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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
An Empirical Study of the Chi-Square Probability Plot
for Assessing Multivariate Normality
by
M.A. Wong and W.Mo Lebow
Working Paper
# 1132-80/1
June 30, 1980
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
An Empirical Study of the Chi-Square Probability Plot
for Assessing Multivariate Normality
by
M.A. Wong and W.M„ Lebow
Working Paper
#
1132-80/^
June 30, 1980
An Empirical Study of the Chi-Square Probability Plot
for Assessing Multivariate Normality
by
M.A. Wong and W.M. Lebow
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
MA 02139,
U.S.A.
SUr/IMARY
The chi-square probability plot of the ordered generalized
distances from the sample mean vector has been suggested for use in
checking the normal assumption for a given body of multivariate
observations. Here, the results of an empirical study are presented
to illustrate the small and large sample behavior of this plot for
samples from both normal and non-normal populations. Our sample
sizes range from 10 to 200.
As might be expected, samples of 10
tell us almost nothing about multivariate normality, whereas samples
of 200 seem very stable except for their few highest points. In
general, for samples of size 30 or more, the chi-square plot appears
to be an effective graphical aid for assessing multivariate normality.
Keywords:
Multivariate Normality; Generalized Distances; Chi-Square
Probability Plot;
Computer Simulations
.
1.
INTRODUCTION
Since the assumption of multivariate normality underlies
much of the classical multivariate statistical methodology, it
would be useful to have procedures for checking the validity of the
normal assumption for a given set of multivariate data. As pointed
out by Gnanadesikan (1977) and Everitt (1978)
,
such checks would
be helpful in guiding the subsequent analysis of the data, perhaps
by suggesting the need for and nature of a transformation of the
data to make them more nearly normally distributed, or perhaps by
indicating appropriate modifications of the models and methods for
analyzing the data. Numerical methods for testing joint normality
have been developed by Mardia (1970, 1975)
(1973), and Andrews et„ al.
Afifi, A. A.
,
Malkovich, J.F. and
(1971,
1972, 1973); a
detailed summary of these techniques is given in Gnanadisikan (1977)
Healy (1958) first proposed an extension of the normal
probability plot to assess multivariate normality graphically. His
method uses the generalized distance,
d.
,
of each point from the
sample mean vector, where
d.
1
and
X.
=
(x.
~i
_ T
- x)
~
-1
S
(x.
-^
- X)
~
is the i th observation vector, x is the p-variate sample mean
vector, and S is the sample variance-covariance matrix. If the sample
data are from a multivariate normal distribution, then these distances
have approximately a chi-square distribution with p degrees of freedom.
(The exact marginal distribution of d.
is known to be a constant
multiple of a beta rather than a chi-square distribution; see
Gnanadesikan and Kettenring, 1972).
This distributional property
suggests that a chi-square probability plot of the ordered
generalized distances may be useful for assessing joint normality.
(i =
1,..., n), are
Specifically, the n generalized distances,
d.
ordered in magnitude, and the ith ordered
value is plotted
against the quantile of a chi-square distribution with p degrees of
freedom corresponding to a cumulative probability of
i
= l,...,n.
(i - *5)/n,
for
For multivariate data, the resulting plot should resemble
a straight line passing through the origin;
departures from normality
will be indicated by departures from linearity in this plot. In 1973,
Andrews et. al. suggested a graphical procedure that utilizes a
generalized distances - and - angles representation of multivariate
data, a procedure which enconpasses Healy's technique.
Like other probability plotting procedures, the proposed
chi-square probability plot can be considered as an informal but
informative graphical aid for data analysis. However, unless the users
acquire some feeling for the behavior of this plot for samples from
both normal and non-normal distribution, its effectiveness would be
limited.
Some examples illustrating the usefulness of the chi-square
probability plot for assessing joint normality are given in Gnanadesikan
(1977) and Everitt
(1978); but, no systematic study has been done to
examine the small and large sample behavior of this plot for samples
from both normal and non-normal populations.
In this study, we use computer simulated random samples to
examine the behavior of the chi-square probability plot for both the
null case (normal population) and the non-null case (non-normal population)
In the null case, we generated samples of size
n = 10,
20,
30,
50,
and
200 from 2- and 5- variate normal distributions, and obtained the
corresponding chi-square probability plots.
These generated plots
provide a reference for judging the normality of a given sample; they
are given in Section 2.
The results obtained for the non-null case are given in
Section
3.
There, generated plots are produced to illustrate the
power of the chi-square probability plot when samples are drawn from
2- and 5- variate
(independently distributed variates) non-normal
distributions including chi-square
(10
degrees of freedom)
,
10%
contaminated normal, and Cauchy distributions. Concluding remarks are
given in the final section.
.
CHI-SQUARE PROBABILITY PLOTS FOR NORMAL DATA
2.
All the probability plots presented in this paper were
produced on an IBM 370/168 computer, and the plotting routine used
is a modification of Spark's
(1971)
algorithm. In each of the plots,
a line with unit slope which passes through the origin is plotted for
reference purposes; moreover, multiple points are represented by the
letter "0" and single point by the asterisk "*". And the required
chi-square quantiles were computed using the methods of Kuo (1965)
and Goldstein (1973)
The random number generator used is the one given in Lewis
et. al.
(1969)
,
In the null case, normal deviates were computed from
the generated random numbers
using Cunningham's (1969) algorithm.
For each of the two number of dimensions, p =
2
and 5, four chi-square
probability plots were produced for each of the sample sizes n= 10, 20,
30,
50, and 200;
where
MVN
I
(0,
matrix.
half of the normal samples were drawn from MVN(0, I),
is the identity matrix of rank p, and half were drawn from
E
)
where
E
The plots are given in Figures lA - 5A (p=
Figures IB - 5B (p=
30,
is some randomly generated positive - definite
50 and 200.
