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Applications of nonparametric statisitcs ot multicomponent solids mixing
Powder Technology, 26 (1980) 131- 146 @ Elsetier Sequoia S A.. Lausann e - Printed in the Netherlands
131
Applications of Nonparametric Statistics to Multicomponent
Solids Mixing
J. R. TOO, L. T. FAN Department of Chemical Engineering, Kansas State Uniuersity. Manhattan. KA
66506 (U.S.A.)
R. M. RUBISON
Department of Sfafistics,Kansas State University. Manhattan, KX 66506 (iJ.S_A.) and F_ S_ LAI U.S. Grain Marketing Research Center, Science & Education Administration, US. Department of Agriculfure,
1515 CoZZegeAvenue. Manhattan. KA
(Received
September
66502 (U.S.A.)
26,1979)
SUMMARY
This study demonstrates the applicability of nonpamxetric procedures to the analysis of mixingprocesses. fn part&u&r, multivariate nonparametric methods are used to eualuate the properties of a mutticomponent solids mixture. Specific problems considered are: (1) test of sampling techniques, (2) test of treatment effects, (3) test of the completely mixed state, and (4) test of a quality standard_ The usefulness of the proposed nonparametric techniques is amply demonstrated with both homogeneous and heterogeneous mixtures generated by a drum mixer. The techniques presented in this paper are also applicable to other mixers.
1. INTRODUCTION
Solids mixing or blending is an operation by which two or more particulate solid materials are scattered chaotically by the random movement of the particles in a mixer. Solids mixing tends to eliminate existing inhomogeneities, or to reduce gradients. Solids mixing is an essential operation in plastic processing, ore smelting, pharmaceutical preparation, fertilizer production, food manufacture, and catalytic synthesis of chemicals. Solids mixing operations are multicomponent in nature whenever these operations involve the blending of more than tie ingredients- Thus, the study of solids mixing for multicomponent mixtures is of practical importance.
The theory of nonparametric statistical methods is essentially concerned with the development of statistical inference procedures without the explicit assumptions regarding the functional form of the probabiIity distribution of the sample observations_ Since the distributions of the components during mixing are usually unknown, nonparametric statistical methods should provide a class of appropriate and effective techniques for the analysis of mixing systems. The applications of certain nonparametric tests for solids mixing for binary mixtures have been previously demonstrated by Lai et al_ [1] _ The statistical properties of a multicomponent heterogeneous solids mixture have been of intense interest to researchers in the field of solids mixing, yet a systematic approach to this problem is still lacking. The object of this study is to demonstrate the applicability of nonparametric statistics to the analysis of mixing processes of multicomponent mixtures and the characterization of such mixtures. As specific examples, the mixing processes carried out in a drum mixer and several mixtures generated by the processss are considered_
2. THEORY
Consider a mixture containing (m + 1) For a trivial case of m = 1, the mixture is called binary. In this paper, we are particularly interested in nontrivial cases for whichm> l_ components_
132
Let Xii be a random variable denoting the weight Caction of the ith component in the ith spot sample from a mixture (i = 1,2, _. . , m f l;i= 1,2, . . . . n). Since m+l z Xij = 1
rr,:r =crv
(4)
against the alternative hypothesis H1:Irflro where cro = [cocks,, . . . P,,,] vector. Let
i=l
only m of (m + 1) weight fractions need to be determined_ Thus, Xj=[XliX2i.._Xmi]‘,
j=1,2
,_..,
n
(1)
denotes an arbitrary seIection of m weight fractions from a given sample. Several nonparametric statistical methods, which are applicable to the analysis of a variety of sampling resuhs from multicomponent solids mixing, are presented in this section.
’ is a specified
Yii = Xii - Pia
(5)
denote the adjusted samples values. Then the random sample matrix becomes ry,,
I y2,
Y”’
_
y12
Y 22 _
--- Y1n ---
7
Y 2n
_. . _-
Y Y : m2 --- mn 1 Ranking the n elements in each row of Y, in an increasing order of their absolute values, we obtain an (m X n) rank matrix Y ml
2.1_ On+uzmple Zocation problem In many mixing problems, the true proportions of components in a mixture are known. The problem of interest then becomes a test of the sampling procedure. If sampling is random throughout the mixture, the sampIe mean vector should be representative of the population and the sample mean vector should not be significantly different from the true proportions of components. Multivzn5ate rank tests for the one-sample location problem [2] are thus appropriate for a test of the sampling procedure. Suppose that n spot samples are taken from amixture.LetXi~=1,2,...,n)bearandom sample (vector-v&red) with a continuous cumulative distribution F(x), x E R” , where Rm is the set of all m-tuples x = ]xlrz ___ x, ] ‘_ F(x) may be written as
whereRji istherankof IY,I u= 1,2, __., n) amongtheset CIY~~I, IY,,I, ..-, IYi,IISince the population distributions are assumed continuous, the probability of a tieiszero_Thenforeachi(i=1,2,.._, m), we replace the ranks 1, 2, . _. , n in the ith row of R, by a set of general scores denoted by
F(x) = q-v, P)
{Ey),
(2)
where ,Q= [P+~ __. P, ] ’ is a location (vector) parameter_ The random (m X n) matrix takes the form cx,
x*
(‘3
R II 221
R,
=
. .
. I
R
-
(7)
ml
j=
1, 2, ___, n’J
Hence, we obtain an (m X n) matrix of general scores E, corresponding to R, :
-..Xrz] E,, =
(8)
(3)
We now test the null hypothesis
We refer the reader to Chapter 4 of Puri and Sen [2] for a detailed discussion of the
133
regularityconditions on a score generating function which determines the constants El’). We now consider a univariats rank order statistic for each coordinate (compcnent) of the form T(i)
=
Ebb Cij
~ i=l
where Cij = 1 when Yii > 0 or -1 when Yii < 0. Therefore, T<‘) is the difference between the sum of the scores Ei<‘) for which i’, > 0 and the Sam of those for which Yij < 0. Let T=
[~<1'~<2'___
2oo]
WV
Under the null hypothesis, eqn. (4), the mean and dispersion matrix of T are respectively E[zJ
=o
(11)
and E[z?-)
=nY
(12)
where V is an (m X m) matrix with elements
J
,+i> _ J
i=l,&...,m;
n+l’
then S reduces to the multivariate generaliza-
tion of the one-sample Wilcoxon statistic_
Note that
gl CE'Eijl
2J i=l,2,...,m
(14)
J
The test statistic, S, formed by S = ;
CTV-lT’]
(15)
i= 1, 2, ___,m;
the test statistic, test. Second, if
in [23
is used
i=
1,2,
_..,n
S, is the multivariate
sign
(16)
___Xif]‘,
j=
1,2,
___,nk (18)
be nk independent and identically distributed (vector-valued) random variables having an m-variate absoIutely continuous cumulative distribution function (cd-f.) FCk’(x) for k = 1, 2, . . . , t. Assuming the identity of locations, we test the hypothesis of the equality of dispersion matrices, Le..
against the alternative f,I
_
-p>
a-
= 1,
discussed
k is denoted by nk_ Let X5-” be a random variable representing the &ight fraction of the ith component in the jth sample for the kth treatment. Also let Xgk’ = [X$‘X$’ J
is asymptotically distributed as a chi-square random variabIe with m degrees of freedom for large samples_ If T is stochastically different from 0, S will be large, which will lead us to reject the null hypothesis. The appropriate P-value (see Appendix) is the probability that a chi-square variable is greater than or equal to the observed vaIue of S, that is, a rig&Mail probability. Two special cases are considered in rnis paper for the one-sample location problem. First, setting
Gi’
matrices
to assess treatment effects. Suppose that we wish to compare the effects among t treatments. (Ah f treatments are assumed to be mutually independent.) The number of spot samples for the treatment
i, Q = 1, 2, __., m
f
signed-rank
2.2 Test of homogeneity of dispersion matrices The degree of mixedness is used to judge the difference between various experimental situations (treatments). For example, we may be interested in comparing different types of mixers, mixing speeds or mixing times. The covariance (dispersion) matrix characterizes the degree of dispersion of each component proportion. Therefore, by testing the homogeneity (or equality) of several dispersion matrices, we may be able to judge whether their degrees of mixedness are significantly different. In other words, we may determine if the variation in composition among spot samples of each treatment is identical_ A multivariate nonparam etric test of the equality of dispersion
uii =
lGj
,
~<2’
Let N=
;;rzk
k=l
hypothesis,
, . __ , zCt’
&e not all identical_
Xjk', j=l,2,...,nk;
k=l,2,...,t
anddefinearandommatrixofdimension(mX
N)as
(21)
RankingtheNelements in each row ofX N intheincreasingorderoftheirmagnitudes,we cbtain&(mXN)rankmatrix
RN = . _ _
_ _ _
.
