An Omnibus Test for Univariate and

Multivariate Normality
By JURGEN A. DOORNIKy and HENRIK HANSEN
Nueld College, Oxford University of Copenhagen, Denmark
November 24, 1994

SUMMARY
We suggest an easy to use version of the omnibus test for normality using skewness
and kurtosis based on Shenton and Bowman (1977) which controls well for size. A
multivariate version is introduced. Size and power are investigated in comparison
with four other tests for multivariate normality. The alternative hypothesis in the
power simulations is the whole Johnson system of distributions.

Keywords: JOHNSON SYSTEM; KURTOSIS; MULTIVARIATE NORMALITY TEST; SKEW-

NESS; UNIVARIATE NORMALITY TEST; WILSON-HILFERTY TRANSFORMATION

1. INTRODUCTION
Testing for normality is a common procedure in much applied work and many tests
have been proposed, see for example the overviews in Mardia (1980), D'Agostino (1982)
and Small (1985). The need for testing normality in a multivariate setting is discussed,
inter alia, by Gnanadesikan (1977, x5.4.2), Cox and Small (1978) and recently Cox and
Wermuth (1994).
A test frequently used is the sum of squares of the standardized sample skewness
and kurtosis, which is asymptotically distributed as a 2 -variate. For small samples the
transformation to approximate 2 as given in Bowman and Shenton (1975) is cumber-
some, and many computer programs only report the asymptotic version of the test.
The next section bases a more convenient test statistic on work by Shenton and
Bowman (1977) (a possibility already mentioned by Pearson, D'Agostino and Bowman,
1977), who give the kurtosis a conditional gamma distribution. The test statistic uses
transformed skewness and kurtosis, and is simple to implement. A multivariate version of
the test is introduced in x3. Size and power comparisons with several other multivariate
tests are given in x5 and x6.
All results in this paper are based on random samples. In practice, however, the
tests will also be applied to regression residuals and residuals from time series models.
yAddress for correspondence: Nueld College, Oxford OX1 1NF, UK

1
 an omnibus test for normality 2

Limited numerical results in Pierce and Gray (1982), Pierce (1985) and Lutkepohl and
Schneider (1989) suggest that the size of the tests is largely una ected, but there could
be some loss of power.

2. THE UNIVARIATE OMNIBUS TEST

Let (x1; : : :; x ) be a sample of n independent observations on a 1{dimensional random
n

variable with mean  and variance  2 . Write  = E (X , ) , so that  2 = 2 . The

i
i

skewness and kurtosis are de ned as:

p =  and 3
= 24 :
1
 3=2
2
2
2
Sample counterparts are de ned by:

x = n1
X
n

x ; m = n1
X
n
p
(x , x) ; b1 = m332 and b2 = m42 :
i

i =1
i i

i=1
i
m2 =
m2
Bowman and Shenton (1975) consider the test ( f denotes `asymptotically dis-
a

tributed as'): pb )2 n (b , 3)2

n (
E = 6 1 + 224 f  (2)
a 2
p a

p
unsuitable except in very large samples. The statistics b and b are not independently
1 2
distributed (although uncorrelated), and the sample kurtosis especially approaches nor-
mality
p very slowly. They proceed to derive a test based on approximating the distribution
of b1 and b2 by the Johnson system (see e.g. Kendall, Stuart and Ord, 1987, x6.27{36,
or Pearsonpand Hartley, 1972, x18), assuming independence. The Johnson S approxi- U

mation to b1 is given in D'Agostino (1970). The kurtosis part is less convenient, as it

involves iteration to nd the parameters of the tting distribution (owing to a Johnson
S being used for n < 25).
B

The test considered here derives from Shenton and Bowman (1977), p who give b2
(conditional on b2 > 1 + b1 ) a gamma distribution. The distribution of b1 is still based
on Johnson S . Let z1 and z2 denote the transformed skewness and kurtosis, where the
U

transformation creates statistics which are much closer to standard normal. The test
]
statistic is ( denotes `approximately distributed as'):
]  (2) :
app