5)
2)
and
respectively for the five sample sizes n=10, 20,
As might be expected, samples of 10 tell us almost
nothing about normality, whereas samples of 200 seem very stable
except for their few highest points.
Samples of 20 show wild
fluctuations; samples of 30 are better behaved; and samples of 50
nearly always appear linear but fluctuate at their upper ends.
Neither the population covariance structure nor the number of
dimension appears to affect the behavior of the plot.
(Insert Figures lA-B to 5A-B here.)
.
.
3.
CHI-SQUARE PROBABILITY PLOTS FOR NON-NORMAL DATA
In order to examine the power of the chi-square probability
plot, we repeated the simulation study using samples from non-normal
distributions
3.1
10% Contaminated Multivariate Normal Distribution
Here the generated data are from 90% MVN (0,
For each of the two number of dimensions, p=
2
I)
+ 10% MVN
(0,101)
and 5, two chi-square
probability plots were produced for each of the sample sizes n= 10,30 and
200.
At n= 200, the chi-square plots clearly indicate non-normality
(see Figure 6)
;
at n=30, there is some evidence that the distribution
(see Figure 7)
is not normal
can be observed (see Figure
;
8)
but at n= 10, no indication of contamination
.
Thus as expected, large sample size is
needed to detect contamination.
(Insert Figures 6 to 8 here)
3.2
Multivariate Chi-Square Distribution
The generated data consists of p independent (p=
2
and
5)
variates, each of which has a chi-square (10) distribution. This is a
mildly skewed distribution and at n= 200, there is again a clear
indication of non-normality (see Figure
(n=10 and 30)
,
9)
;
but for smaller samples
we find it hard to distinguish the plots given in
Figures 10 and 11 from those given in Figures lA-B and 3A-B respectively.
Hence, a rather large sample is needad to detect the moderate skewness
in the chi-square population, and it should be noted that the chi-square
probability plots obtained from a "skewed" distribution are convex.
(Insert Figures 9 to 11 here.)
3.3
Multivariate Cauchy distribution
This is an extremely heavy-tailed distribution. Even at n= 10,
the probability plots are far from linear
(see Figure 12)
.
Apparently,
the chi-square probability plot has good power for rejecting normality
when the time distribution is heavy-tailed. The plots for the sample
sizes n= 20 and 100 are given in Figures 13 and 14 respectively.
(Insert Figures 12 to 14 here.)
3.4
Multivariate Normal Distribution with an Outlier
Since the commonest type of non-normality is a single oversized
outlier, it would be interesting to find out if the chi-square probability
plot would be able to detect multivariate outliers. Realizing that there
may be various types of multivariate outliers (see Gnanadesikan and
Kettenring, 1972)
,
we confine our effort to studying outliers in one
particular dimension.
In each of the plots given in Figure 15, one
observation has been contaminated by adding 10 standard deviations to
the first variate.
For samples of 30 and more, the chi-square plots
consistently identify the outliers; however, the plots are not that
effective for small samples.
(Insert Figure 15 here.)
.
4.
DISCUSSION
The object of this simulation study is to illustrate the
usefulness of the chi-square probability plot for assessing multivariate normality.
The results indicate that the minimum sample
size needed to detect non-normality depends on the degree to which the
underlying distribution is "non-normal";
but, a sample size of 30 is
usually sufficient to detect all but slight deviations from normality.
For normal samples, chi-square plots for various sample sizes were
generated to provide a reference for the "expected" behavior. However,
it should be pointed out that it is only through routine usage that
an user can become more adept at interpreting the chi-square plots
10
REFERENCES
Andrews, D.F., Gnanadesikan, R. , and Warner, J.L. (1971). Transformations
of multivariate data. Biometrics , 27 , 825-840.
Bell
Methods for assessing multivariate normality.
(1972) .
Laboratories Memorandum.
Methods for assessing multivariate normality. In
(1973) .
Multivariate Analysis III (P.R. Krishnaiah, ed.). Academic
Press, New York, pp. 95-116.
Cunningham, S.W. (1969). Algorithm AS 24. From Normal Integral to
Deviate. Applied Statistics , 18 , 290-293.
Everitt, B.S. (1978). Graphical Techniques for Multivariate Data
Holland, New York.
.
North-
Gnanadesikan, R. (1977). Methods for Statistical Data Analysis of
Multivariate Observations. John Wiley and Sons, New York.
Robust estimates, residuals
Gnanadesikan, R. and Kettenring, J.R. (1972).
and outlier detection with multiresponse data. Biometrics , 28 ,
81-124.
Goldstein, R.B. (1973). Algorithm 451. Chi-Square Quantiles. Communications
of the ACM , 16 , 483-485.
Healy
,
M.JoRo
17_,
Kuo, S.S.
(1968). Multivariate Normal Plotting. Applied Statistics ,
157-161.
Numerical Methods and Computers Add i son-Wesley
Publishing Company, Reading, Massachusetts, pp. 83-89.
(1965).
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Lewis, P.A.W., Goodman, A.S., and Mills, JoM. (1969). Pseudo Random Number
Generator for the System /360. IBM Systems Journal 2_.
,
Malkovich, J.F. and Afifi, A. A. (1973). On tests for multivariate normality.
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Mardia, K.V. (1970). Measures of multivariate skewness and kurtosis with
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applications. Biometrika ,
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Hotelling's test. Applied Statistics , 24 , 163-171.
Sparks, D.N. (1971). Algorithm AS 44. Scatter Diagram Plotting. Applied
Statistics, 20, 327-331.
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