. _ _ __
(22)
. __ R:At
wherethepossibilityoftiesmaybeignoredinprobabiZitybyvirtueofthecontinuityofthe byasetofscores c.d.f_'s. For each i (i= 1, 2, ___,m),wereplacetheranksintheithrowofR, jp
=
1
-LJ--t12
N+l-y
1 ,
i=l,%...,
N
andobtainthecorrespondingsccrematrix @I’ R:‘;
E
ml
L
=,/+&N
EC”’
R
i
---
E(l) R<” -t
1
---
-
iJN)
where JN =
jp) R:‘2’
1 1 __.1 l-
1 1 1 ...1 1
mXN
(24)
135 Now,letusdefine &Tp = _ nk
1 (25)
-1
iGQ=l,2,...,m;
k=l,2,...,t
and
(26) where (27) Furtbermore,let
UiQ.i'Q'(RN)
j+Q' R$'
=
~"'1 Ri%)
ECQ') Rf?;
_--vi*,q,,
(23)
i,i',Q,Q'= 1, 2,...,m Setting r= $(i--1)(2m-i)+Q
foriGQ=1,2,.._,m
wecanrewrite i=GQ=1,2,_..,m)
WiK as U@' =
1
u!k', ?-= 1,2,.._, +Iz+l)
I
(29)
and E&Y
iGQ=1,2,...,m)
as im(m+l)
CT;, r-1,2,_..,
(30)
and
~Nmv) = C~r,(RN)I
9
r, s = L2,
- - - , $m(m+l)
(31)
Thus,thet.estst.atisticcanbeexpressedas L=
k$lnkCu
~k'-u*]v~l(RN)[U~k'
-u*]'
Under the nullhypothesis,eqn.(19),thetestskAisfkL(forlargesamples)isasymptotically distributedasachi-squarerandomvariablewith~ degreesoffreedom,where
(32)
136
V=
+n
+
l)(t-l)
(33)
The P-value for this test is a right-tail probability from a chi-square distribution table with v degrees of freedom.. 2.3 Distribution-free tests of fit Besides testing hypotheses concerning parameters of location and dispersion we are often interested in the validation of a specified distribution. The goodness-of-fit problem in this multivariate setting may be described as follows: Let XI, X2, _ . . , X, be independent multivariate random variables with the distribution function F(x). We wish to test the hypothesis E&J I F(X) = Fo (X)
(34)
component in the jth spot sample. If the samples are taken from a specified distribution, the expected number of particIes of the ith component in the jth sample wiII be known and denoted by eii. Furthermore, under the null hypothesis, eqn. (34), there should be close agreement between these corresponding frequencies. The deviations (fii - eii) measure lack of agreement_ We eliminate the signs by squaring each difference and reduce that value to original units by dividing by the respective eii. Therefore, tfij
-eij12/eij
measures lack of agreement for the ith component in the jth sample_ An overall measure of the lack of agreement is the sum of these individual measures. Thus, the test statistic Q is defined as
against the alternative rr,: F(x) # F,(X)
Q
where F,(X) is some particular distribution function (either continuous or discrete). We can distinguish two special cases for tests of fit [S] : (i) Simple null hypotheses Under a simple null hypothesis, the distribution of the random variable is completely specified by FO(x). (ii) Composite null hypotheses Under a composite nulI hypothesis, the distribution of the random variable is not completely determined by F,(x). If a composite nulI hypothesis depends upon unknown parameters, their maximum likelihood estiators [4] are usually used to derive the appropriate test. When samples are obtained in a multicomponent solids mixing problem, the data can be expressed as component sample
1
2
1
f,l
fl2 --- fl?n flOn+l~
2
f2l
f22
.
..-
L
k2
n
. ..m
--- f2m
m+l
f2
..:hm
fn(mcl,
wherefii (i= 1,2, . . . . (m + l);j= 1, 2, ._., n) denotes the number of particies of the ith
=yl i=l
5 (fij j=l
-eij)2
eii
(35)
A small value of Q supports the null hypothesis EiO, whereas a large value reflects a general incompatibility between the frequencies observed and those expected under HO. The asymptotic distribution (large n) of Q is independent of the UnderIying distribution_ For a simple null hypothesis, Q is asymptotically distributed as a chi-square random variable with m X n degrees of freedom under Ho 15,61As mentioned earlier, it is sometimes necessary to estim2te some parameter values before the test can be performed_ Once the parameters are estimated and subsequently used to estimate eij, Q is calculated as previously according to eqn. (35). For a composite null hypothesis, the distribution of Q is approximately chi-square but with n(m - w) degrees of freedom, where w denotes the number of independent unspecified parameters. Reduction in the number of degrees of freedom shifts the boundary of the critical region so that Q has to be smaller for acceptance at a given level. 2.4 Binomial test A common problem in solids mixing involves the blending of an active ingredient with several dilutents. The homogeneity of this active tigredient in the entire mixture is of primary importance. Given a prescribed
137
quality standard, we are interested’in testing the hypothesis that the proportion of the mixture, 8, which meets the quality standard exceeds a fixed level_ In general, the hypothesis may take one of the following forms for some specified value of&,(O
ecr
(36)
e -c a,
(37)
(a) E&r 0 G (-I0uersus NC: 0 > (b) Ho,:
8 3 eO uersus H-:
(ii) Two-sided alternative Ho3: B = e. versus H,: 0 # eO
(38)
Suppose that n spot samples are drawn from the mixture with each spot sample being classified as satisfactory or unsatisfactory_ Denote the numbers of satisfactory and unsatisfactory samples by S, and S-, respectively. We first consider a test of hypothesis (i(a)). The hypothesis PI,,, is rejected at the Q Ievel whenever S* > c,,
pre,cs+
< G,l
&?JS+
( Gil
and %,lS*
> Gt21
are approximately
equal. For large samples, we define the standardized variables (wish a continuity correction of 0.5) to be s+-neo-0.5 neo(l-eo)
and s_-~~n(l-e~)
-0.5
(40)
2_= neo(l-eo)
The P-value associated with the tests of the above three hypotheses are obtained from the standard normal table as [Z] : P-value Hypothesis
Ho3 vs. Ho
Note that the distribution of S,, when 0 = eO, is binomial with parameters x and t+,; hence,
(39)
z+.=
%3JS+
=a
= a
test selects critical values Car such that
Ho1 vs. H+
G,l
> &,I
An equaMa&
and C,,
where the critical vahre Cal is determined such that >
+ pr,JS*
Ho2 vs. H-
ficz Rqz
> z*l > z-1
W==(fiCZ > WZ > z-1 1)
z+I ,
where 2 has a standard normal distribution_
Go C&l = n@o Similarly, we reject the hypothesis Ho2 at the 0: level whenever s_>
c,,
where pre,[S_>
C,,]
= OL
Here, the distribution of S_, when B = Bo, is binomial with parameters n and (1 - 13~); hence, E&L]
= n(1 - 6,)
Since S+i-S_=n the rejection region of an a level test of the hypothesis Ho3 is decermiued by S, < C,, where
or S, > C=2
3. EXPERIMENTAL
The experimental apparatus, mate&& and procedures employed are described in this section. To minimize experimental and computational effort, ternary particles systems were chosen to demonstrate the analysis of multicomponent solids mixing by nonparametric statistical methods. 3-1 Apparatus and materials The apparatus used in this experiment was a cyhndricai plexigkss mixer of the following dimensions: internal length 38.1 cm (15 in.), diameter 14.0 cm (5.5in.)and end flanges diameter 25.4 cm (10 in.). The tube was split axially so that the upper portion could be removed for loading and sampling. The end flanges were accurately made to insure that during mixing the axis of rotation coincided with the geometric axis of the mixer. The
138
plexiglass cylinder was set horizontally on a jar mill whose rotational speed was accurately maintained at a speed between 10 and 50 r.p.m. Particles used in this experiment were Lucite spheres with average diameters of 0.16 cm (smah), 0.32 cm (medium) and 0.48 cm (huge) and with an average density of 1.156 g/cm3 _ 3.2 Procedure Prior to mixing, two thin semicircular partitions were placed between the two ends of the mixer normal to the mixer axis dividing it into three equal compartments. One hundred and seventy grams ezch of 3 types of particles were loaded in each compartment. Approximately 30% of the overall volume of the mixer was occupied by the particIes. The bed was then leveled, the partitions were removed, the cover was put in place, and the mixer was rotated. Two types of systems were created: (i) Heterogeneous Three types of particles of different sixes (small, medium and large) were employed in this system. (ii) Homogeneous Three types of (large) particles, which had identical properties except color, were used in this system. Table 1 summarizes the experimental conditions of each run. After a predetermined TABLE
1
Summary
of particle systems
and experimental
Conditions
(1) Heterogeneous particles) Experimentzl
Nn 1 2 3 4 5
(2) Homogeneous
system
(small, medium
mixing time, twelve spot SampieS were randomly drawn from the mixture for each experimental run and the weight fractions of ah three types of particles in the sample were recorded_ For the homogeneous system, the number of particles of each type in the sampie was also counted.
4. RESULTS
(r.p.m.)
speed
tests
are expkined.
4.1 Test of sampling techniques In a multicomponent solids mixing problem, the sample mean vector should not deviate greatly from its known population mean vector. Ou the other hand, the sample mean vector by itself should not be used as a measure of the degree of mixedness, since, if the batch is properly sampled, the only variation between sample mean vectors should be the sampling variation, regardless of how well mixed the batch is. If the sample mean vector differs significantJy from the population mean vector po, the sampling may have been biased due to Iocation or method [9]. If so, this bias should be eliminated before further samplingTo accomplish this, we have to test the hypothesis that the mean vector is specified, i.e., to test the null hypothesis
and large
Mixing time
(min)
30 3G 30 20 45
2 10 30 10 10
system (large particks)
Experimental
Rotating
Nil
(r-pm.)
