ERRATA SHEET
SCME CCMPARISONS OF SENSITIVITIES FOR
FOR TWO METH rns OF MEASUREMENT
vI. L. Hatley
Page 17
..
Line preceding equation (3.12) .. w
Page 18
..
Line preceding equation (014) .. (3.,11) read (3.12) •
1
read w 1 •
Line 2 of Section 3.2.2.1 .. (3.3) read (3~4)
Page 26
.. Equation at bottom of page
Page 29
.. Right hand side of last equation read
3 a29
e
Page
49 ,.
..
C>
(3.4) read (3.6)
0
~~~ + .~: ) 1/2
Equations (4.9) .. Subscript on
read 1-.-.< ~ 1-..</2 •
Page 55
..
4th line from bottom .. Sf read HI •
Page 59
-
Lines 1 and 2 of 2nd paragr'aph - "relati ve sensitivityll read "relative
sample sip ~I'
Page 62
-
Insert atter Patnaik Pitman, E.J .G q
1939. A note on normal correlations. Biometrika,
~:9-12.
11
f"
..."
scm: CCMPWSONSOF SBNSrrmTIES' FeE
TWO METHODS OF
~
by
William LeJioy Hatley
•.J'....
~,
..
•
Institute of Statistics
Mimeograph Series No. 253
April, 1960
iv
TABlE OF CONTENTS
Page
LIST OF TABLES • • • • • • • • • • • • • • • • • • • • • • • • • • •
Ir
v
f'
..
LIST OF FIGURES
. . . . . . . . . . . . . . . . . . . ; . . . . . . . vi
CHAPl'ER
I.
mTRODUCTION • • • • • • • • • • • • • • • • • • • • • • • •
1
General • • • • • • • • • • • • • • • • • • • • • • •
Statemebt of the Problem • • • • • • • • • • • • • •
1
DISCOSSIONOF L1TER.A.TURE • • • • • • • • • • • • • • • • • •
5
2.1 General • • • • • • • • • • • • • • • • • • • • •••
2.2 Multivariate Approach • • • • • • • • • • • • • • • •
5
2.3
2.4
2.5
III.
3
5
Components of Measurement • • • • • • • • • • • • • •
6
8
Sensitivity Criterion •• • • • • • • • • • • • •••
I
Compariscn of Similar Experiments • • • • • • • • • • 11
THEORETICAL
:IE'VELOHmlT.·.·. • • • •
3.1 IntrodUction
• • • • • • • • • • • •
• • • • • • • • • • • • • • • • • • ••
3.2
Case I: Both Measurements'Taken on the Same
Experimental Units • • • • • • • • • • • • • • • •
3.3
case II: Each Measurement Taken on Separate
Snbsamples from the Same Experimental Unit
14
14
26
3.4 Determination of Relative Sample Size • • • • • • • • 29
,
3.5
Power of the Tests
3.6 Case·· Itt:
TeSt
IV.
;
••••
..
•••••••••••••••••
Approximation for the Schuma.nn-Bradley
••••••••••••••••
0
•
•
•
•
•
•
36
'.. .A.FP:LICll'rION • • • • • • • • • • • • • • • • • •
0
•
•
•
•
•
•
46
4.1
4.2
Test Procedures • • • • • • • • • • • • • • • • • • •
Example of a Case I Experiment • • • • • • • • • • • •
• •
••••••••••
00
.
~
5.1
S1lIJlII1&:ry'. •
5.2
Suggestions for Further Research
•
•
•
•
•
•
•
•
•
•
•
46
51
53
59
59
4.3 Example of a Case II Experiment • • • • • • • • • • •
v.
32
•
•
•
•
•
•
•
•
LIST OF BmFEBENCES • • • • • • • • • • • • • • • • • • • • • • • • • 61
CHAPTER I
IN1.RODUCTION
1.1 General
In many experimental situata.ans the question frequently arises as
1:0 whioh of two ar mare methods of measurement should be
some oharaoteristic or response of interest.
U!l
ed 1:0 observe
In such caSes it is neees-
sary 1:0 have a Dans of compar:mg the relative merits of the various
methods. When the true value of the characteristic of interest is known,
the customary procedure is to determine the accuracy of a measurement
method by comparing the measured value with the true value, and to determine its standard deviation as a measure of the precision or reproducibiiiiU of the method.
Aoouraoy and precision are then used as a basis
for comparison of alternative methods.
However, there are many situa-
tions in whioh the oharaoteristic of interest is either ill""Ciefined;
suoh as, quality, performance, etc., or can only be measured by means of
the instruments being oompared; SIlch as, strength, smoothness, etc.