E = z12 + z22
p app
2

The transformation for the skewness is as in D'Agostino (1970); the kurtosis is trans-
formed from a gamma distribution to 2 , which is then translated into standard normal
using the Wilson-Hilferty cubed root transformation. The precise form of z1 and z2 is
given in the appendix, together with a numerical example. The advantage of this statis-
tic is that is easy to implement and requires only tables of the 2 distribution. Note
that the formulae break down for n  7.
 an omnibus test for normality 3

3. THE MULTIVARIATE OMNIBUS TEST

Let X 0 = (X1 ; : : :; X ) be a p  n matrix of n observations on a p-dimensional vector
n

with sample mean and covariance X = n,1 (X1 + : : : + X ) and S = n,1 X 0X where
n

X 0 = (X1 , X;
 : : :; X , X ).
n

Create a matrix with the reciprocals of the standard deviation on the diagonal:
 
V = diag S11,1 2; : : :; S ,1 2 ;
=
pp
=

and form the correlation matrix C = V SV . De ne the p  n matrix of transformed

observations:
R0 = H ,1 2H 0V X 0;
=

with  = diag (1; : : :;  ), the matrix with the eigenvalues of C on the diagonal. The
n

columns of H are the corresponding eigenvectors, such that H 0H = I and  = H 0CH . p

Using population values for C and V , a multivariate normal can thus be transformed
into independent standard normals; using sample values this is only approximately so.
We may now compute univariate p skewness
p and kurtosis of each transformed n{vector
of observations. De ning B1 = ( b11; : : :; b1 ), B20 = (b21; : : :; b2 ) and  as a p-vector
0 p p

of ones, the test statistic:

nB10 B1 + n (B2 , 3)0 (B2 , 3) 2 (2p)
6 24 f
a

will again require large samples.

The proposed multivariate statistic is:
E = Z10 Z1 + Z20 Z2
p ]  (2p) ;
app
2

where Z10 = (z11 ; : : :; z1 ) and Z20 = (z21; : : :; z2 ) are determined by (1) and (2) given
p p

in the appendix. So after transformation of the data to approximately independent

standard normals, the univariate test is applied to each dimension. Using the correlation
matrix, rather than the covariance, makes the test scale invariant. Basing the square
root of C on the eigenvectors gives invariance to ordering (using Choleski decomposition
to compute the square root is inadvisable, as it makes the test sensitive to the ordering
of the variables). In practice it will be useful to inspect all elements in Z1 and Z2 and
maybe even compute the transformed skewness and kurtosis of the original data vectors.
If the rank of C is less than p, some eigenvalues will be zero. In that case the
following procedure can be used. Select the eigenvectors corresponding to the p non-
zero eigenvalues, G say, and create a new data matrix X  = XG. This will be an
n  p matrix. Now compute E using X  and base the tests on p degrees of freedom.
p

Simulations showed that this works satisfactorily.

4. FOUR OTHER MULTIVARIATE TESTS

Four other tests for multivariate normality are used in our size and power comparisons.
Whereas x3 suggests to rst transform to approximate independent normality and then
 an omnibus test for normality 4

compute marginal skewness and kurtosis coecients, Small (1980) took the opposite
route: weight the marginal skewness and kurtosis coecients of the raw variables by
their approximate correlations. Let B1 and B2 denote the vectors of sample skewness
and kurtosis of the original data, which has sample correlation matrix C = (c ). Small ij

(1980) used the Bowman and Shenton (1975) correction, but here we use x2 to compute
the transformed skewness and kurtosis of the original data vectors obtaining Z1 and Z2,
and:
Q = Z10 U1,1 Z1 + Z20 U2,1Z2
p 2 (2p) ; ]
app

where U1 = (c3 ) and U2 = (c4 ).