(~4
6 7 8 9
30 45 45 45
30 30 60 150
speed
Mixingtime
DISCUSSION
The theories and procedures presented in Section 2 are employed to analyze the experimental data obtained. Implications of various
Ho:p=po= Rotating
AND
11’ [
3
3
1
(41)
against the alternative hypothesis
An example of this calculation is shown below for the first. experimental run. The experimental data for the 3 particle sizes (small, medium and large) in 12 random spot samples are tabulated in Table 2. By selecting the small and medium sized particles, we express the sample data matrix, X, as
139 TABLE2 Experiment.aldataexpressedinweightfi-actionforexperimentalruns1through3 spot sample
1 2 3 4 5 6 7 8 9 10 11 12
Experimentalrun (2min,30r_p.m.) small medium
large
Experimentalrun (lOmin,30r.p.m.) small medium
0.127 0.968 0.004 0.000 0.022 0.992 0.869
0.797 O-032 0.370 0.410 0.695 0.008 0.123
0.076 0.000 0.625 0.590 0.283 0.000 0.008
0.012 O-435 0.115 0.017 0.271 OS46 0.089
0.241
0.752
0.007
0.929
0.000 0.018 0.987 0.969
0.000 0.658 0.013 0.031
1.000 0.324 0.000 0.000
0.272 0.600 0.067 0.431
0.259 0.565 0.606 0.771 0.611 0.836 O-250 0.071 0.517 C-225 0.202 0.419
large
Experimentalrun (30min,30r.p.m.) medium
large
0.729 0.000 0.279 0.212 0.118 0.017 0.661 0.000 0.211 0.175 O-731 0.150
0.035 1.000 O-029 0.000 0.109 0.085 0.008 0.064 0.064 0.136 0.052 0.050
0.539 o_ooo 0.477 0.951 0.409 0.408 0.792 O-473 0.353 0.336 0.444 0.476
0.426 0.000 0.493 0.049 0.482 o-507 0.200 0.463 0.583 0.529 0.504 0.474
2 9 5 6.5 3 12
8
1 6.5 4
11 10'
I?,= 12 4 1 2
10
7.5 3 11 9
7.5
6
Two multivariaterank testswere developed to test the hypothesis of a prescribed mean vector: (i)Amultivariatesigntest Thescore matrix takesthe form(seeeqn_ (16)) 111111111111
E, =
[ 111111111111
y=x-po=
-0.206 0.635 -0.329 -0.333 -0.311 0.659 0.536 -0-092 -0.333 -0.315 0.654 0.636
0.464-0-3010.037 0.077 0.362 -0.325 -0.210 O-419 -0.333 ,
Since tiesoccur in the apphcation ofrank tests,weuseamidra.nkprocedurethatassigns thesimpleaverageoftherankswhichwould havebeen assignedtotheobservationsifthey werenottied.Thus,rankingtheelementsof eachrowofy inincreasingorderoftheir absolutevahms,weobtain
Accordingto T'1' =_2,W
1
eqn.(g),wehave =O
and T=[-2
O]
Thus,from
v=
and
eqns.(13)and(14),weobtain
5,
Therefore, the test statistic, S, defined in eqn. (15), is calc*iated as
Thus, the test statistic, eqn. (15), is
S= ;[TV-'T'] 36
36
2111 = &-2
O]
30
= 1.5476
r-2
36
The associated P-value is P = 0.4613
0 1.
Since S is asymptotically distributed as a chi-square random variable with two degrees of freedom, we can calculate the P-value 2s P
=
0.5796
Such a large P-value supports HO; hence, the sampling technique is judged to yield representative samples of the mixture. (ii) A multivariate generalization of the Wilcoxon signed-rank test In this case, the score matrix E,=
&R,
BY eqn- (9), T(I) = 1.692,
4-2 Test of treatment effects The data from the first three experimental runs are used to illustrate a test of significance of treatment effects. This is accomplished by testing the homogeneity of their covariance matrices_ Thus, we test the hypothesis rr,: x;
!I”” = 0.692
H a-d - T’(l) , cc2) and Z(3) are not identical
and T = Cl.692
Therefore, use of the multivariate Wilcoxon signed-rank procedure leads to the same conclusion as the multivariate sign test, and the sampling technique is judged to be representative of the mixture. Table 3 lists the P-values for all pairs of particles considered and the two multivariate rank tests. Calculations have been carried out for experimental runs 2 through 5. The results from these runs are also shown in Table 3.
0.6921
According to eqns. (13) and (14), 0.320 -0.158 -, v= 0.320
we have
In other words, I& hypothesizes no difference in dispersion for mixing 2,10 and 30 minutes. The experimental data are listed Using the small and medium sized for illustration, we define
significant times of in Table 2. particles
TABLE3 Resultsoftestingthesamplingtechnique Experimental run
Calculationbasedonthepairof (smaU,medium) (medium,large)
1 2 3 4 5
0.5796 OS.653 OS054 0.2231 0.1653
1 Multivariate signed- 2 3 rank test 4 5
0.4613 0.3293 0.3392 0.3013 0.5732
Multivariate signtest
Wilcoxon
Inference:HO, eqn. (41), (smaU.large)
isrejected
0.2138 0.4758 0.1054 0.1353 0.7788
0.1054 0.0765
no no
0.3285 0.3196 0.3443 0.2808 0.6985
0.2712 0.2366 0.7649 0.3488 0.8583
0.2231
0.1353 0.1653
IlO
no Xl0 IlO
no no no no
141 XN
[x<1’x<2’x<3q
=
2X36
where
,
=
0.797
o-127 0.9CS 0.004 0.000 0.022 0.992 0.869 0.241 o_ooo 0.018 0.987 0.969 i
1
0.032 0.370 0.410 0.695 0.008 0.123 0.752 0.0.00 0.658 0.013 0.031
I
x(2)
=
I
’
0.259
0.012 0.435 0.115 0.017 0.271 0.146 0.089 0.929 0.272 0.600 0.067 0.431
0.565 0.606 0.771 0.611 0.836 0.250 0.071 0.517 0.225 O-202 0.419
and
&3)
f
0.426 0.000 0.493 0.049 0.482 0.507 O-200 0.463 0.583 0.529 0.504 9.474
0.085 O-008 0.064 0.064 0.136 O-052 0.050
Rankingthe36elements obtaintherankmatrix RN
= [R
,
ofeachrowofthematrix~~
inincreasingorderofmagnitude,we
1.5
36
where 16
31
3
35
7
16
4
26
13
7
36
29
21
1.5
20
32
2
9
33
15
5
22
17
12
28
29
34
30
36
23 13
17 -20
26 14
21 18
18 10.5
6
35
32
1
31
3
6
30
23
27
10.5
12
8
27
11
10
19 8
25 19
249
4 34
15 28
R
25
R(2)=
1
22J
and
R<3'=
335
Accordingtoeqn.(24),thegeneralscorematrixisoftheform
14 24
1
142
pairs (small, medium), (small, large) and (medium, large) are 0.720,0.360 and 0.753, respectively. Because the P-values are quite large, we conclude that the difference in dispersion among rotating speeds of 20,30 and 45 r.p.m. is not significant.
RN-
where
JN =
I:
: IIIIII
:I,,,.
4.3 Test of the completely miked state In solids mixing, the completely random (or mixed) state is characterized by the property that the probability of selecting a particle of a given component at any location in the mixture is identical. When the population proportions are known for the components of a mixture, the chi-square goodnessof-fit test can be used to test the hypothesis that the mixture is in the completely mixed state. In other words, we test
Thus, in our example,
Using eqns. (25) WI’
through (31),
= [l-7099
-0-7724
1.56681
UC” = CO-6743
-0.0612
0.92011
u
-0.4645
0.45651
u’
-0.4126
0.97301
= 10.9727
-0.2705
-0.1870
0.1870 -0.2705
-0.2442 0.9632
0.6623 -0.2442
1.7878
0.4113
[ -0.3515 0.4113
0.3411 1.2402
0.6606
vN(RN)
L
=
and
Vi’(RN)
=
-0-3515 1.7348 0.3411
According to eqn. (32), the test statistic is L=
g EZ,[CJ’~‘--U*]
I 1
I-r,:
the mixture is in the completely
(44)
against
H,:
the alternative
E-I, is not true
The data generated in run 6 of the experiments are shown in Table 4_ From eqn_ (35), the test statistic is computed as
= (47 -
V~‘(RN)[UCk’--U*]
(22 -
21.6334
Since L is asymptoticahy distributed as a chi-square random variable with six degrees of freedom, the associated P-vaIue is P = 0.001 Therefore, the nuII hypothesis H,, eqn. (42), is rejected, and we conclude that there exists a significant difference in dispersion between the mixing times of 2,10 and 30 minutes. The P-vaIue based on smah and large sized particles and that based on medium and large sized particles are 0.001 and O-006, respectively, In the second
34.333)2
+ (34 -
34.333
k=l =
experiment,
we test
the effect
of rotating speeds (20,30 and 45 r-p-m.) on dispersion for a fixed mixing time of 10 minutes (experimental runs 2,4 and 5). The P-vahies corresponding to test statistics for
mixed
State
34.333
34.333)2
34.333 (43 40
40)2
34.333)s
+
---
+
+ (37 -40)2 40
(40 40
+ 40)s
-I-
= 162.280
Under the nuU hypothesis, eqn. (44), Q is asymptotically distributed as the chi-square random variable with 24 degrees of freedom. SinceP < 0.001, the nuII hypothesis is rejected (at the usual levels) ; we conclude that the mixture has not reached the completely mixed state. Table 5 summarizes the results of completely mixed state for experimental runs 6 through 9. 4.4 Test of a quality standard In quality control decisions involving multicomponent solids mixing,we may be concerned wifih the fraction of a population which meets
143 TABLE4 Thedatageneratedinthe6tbrunoftheexperiment spot samde
Colordistributionf green red
white
Totalnumber of particles
Expectednumberof particles foreach
1 2 3 4 5 6 7 8 9 10 11 12
47 39 26 24 38 33 35 31 32 36 33 40
22 31 42 50 17 35 9 54 31 27 11 37
103 98 84 84 97 107 91 111 102 83 120 120
34.333 32.667 28. 28. 32.333 35.667 30.333 37_ 34_ 27.667 40_ 40.