In
this oase it is neoessary to have a means of comparing the relative merits
of methods of measurement which is independent of any knowledge of tba
true value of the charaoteristic of interest.
Both of tbase situations, where the true value of the oharacteristic
is either known or unknown, have been disoussed by Cochran (1943) and
"of'
.•
Mandel and Stiehler (1954), and they suggested the conoept of sensitivity
as a .qualitative measure of merit of measurement methods.
Basically, the
sensitivity of a measurement is a oanbination of its ability to detect
2
small variations in the characteristic of interest, or small differences
between treaiments; and its reproducibility.
As Cochran put it:
"It is clear that the answer (to the sensitivity question)
depends both on the experinental errors associated with
the scales and on the magni'bldes of the treaiment effects
in the two scales."
I.
Tlms,. the most sensitive method will be that method which provides the
op'ti.mu.m combination of high discrimination of trea'bnent effects and high
reproducibility or min:i.mum error variance.
When the sensitivity problem is to be investigated, experiments may
be carried out specificaJ.ly for the pIlrpGse of compar1ngtwo methods or
scales of measurElllEmt..
It is assumed that it is possible to seleot
treatments that will effect changes in the characteristic of interest,
or that materials can be selected whieh are known to span the rage of
values of this characteristic; for example, see !.ashof, Mandel and
Worthington (1956)
0
It is not necessary to know the true values of the
characteristic, only that differences, be they large or small, exist.
Wl¥3:n at least one of the measurement processes leaves the characteristic
of interest unaltered, both methods of measurement may be applied to the
same experimental units.
In the si'blation where both measurement processes
alter the characteristic of interest, it is often possible to apply the
two methods of measurement to separate sets of subsamples from the same
eJCperlmentaJ. units.
,.
There mq be si-wations, however,
were it will be
necessary to apply the methods to be compared to independent experiments,
the onlJr common feafure being that the same set of treatments will be
used :in bo-th experimemts.
This last situation has been considered bjr
SchumaXlD and Bradley (1957) aId (1959).
3
Provided the techniques of analysis of variance are applicable, it
is possible to obtain a comparison of sensitivities of methods of measarement by comparing the F-ratios for trea1lnents from the analysis for each
method.
The most sensitive method is that one which better demonstrates
trea1lnent effects (Model I of the analysis of variance) or the existence
of a between-treaiments component of variance (Model II.. of the anaJ.ysis
of variance).
This definition of the most sensiti.ve maasurement mathod
conforms to that given previously.
The comparison of F-ratios is the approach considered by Schumann
and Bradley.
They
deal with the situation where independente:xperiments
are conducted; and, hence, the F-ratios to be compared are independent.
In the siination where one experiment is condlcted and both methods of
:meaSlrement are applied to the same experimental units, or to subsamp1es
of the same experimental units, the comIBrison of the F-ratios far treatments from the analyses of variance will involve the comptrison of correlated F-ra. tios.
It is this 1a tter sitaation with which this paper is
concerned.
1.2 Statement of the Problem
The prima.ry objective of the study reported in this paper was to
develop a stati.stical test for the comparison of the sensitivities of
two methods of measurement applied to the
~
e:xperiment.
The approach
to be considered was the canparison of the F-ratios for trea1lnents from
the analyses of variance.
Two s iinations were of primary interest:
first being the ease where both measurement methods can be applied to
the same experimental units, ani the second being the case where the
the
4
measurement nethods must be applied to separate samples from the same
e:xperimental units.
.
The secondary objectiva was to investigate the
preperties of the tests that were devalopad, particuJArly with regard
iD tlB power of the tests.
As an outgrowth of the sindy it was possible to develop am apprexi-
rna te test for the case considered by Schumann and Bradley (1959). They
developed an exact test for the case where two ind8peDdent e:x;periments
are conductedo A further developllBnt was a solution for the problem of
obtairt:i.mg ma:x:imwn sensitivity with minillmm cost.
The theoretical
development of these and the ori;g:inal objectives is presented iD
Chapter IIIo
Ulustrative eDlllples of the application of the tests are
givem in Chapter IV 0
1"'
CHAPTER II
2.1 General
1'1» problem of determining the appropriate scale or nethod of measure-
ment to be employed for the determination of a part:Lcular characterist:Lo,
or the estimat:Lon of a part:Lcular response, has been d:Lscussed throughout
the literature.
