On Testing Normality Using Several Samples:
An Analysis of
Pe~:'cnut!\f1atoxin
Data
by
c~
p ~ Quesenberr'y, ill ~ B. Wni taker an.d ~T . VJ. Dic~ken3
Institute of Statistics
Mimeo Series #1013
Raleigh - 1975
ON TESTING NORMALI'rY USING
SEV'~RAL
SAMPLES:
AN ANALYSIS OF PE.fu1\lUT AnA'l'0XIN DA'I'f,
C. P, Quesenberry
Department of Statistics,
North Carolina State University,
Raleigh, N. C. 27607
T. B. Whitaker and J. W. Dickens
USDA-ARS, Biological and Agricultural Engineering Department
North Carolina State University
Raleigh, N. C. 27607
SUMMARY
Eight samples of size 16 and 3 samples of size 15 consisting of
replicate determinations of aflatoxin in peanut subsamples are considered.
'luis data is analyzed to decide whether the samples could have arisen from
two-parameter normal parents, or from two-parameter lognormal parents.
The
method of analysis consists of first transforming the individual samples
in such a way that the transformed values have uniform distribution:;; on
the unit interval under the null hypothesis, the transformed values are then
pooled and tested for uniformity.
'The analysis leads to the conclusion
that the normal model fits the data quite well, and definitely better than
the lognormal model.
1.
1.1
INl'RODUC'I'ION
The Biological Problem
Aflatoxin is a toxic material produced in peanuts by the fungus
Aspergillus flavus, Bampton [1963].
As a precautionary measure al1
commercial lots of peanuts in the U. S. (approximately 20,000 each
crop year) are tested for aflatoxin, Duggan [1970J.
.A lot of
shelled peanuts may vary in size up to 100,000 pounds.
The concen-
tration of aflatoxin in a lot of shelled peanuts is estimated by
measuring the aflatoxin concentration in a random sample of kernels
drawn from the lot.
The sample is comminuted in a subsampling mill,
Dickens and Satterwhite [1969J, and a subsample of approximately 280 g
is analyzed by a chemical procedure using thin layer chromatography
(TLC), Waltking, et a1. [1968J.
To facilitate an adequate quality
control and consumer protection program, it is desirable to design
a sampling plan that will provide a high level of protection for the
consumer with reasonable assurance to the processor that lots of
good peanuts will not be rejected by the testing program.
Because aflatoxin is often highly concentrated in a small percentage of the kernels, variation among aflatoxin determinations is
large, and accurate estimates of the average concentration in a lot is
exceedingly difficult, Whitaker, et a1. [19 (2, 1974 J.
r
In order to
predict the risk levels and costs associated with an anatoxin sampling
plan, computer simulation methods have been develuped to evaluate
sampling programs, w'bi taker, et al. [19'70].
One important aspect of' the computer model is estimation of
the distribution of analytical results about the 8ubsample concentra-
-
tion.
"c.
-
It is asstuned that replicated anal;/L;es frum 8'_;"bsan,p18s with
different levels of aflatoxin can all be described bJ distributions
of th,e same :fLl!lc"tional t'ornl, but 9
~pcs3ibl~{"
1'vT i ~:,r~ ~J.:ifferent para..'TIetel'llS.,
T'he objectives of the present study are to answer the questions:
(a)
Does the normal or the lognormal model give a better fit for
the numbers obtained by replicated analy::;esYf peanut subsamples?
(b)
IJoes either the normal or the lognormal model provide an
adequate fit for this data.'?
2.2
'I'J:le StatIstical Problem
Suppose tllere are available k independent samples:
X
"II'
"'~l~~
n
X
X,) 1
l -. . L
22
·..,
· . .,
Xln ;.
l
xI")
Ln ;
2
(1)
, ·.. ,
The random variables in (1) will be <"aid to satisfy a normal
model if all members of the jth sample are independeLtly distributed
with a N(J.L
.,el)
J
J
distribution for j
-..