*Tberatio
34 23 16 10 42 39 47 26 39 20 76 43
coior
is1:1:1forcategoriesgreen1red:white,respectively.
TABLE5 Resultsoftestingcompletelymixedstate Experimental run
Mixing time (rum)
Rotating speed (r.p.m.)
forexperimentalruns6through Expected distribution
9
Test statistic
Associated probability
Q
P
Inferenc2 aboutHo, eqn(44)
6
30
30
111 -_:__:3 3 3
162.280
~0.001
rejected
7
30
45
111 --f-_I3 3 3
56.246
<0_001
rejected
8
60
45
111 -:-_L3 3
43-429
0.009
rejected
13.202
0.963
accepted
9
150
45
111 -_I__:_ 3 3 3
a quality criterion. The binomial test can be used to solve this multicomponent solids mixing problem. Again, let xii denote the weight fraction of the ith component in the jth sample and pi0 the population weight G?lction of the ith component. We may set the criterion as m+l ;z_
Mxij
-PiO)2
G
E
3
(45)
Assume that three components are equally important. Hence, let h,=Xs=Xs=l We wish to test whether the satisfactory proportion of a mixture has reached 95% for a pre-selected value of 0.015 for E. In the following examples, we test the null hypothesis H,,: 0 > 0.95 against
where Xi = arbitrary positive constant which might reflect the relative importance of the ith component being mixed, and E = preselected positive real number. We say that a spot sample is satisfactory if it satisfies this criterion, otherwise, it is unsatisfactory_
H_:
(461
the alternative hypothesis
6 < 0.95
The calculation for run 7 of the experiments is shown in Table 6_ The results are S+=7,S_=5,n=12
144 TABLE6 ThecalculationforexperimentaI-7
Snot sample
Xii
1 2 3 4 5 6 7 8 9 10 11 12
0.3333 O-3537 0.2941 0.3196 0.3113 0.2857 0.2785 0.3678 0.3229 0.3182 0.3367 0.3636
*cl10=l&-J =
x2j
p30
=
fortestingthequ~tystandardasdef~edineqn_(45)
i=I
Is it satisfactory?
0.0018 0.0174 0.0034 0.0175 0.0382 0_0050 0.0334 0.0129 0.0046 0.0014 0.0567 0.0021
Yes no Yes no no Yes no Yes Yes Yes no Yes
5
=3i
0.2564 O-4146 0.3765 0.2474 O-4811 0.3286 0.2405 0.3908 0.2917 0.3636 0.5000 0.2987
0.4103 0 2317 c-3294 b-4330 0.2075 0.3587 3.4810 0.2414 0.3854 0.3182 0.1633 0.3377
-&O)2f
o-3333_
The guide indicates that the appropriate P-value is a left-tail probability for S, = 7 arameter of 0.95, which from a withap binomial table gives P = 0.0002. Since this P-value is so small, we conclude that the data reject Ho in favor of EL. Hence,
we conclude that mixing is not adequate_ The results of runs 6 through 9 are summarized in Table 7. Note that this test may also be used in the analysis of a continuous mixing process.
5.CONCLUSIONS
Statistical analysis is recognized as a major too1 in solids mixing investigations_ Traditionally, results of sampling have been analyzed using normal [S, lC].
Cxij
theory
statistical
techniques
TABLE7 Resultsoftestingthesatisfactory proportionof mixturesforexperimentalnms6 through 9 Experimental run
S+
s-
n
P
Inference concemingHO,eqn.(46)
6 7 8 9
5 7 10 12
7 5 2 0
12 12 12 12
0.0000 0.0002 0.1184 1.0000
rejected rejected accepted accepted
This study proposes the applicability of several nonparameteric statistical techniques to problems in multicomponent solids mixing_ The most imporknt feature of a nonparametric procedure is its lack of dependence on a particular distribution type, e.g., normal. Since the distributions of components during mixing are usually unknown, nonparametric procedures comprise a substantial collection of alternatives to the classical parametric procedures. Recently, the extension of nonparametric techniques from the univariate to the multivariate case has been pursued [ 21. The present study demonstrates the applicability of multivariate tests of location and dispersion to test hypotheses concerning the sampling technique and the significance of treatment effects in multicomponent solids mixing problems. The proposed nonparametic procedures were tested with actual homogeneous and heterogeneous ternary mixtures generated by a drum mixer_ In spite of the small number of the samples drawn (n = 12), the results tend to support the practical significance of nonparametric statistics in the evaluation of mixing systems. Resides the robustness of the nonparametric methods against the assumption of a specified distributional form, it is important to note their simplicity in application. An effort will be made in the future to study the performance of the proposed nonparametric methods for a larger number of samples.
145 LIST
G cij
E(X)
EN E
j$il eij
fij
HL
rl Rk
P Q
RN
Rij
Rm R$ sS
T
W
OF
SYMBOLS value at the significance level o[ sign indicator of Yij expected value of random variable X score matrix as defined in eqn. (24) score matrix as defined in eqn. (8) mean score of the ith variate expected number of particles of the ith component in the jth sample cumulative distribution function (c.d.f.) of X number of particles of the ith component in the jth sample null hypothesis two-sided alternative one-sided alternative with positive direction one-sided alternative with negative direction test statistic for testing homogeneity of dispersion matrices number of variates total number of spot samples for t treatments number of spot samples number of samples of treatment k associated probability test statistic for goodness of fit test (m X IV) rank matrix as defined in eqn. (22) (m X n) rank matrix as defined in eqn. (7) rankof Yij among(Yil, ---, Yin) set of ell order m-tuples [x1x2 _ __ x, ] ’ rank of Xl!) among (X$1, _I_, Xi:,‘) fork=l,z,._.,t number of unsatisfactory samples test statistic for testing equality of mean vectors univariate rank order statistic as defined in eqn. (9) row vector as defined in eqn. (10) number of treatments row vector as defined in eqn. (29) row vector as defined in eqn. (30) as defined in eqn. (25) as defined in eqn. (26) as defined in eqn. (13) as defined in eqn. (12) as defined in eqn. (31) number of unspecified parameters estimated from data critical
row
vector = [XGX*j
..- Xmj]‘,
j=l,2,.._,n
row vector = [X159@’ . . . X(mk] ‘, j= 1, 2, ___,nk a random variable denoting the weight fraction of the ith component in the jth spot sample a random variable representing the weight fraction of the ith component in the jth sample for the kth treatment realization of Xj = Xij -fliO (m X n) random matrix (m X IV) pooled random matrix standardized variable as defined in eqn. (39) standardized variable as defined in eqn. (40) a parameter representing the probability of satisfaction location vector parameter weight proportion of the ith component in the population degrees of freedom as defined in eqn. (33) chi-square distribution dispersion matrix of treatment k significant level for testing a hypothesis an arbitrary positive real number an arbitrary positive constant to reflect the relative importance of the ith component APPENDIX
On the use of P-values in hypothesis
testing
The traditional method of testing a hypothesis is the determination of a rejection region and a corresponding rejection rule for which the probability of making a Type I error does not exceed some preselected value called the level of the test. In many cases, the choice of the level of the test is arbitrary and in some testing situations the chosen level may not even be attainable. These problems are circumvented by the reporting of P-values. The P-value is defined as the probability under the null hypothesis of a sample outcome equal to or more extreme than that observed. The reporting of P-values clearly contains more information than merely report-
146
ing the decision made on a hypothesis at a possibly arbitrary IeveI. The use of P-vaIues is clear for those tests in which the outcomes can be ordered relative to the expected outcome under the null hypothesis. In those unambiguous cases the P-value is the probability associated with a corresponding right- or left-tail probability. In the more complex situations where ‘more extreme’ is an ambiguous relation, conventions must be defined for the reporting of P-vaIues_ REFERENCES
4
5
6
7 8 9
1
F. S. Lai, R. H. Wang and L. T. Fan, Powder
Technol_. IO (1974) 13_ 2 M_ L. Puri znd P. K. Sen, Nonparametric Methods in Multivariate Analysis. Wiley, New York, 1971.
and A_ Stuart, The Advanced Vol_ 2, Charles GriftZm, London. 4th edn_, 1977. A. M. Good, F. A. Graybii and D. C. Bees, rntroduction to the Theory of Statistics, McGrawHill, New York, 3rd edn., 1970. E. L. Lehmann, Nonpammeticsr Statistical Methods Based on Ranks, McGraw-Hii, New York, 1975. J. D. Gibbons, Nonparametric Methods for Quantitative Analysis, Holt, Rinehart and Winston, New York, 1976. M. Hollander and D_ A_ Wolfe, Nonparametric Statisticat Methods. Wiley, New York, 1973. W. J. Conover, Practical Nonparametric Statistics. Wiley, New York, 1971. S. S. Weidenbaum, in T. B. Drew and J. W. Hoopes, Jr. (Eds.), Advances in Chemical Engineering, Vol. II, Academic Press, New York, 1958. R. H. Wang, L. T. Fan and J. R. Too, Powder Tecknol. 21 (1978) 171.