ij ij

The next test is that proposed by Mardia (1970). Create the n  n matrix:
 ,1X 0;
D = (d ) = XS
ij

and de ne multivariate measures of skewness and kurtosis as:

b = 1 d3 and b = 1 d4 :
XX n n
X n

1p
n2 i=1 j =1
ij 2p
n i=1
ii

An omnibus test based on these measures is:

 
nb n b  , p (p + 2) 2  
M = 6 + p (p + 1) (p + 2)
f  +1 :
1 p 2 p
2
p
8p (p + 2) 6a

A large sample will be required for the 2 distribution to hold.

Recently, Mudholkar, McDermott and Srivastava (1992) proposed a new test of mul-
tivariate normality based on the correlation between the normalized diagonal of D and
jackknifed estimates of its variance. At various stages the statistic is transformed and
empirically adjusted, resulting in a statistic which is approximately normally distributed.
We square this statistic to obtain the 2 equivalent:

]
!
z + A1 (p)=n , A2 (p)=n
2 2
Z = p
 2
(1) :
fB1(p)=n , B2(p)=n2g 21
p app

The original paper shows how to obtain A and B . To obtain a correct size for the test
i i

statistic, we found that we had to add  = A1 (p)=n , A2 (p)=n2 to z rather than

n;p p

subtract it.
Finally, we consider the multivariate Shapiro-Wilk test as proposed by Royston
(1983). This combines the univariate tests (see Royston, 1982) creating a statistic H
which has an approximate 2 distribution with non-integer degrees of freedom e, where
1  e  p. For convenience we translate this into a 2 (1) using the Wilson-Hilferty
approximation:

]  (1) :
( )
w= H  13 , 1 + 2 9e  21 ; W = w2 2
e 9e 2 p app
 an omnibus test for normality 5

5. SIZE OF THE TESTS

To investigate the size of the E test we simulated rejection frequencies under the null
p

of normality, using 2 (2p) critical values. The reported simulations were done in Ox
(a C++ like matrix language), using the internal random number generator. The com-
putations were repeated in Gauss, resulting in minor descrepancies owing to a di erent
random number generator. Each table is based on common randomp numbers. We report
the approximate Monte Carlo standard error as MCSE = fq (1 , q )=M g where M is
the number of replications and q the tail probability of interest.
Table 1 contrasts E with its asymptotic version (E ) and the form suggested by
p
a
p

Anscombe and Glynn (1983), labelled E . The latter uses the Pearson type V distri-
V
p

bution for the kurtosis and a Johnson S for the skewness. Table 1 shows that this
U

does not control the size as well as E . Figure 1 illustrates the bene ts of taking the
p

dependence into account.

Tables 2 and 3 give the size for some multivariate cases. In Table 2 the simulations
are under the null of independent normality, with all tests showing acceptable size,
although W shows that the Wilson-Hilferty approximation which we adopted only works
p

moderately well in the bivariate case. Table 3 is for the bivariate normal case, with
various correlation coecients. It shows that Q is more sensitive to  than E . Both
p p

M and Z are invariant to .

p p

6. POWER COMPARISONS
For the power simulations we compute rejection frequencies at 5% under the alternative
distribution, using a sample size of 50 observations. The critical values for E , Z and
p p

W are taken from the 2 table; the 5% critical values for M are found by simulation:
p p

18.2 for p = 3, n = 50 and 10.0 for p = 2, n = 50.

