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
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.
C
opyright © 2012 SciRes. OJS

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
,

 33 ,


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 LR
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
LR
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.
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