3 M. G. Theory
10
Kendall of
Statistics,
131
Applications of Nonparametric Statistics to Multicomponent
Solids Mixing
J. R. TOO, L. T. FAN Department of Chemical Engineering, Kansas State Uniuersity. Manhattan. KA
66506 (U.S.A.)
R. M. RUBISON
Department of Sfafistics,Kansas State University. Manhattan, KX 66506 (iJ.S_A.) and F_ S_ LAI U.S. Grain Marketing Research Center, Science & Education Administration, US. Department of Agriculfure,
1515 CoZZegeAvenue. Manhattan. KA
(Received
September
66502 (U.S.A.)
26,1979)
SUMMARY
This study demonstrates the applicability of nonpamxetric procedures to the analysis of mixingprocesses. fn part&u&r, multivariate nonparametric methods are used to eualuate the properties of a mutticomponent solids mixture. Specific problems considered are: (1) test of sampling techniques, (2) test of treatment effects, (3) test of the completely mixed state, and (4) test of a quality standard_ The usefulness of the proposed nonparametric techniques is amply demonstrated with both homogeneous and heterogeneous mixtures generated by a drum mixer. The techniques presented in this paper are also applicable to other mixers.
1. INTRODUCTION
Solids mixing or blending is an operation by which two or more particulate solid materials are scattered chaotically by the random movement of the particles in a mixer. Solids mixing tends to eliminate existing inhomogeneities, or to reduce gradients. Solids mixing is an essential operation in plastic processing, ore smelting, pharmaceutical preparation, fertilizer production, food manufacture, and catalytic synthesis of chemicals. Solids mixing operations are multicomponent in nature whenever these operations involve the blending of more than tie ingredients- Thus, the study of solids mixing for multicomponent mixtures is of practical importance.
The theory of nonparametric statistical methods is essentially concerned with the development of statistical inference procedures without the explicit assumptions regarding the functional form of the probabiIity distribution of the sample observations_ Since the distributions of the components during mixing are usually unknown, nonparametric statistical methods should provide a class of appropriate and effective techniques for the analysis of mixing systems. The applications of certain nonparametric tests for solids mixing for binary mixtures have been previously demonstrated by Lai et al_ [1] _ The statistical properties of a multicomponent heterogeneous solids mixture have been of intense interest to researchers in the field of solids mixing, yet a systematic approach to this problem is still lacking. The object of this study is to demonstrate the applicability of nonparametric statistics to the analysis of mixing processes of multicomponent mixtures and the characterization of such mixtures. As specific examples, the mixing processes carried out in a drum mixer and several mixtures generated by the processss are considered_
2. THEORY
Consider a mixture containing (m + 1) For a trivial case of m = 1, the mixture is called binary. In this paper, we are particularly interested in nontrivial cases for whichm> l_ components_
132
Let Xii be a random variable denoting the weight Caction of the ith component in the ith spot sample from a mixture (i = 1,2, _. . , m f l;i= 1,2, . . . . n). Since m+l z Xij = 1
rr,:r =crv
(4)
against the alternative hypothesis H1:Irflro where cro = [cocks,, . . . P,,,] vector. Let
i=l
only m of (m + 1) weight fractions need to be determined_ Thus, Xj=[XliX2i.._Xmi]‘,
j=1,2
,_..,
n
(1)
denotes an arbitrary seIection of m weight fractions from a given sample. Several nonparametric statistical methods, which are applicable to the analysis of a variety of sampling resuhs from multicomponent solids mixing, are presented in this section.
’ is a specified
Yii = Xii - Pia
(5)
denote the adjusted samples values. Then the random sample matrix becomes ry,,
I y2,
Y”’
_
y12
Y 22 _
--- Y1n ---
7
Y 2n
_. . _-
Y Y : m2 --- mn 1 Ranking the n elements in each row of Y, in an increasing order of their absolute values, we obtain an (m X n) rank matrix Y ml
2.1_ On+uzmple Zocation problem In many mixing problems, the true proportions of components in a mixture are known. The problem of interest then becomes a test of the sampling procedure. If sampling is random throughout the mixture, the sampIe mean vector should be representative of the population and the sample mean vector should not be significantly different from the true proportions of components. Multivzn5ate rank tests for the one-sample location problem [2] are thus appropriate for a test of the sampling procedure. Suppose that n spot samples are taken from amixture.LetXi~=1,2,...,n)bearandom sample (vector-v&red) with a continuous cumulative distribution F(x), x E R” , where Rm is the set of all m-tuples x = ]xlrz ___ x, ] ‘_ F(x) may be written as
whereRji istherankof IY,I u= 1,2, __., n) amongtheset CIY~~I, IY,,I, ..-, IYi,IISince the population distributions are assumed continuous, the probability of a tieiszero_Thenforeachi(i=1,2,.._, m), we replace the ranks 1, 2, . _. , n in the ith row of R, by a set of general scores denoted by
F(x) = q-v, P)
{Ey),
(2)
where ,Q= [P+~ __. P, ] ’ is a location (vector) parameter_ The random (m X n) matrix takes the form cx,
x*
(‘3
R II 221
R,
=
. .
. I
R
-
(7)
ml
j=
1, 2, ___, n’J
Hence, we obtain an (m X n) matrix of general scores E, corresponding to R, :
-..Xrz] E,, =
(8)
(3)
We now test the null hypothesis
We refer the reader to Chapter 4 of Puri and Sen [2] for a detailed discussion of the
133
regularityconditions on a score generating function which determines the constants El’). We now consider a univariats rank order statistic for each coordinate (compcnent) of the form T(i)
=
Ebb Cij
~ i=l
where Cij = 1 when Yii > 0 or -1 when Yii < 0. Therefore, T<‘) is the difference between the sum of the scores Ei<‘) for which i’, > 0 and the Sam of those for which Yij < 0. Let T=
[~<1'~<2'___
2oo]
WV
Under the null hypothesis, eqn. (4), the mean and dispersion matrix of T are respectively E[zJ
=o
(11)
and E[z?-)
=nY
(12)
where V is an (m X m) matrix with elements
J
,+i> _ J
i=l,&...,m;
n+l’
then S reduces to the multivariate generaliza-
tion of the one-sample Wilcoxon statistic_
Note that
gl CE'Eijl
2J i=l,2,...,m
(14)
J
The test statistic, S, formed by S = ;
CTV-lT’]
(15)
i= 1, 2, ___,m;
the test statistic, test. Second, if
in [23
is used
i=
1,2,
_..,n
S, is the multivariate
sign
(16)
___Xif]‘,
j=
1,2,
___,nk (18)
be nk independent and identically distributed (vector-valued) random variables having an m-variate absoIutely continuous cumulative distribution function (cd-f.) FCk’(x) for k = 1, 2, . . . , t. Assuming the identity of locations, we test the hypothesis of the equality of dispersion matrices, Le..
against the alternative f,I
_
-p>
a-
= 1,
discussed
k is denoted by nk_ Let X5-” be a random variable representing the &ight fraction of the ith component in the jth sample for the kth treatment. Also let Xgk’ = [X$‘X$’ J
is asymptotically distributed as a chi-square random variabIe with m degrees of freedom for large samples_ If T is stochastically different from 0, S will be large, which will lead us to reject the null hypothesis. The appropriate P-value (see Appendix) is the probability that a chi-square variable is greater than or equal to the observed vaIue of S, that is, a rig&Mail probability. Two special cases are considered in rnis paper for the one-sample location problem. First, setting
Gi’
matrices
to assess treatment effects. Suppose that we wish to compare the effects among t treatments. (Ah f treatments are assumed to be mutually independent.) The number of spot samples for the treatment
i, Q = 1, 2, __., m
f
signed-rank
2.2 Test of homogeneity of dispersion matrices The degree of mixedness is used to judge the difference between various experimental situations (treatments). For example, we may be interested in comparing different types of mixers, mixing speeds or mixing times. The covariance (dispersion) matrix characterizes the degree of dispersion of each component proportion. Therefore, by testing the homogeneity (or equality) of several dispersion matrices, we may be able to judge whether their degrees of mixedness are significantly different. In other words, we may determine if the variation in composition among spot samples of each treatment is identical_ A multivariate nonparam etric test of the equality of dispersion
uii =
lGj
,
~<2’
Let N=
;;rzk
k=l
hypothesis,
, . __ , zCt’
&e not all identical_
Xjk', j=l,2,...,nk;
k=l,2,...,t
anddefinearandommatrixofdimension(mX
N)as
(21)
RankingtheNelements in each row ofX N intheincreasingorderoftheirmagnitudes,we cbtain&(mXN)rankmatrix
RN = . _ _
_ _ _
.