1
.J... .,
..
('>
..
k
,
.
They win be
said to satisfy a lognormal model if all mem-bers of the j-th sarnple
.,
are independently distributed with a LN(J.L.,cr~) distribution for
J
j ,.:: 1, ... , k
J
"
(A random variable y has a IJDT( J.L,c/:) distribution
2
if' In y has a N(J.L,cr ) ciist:c'ibution.)
Tn order to ar18wer questj ems
(a) and (b) above for a particular set of data of the stru'?ture of
we require k-sample compcsi te goodrtess-o:f'-fit and model comparison
techniques.
There is a small statistical literatllTc that eonsiders
types of problems.
Pet-roY
the~!e
[1956J cot18idersshi s model for normal
(1),
distributions and samples of tbe
tional results.
Prohorov
[.L'jtJ6]
~
siz,::,: and gives
SUItie
distribu.-
!1ien~-L(m8 cui: d.ces !lot, d.C"relop ty,p
type of transformation-pooling appruadl
tl;~.1t
,rill te 'J.sed. iJel'e.
The work of a number of 'iNTi terswLc have cGnsidered the problem
of transforming a sir.gle sample 'under :so composite null bYTotbesis
assumption is of some interest here.
and Johnson
'J'hese iEclude papers by Dav-id
[1948J, Sarkadi [1960, 1965J, Durbin [1961J, Stormer
[1964J, Seshandri, et a1, [1969J, and 0 1 Reilly and Quesenberry
[1973J.
2•
2.1
ANALYSIS OF AF1)\''lOXIN
~l\T/'
Experimental Procedure
Approximately
6200 g of raw pean-u tkclT!e12 cor:.tan:.ina r~.ed
1JJitL
aflatoxin were cormninuted in a mill 8il:',i1::.;.;: to tbat t:sed by the
spection service.
The ground meal was t1> ::':C~ divided into eleven
subsamples weighing approximately
minuted peanuts was blended with
for 2 minutes.
in~
560 g e:qr::l-:.
Each subsaIT~ple of com-
2800 ml n:etLanol=water=hexane
3(11).tio~1
The blended material was :.~ivided e:,paJly among 1 i)
centrif'uge bottles.
The content of eact centrifuge bottle was 2.tialyze'd
by a modified version of the Walt-king metr'.od, Wal tking, et 8.1.,
[1968J.
One observation was lost from eac}) of
3 sutsamr:-les lemring
8 subsamples with 16 determinations and 3 Gutsa::nples witb 15 determina:;.
tions.
The data is shown in Tatle 1,
Table 1 near bere
TABLE 1
REPLICATED P-..FLATOXTN DETE"Lli'1INATIONS
S~:\I~fPLE
,.1.
:~u~'ffiER
2
3
4
5
6
121.23
95.33
20.04
,') ~ c:
L ...... ...JJ
71. 69
117,,9:1..
91. 09
55.9!.
72 .01
53.96
114.62
41. 64
98 . 76
53.62
2C~30
32.54
23.23
30.02
26.26
26.48
10.84
19.31
68.31
,)0.,)0
"\ r
13.20
--;?
1C4.86
151.00
125.40
83.94
137.53
83. L,9
116.73
90.72
100.54
74.59
137.19
146.25
24.69
22.26
34.30
27.90
34.87
32.56
31.67
29.06
32.00
16.71
18.11
11.19
6.37
11~S'8
5~28
24.8,S
t:.
12.61
28.60
32.59
37.24
2~~. 61
24.71
35.83
48.93
20 1e
1'J9
E. 74
l.? . 06
..: ,.
33 .. ~:::32. ci8
18.07
3~.99
33,,40
('
'j ';
;J .. - - i
7.53
'::: l.-f. .. ':..!:J
13.19
16.65
15.85
13.53
2Si . 10
30.55
28,,89
·31~87
32" L;o
4CL 31
26.05
is .35
21.12
17.23
13.67
51.61
2.1,06
35.88
36.50
27.53
36.71
15.25
16.25
31;»04
')7
:;1
..................
