Critical values of Mandel’s h and k, the Grubbs and the Cochran test statistic

AStA Adv Stat Anal(2013)97:1–10

DOI10.1007/s10182-011-0185-y

O R I G I NA L PA P E R

Critical values of Mandel’s h and k,the Grubbs

and the Cochran test statistic

Peter-T.Wilrich

Received:28July2011/Accepted:17November2011/Published online:7December2011

?Springer-Verlag2011

Abstract According to ISO5725-2(1994),measurement results obtained in an in-terlaboratory experiment are inspected for consistency by plotting Mandel’s h and k statistics and for outliers by application of the Grubbs test and the Cochran test. Critical values of these statistics for signi?cance levelsα=5%andα=1%and for some numbers p of laboratories and n of repeated measurements in the laboratories are supplied in ISO5725-2without reference to methods for their calculation.In this paper,exact formulae for the critical values of Mandel’s h and k and approximate formulae for the critical values of the Single Grubbs test,the Double Grubbs test and the Cochran test are derived.

Keywords Cochran test·Grubbs test·Mandel’s h and k statistics·Outlier tests

1Introduction

According to ISO5725-2(1994)measurement results obtained in an interlaboratory experiment are inspected for consistency by plotting Mandel’s h and k statistics and for outliers by application of the Grubbs test and the Cochran test.

Mandel’s h statistics are the differences between the laboratory means and their mean divided by the standard deviation of the laboratory means.

The Single Grubbs test for the detection of an extreme laboratory mean as an outlier uses the maximum and the minimum of Mandel’s h statistics as test statistic.

The Double Grubbs test for the detection of two extreme laboratory means as outliers uses the sum of squared deviations of the laboratory means from their average P.-T.Wilrich( )

Institut für Statistik und?konometrie,Freie Universit?t Berlin,Garystrasse21,14195Berlin, Germany

e-mail:wilrich@wiwiss.fu-berlin.de

2P.-T.Wilrich without the two extreme means divided by that with the two extreme means as test statistic.

Mandel’s k statistics are the within-laboratory standard deviations divided by the square root of the mean of the within-laboratory variances.

The Cochran test for the detection of an extremely large within-laboratory stan-dard deviation as an outlier uses the largest within-laboratory variance divided by the sum of the within-laboratory variances as test statistic,i.e.the maximum of the squares of the k statistics multiplied by the number of laboratories.

Critical values of these statistics for signi?cance levelsα=5%andα=1%and for some numbers p of laboratories and n of repeated measurements in the laborato-ries are supplied in ISO5725-2without reference to methods for their calculation.

If the statistical analysis of the data obtained in an interlaboratory experiment shall be performed by a computer program it would be more convenient to have formulae for the calculation of the critical values instead of picking them from a table and interpolating if a tabled value does not exist for a particular case.

In the following,formulae for the calculation of these critical values are derived. 2Critical values of Mandel’s h statistics

Let(x1,x2,...,x n)be a sample of n observations.The x i;i=1,...,n are modelled as realisations of random variables X i;i=1,...,n being identically and indepen-dently distributed according to the normal distribution N(μ,σ2).We denote:

ˉX=1

n

n

i=1

X i(1)

is the mean of the X i,

Q=

n

i=1

(X i?ˉX)2(2)

is the sum of the squared deviations of the X i from their meanˉX and

S2=

Q

n?1

(3)

is the sample variance of the X i.

Mandel’s h statistics(Mandel1985)

H i=X i?ˉX

S

;i=1,...,n(4)

have the same distribution for all i=1,...,n.As one of them we pick

H n=X n?ˉX

S

=

S

with =X n?ˉX(5)

Mandel,Grubbs and Cochran statistics3 and split the sample(X1,...,X n)into the subsample(X1,...,X n?1)and X n.For the subsample we denote the mean and the sum of squared deviations from the cor-responding mean as

ˉX

1=

1

n?1

n?1

i=1

X i(6)

and

Q1=n?1

i=1

(X i?ˉX1)2,(7)

respectively.X n andˉX1as well as X n and Q1are independent,and under normality alsoˉX1and Q1are independent(see Mood et al.1974,p.245).Hence,X n?ˉX1is independent of Q1and normally distributed with expectation

E(X n?ˉX1)=0(8) and variance

V(X n?ˉX1)=V(X n)+V(ˉX1)=σ2+

σ2

n?1

=n

n?1

σ2.(9)

The standardised random variable

U=X n?ˉX1

σ

n?1

n

(10)

follows the standard normal distribution N(0,1).