. _ _ __
(22)
. __ R:At
wherethepossibilityoftiesmaybeignoredinprobabiZitybyvirtueofthecontinuityofthe byasetofscores c.d.f_'s. For each i (i= 1, 2, ___,m),wereplacetheranksintheithrowofR, jp
=
1
-LJ--t12
N+l-y
1 ,
i=l,%...,
N
andobtainthecorrespondingsccrematrix @I’ R:‘;
E
ml
L
=,/+&N
EC”’
R
i
---
E(l) R<” -t
1
---
-
iJN)
where JN =
jp) R:‘2’
1 1 __.1 l-
1 1 1 ...1 1
mXN
(24)
135 Now,letusdefine &Tp = _ nk
1 (25)
-1
iGQ=l,2,...,m;
k=l,2,...,t
and
(26) where (27) Furtbermore,let
UiQ.i'Q'(RN)
j+Q' R$'
=
~"'1 Ri%)
ECQ') Rf?;
_--vi*,q,,
(23)
i,i',Q,Q'= 1, 2,...,m Setting r= $(i--1)(2m-i)+Q
foriGQ=1,2,.._,m
wecanrewrite i=GQ=1,2,_..,m)
WiK as U@' =
1
u!k', ?-= 1,2,.._, +Iz+l)
I
(29)
and E&Y
iGQ=1,2,...,m)
as im(m+l)
CT;, r-1,2,_..,
(30)
and
~Nmv) = C~r,(RN)I
9
r, s = L2,
- - - , $m(m+l)
(31)
Thus,thet.estst.atisticcanbeexpressedas L=
k$lnkCu
~k'-u*]v~l(RN)[U~k'
-u*]'
Under the nullhypothesis,eqn.(19),thetestskAisfkL(forlargesamples)isasymptotically distributedasachi-squarerandomvariablewith~ degreesoffreedom,where
(32)
136
V=
+n
+
l)(t-l)
(33)
The P-value for this test is a right-tail probability from a chi-square distribution table with v degrees of freedom.. 2.3 Distribution-free tests of fit Besides testing hypotheses concerning parameters of location and dispersion we are often interested in the validation of a specified distribution. The goodness-of-fit problem in this multivariate setting may be described as follows: Let XI, X2, _ . . , X, be independent multivariate random variables with the distribution function F(x). We wish to test the hypothesis E&J I F(X) = Fo (X)
(34)
component in the jth spot sample. If the samples are taken from a specified distribution, the expected number of particIes of the ith component in the jth sample wiII be known and denoted by eii. Furthermore, under the null hypothesis, eqn. (34), there should be close agreement between these corresponding frequencies. The deviations (fii - eii) measure lack of agreement_ We eliminate the signs by squaring each difference and reduce that value to original units by dividing by the respective eii. Therefore, tfij
-eij12/eij
measures lack of agreement for the ith component in the jth sample_ An overall measure of the lack of agreement is the sum of these individual measures. Thus, the test statistic Q is defined as
against the alternative rr,: F(x) # F,(X)
Q
where F,(X) is some particular distribution function (either continuous or discrete). We can distinguish two special cases for tests of fit [S] : (i) Simple null hypotheses Under a simple null hypothesis, the distribution of the random variable is completely specified by FO(x). (ii) Composite null hypotheses Under a composite nulI hypothesis, the distribution of the random variable is not completely determined by F,(x). If a composite nulI hypothesis depends upon unknown parameters, their maximum likelihood estiators [4] are usually used to derive the appropriate test. When samples are obtained in a multicomponent solids mixing problem, the data can be expressed as component sample
1
2
1
f,l
fl2 --- fl?n flOn+l~
2
f2l
f22
.
..-
L
k2
n
. ..m
--- f2m
m+l
f2
..:hm
fn(mcl,
wherefii (i= 1,2, . . . . (m + l);j= 1, 2, ._., n) denotes the number of particies of the ith
=yl i=l
5 (fij j=l
-eij)2
eii
(35)
A small value of Q supports the null hypothesis EiO, whereas a large value reflects a general incompatibility between the frequencies observed and those expected under HO. The asymptotic distribution (large n) of Q is independent of the UnderIying distribution_ For a simple null hypothesis, Q is asymptotically distributed as a chi-square random variable with m X n degrees of freedom under Ho 15,61As mentioned earlier, it is sometimes necessary to estim2te some parameter values before the test can be performed_ Once the parameters are estimated and subsequently used to estimate eij, Q is calculated as previously according to eqn. (35). For a composite null hypothesis, the distribution of Q is approximately chi-square but with n(m - w) degrees of freedom, where w denotes the number of independent unspecified parameters. Reduction in the number of degrees of freedom shifts the boundary of the critical region so that Q has to be smaller for acceptance at a given level. 2.4 Binomial test A common problem in solids mixing involves the blending of an active ingredient with several dilutents. The homogeneity of this active tigredient in the entire mixture is of primary importance. Given a prescribed
137
quality standard, we are interested’in testing the hypothesis that the proportion of the mixture, 8, which meets the quality standard exceeds a fixed level_ In general, the hypothesis may take one of the following forms for some specified value of&,(O
ecr
(36)
e -c a,
(37)
(a) E&r 0 G (-I0uersus NC: 0 > (b) Ho,:
8 3 eO uersus H-:
(ii) Two-sided alternative Ho3: B = e. versus H,: 0 # eO
(38)
Suppose that n spot samples are drawn from the mixture with each spot sample being classified as satisfactory or unsatisfactory_ Denote the numbers of satisfactory and unsatisfactory samples by S, and S-, respectively. We first consider a test of hypothesis (i(a)). The hypothesis PI,,, is rejected at the Q Ievel whenever S* > c,,
pre,cs+
< G,l
&?JS+
( Gil
and %,lS*
> Gt21
are approximately
equal. For large samples, we define the standardized variables (wish a continuity correction of 0.5) to be s+-neo-0.5 neo(l-eo)
and s_-~~n(l-e~)
-0.5
(40)
2_= neo(l-eo)
The P-value associated with the tests of the above three hypotheses are obtained from the standard normal table as [Z] : P-value Hypothesis
Ho3 vs. Ho
Note that the distribution of S,, when 0 = eO, is binomial with parameters x and t+,; hence,
(39)
z+.=
%3JS+
=a
= a
test selects critical values Car such that
Ho1 vs. H+
G,l
> &,I
An equaMa&
and C,,
where the critical vahre Cal is determined such that >
+ pr,JS*
Ho2 vs. H-
ficz Rqz
> z*l > z-1
W==(fiCZ > WZ > z-1 1)
z+I ,
where 2 has a standard normal distribution_
Go C&l = n@o Similarly, we reject the hypothesis Ho2 at the 0: level whenever s_>
c,,
where pre,[S_>
C,,]
= OL
Here, the distribution of S_, when B = Bo, is binomial with parameters n and (1 - 13~); hence, E&L]
= n(1 - 6,)
Since S+i-S_=n the rejection region of an a level test of the hypothesis Ho3 is decermiued by S, < C,, where
or S, > C=2
3. EXPERIMENTAL
The experimental apparatus, mate&& and procedures employed are described in this section. To minimize experimental and computational effort, ternary particles systems were chosen to demonstrate the analysis of multicomponent solids mixing by nonparametric statistical methods. 3-1 Apparatus and materials The apparatus used in this experiment was a cyhndricai plexigkss mixer of the following dimensions: internal length 38.1 cm (15 in.), diameter 14.0 cm (5.5in.)and end flanges diameter 25.4 cm (10 in.). The tube was split axially so that the upper portion could be removed for loading and sampling. The end flanges were accurately made to insure that during mixing the axis of rotation coincided with the geometric axis of the mixer. The
138
plexiglass cylinder was set horizontally on a jar mill whose rotational speed was accurately maintained at a speed between 10 and 50 r.p.m. Particles used in this experiment were Lucite spheres with average diameters of 0.16 cm (smah), 0.32 cm (medium) and 0.48 cm (huge) and with an average density of 1.156 g/cm3 _ 3.2 Procedure Prior to mixing, two thin semicircular partitions were placed between the two ends of the mixer normal to the mixer axis dividing it into three equal compartments. One hundred and seventy grams ezch of 3 types of particles were loaded in each compartment. Approximately 30% of the overall volume of the mixer was occupied by the particIes. The bed was then leveled, the partitions were removed, the cover was put in place, and the mixer was rotated. Two types of systems were created: (i) Heterogeneous Three types of particles of different sixes (small, medium and large) were employed in this system. (ii) Homogeneous Three types of (large) particles, which had identical properties except color, were used in this system. Table 1 summarizes the experimental conditions of each run. After a predetermined TABLE
1
Summary
of particle systems
and experimental
Conditions
(1) Heterogeneous particles) Experimentzl
Nn 1 2 3 4 5
(2) Homogeneous
system
(small, medium
mixing time, twelve spot SampieS were randomly drawn from the mixture for each experimental run and the weight fractions of ah three types of particles in the sample were recorded_ For the homogeneous system, the number of particles of each type in the sampie was also counted.
4. RESULTS
(r.p.m.)
speed
tests
are expkined.
4.1 Test of sampling techniques In a multicomponent solids mixing problem, the sample mean vector should not deviate greatly from its known population mean vector. Ou the other hand, the sample mean vector by itself should not be used as a measure of the degree of mixedness, since, if the batch is properly sampled, the only variation between sample mean vectors should be the sampling variation, regardless of how well mixed the batch is. If the sample mean vector differs significantJy from the population mean vector po, the sampling may have been biased due to Iocation or method [9]. If so, this bias should be eliminated before further samplingTo accomplish this, we have to test the hypothesis that the mean vector is specified, i.e., to test the null hypothesis
and large
Mixing time
(min)
30 3G 30 20 45
2 10 30 10 10
system (large particks)
Experimental
Rotating
Nil
(r-pm.)