32.05
30.67
15.4-3
4.39
31. 76
16.82
11.85
"') ()
13~12
27~51
L:-8.01
')-(1
..... - . - 1 . /
21.45
27.44
17.20
1:::.44
78.53
29.(9
49_.S0
15.68
30.79
....{
.... ('u • ')
~_--'
24.40
91.92
66.88
91. 81
100.37
29.85
11.88
77.26
91.56
66.25
14.85
23.10
22.21
.~
"';)
24.92
2L:.OLf
24,34
15.32
51.98
2E .87
') ')
('\"~
:-_.U-J
26.54
21.80
22.74
29.12
36.65
35.36
~O.76
23.81
36.19
.
,,
8
12.39
7
29.17
~
10
l
...,
{,":<
I .. v .....
••
,r., • ..,
G~'ji
O.
r
-,:
J
q.
.,
.., ..........{...
~
)~
~~~2 ~
flO
69~S9
.... ",.....
~,'
_~):"..J •
J... J
,-
1~:'
.
..-r...
OJ..
73 . 98
12.58
112. ~.l
s~ 7 . S:;
;'"
~--.
l::.c·~
"ro,
71
-Lv • •
..£-
-....: ......
,",,-"
""..,.;. ",)0
?...., "1'"
-~_/~~!
3.41
32~19
O=)
7.46
94.55
60.24
if
..;....:
;j
'-
-
- 42.2
Statistical Analysis
To simplify notation we shall first consider a single sample
... , x
Let
n
s
= 3,
for r
2
r
r
=
(l/r)
I: (x. - -x )2 ,
l
r
i=l
4, ..• , n; and G\) denote the distribution function of a
Student-t distribution with \) degrees of freedom.
Then put
(2)
for r = 3, ••. , n •
It follows from O'Reilly and Quesenberry [1973J, Corollary 2.1
and example 4.1, that if xl' ..• , x
n
is a random sample from a member
of the N(~,a2) family then u ' ••• , u _ are independently and
n 2
l
identically distributed as uniform random variables on the unit
interval (O,l)--i.i.d. U(O,l) rv's.
Let each sample in (1) be transformed using the transformations
in (2).
Then from the j!h sample (j
= 1,
... , k) we obtain n. - 2
i.i.d. U(O,l) random variables, and a total of N
pooled i.i.d. U(O,l) random variables.
=n l
J
+ ~-2k
+
Let U(l) ~ U(2) ~
~ U(N)
denote the ordered values of this pooled set of random variables.
Under the normal model hypothesis then E(U(j))
j
= 1,
=
j/(N+l) for
••• , N; and if the pairs (U(j)' j/(N+l)) are plotted on
Cartesian axes the points should, approximate the line g(u)
for
° < u < 1.
=u
This gives a graphical procedure for judging
- 5 -
whether the parent distribution is normal.
(See Epstein [1960J,
section 1, for a similar graphical procedure.)
As mentioned earlier in subsection 2.1, Table 1 gives the data
obtained from repeated determinations of aflatoxin from 11 subsamples.
The physical interpretation of the parent distribution of interest
is that it is the distribution of analytical "errors" of the determination process.
The transformations of (2) have been performed on each of the
samples in Table 1.
The u-values obtained for each sample have been
pooled and ranked and are given in Table 2.
Table 2 near here
The 151 values in Table 2 have been used to plot the 151 points
(U(j),j/(N+l)) in Figure 1.
Figure 1 near here
In addition to this graphical procedure, we compute two
goodness~
of-fit statistics as measures of the uniformity of the u-data of
Table 2.
We compute the Pearson ~2 statistic using 10 cells of equal
probability, and the Watson
uf
statistic, Watson [1961J.