In the rst set of experiments we use the entire Johnson system as the set of alterna-
tive distributions. This results in power simulations forming a two-dimensional grid over
the skewness{kurtosis plane. A similar approach was adopted by Baringhaus, Danschke
and Henze (1989) who considered four univariate tests, one of them the test of Bowman
and Shenton (1975). Because of symmetry we only consider positive skewness. Remem-
ber that 2 > 1 + 1 always; power in that part of the skewness{kurtosis plane which
does not satisfy this relation are set to zero, resulting in the at area in the far corner.
Along
p the edge 2 = 1 + 1 are the bimodal distributions. The normal distribution is at
( 1 ; 2) = (0; 3); the exponential at (2; 9), just outside the graphs.
The experiments use three independent random variables, and can be summarized
as follows:
label n M p X1 X2 X3
P12 50 1000 3 Johnson standard normal standard normal
P30 50 1000 3 Johnson Johnson Johnson
The graphs are presented for P12 only, for the test statistics E , Z and M ; P12 for
p p p

Q very closely resembles that of E and is omitted. We see from Figs. 2{4 that E has
p p p
 an omnibus test for normality 6

the best power properties. Z has virtually no power against any of these alternatives.
p

Z focuses only on the elements of D which form the kurtosis component of M , so low
p p

power against skewed alternatives is expected, however, power in the kurtosis dimension
is absent too. Figure 5 shows the di erence of power between E and W , and con-
p p

rms an established result for the univariate case, namely that the Shapiro-Wilk test
has higher power against platykurtic distributions, and lower power against leptokurtic
distributions. Or more precisely: Shapiro-Wilk has higher power against alternatives in
the bimodal area of the Johnson S system, but lower against S . The graphs for the
B U

P30 simulations are not presented, but give the same overall picture: all powers have
improved, but the relative ranking remains the same.
We experimented with an alternative form of the proposed test statistic E using p

only one degree of freedom for the kurtosis component: the p 2 components P
of the
kurtosisPare added up, giving (non-integer) degrees of freedom = pa + c b1 for i

 = 2k (b2 , 1 , b1 ), to which the Wilson-Hilferty transformation is applied (these

i i

replace the and  in (2), given in the appendix). This alternative test results in
somewhat better size properties, but a loss of power in low kurtosis directions (a property
shared by M which also has only one kurtosis component). This form could be useful
p

if all kurtosis components show departure in the same direction.

In the next set of power experiments, the alternative is the generalized Burr-Pareto-
logistic distribution with normal marginals, see Cook and Johnson (1986). The param-
eters and correspond to the notation used there. The case of ! 1 and = 0
corresponds to independent normals, whereas ! 1 for 6= 0 is Morgenstern's bivari-
ate uniform distribution, still with normal marginals. We see in Table 4 that Mardia's
test performs best here, followed by E . The most salient feature, however, is the com-
p

plete lack of power of the Q and W tests against a non-normal bivariate distribution
p p

with normal marginals. It is interesting to note that the test suggested by Malkovich and
A (1973), which consists of that linear combination of the variables which maximizes
skewness or kurtosis, will also fail in this situation. The W  statistic is an alternative
p

form of the multivariate Shapiro-Wilk test, where the data are rst transformed to ap-
proximate independent normality as for E ; then the p univariate Shapiro-Wilk tests are
p

squared and added up, resulting in a 2 (p) test statistic.

We conclude that E is the preferred test of the four considered. It is both simple,
p

has correct size and good power properties. However, E could potentially be used in
p

conjunction with M .p

ACKNOWLEDGEMENTS
We wish to thank Sir David Cox, David Hendry, Chris Orme and Neil Shephard for
helpful comments. Financial support from the UK Economic and Social Science Research
Council is gratefully acknowledged by Jurgen Doornik.
 an omnibus test for normality 7