(~4
6 7 8 9
30 45 45 45
30 30 60 150
speed
Mixingtime
DISCUSSION
The theories and procedures presented in Section 2 are employed to analyze the experimental data obtained. Implications of various
Ho:p=po= Rotating
AND
11’ [
3
3
1
(41)
against the alternative hypothesis
An example of this calculation is shown below for the first. experimental run. The experimental data for the 3 particle sizes (small, medium and large) in 12 random spot samples are tabulated in Table 2. By selecting the small and medium sized particles, we express the sample data matrix, X, as
139 TABLE2 Experiment.aldataexpressedinweightfi-actionforexperimentalruns1through3 spot sample
1 2 3 4 5 6 7 8 9 10 11 12
Experimentalrun (2min,30r_p.m.) small medium
large
Experimentalrun (lOmin,30r.p.m.) small medium
0.127 0.968 0.004 0.000 0.022 0.992 0.869
0.797 O-032 0.370 0.410 0.695 0.008 0.123
0.076 0.000 0.625 0.590 0.283 0.000 0.008
0.012 O-435 0.115 0.017 0.271 OS46 0.089
0.241
0.752
0.007
0.929
0.000 0.018 0.987 0.969
0.000 0.658 0.013 0.031
1.000 0.324 0.000 0.000
0.272 0.600 0.067 0.431
0.259 0.565 0.606 0.771 0.611 0.836 O-250 0.071 0.517 C-225 0.202 0.419
large
Experimentalrun (30min,30r.p.m.) medium
large
0.729 0.000 0.279 0.212 0.118 0.017 0.661 0.000 0.211 0.175 O-731 0.150
0.035 1.000 O-029 0.000 0.109 0.085 0.008 0.064 0.064 0.136 0.052 0.050
0.539 o_ooo 0.477 0.951 0.409 0.408 0.792 O-473 0.353 0.336 0.444 0.476
0.426 0.000 0.493 0.049 0.482 o-507 0.200 0.463 0.583 0.529 0.504 0.474
2 9 5 6.5 3 12
8
1 6.5 4
11 10'
I?,= 12 4 1 2
10
7.5 3 11 9
7.5
6
Two multivariaterank testswere developed to test the hypothesis of a prescribed mean vector: (i)Amultivariatesigntest Thescore matrix takesthe form(seeeqn_ (16)) 111111111111
E, =
[ 111111111111
y=x-po=
-0.206 0.635 -0.329 -0.333 -0.311 0.659 0.536 -0-092 -0.333 -0.315 0.654 0.636
0.464-0-3010.037 0.077 0.362 -0.325 -0.210 O-419 -0.333 ,
Since tiesoccur in the apphcation ofrank tests,weuseamidra.nkprocedurethatassigns thesimpleaverageoftherankswhichwould havebeen assignedtotheobservationsifthey werenottied.Thus,rankingtheelementsof eachrowofy inincreasingorderoftheir absolutevahms,weobtain
Accordingto T'1' =_2,W
1
eqn.(g),wehave =O
and T=[-2
O]
Thus,from
v=
and
eqns.(13)and(14),weobtain
5,
Therefore, the test statistic, S, defined in eqn. (15), is calc*iated as
Thus, the test statistic, eqn. (15), is
S= ;[TV-'T'] 36
36
2111 = &-2
O]
30
= 1.5476
r-2
36
The associated P-value is P = 0.4613
0 1.
Since S is asymptotically distributed as a chi-square random variable with two degrees of freedom, we can calculate the P-value 2s P
=
0.5796
Such a large P-value supports HO; hence, the sampling technique is judged to yield representative samples of the mixture. (ii) A multivariate generalization of the Wilcoxon signed-rank test In this case, the score matrix E,=
&R,
BY eqn- (9), T(I) = 1.692,
4-2 Test of treatment effects The data from the first three experimental runs are used to illustrate a test of significance of treatment effects. This is accomplished by testing the homogeneity of their covariance matrices_ Thus, we test the hypothesis rr,: x;
!I”” = 0.692
H a-d - T’(l) , cc2) and Z(3) are not identical
and T = Cl.692
Therefore, use of the multivariate Wilcoxon signed-rank procedure leads to the same conclusion as the multivariate sign test, and the sampling technique is judged to be representative of the mixture. Table 3 lists the P-values for all pairs of particles considered and the two multivariate rank tests. Calculations have been carried out for experimental runs 2 through 5. The results from these runs are also shown in Table 3.
0.6921
According to eqns. (13) and (14), 0.320 -0.158 -, v= 0.320
we have
In other words, I& hypothesizes no difference in dispersion for mixing 2,10 and 30 minutes. The experimental data are listed Using the small and medium sized for illustration, we define
significant times of in Table 2. particles
TABLE3 Resultsoftestingthesamplingtechnique Experimental run
Calculationbasedonthepairof (smaU,medium) (medium,large)
1 2 3 4 5
0.5796 OS.653 OS054 0.2231 0.1653
1 Multivariate signed- 2 3 rank test 4 5
0.4613 0.3293 0.3392 0.3013 0.5732
Multivariate signtest
Wilcoxon
Inference:HO, eqn. (41), (smaU.large)
isrejected
0.2138 0.4758 0.1054 0.1353 0.7788
0.1054 0.0765
no no
0.3285 0.3196 0.3443 0.2808 0.6985
0.2712 0.2366 0.7649 0.3488 0.8583
0.2231
0.1353 0.1653
IlO
no Xl0 IlO
no no no no
141 XN
[x<1’x<2’x<3q
=
2X36
where
,
=
0.797
o-127 0.9CS 0.004 0.000 0.022 0.992 0.869 0.241 o_ooo 0.018 0.987 0.969 i
1
0.032 0.370 0.410 0.695 0.008 0.123 0.752 0.0.00 0.658 0.013 0.031
I
x(2)
=
I
’
0.259
0.012 0.435 0.115 0.017 0.271 0.146 0.089 0.929 0.272 0.600 0.067 0.431
0.565 0.606 0.771 0.611 0.836 0.250 0.071 0.517 0.225 O-202 0.419
and
&3)
f
0.426 0.000 0.493 0.049 0.482 0.507 O-200 0.463 0.583 0.529 0.504 9.474
0.085 O-008 0.064 0.064 0.136 O-052 0.050
Rankingthe36elements obtaintherankmatrix RN
= [R
,
ofeachrowofthematrix~~
inincreasingorderofmagnitude,we
1.5
36
where 16
31
3
35
7
16
4
26
13
7
36
29
21
1.5
20
32
2
9
33
15
5
22
17
12
28
29
34
30
36
23 13
17 -20
26 14
21 18
18 10.5
6
35
32
1
31
3
6
30
23
27
10.5
12
8
27
11
10
19 8
25 19
249
4 34
15 28
R
25
R(2)=
1
22J
and
R<3'=
335
Accordingtoeqn.(24),thegeneralscorematrixisoftheform
14 24
1
142
pairs (small, medium), (small, large) and (medium, large) are 0.720,0.360 and 0.753, respectively. Because the P-values are quite large, we conclude that the difference in dispersion among rotating speeds of 20,30 and 45 r.p.m. is not significant.
RN-
where
JN =
I:
: IIIIII
:I,,,.
4.3 Test of the completely miked state In solids mixing, the completely random (or mixed) state is characterized by the property that the probability of selecting a particle of a given component at any location in the mixture is identical. When the population proportions are known for the components of a mixture, the chi-square goodnessof-fit test can be used to test the hypothesis that the mixture is in the completely mixed state. In other words, we test
Thus, in our example,
Using eqns. (25) WI’
through (31),
= [l-7099
-0-7724
1.56681
UC” = CO-6743
-0.0612
0.92011
u
-0.4645
0.45651
u’
-0.4126
0.97301
= 10.9727
-0.2705
-0.1870
0.1870 -0.2705
-0.2442 0.9632
0.6623 -0.2442
1.7878
0.4113
[ -0.3515 0.4113
0.3411 1.2402
0.6606
vN(RN)
L
=
and
Vi’(RN)
=
-0-3515 1.7348 0.3411
According to eqn. (32), the test statistic is L=
g EZ,[CJ’~‘--U*]
I 1
I-r,:
the mixture is in the completely
(44)
against
H,:
the alternative
E-I, is not true
The data generated in run 6 of the experiments are shown in Table 4_ From eqn_ (35), the test statistic is computed as
= (47 -
V~‘(RN)[UCk’--U*]
(22 -
21.6334
Since L is asymptoticahy distributed as a chi-square random variable with six degrees of freedom, the associated P-vaIue is P = 0.001 Therefore, the nuII hypothesis H,, eqn. (42), is rejected, and we conclude that there exists a significant difference in dispersion between the mixing times of 2,10 and 30 minutes. The P-vaIue based on smah and large sized particles and that based on medium and large sized particles are 0.001 and O-006, respectively, In the second
34.333)2
+ (34 -
34.333
k=l =
experiment,
we test
the effect
of rotating speeds (20,30 and 45 r-p-m.) on dispersion for a fixed mixing time of 10 minutes (experimental runs 2,4 and 5). The P-vahies corresponding to test statistics for
mixed
State
34.333
34.333)2
34.333 (43 40
40)2
34.333)s
+
---
+
+ (37 -40)2 40
(40 40
+ 40)s
-I-
= 162.280
Under the nuU hypothesis, eqn. (44), Q is asymptotically distributed as the chi-square random variable with 24 degrees of freedom. SinceP < 0.001, the nuII hypothesis is rejected (at the usual levels) ; we conclude that the mixture has not reached the completely mixed state. Table 5 summarizes the results of completely mixed state for experimental runs 6 through 9. 4.4 Test of a quality standard In quality control decisions involving multicomponent solids mixing,we may be concerned wifih the fraction of a population which meets
143 TABLE4 Thedatageneratedinthe6tbrunoftheexperiment spot samde
Colordistributionf green red
white
Totalnumber of particles
Expectednumberof particles foreach
1 2 3 4 5 6 7 8 9 10 11 12
47 39 26 24 38 33 35 31 32 36 33 40
22 31 42 50 17 35 9 54 31 27 11 37
103 98 84 84 97 107 91 111 102 83 120 120
34.333 32.667 28. 28. 32.333 35.667 30.333 37_ 34_ 27.667 40_ 40.