To compute the chi-squared statistic the interval (0,1) is
partitioned into ten intervals of length 0.1 each and the vector
of cell frequencies from the data of Table 2 is (12, 14, 10, 20, 19,
15, 16, 13, 15, 17).
This gives a chi-squared value
of
rq
'(..:J
C"'1
....:1'
C--.1 \0
(q (') ("'·1
(Y)
'--,.n co (';'\ ':----i
C'-J ('"',I '>J ""
0\ 0\ C'\ C'"
~-: \.0 C') ",,0 \.i...)
(>,J (.) "_ ') C J If'~ '"n r'-l
(' ,j (q C) IJI (7'\ <i'
\,~)
tt'j
\.'..i
,~',:\
1'-;'1 r.J\
\~)
':f'
\.(:~
r ~
("----I ..< t
('1
l('"
I r'l
"r")
,(
(j)
{J Ch
'--.:")
f· ...
(1.)
-:,0 l,"() (I',
(J)
C.J"
CJ\
C)"
G'" C.h
(;'\
(j\
-.:r
f', If) "X) ,-1 r-··
n' r- r . . . (''I ,-{.
1.1' CO 0\
'0 In -1' C) '·0 ('1 c"l 0-1
....:.j m :",0 l(j (I) r"'" \t) If; (""-J ''';''f' 1''''"" \(")
("-·1 Lrj CO r·',1 ,.~j. U"l (") ()") C ...) r.. . . U') ':U \""'1 r-.. . . Or'''''' CO'
(:) C) y-\ I:'-J ("·1 (.""1 C·) ('I'"', 'I· U"l 1
1,,0 f-' CO
0
()
CU (,:::J CO CO :) {;) ,:,.() CO I)) U~ 0::.) CO ':I:) 'f) CO (J"l (7'\
,j,
n
Il") .... 0
(YI ;,-1 0.) I "j C) r,," \.c> 0
0 l.......J 0 ',",I
('--1 \D CO Cf'') ('~,; G'. r-'~ \.0 I") ( ) rf"i r- .... t'-t v") co aJ
,-.~ <J \,0 1i") 0"', ,"1 ....: j" ,..:t C) rY) <~~l <"1 If) t() G\ 0:..)
0' (71 C;\ C) C) ,·1 ,--i ,-j " j ('I (', ('I) (', I.f) I', CO
\.0 \.0 \,() \[) f ....,. r~.... r.. . . }'--"
r·.. . . f~-. r-.... r-... f"""... r.. . . f~ .. f"'.
(1)
o
(J"\ ,,,.. -1
r---{
(j,
(:J C) ('()
() "I r--
Ct.)
Cf)
~ i) r-..... ('·1 C\
C) C) 1.--1 Vi
I· ":
C·l
r--~
......1"'
In
Lf') If) l() l[) \.0 \,.::) \0 . . .0
r"~
...:to cO CO ....; .
0" l.f) In
C) (J\ r-l \/) r-.
\() I.[) \0 f·..... 0..) 0
0 ,-'-1 C'l 0'1
o
(l)
\S,)
r-...
C) r--l
<:J [-,I 0
Lf")
r.....
lrJ
r'-~
N
C"-J C)
,.--.,
(,J
\.0 T"'~. \() r--~ (.:J <t' \·,,1 c..-) 0 ...., ',0
\.0 \0 ~........ t........ OJ (0 0.... (:) 0
r-l
. . .J <:1 ·<t ......::t . . .:.t· ...:'t' ....: j In In tf1
o
(j\
(Y)
\.,0
l"~'l
r.j\ \ __.D
r-.. ,
«(')
(Y")
er'! 1.>1
.....1"
\..0 0
r-..... -..:t ....:..1'
0·1 0,
f~"l
0) .'J\ () r-l ,..-1
c·',
-~1'
<1' .....-j.
o("1')
r:-. (')
r......
iii \.0 r;o CO G'\
\.0 "0 \0 \0
~.o
(".j \,0 '-J.)