APPENDIX
p
The transformation for the skewness b1 into z1 is as in D'Agostino (1970):
27n , 70)(n + 1)(n + 3) ;
= 3((nn ,+2)(
2

n + 5)(n + 7)(n + 9)
! 2 = ,1 + f2( , 1)g 2 ;
1

 = n p1 o 1 ; (1)
log( ! 2) 2
p
y = b1 2

! 2
, 1
1
(n + 1)(n + 3) 2
;
 6(n , 2)
,
1
z1 =  log y + y 2 + 1 2 :
The kurtosis b2 is transformed from a gamma distribution to 2 , which is then translated
into standard normal z2 using the Wilson-Hilferty cubed root transformation:
 = (n , 3)(n + 1)(n2 + 15n , 4);
a = (n , 2)(n + 5)(n + 7)(n2 + 27n , 70) ;
6
c = (n , 7)(n + 5)(n + 7)(n2 + 2n , 5) ;
6
k = (n + 5)(n + 7)(n3 + 37n2 + 11n , 313) ; (2)
12
= a + b1c;
 = (b2 , 1 , b1)2k;
 
z2 =   13 , 1 + 1 (9 ) 12 :
2 9
We base a numerical example on the iris data of Fisher (1936), which was used for
the same purpose by Small (1980) and Fang and Wang (1994, example 6.5). Using the 50
observations for Iris Setosa on sepal length, sepal width, petal length and petal width,
we nd for each individual variable: skewness (0.11645, 0.039921, 0.10318, 1.2159) and
kurtosis (2.6542, 3.7442, 3.8046, 4.4343). Applying the transformation to approximate
normality discussed in x3 yields skewness B10 = (0.19965, ,0.17132, 0.15837, 1.1610) and
kurtosis B20 = (2.8221, 4.1994, 3.9722, 4.5793). Application of (1) and (2) then yields
Z10 = (0.63839, ,0.54876, 0.50762, 3.1862) and Z20 = (0.24687, 2.5423, 2.2381, ,1.3278).
We see that after rotation, skewness is a problem in one dimension, whereas excess
kurtosis is evident in two dimensions. The resulting test statistics are: E = 24:4145,
p

Q = 24:0387, M = 27:3413, Z = 1:12299. Both E and Q are signi cant at less

p p p p p

than 1%, whereas the Z value corresponds to the 29 per cent point, and M to 10%
p p

(based on simulation results). In this case the non-normality of petal width seems to be
the main factor, as without that variable non-normality is not rejected at the 5% level,
with test-statistics of 10.8174 (9%), 8.74419 (19%), 12.9066 (20%) and 1.06251 (30%)
respectively. The same conclusion was reached by Royston (1983).
 an omnibus test for normality 8

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 an omnibus test for normality 9

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 an omnibus test for normality 10

Table 1. Empirical size of E , E and E for p = 1, M = 10 000.

a
p
V
p p

E a
p
E
V
p
Ep

n 20% 10% 5% 1% 20% 10% 5% 1% 20% 10% 5% 1%

10 .033 .017 .010 .002 .158 .093 .057 .024 .204 .101 .048 .007
20 .062 .036 .022 .010 .165 .091 .055 .018 .185 .092 .046 .011
50 .094 .055 .035 .012 .177 .095 .055 .017 .178 .088 .046 .011
150 .138 .070 .044 .018 .191 .098 .055 .016 .182 .092 .047 .012
MCSE .004 .003 .002 .001 .004 .003 .002 .001 .004 .003 .002 .001

Table 2. Empirical size of E , Q , Z and W for p = 2; 3; 6, M = 10 000.

p p p p

Ep Qp Z
p Wp

n; p 20% 10% 5% 1% 20% 10% 5% 1% 20% 10% 5% 1% 20% 10% 5% 1%

20,2 .184 .097 .049 .011 .183 .097 .046 .009 .225 .110 .051 .007 .190 .089 .038 .005
50,2 .179 .092 .051 .015 .177 .096 .050 .014 .208 .104 .054 .009 .192 .089 .039 .007
150,2 .181 .096 .049 .013 .181 .096 .049 .014 .204 .103 .051 .010 .192 .091 .038 .006
20,3 .184 .093 .048 .011 .185 .093 .046 .010 .220 .109 .049 .007 .195 .094 .044 .008
50,3 .180 .093 .051 .015 .180 .095 .052 .015 .215 .107 .050 .006 .201 .093 .045 .009
150,3 .182 .095 .054 .016 .183 .097 .053 .016 .200 .102 .050 .010 .205 .103 .047 .008
20,6 .179 .093 .048 .012 .179 .091 .048 .011 .217 .109 .052 .008 .200 .102 .052 .010
50,6 .177 .092 .052 .012 .180 .096 .051 .013 .218 .106 .052 .008 .208 .099 .051 .009
150,6 .182 .094 .052 .015 .182 .096 .053 .016 .198 .097 .045 .009 .207 .105 .053 .011
 an omnibus test for normality 11