*Tberatio
34 23 16 10 42 39 47 26 39 20 76 43
coior
is1:1:1forcategoriesgreen1red:white,respectively.
TABLE5 Resultsoftestingcompletelymixedstate Experimental run
Mixing time (rum)
Rotating speed (r.p.m.)
forexperimentalruns6through Expected distribution
9
Test statistic
Associated probability
Q
P
Inferenc2 aboutHo, eqn(44)
6
30
30
111 -_:__:3 3 3
162.280
~0.001
rejected
7
30
45
111 --f-_I3 3 3
56.246
<0_001
rejected
8
60
45
111 -:-_L3 3
43-429
0.009
rejected
13.202
0.963
accepted
9
150
45
111 -_I__:_ 3 3 3
a quality criterion. The binomial test can be used to solve this multicomponent solids mixing problem. Again, let xii denote the weight fraction of the ith component in the jth sample and pi0 the population weight G?lction of the ith component. We may set the criterion as m+l ;z_
Mxij
-PiO)2
G
E
3
(45)
Assume that three components are equally important. Hence, let h,=Xs=Xs=l We wish to test whether the satisfactory proportion of a mixture has reached 95% for a pre-selected value of 0.015 for E. In the following examples, we test the null hypothesis H,,: 0 > 0.95 against
where Xi = arbitrary positive constant which might reflect the relative importance of the ith component being mixed, and E = preselected positive real number. We say that a spot sample is satisfactory if it satisfies this criterion, otherwise, it is unsatisfactory_
H_:
(461
the alternative hypothesis
6 < 0.95
The calculation for run 7 of the experiments is shown in Table 6_ The results are S+=7,S_=5,n=12
144 TABLE6 ThecalculationforexperimentaI-7
Snot sample
Xii
1 2 3 4 5 6 7 8 9 10 11 12
0.3333 O-3537 0.2941 0.3196 0.3113 0.2857 0.2785 0.3678 0.3229 0.3182 0.3367 0.3636
*cl10=l&-J =
x2j
p30
=
fortestingthequ~tystandardasdef~edineqn_(45)
i=I
Is it satisfactory?
0.0018 0.0174 0.0034 0.0175 0.0382 0_0050 0.0334 0.0129 0.0046 0.0014 0.0567 0.0021
Yes no Yes no no Yes no Yes Yes Yes no Yes
5
=3i
0.2564 O-4146 0.3765 0.2474 O-4811 0.3286 0.2405 0.3908 0.2917 0.3636 0.5000 0.2987
0.4103 0 2317 c-3294 b-4330 0.2075 0.3587 3.4810 0.2414 0.3854 0.3182 0.1633 0.3377
-&O)2f
o-3333_
The guide indicates that the appropriate P-value is a left-tail probability for S, = 7 arameter of 0.95, which from a withap binomial table gives P = 0.0002. Since this P-value is so small, we conclude that the data reject Ho in favor of EL. Hence,
we conclude that mixing is not adequate_ The results of runs 6 through 9 are summarized in Table 7. Note that this test may also be used in the analysis of a continuous mixing process.
5.CONCLUSIONS
Statistical analysis is recognized as a major too1 in solids mixing investigations_ Traditionally, results of sampling have been analyzed using normal [S, lC].
Cxij
theory
statistical
techniques
TABLE7 Resultsoftestingthesatisfactory proportionof mixturesforexperimentalnms6 through 9 Experimental run
S+
s-
n
P
Inference concemingHO,eqn.(46)
6 7 8 9
5 7 10 12
7 5 2 0
12 12 12 12
0.0000 0.0002 0.1184 1.0000
rejected rejected accepted accepted
This study proposes the applicability of several nonparameteric statistical techniques to problems in multicomponent solids mixing_ The most imporknt feature of a nonparametric procedure is its lack of dependence on a particular distribution type, e.g., normal. Since the distributions of components during mixing are usually unknown, nonparametric procedures comprise a substantial collection of alternatives to the classical parametric procedures. Recently, the extension of nonparametric techniques from the univariate to the multivariate case has been pursued [ 21. The present study demonstrates the applicability of multivariate tests of location and dispersion to test hypotheses concerning the sampling technique and the significance of treatment effects in multicomponent solids mixing problems. The proposed nonparametic procedures were tested with actual homogeneous and heterogeneous ternary mixtures generated by a drum mixer_ In spite of the small number of the samples drawn (n = 12), the results tend to support the practical significance of nonparametric statistics in the evaluation of mixing systems. Resides the robustness of the nonparametric methods against the assumption of a specified distributional form, it is important to note their simplicity in application. An effort will be made in the future to study the performance of the proposed nonparametric methods for a larger number of samples.
145 LIST
G cij
E(X)
EN E
j$il eij
fij
HL
rl Rk
P Q
RN
Rij
Rm R$ sS
T
W
OF
SYMBOLS value at the significance level o[ sign indicator of Yij expected value of random variable X score matrix as defined in eqn. (24) score matrix as defined in eqn. (8) mean score of the ith variate expected number of particles of the ith component in the jth sample cumulative distribution function (c.d.f.) of X number of particles of the ith component in the jth sample null hypothesis two-sided alternative one-sided alternative with positive direction one-sided alternative with negative direction test statistic for testing homogeneity of dispersion matrices number of variates total number of spot samples for t treatments number of spot samples number of samples of treatment k associated probability test statistic for goodness of fit test (m X IV) rank matrix as defined in eqn. (22) (m X n) rank matrix as defined in eqn. (7) rankof Yij among(Yil, ---, Yin) set of ell order m-tuples [x1x2 _ __ x, ] ’ rank of Xl!) among (X$1, _I_, Xi:,‘) fork=l,z,._.,t number of unsatisfactory samples test statistic for testing equality of mean vectors univariate rank order statistic as defined in eqn. (9) row vector as defined in eqn. (10) number of treatments row vector as defined in eqn. (29) row vector as defined in eqn. (30) as defined in eqn. (25) as defined in eqn. (26) as defined in eqn. (13) as defined in eqn. (12) as defined in eqn. (31) number of unspecified parameters estimated from data critical
row
vector = [XGX*j
..- Xmj]‘,
j=l,2,.._,n
row vector = [X159@’ . . . X(mk] ‘, j= 1, 2, ___,nk a random variable denoting the weight fraction of the ith component in the jth spot sample a random variable representing the weight fraction of the ith component in the jth sample for the kth treatment realization of Xj = Xij -fliO (m X n) random matrix (m X IV) pooled random matrix standardized variable as defined in eqn. (39) standardized variable as defined in eqn. (40) a parameter representing the probability of satisfaction location vector parameter weight proportion of the ith component in the population degrees of freedom as defined in eqn. (33) chi-square distribution dispersion matrix of treatment k significant level for testing a hypothesis an arbitrary positive real number an arbitrary positive constant to reflect the relative importance of the ith component APPENDIX
On the use of P-values in hypothesis
testing
The traditional method of testing a hypothesis is the determination of a rejection region and a corresponding rejection rule for which the probability of making a Type I error does not exceed some preselected value called the level of the test. In many cases, the choice of the level of the test is arbitrary and in some testing situations the chosen level may not even be attainable. These problems are circumvented by the reporting of P-values. The P-value is defined as the probability under the null hypothesis of a sample outcome equal to or more extreme than that observed. The reporting of P-values clearly contains more information than merely report-
146
ing the decision made on a hypothesis at a possibly arbitrary IeveI. The use of P-vaIues is clear for those tests in which the outcomes can be ordered relative to the expected outcome under the null hypothesis. In those unambiguous cases the P-value is the probability associated with a corresponding right- or left-tail probability. In the more complex situations where ‘more extreme’ is an ambiguous relation, conventions must be defined for the reporting of P-vaIues_ REFERENCES
4
5
6
7 8 9
1
F. S. Lai, R. H. Wang and L. T. Fan, Powder
Technol_. IO (1974) 13_ 2 M_ L. Puri znd P. K. Sen, Nonparametric Methods in Multivariate Analysis. Wiley, New York, 1971.
and A_ Stuart, The Advanced Vol_ 2, Charles GriftZm, London. 4th edn_, 1977. A. M. Good, F. A. Graybii and D. C. Bees, rntroduction to the Theory of Statistics, McGrawHill, New York, 3rd edn., 1970. E. L. Lehmann, Nonpammeticsr Statistical Methods Based on Ranks, McGraw-Hii, New York, 1975. J. D. Gibbons, Nonparametric Methods for Quantitative Analysis, Holt, Rinehart and Winston, New York, 1976. M. Hollander and D_ A_ Wolfe, Nonparametric Statisticat Methods. Wiley, New York, 1973. W. J. Conover, Practical Nonparametric Statistics. Wiley, New York, 1971. S. S. Weidenbaum, in T. B. Drew and J. W. Hoopes, Jr. (Eds.), Advances in Chemical Engineering, Vol. II, Academic Press, New York, 1958. R. H. Wang, L. T. Fan and J. R. Too, Powder Tecknol. 21 (1978) 171.
3 M. G. Theory
10
Kendall of
Statistics,