A great majority of the art:Loles in whioh th:Ls problem
has beEll d:LSalssed have made no attempt at presenti.ng a general approach
to the problem, b1 t rather have dealt with the topic only in so far as it
pertained to the particular s1l:ldy being presentedo A few examples are
Dillon (1936), Roth and Stiehler (194e), Carlin, Kempthorne and Gordon
(1956), Lashof,
!!!!l,. (1956), Teichman,
~!1 .. (1957), and Hart (1958)"
Tl'e only artic les to be dis cussed in this chapter, however, are those :in
which an attempt has been made to present a valid statistical. test for
the general problem of oompar:Lng sensit:Lvit:Les of measurements"
These
latter art:Loles are oanparable to the type of s1l:ldy presented in this paper.
2" 2 Mo.lti:va.r:Late Appro a~h
The statis t:Loal aspects of the problem of eompai':Lng different scales
of measurement for experim.ental results were discussed in cClls:Ld.erable
detail by Cochran (1943)"
He aBeumed that the teehniqaes of analysis of
variance were applioable and conf:Lned his attention to tm case in which
all scales measure the same replicated experiment" What was at that time
recerrli work in nmlt:Lvariate
ana~is
was used to prov:Lde tests of the
hypothesis that the treatment effects are the same :in all scales, and of
the hypothesis that tba scalas are limarly related.
All of the tests
6
presented were for large sample results. Aside from :lndicating the
problems involved, no attempt was made to develop small sample tests.
A brief discussion was presEllted of methods for comparing the
"relative sensitivity" of two scales. What Cochran refers to as relative
sensitivity corresponds to what bas been ealled the
by otlBr authors.
lts~nsitivity ratiolt
Tests were suggested for comparison of two scales when
0nly tW0 tl'ea'bnents are used and the scales are either equivalent or
l:1nearly related.
For the case where more than Wo treatments are used it was indicated
that the comparis0n of two scales would inwlve a test of the lr3PothesiB
that two n0n-eentral variance ratios are equal. However, n0 such test
was ac1ually developedo
2.3 Components of Measurement
Several publications have appeared in which the proposed technique
for comparing two or more methods of measurement assumes that a measurement is made up of several components.
In an article presented by
Grubbs (1948) a measurement or observed value was considered to be the
sum of two components -
one the absolute valu e of the characteristic
measured and the other an error of measurement. He called the first
component, product variability; 1dlereas, the second component was referred to as tlB precision of the measurElllento
Techniques were given for
separat:lng and estimating product variability and precision of the
measurElllent, and the application of these techniques to cases :involving
the oompar.ison of two or more measurement methods were discussed.
significance tests were presented.
No
The decision as· to which measur:1ng
7
method is superior was determ:ined by comparing the relative order of
magni1ude of the estima. tes of tile canponents.
Later, Smith (199:»used GrUbbs' devalopment to consider the problem of oomparing two instruments for measuring a character which can
only be observad via these or similar instruments; i.eo, there are no
theoretically absolute values against which the instruments can be
calibrated.
He presents a technique for campu:hing the precision of two
measu:ring instruments men there is a linear relation between the scales
of the two instruments.. A technique is presented for obtaining estimates
of the components of neasuranent as defined by Gro. bbs, which re<:plires an
estimate of the regression of one scaJ.e on the other..
S:ince Smith uses
the linear relationship betweEll the scales only as a means of obtaining
estimates of the canponents of measurement, the outcane is :independent
of
~ich
regression is estimated ..
Further 'WOrk 1n the use of components was presented by Mandel (1959)
a1'd by Mandel and Lashof (1959).
Both pap:lrs gave particular attentim
1:0 interlab<ratory studies of test methods.
Tm assumption was made that
systanatic differences exis t between sets of measurements made by the
same observar at different times or on different instruments, or by
different observers in the same or different laboratories, a.rrl that these
systematic differences are linear functiClls of the magnitude of the
measurements ..
Tm first pap:lr by Mandel proposes a theoretical framework for the
mathematical expression of the sources of variation in measuring methods
and develops a suitable method of statisticaJ. analysis.
The latter
paper deals with the practical aspects of the scheme, which is called
8
"the linear mQdel", and demonstrates its use :ill the design and analysis
for 1'OIlnd-robin tests. A. further discussion of the use of :illterlabaratory
studies f'orthe pirpose of compar:illg methods of' measurement was also
presented •
2.4 Sensitivity 6riterion
An alternative approach to the preceding techniques falls :into a
category of its own, which will be referred to as the sensitivity criterion.
This crtterion was presented by Mandel and Stiehler (19,4).