Q1/σ2follows theχ2ν-distribution withν=n?2degrees of freedom so that

T n?2=

U

2

ν

=X n

?ˉX1

√

Q1/(n?2)

n?1

n

=X n

?ˉX1

√

Q1

(n?2)(n?1)

n

(11)

follows Student’s tν-distribution withν=n?2degrees of freedom.We can write

ˉX=1

n

n

i=1

X i=

(n?1)ˉX1+X n

n

,

ˉX

1=nˉX?X n n?1

,

X n?ˉX1=

n

n?1

(X n?ˉX)=

n

n?1

,(12)

ˉX

1?

ˉX=?1

n?1

(X n?ˉX)=?

1

n?1

,(13) Q=

n?1

i=1

(X i?ˉX)2+(X n?ˉX)2

=

n?1

i=1

(X i?ˉX1)2+(ˉX1?ˉX)2

+(X n?ˉX)2

4P.-T.Wilrich

=Q 1+(n ?1)(ˉX

1?ˉX)2+(X n ?ˉX)2=Q 1+ 2n ?1+ 2=Q 1+n n ?1

2,Q 1=Q ?n n ?1

2,(14)insert (12),(14)and =H n S =H n √Q/(n ?1)according to (5)in (11),

T n ?2=n n ?1 Q ?

n n ?1 2

(n ?2)(n ?1)n =n (n ?1)3/2H n √Q Q ?n (n ?1)2H 2n

Q (n ?2)(n ?1)n =H n 1?n (n ?1)2H 2n (n ?2)n (n ?1)2(15)

which,solved for H n ,gives

H n =(n ?1)T n ?2 n(n ?2+T 2n ?2).(16)

Thompson (1935)derived the exact distribution of a statistic that is closely related to H n .However,we only need the quantiles of this distribution.With T n ?2also H n is symmetrically distributed around zero.In ISO 5725-21the (two-sided)critical value for H n ,h n ;1?α/2,is de?ned by

P (h n ;α/2≤H n ≤h n ;1?α/2)=P (?h n ;1?α/2≤H n ≤h n ;1?α/2)

=P (|H n |≤h n ;1?α/2)=1?α.(17)

Since H n according to (16)is an increasing function of T n ?2,we get the critical value as h n ;1?α/2=(n ?1)t n ?2;1?α/2

n(n ?2+t 2n ?2;1?α/2),(18)

where t n ?2;1?α/2is the (1?α/2)-quantile of the t -distribution with ν=n ?2degrees of freedom.

1In ISO 5725-2Mandel’s h and also the Grubbs tests of Sects.3and 4are applied to laboratory means ˉx i ;

i =1,...,p ,each being based on n observations.Hence,n has to be substituted by p ,x i by ˉx

i ,ˉx by the mean of the ˉx i and s by the standard deviation of the ˉx i .

Mandel,Grubbs and Cochran statistics 53Critical values of the Single Grubbs test

If we want to determine with the Grubbs test (Grubbs 1950)whether the largest ob-servation X max =max (X 1,...,X n )is an outlier we use the test statistic

G =max i =1,...,n H i =max i =1,...,n X i ?ˉX S =X max ?ˉX S

.(19)Analogously,if we want to determine whether the smallest observation X min =min (X 1,...,X n )is an outlier we use the test statistic

G =max i =1,...,n (?H i )=max i =1,...,n ˉX ?X i S =ˉX ?X min S

.(20)The null hypothesis of the Single Grubbs test and as well of the Double Grubbs test of Sect.4is H 0:The X i ;i =1,...,n are i.i.d.according to a normal distribu-tion.This is also the null hypothesis of any goodness of ?t test for normality or any speci?c test for normality,e.g.the Shapiro–Wilk test.However,whereas the tests for normality are powerful against the alternative hypothesis that the X i are i.i.d.accord-ing to a non-normal distribution the Grubbs tests as outlier tests are powerful against the alternative hypothesis that all X i are i.i.d.normally distributed except the ran-dom variables modelling the extreme values of the sample,and that these follow a distribution with larger (smaller)mean.

We look for the critical value g n ;1?αof G de?ned by P (G ≤g n ;1?α)=P max i =1,...,n H i ≤g n ;1?α =P n i =1(H i ≤g n ;1?α) =1?α(21)

or

P (G >g n ;1?α)=P n i =1

(H i >g n ;1?α) =α.(22)

Since the H i ;i =1,...,n are weakly negative correlated,we cannot simplify (22),however,by application of Bonferroni’s inequality that is recommended in Hawkins (1980,p.31),P n i =1(H i >g n ;1?α) ≤n i =1

P (H i >g n ;1?α)=nP (H n >g n ;1?α),(23)

and by setting nP (H n >g n ;1?α)=αor P (H n >g n ;1?α)=α/n we see that the (1?α)-quantile g n ;1?αis approximated by the (1?α/n)-quantile h n ;1?α/n .With

(18)we ?nally ?nd g n ;1?α≈(n ?1)t n ?2;1?α/n n(n ?2+t 2n ?2;1?α/n ),(24)

where t n ?2;1?α/n is the (1?α/n)-quantile of the t -distribution with ν=n ?2degrees of freedom

6P.-T.Wilrich as a slightly conservative approximation of the critical value of the Single Grubbs test.