~,O I....... ('1")
..·i (--! ...j'
~".! C'-l (f'")
~()
II)
('.-'., (.il
'.1)
c~.,
0, <.f)
N
r·"
n
'.0 <t '-0 ,...j ''"' C) ,:0
C) 1I) (I") ,.c; f"..... -..:.t -.::t
'-0 (I) <I" r'~ (I) If)
CO ,::1) 0
V')
r·... r-t
t..f"') (',0
""'::1 If) u')
ul ('-1 It) rl ':"'1 ("') (J'\
C) I n 00 l t l f-- .......1" r--{
~..:..j'
r'~'1
, .. J
C"~
-~J
"':T
0,-.:r- -;)",.,. ., (') ...::t
U') CG C) (TJ
0'1
"r
,j" ~-t
('J N
~T
'.0
<;" ('I,
. . .-1" "
co
."
if} ........ t
!~J~'
~·T
C1) (X) 0"\
Ul If) VI
C') (', IV)
r·-j ':) c'l
r"! (..: (J 0)
,.0 0\
Cl' ('I ,j"
n
m
-<1"
In \.,()
lf')
\..0
('1 Cy )
(Y)
o
0,
,I)
M Cl> N
In ,0 -.j0
If) '-i.J ,'X)
Col ('I ('I
If,
0\ ,...(
c> r.... C' co I.n '0 n ('01 0\ M
C') \0 00 N 1-, ,.:)"\ N 0' \.!) CO r~ Cf) 0
r -1 r..... 1-1 r...... r-.. ,,~O U'\ \..0 0
I-wi ....::t \.Q
.--.. j-""" co W ,::..x.) G\ ':)...... C) N ('1 C'l (""1
Cf'~
~q
1:7\
If) ~(1
\.!) '..0
(',I
.-l
C) ....... ! \.0
(I')
\..0
\1""1 Lll l{') II') u")
~1 ~i ~ ~~,~ ~~ ~:~ ~~ ::~ ;~1 ~: ~~ ~:,] ~
o
(f')
~-Y,
n "") ,') ('/
0\ 0
OJ
l{")
C·'" v'")
co \.0
1"" 0\ u) (I") V"I ll) <:t '<1" \!) j.............1" ("f') cf) 1·--{
<:/' r-I (,X) r;-....) \-0 l l j 0\ ("--I N N .....1 · ....:t 0"\
0')
0'\ (.1'\ ( ) C) ;'-1 ';-', ....:--1'" .....1·
c·l
l[)
ct")
(J\ (y';
(Yl 1.(1 c.'") (.1"). ,_'0 ........ t '.... (;:.:J C\
\.0 \L'J ("',J ....0 C"" C~~l L'") r--.. 0"
\..:.J \.0
'J.
~D C"1 0'\ ~Y)
.-1 l.f) ..j- ,;-4 ,-·1 0
\D \0 C', <j' CO 0
\..0 '-0 \.0
01 CO
1-;
r
OOOOOOOOOOOO~rlrlrlM
IT
,9 -
.8
.7
-
if)
liJ
::>
-l
<r
>
I
::>
0
w
r"
0
L!J
0...
><
w
T
.0,-
?fI-
.3L
,2
.I
YTCUV:: 1
- 6 ~
2
= (k/N)
k
2
t N. - N ,
j=l J
= (10/151)(2365) - 151 ,
=
The Watson
uf
5.62 .
statistic has been shown by Miller and Quesenberry
[1975J to have attractive power properties as a test of uniformity
and is given by
.2
U-
N
= (1/12N) + t [U(.) - (2j-l)/2N]
j=l
J
where u = (l/N)(u(l) + ..• + u(N))'
2
-
- N(U-l/2)
2
The value of
,
uf
for the data
of Table 2 is 0.0785.
To study the fit of the lognormal model to the data of Table 1,
we proceeded as follows.
First, the natural logarithm of each
number in Table l'.was taken, and these logarithms were then analyzed
for normality just as described above.