They' suggested as a measure of tl'e sensitivity of a test
where M is an obtainable measure of some property of interest, Q, and
is its standard deviation.
"M
It was pointed out that the advantage of this
criterion :is the. t it takes into account, not only the reporducibility of
the testing procedure, bIt also its ability to detect small variations
:in the characteristic to be measured.
For the case where we wish to oompare the sensitivities of' two alternative methods of neasurement applied to the same experiment, the sensitivi ty ratio was presented :ill the form
'I'M
11J'
dJii1
'PN·
iWAQ'.
O"N
~
,
where M and N are the alternative measuring methods.
ratio was then reduced to
•
This sensitiVity
Hence, it was pointed OI1t that it :is not necessary to bave a knowledge of
the relation of M and N to the theoretical Q.
All that is recp,ired is a
knowledge of the mutual reJAtionship 'between M ad N. Mandel ad Stiehler
indicated tbiLt lilA/AN will be the slope of the curve ot Mplotted as a
f1mctia of N.
The superior metACJd 1$ dete1'DliDed bY' observ11'lg :1.11 waiek
direction the sensitivity' ratio dev:lates from GIIe. Wl»n there is no
ditter••e :bD the sensitiT.i. ties this ratio will eql1al one.
Mandel
am
Stiebler fIlrther stated that the functional relatiC!BIship
asswned to exist between the methods M and N need not be knOWD for the
application of the sensttivity ratio approach.
on~ necessary-
to obtain the regression
1'hI:ty .implied that it is
of M on N; and then Ii1:!/AW will 'be
estima ted bY' the slope of this regression.
Let us investigate this poiDt.
If £ (~IN) denotes the conditional
c·
distri'hution of M given N, the re gression of M on N :is defiDed as
CD
E(MIN) =
.I Mt£c(M/N)]dM ,
(2 •. 4)
-00
and, si.mi1arly, the regression
of N on M is
00
E(NIM)=
.I Nlic(NIM)]dN ,
""00
Their respect!va slopes will therefore be
N1
AN=
and
ICD.(~[fc.(MIN)])
-(!)
M
b.N
.
dM,
(2.6)
10
Ncm,
..L
-
i'N
implies
I~I
(2.8)
'I'M
wbich implies
1M Carte;:l!/) J) elM/. fIN caite:'
I)!) J) cDI
•
(2.9)
It is well known that this last relationship is not necessarily true.
Far example, i f M and N eo. from a bivariate normal population,
a.
E(MIN) ==
lJ...
'J.Vl
+
p.s:!
'!N
(N - Po )
N'
amd, hence
•
(2.11)
By. the same token
J!.
III
fjJ!1
aE(NlMl
3M
III
p
51
o
(1M
Therefore, su'bst:i:b1 ting (2 ..11) and (2012) in (2 ..8) gives
which is a contradiction except when p
III
1.
Heme, the determination of
tlB mest seDsitive method of neasuranent, using the slope of the regL"u-
sion of one nethod em the other as EAlggested bY' Mandel aDd Stiehler, is
dependent on which regression is estimated.
A proposal for obtaining eanfidEmce limits for the sensitiv:i:ty ratio
bY' use ef the relation
(2.14)
11
where F is distriblted as Snedecorfs F and 8fllt»fI is assumed to be determined without error, was presented.
This relation may be incorrect since
the variances which appear in the ratio under consideration are correlated
when both methods are used on the same expe:r:iment.
Im this case, the
relation that should be used for confidence bounds on
IAMIANI (aN/~)
is
t1la t gi.ven by Pitman (:1.939), namely:
F •
A'M)2 (SN
aN) 2
('A\t
s; - ~
(1'1-2)
4(1-r~) (t) 2 (~)
,
wEe F is distribu. ted as Snedecorfs F with (1, 1'1-2) degrees of freedom,
22/··
.
r MN • BMN YN' n:is tba number of samples measured, and sN' ~ and sMN
are the sample standard deviations and covar:iancei" respectively.
Aside from these points, if the functional relationship between the
two me thods of measurement is known, or the C)lali ty Q can be measured with-
out error, the sensitivity criterion appears to be a satisfactory approach.
2.5 Comparison of Similar Experiments
The comparison of variance ratios from :independent but similar experiments as an 'approach to comparing sensitivities was introduced by Scl1J.1m.ann
and Bradley (195'7)
0
They suggest that in order
to compare two methods of
measurement, two independent bu.t similar experiments should be set up, and
that the two scales of measurement be applied to separate experiments.