The extreme value(that is either the largest or the smallest value)of the sample is detected as an outlier at the signi?cance levelαif G>g n;1?α.

The critical values obtained with the approximation(24)deviate from the tabulated values for n=3,4,...,147andα=0.001,0.005,0.01,0.025,0.05,0.1presented in Grubbs and Beck(1972)by not more than0.009.Because in ISO5725-2the Grubbs test is applied simultaneously to the largest and the smallest observation of the sample its Table5presents the two-sided critical values g n;0.005forα=1%and g n;0.025for α=5%.

4Critical values of the Double Grubbs test

The sample(X1,X2,...,X n),ordered from the smallest to the largest observation, is denoted by X(1)≤X(2)≤X(n?1)≤X(n).If we want to determine with the Double Grubbs test(Grubbs1950)whether the two largest observations(X(n?1),X(n))of the sample are outliers we use the test statistic

D=Q2/Q(25)

where Q is given in(2)and

Q2=n?2

i=1

(X(i)?ˉX n?2)2(26)

with

ˉX

n?2=

1

n?2

n?2

i=1

X(i)(27)

is the sum of squares without the two largest observations.

Analogously,if we want to determine whether the two smallest observations (X(1),X(2))of the sample are outliers,we use the test statistic D=Q2/Q where

Q2=

n

i=3

(X(i)?ˉX2)2(28)

with

ˉX

2=

1

n?2

n

i=3

X(i)(29)

is the sum of squares without the two smallest observations.

In order to obtain the distribution of D we look at

D?=Q?2/Q(30)

where Q?

2is the sum of squares,analogous to Q2,if any two observations are deleted.

Q according to(2)is written as the sum

Q=Q?2+R.(31)

Mandel,Grubbs and Cochran statistics 7Under the assumption that the X i are independent and normally distributed as N(μ,σ2),the sum Q divided by σ2is χ2-distributed with n ?1degrees of freedom and Q ?2and R divided by σ2are independently χ2-distributed with n ?3and 2degrees of freedom,respectively.The reciprocal of the test statistic D ?is 1D ?=Q Q ?2=Q ?2+R Q ?2=1+R Q ?2(32)

where R/Q ?2is the ratio of the χ2-distributed random variable R with ν1=2degrees

of freedom and the independent χ2-distributed random variable Q ?2with ν2=n ?3degrees of freedom.Hence,

R/2Q ?2/(n ?3)

=F ν1,ν2(33)is F -distributed with ν1=2and ν2=n ?3degrees of freedom,and we get 1D ?=1+2n ?3F ν1,ν2

(34)or

D ?=1

1+2n ?3F ν1,ν2.(35)

We look for the critical value d ?n ;αof D ?de?ned by P (D ?≤d ?n ;α)=P 11+2n ?3F ν1,ν2≤11+2n ?3F ν1,ν2;1?α =α.(36)

There are n(n ?1)/2different ways to select two out of the n observations in the sample;each of them de?nes a random variable D ?=Q ?2/Q ,and they are identically distributed.If the n(n ?1)/2random variables D ?were independent the critical value of the smallest of these random variables,min (D ?),would be given by

P min (D ?)≤d n ;α =P D ≤d ?1?(1?α)1/(n(n ?1)/2) =P D ≤11+2n ?3F ν1,ν2;(1?α)

1/(n(n ?1)/2) ,(37)i.e.

d n ;α=1

1+2n ?3F ν1,ν2;(1?α)1/(n(n ?1)/2).(38)

However,the random variables D ?are strongly dependent and min (D ?)is not necessarily equal to D =Q 2/Q .Therefore we substitute n(n ?1)/2in (38)by

f (n)=an 2+bn +c (39)

in order to obtain approximate critical values d n ;α.We determine the coef?cients a,b,c in (39)by ?tting the exact critical values given in Grubbs and Beck (1972)for n =4,...,149with the method of least squares.Table 1presents the results.For each αwe ?nd a different set of coef?cients a,b,c .The last column of Table 1gives

8P.-T.Wilrich Table1Coef?cients of f(n)

for the calculation of the critical

values of the Double Grubbs test

αa b c Maximal absolute error

0.00100.0443 1.0012?4.24930.0007

0.00500.03880.9558?3.66130.0012

0.01000.03620.9250?3.31010.0014

0.02500.03220.8833?2.85800.0017

0.05000.02890.8501?2.50750.0023

0.10000.02510.8169?2.16150.0029

the maximal absolute errors of the approximation in the range of n=4,...,149;they are all smaller than0.003.

