probability and non-zero

mean and thus is better than the CLT test and robust for

most cases. In addition, high Type I errors caused by the

MCLT test can be adjusted by the permutation test for

small sample size. Therefore, the MCLT test can be

recommended for general use regardless whether the data

distribution is known or unknown. Numerical investiga-

tion on other types of distributions should help gain more

information regarding the MCLT method.

Even though the permutation test showed several ma-

jor advantages, the LR and Wald test still sustain desir-

able Type I errors and testing powers and are not as

computationally intensive when the distribution for a

large data set is known or the assumed distributions are

appropriate. Nevertheless, the permutation test could be a

valuable addition to the current statistical tests especially

when a data set is small or the distribution is unknown.

REFERENCES

[1] D. M. Titterington, A. F. Smith and U. E. Makov, “Statis-

J. X. WU ET AL.

280

tical Analysis of Finite Mixture Distributions,” John Wiley

and Sons, New York, 1985.

[2] J. Lawless, “Negative Binomial and Mixed Poisson Re-

gression,” Canadian Journal of Statistics, Vol. 15, No. 3,

1987, pp. 209-225. doi:10.2307/3314912

[3] D. C. Heibron, “Generalized Linear Models for Altered

Zero Probability and Overdispersion in Count Data,” SIMS

Technical Report No. 9, University of California, San

Francisco, 1989.

[4] R. Schall, “Estimation in Generalized Linear Models with

Random Effects,” Biometrika, Vol. 78, No. 4, 1991, pp.

719-727. doi:10.1093/biomet/78.4.719

[5] C. E. McCulloch, “Maximum Likelihood Algorithms for

Generalized Linear Mixed Models,” Journal of American

Statistical Association, Vol. 92, No. 437, 1997, pp. 162-

170.

[6] D. B. Hall, “Zero-Inflated Poisson and Binomial Regres-

sion with Random Effects: A Case Study,” Biometrics,

Vol. 56, No. 4, 2000, pp. 1030-1039.

doi:10.1111/j.0006-341X.2000.01030.x

[7] L. Zhang, J. Wu and W. D. Johnson, “Empirical Study of

Six Tests for Equality of Populations with Zero-Inflated

Continuous Distributions,” Communications in Statistics

—Simulation and Computation, Vol. 39, No. 6, 2010, pp.

1196-1211. doi:10.1080/03610918.2010.489169

[8] G. Casella and R. L. Berger, “Statistical Inference,” Dux-

bury Inc., San Francisco, 2002.

[9] A. Wald, “Tests of Statistical Hypotheses Concerning

Several Parameters When the Number of Observations Is

Large,” Transactions in American Mathematical Society,

Vol. 54, No. 3, 1943, pp. 426-482.

[10] E. S. Edgington, “Statistical Inference and Nonrandom

Samples,” Psychological Series A, Vol. 66, No. 6, 1966,

pp. 485-487. doi:10.1037/h0023916

[11] B. E. Wampold and N. L. Worsham, “Randomization

Tests for Multiple Baseline Designs,” Behavioral As-

sessment, Vol. 8, 1986, pp. 135-143.

[12] R. C. Blair and W. Karniski, “An Alternative Method for

Significance Testing of Waveform Difference Potentials,”

Psychophysiology, Vol. 30, No. 5, 1993, pp. 518-524.

doi:10.1111/j.1469-8986.1993.tb02075.x

[13] D. C. Adams and C. D. Anthony, “Using Randomization

Techniques to Analyze Behavioural Data,” Animal Be-

haviour, Vol. 61, No. 1, 1996, pp. 733-738.

doi:10.1006/anbe.2000.1576

[14] J. Ludbrook and H. Dudley, “Why Permutation Tests Are

Superior to t and F Tests in Biomedical Research,”

American Statistician Association, Vol. 52, No. 2, 1998,

pp. 127-132.

[15] A. F. Hayes, “Randomization Tests and Equality of Vari-

ance Assumption When Comparing Group Means,” Ani-

mal Behaviour, Vol. 59, No. 3, 2000, pp. 653-656.

doi:10.1006/anbe.1999.1366

[16] L. H. Koopman, “Introduction of Contemporary Statisti-

cal Methods,” 2nd Edition, Duxbury Press, Boston, 1981.

[17] J. Aitchison, “On the Distribution of a Positive Random

Variable Having a Discrete Probability Mass at the Ori-

gin,” Journal of American Statistical Association, Vol. 50,

No. 271, 1995, pp. 901-908.

[18] S. C. Wang, “Analysis of Zero-Heavy Data Using a Mix-

ture Model Approach,” Ph.D. Thesis, Virginia Polytech-

nic Institute and State University, Blacksburg, 1998.