Table 3. Empirical size of E , Q and W for p = 2, n = 50, M = 10 000.

p p p

E p Qp W p

 20% 10% 5% 1% 20% 10% 5% 1% 20% 10% 5% 1%

.99 .180 .091 .048 .013 .212 .114 .065 .021 .152 .058 .029 .006
.9 .178 .095 .050 .013 .200 .109 .062 .019 .153 .065 .027 .007
.5 .180 .096 .052 .013 .182 .097 .055 .015 .192 .090 .041 .006
0 .179 .092 .051 .015 .177 .096 .050 .014 .192 .089 .039 .007

Table 4. Power at 5% against generalized Burr-Pareto-logistic with normal marginals

for p = 2, n = 50, M = 5 000.
E Q M Z W W
p p p p p p
E Q M Z W W
p p p p p p

.1 ,1 .955 .139 .993 .544 .027 .912 1 ,1 .053 .054 .248 .064 .046 .062
.1 ,.5 .954 .145 .993 .573 .028 .906 1 ,.5 .070 .052 .238 .066 .046 .069
.1 0 .953 .153 .990 .597 .026 .906 1 0 .085 .053 .251 .065 .044 .083
.1 .5 .952 .162 .991 .621 .025 .905 1 .5 .105 .052 .268 .068 .046 .099
.1 1 .955 .179 .991 .643 .026 .911 1 1 .134 .048 .288 .073 .043 .116
.2 ,1 .774 .068 .941 .260 .039 .685 2 ,1 .041 .052 .144 .058 .041 .051
.2 ,.5 .780 .070 .940 .285 .035 .692 2 ,.5 .042 .052 .133 .060 .040 .050
.2 0 .788 .075 .935 .315 .032 .700 2 0 .047 .050 .132 .059 .041 .052
.2 .5 .799 .084 .936 .349 .029 .715 2 .5 .057 .051 .142 .064 .043 .058
.2 1 .815 .095 .938 .387 .028 .731 2 1 .073 .052 .146 .064 .041 .065
.5 ,1 .212 .053 .543 .084 .049 .184 10 ,1 .058 .048 .053 .050 .039 .056
.5 ,.5 .242 .054 .542 .088 .048 .197 10 ,.5 .047 .047 .055 .050 .039 .053
.5 0 .272 .055 .552 .106 .047 .218 10 .0 .045 .048 .057 .050 .037 .049
.5 .5 .311 .056 .577 .124 .046 .251 10 .5 .050 .051 .061 .053 .037 .051
.5 1 .369 .057 .611 .144 .043 .296 10 1 .062 .053 .057 .053 .039 .053
 an omnibus test for normality 12

Fig. 1. Cross-plots of transformed skewness and kurtosis for 5000 samples of sizes n =
20; 50; 100 from the standard normal.
 an omnibus test for normality 13

Fig. 2. P12 for E : power against two standard normal marginals and the Johnson
p

system for the third marginal.

 an omnibus test for normality 14

Fig. 3. P12 for M : power against two standard normal marginals and the Johnson
p

system for the third marginal.

 an omnibus test for normality 15

Fig. 4. P12 for Z : power against two standard normal marginals and the Johnson
p

system for the third marginal.

 an omnibus test for normality 16

Fig. 5. P12 for W , E : di erence in power against two standard normal marginals and
p p

the Johnson system for the third marginal.