Figure 2 is a graph of the
pooled and ranked u-values obtained from these logarithmB using
equation (2), plotted against their null hypothesis expected values.
The Pearson ~2 and the Watson
~2 = 12.64 and
uf
uf
statistics were found to have values
= 0.1089 for this case.
Figure 2 near here
The values of '1.
2
and
uf
models are given in Table 3.
for both the normal and lognormal
The value in parentheses is the
',level of the obSe:tved :value .
significan~e
1.0-
I'
.9 -
.8 --
U)
UJ
:.:J
-l
<!
>I
:J
o
Ld
}--
o
W
0..
X
Lt.J
.
- 7 For the
2
X values, this is the probability that a chi-squared
random variable with 9 d.L is less than the observed value.
The
2
probabilities for U have been evaluated using equation (22) in
Watson [1961J •
TABLE 3
OBSERVED VALUES OF TEST STATISTICS FOR
NORMAL AND LOGNORMAL MODELS
Statistic
2
2.3
Normal
Lognormal
~
5.62 ( .78)
12.64 ( .18)
2
U
.0785 ( .lJ231)
.1089 (.232')
Analysis Summary
Examination of Figures 1 and 2 shows that the normal model
fits quite well, and that the lognormal model is not as good.
Exactly the same conclusion is reached by examination of 'rable 3.
Both of the test statistics are smaller for the normal model than
for the lognormal model and so the normal must be preferred.
2
the values of X and
uf
While
are not significant at the usual levels
(.1, .05, etc.), the values for the lognormal model are sufficiently
large to cause suspicion.
We feel that this analysis supports the
following responses to questions (a) and (b) of section 1.1:
(a)
the normal model fits the data better than the lognormal model~ and
(b)
the fit of the normal model is quite satisfactory.
- 8 3.
SOME R.EMARKS CONCERNING THE
S'TA'I'I S'IT CATJ PROCE DlJRE
The approach to model testing and goodness-of-fit used here is
to transform eacll individual sample, and then to pool the transformed
sets.
This approach may also be useful for testing models other than
the normal and lognormal.
We call attention to the fact that the transformations of
equation (2) ~ not s~nmetric in the observations.
that the valuer! Xl' •.. , x
n
It is necessary
as they are used in (;2) be i. i. d. random
variables, and must, therefore, be in random order.
They must not,
for example, be arranged according to magnitude.
The computational work described in the previous sections, ineluding the transformations in (2), the pooling and ranking of the
u-values, and the computation of test statistics have been carried
out using a Fortran program -written by the authors for this purpose.
This work was performed at the Triangle Universities Computation
Center (wec).
The authors are grateful to Professor F. G. Giesbrecht for
helpful discussions.
REFERENCES
Bampton, ~3. C. [1963]. Growth of Aspergillus f'lav"'Us and Production
of Aflatoxin in Groundrluts - Part 1. Tropical Science 1, 74-81.
David, F'. N. andIL L. Johnson [1948J. 'The probability integral
transformation when parameters are estimated from the sample.
Biometrika 35, 182-192.
Dickens, J. W. and J.. B. Satterwhite [1969]. Subsampling Mill for
Peanut Kernels. Food Technology 23, 90-92.
- 9 Duggan, R. E.
13-18.
[19'70 J.
Controlling Aflatoxin.
Durbin, J. [1961J. ~30Ine rnetl~lod3 of
Bicmetl'i1';:a 2±Q., 41-55.
]i'LA Papers, April,
C~Ol'l~tr,-~c<3ing
exa,c't tests .
1"'pstein, B. [1960]. '1'esto for the -valiai ty of' the assumption that
the ur:derlying distribD.tion of life is exponential: Part I.
Technometri~s~, 83-10l.
Gibbons, ,J. D. and ,J. W. Pratt [19'""I5J. P-values: interpretation
and methodology. American Statistician .,S2, No.1, 20-25.
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