The two extreme values of the sample are detected as outliers at the signi?cance levelαif D≤d n;αwhere

d n;α=

1

1+2

n?3

Fν

1,ν2;(1?α)1/f(n)

(40)

where Fν

1,ν2;(1?α)1/f(n)is the(1?α)1/f(n)-quantile of the F-distribution with

ν1=2andν2=n?3degrees of freedom and f(n)is calculated according to(39) with coef?cients exempli?ed in Table1.

5Critical values of Mandel’s k statistics

Let(s2

1,s2

2

,...,s2p)be a series of p sample variances each being based on n ob-

served values.Under the assumption that the observed values x ji;j=1,2,...,p, i=1,2,...,n are realisations of random variables X ji which for each j are identi-cally and independently distributed according to a normal distribution N(μi,σ2),the sample variances S2j;j=1,2,...,p divided by their expectationσ2follow the same χ2ν/ν-distribution withν=n?1degrees of freedom.

Mandel’s k statistics(Mandel1985)

K j=

S j

S

;j=1,2,...,p(41)

with

S=1

p

p

i=1

S2i(42)

have the same distribution for all j=1,2,...,p.We look at

p K j =

p

i=1

S2i

S j

=

S2j+

p

i=1,i=j

S2i

S j

=1+

p

i=1,i=j

S2i

S j

.(43)

Mandel,Grubbs and Cochran statistics 9

(n ?1)S 2j /σ2is χ2ν-distributed with ν=n ?1degrees of freedom,(n ?1)× p i =1,i =j S 2i /σ2is χ2ν-distributed with ν=(p ?1)(n ?1)degrees of freedom and

independent of S 2j .Hence, p i =1,i =j S 2i /((p ?1)(n ?1))S j /(n ?1)=1p ?1 p i =1,i =j S 2i S j

(44)follows the F ν1,ν2-distribution with ν1=(p ?1)(n ?1)and ν2=n ?1degrees of freedom.From (43)and (44)we ?nd p K 2j

=1+(p ?1)F ν1,ν2(45)or

K j = p/ 1+(p ?1)F ν1,ν2 .(46)

Since K j according to (46)is a decreasing function of F ν1,ν2we obtain the critical value k p,n ;1?α,de?ned by

P (K j ≤k p,n ;1?α)=1?α

(47)

as k p,n ;1?α= p/ 1+(p ?1)F ν1,ν2;α ,(48)where F ν1,ν2;αis the α-quantile of the F -distribution with ν1=(p ?1)(n ?1)and ν2=n ?1degrees of freedom.

6Critical values of the Cochran test

If we want to determine with the Cochran test (Cochran 1941)whether the largest variance S 2max =max (S 21,...,S 2p )of the series of p variances (S 21,...,S 2p )each be-ing based on n observations is an outlier we use the test statistic

C =S 2max p j =1S 2j =1p max j =1,...,p S 2j S 2=1p max j =1,...,p K 2j (49)

with K j according to (46).

We look for the critical value c p,n ;1?αof C ,de?ned by P (C ≤c p,n ;1?α)=P 1p max j =1,...,p K 2j

≤c p,n ;1?α =P p j =1 1p K 2j ≤c p,n ;1?α =1?α(50)

or

P (C >c p,n ;1?α)=P n i =1 1

p K 2j >c p,n ;1?α =α.(51)

10P.-T.Wilrich Since the K2j/p;j=1,...,p are weakly negative correlated,we cannot simplify (51),however,similar to the approach in Sect.3,by application of Bonferroni’s in-

equality

P

p

j=1

1

p

K2j>c p,n;1?α

≤

p

j=1

P

1

p

K2j>c p,n;1?α

=pP

1

p

K2j>c p,n;1?α

,(52)

and by setting pP(K2j/p>c p,n;1?α)=αor P(K2j/p>c p,n;1?α)=α/p we see that the(1?α)-quantile c p,n;1?αis approximated by the(1?α/p)-quantile k p,n;1?α/p. With(48)we?nally?nd

c p,n;1?α≈1/

1+(p?1)Fν

1,ν2;α/p

,(53)

where Fν

1,ν2;α/p is the(α/p)-quantile of the F-distribution withν1=(p?1)×

(n?1)andν2=n?1degrees of freedom.

as a slightly conservative approximation of the critical value of the Cochran test.

The largest standard deviation in the series of p standard deviations is detected as an outlier at the signi?cance levelαif C>c p,n;1?α.

The critical values obtained with the approximation(53)deviate from the tabu-lated values for p=2(1)10,12,15,20,24,30,40,60,120,n=2(1)11,17,37,145 andα=0.01,0.05presented in Eisenhart and Solomon(1947)by not more than 0.008.

Acknowledgements The author thanks Tilmann Deutler and Hans Schneeweiss,who read the?nal manuscript,and an anonymous referee,for their helpful comments.

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