The term Its:Lmilar exper:i.lll8nts U was used in the sense that the F-ratios
for treaimEl'lt comparisons resulting from the two experiments have the same
degrees of freedom
0
Tba objective was
to select that measuring method
12
which bas .the larger F-ratio, with the assumption that this would be an
indication that that method was more sensitive to differences in the
ireai:zrents than the method yielding the smaller F-ratio.
The model
aSSlmed for the experiments was Model I, the fixed effects model of the
analJlS is of var:iance as defined by Eisenhart (1947), and a test was
developed for testing the hypothesis that tm non-eentrality' parameters
of the two non-eentral variance ratios are eQlal.
Tm distribution of the ratio of two independent non-eentral variance
ratios was obtained and its properties disOl1ssed.
Pai11aik (1949) they showed that the ratio of _
Using the result of
non-eentral variance
ratios mq be approximated adequatel:yby the diStribution of the ratio
of two central variance ratios with appropr:ia. tely adjusted degrees of
freedan..
A table for use in applications of the latter distribution,
wi 1:h exper:iJnents in which the degrees of freedom for the F-ratios are the
same and the treaiment means are based on tm same number of replications,
was given for one-s:ided
t~sts at the
5":% level
of significance.
Simul-
taneously with this paper a companion paper was presented by Bradley and
Schunumn (1959) in which applications of i:b3 similar exper:iment approach
and use of the table were presented.
Later the application of tm simllar e:x;periment approach to Model II,
the rand()m effects model, of the
an~is
of variance "as defined by
Eisenhart (1947), was presented by lb:toss (1959), and Schumann and Bradley
(199J) • Both papers extended the earlier W>rk of Schumann and Bradley
(195'7) on the comparison of sensitivities of experiments based on Model I
of the
an~is
of variance
to experiments based on Model II of the
13
analysis of variance.
In addition, the Schumann and Bradley paper
presented additional tables for tb9 distribu.tion of tb9 ratio of two
central variance ratios with eCJ,1al pairs of degrees of freedom.
new tables are for the 2.5%, 1% and 0.5% levels of significance.
The
CHAPrER TIl
THEORETICAL DEVELOPMENT .
3.1 IntrOdQction
In this chapter procedures are develhoped for a statistical test for
cOUq,aring the sensi,t1vit:i,es oftWoo methods of measuring some quality of
a
ma~erialwhere
both methods are applied to the same experiment.
oases are of primary interest.
Case
It
These are:
Both methods of measurement are applied to the same
exp~rimental
Case III
Two
units,
Each method of measurement is applied to separate
subsamples from the same experimental units.
A procedure is then presented .for determining the relative simple
size of two measurement methods.
From the relative sample size it is
possible to determine the number of additional samples required with the
least sensitive method to make the two methods equal in sensitivity.
Cost considerations· are introduced and a determination made as to which
method is the least expensive to apply • The power of the tests developed
for the two cases of interest is discussed, and power curves are presented
for several values of the parameters.
Finally, an appro:rlmation for the
Schuniann-Bra.dley test for the case of independent experiments is developed,
and the critical values obtained with the approxima.tion are compared with
the exact "Il'aluers obtained by Schumann. and Bradley.
as Case III.
This is referred to
1.5
The following assumptions are made throughout this ohapter:
(1)
The pairs of observations on eaoh experimental unit are
asswned to oome from a bivariate normal parent popu.lati01'1.
(2)
Tl8 applioable model is Model :0:, the random effeots model,
of the anaJJr$is ofvarianoe.
(3)
The assumptions reqllired for tb:3 use of analysis of varianoe
prooedu.res to infer the existenoe of oomp01'1ents of var1anoe,
as described by Eisenhart (1947), are applioable '00 'the
experimental situations under cQ'lsideration.
3.2 Case II Both Measurements Taken on the
.same Experimental Units
3.2,1 Notation and hypothesis.
.
Suppose for simplicity samples 'are
tested from m grades of a material, by two methods or for. wo properties,
. '
.
th
Xa and lb. Let nr be the number of samples fran the r
grade, and let
m
~ (~)
III
Then~ from the analysis of variance and covariance for the
N.
two methods, we obtain the sums of squares and sums of products presented
in Table 3.1.
Table 3010 Analysis of variance and covariance
Souroe
BetMeen Grades
Within Grades
d.f.
~
m~
s:ia
~
a2
SXa~
c1
c2
sX;
bl
b2
16
The necessary upectations are:
E
(~) •
a; + ko~
,
E
(~.
;;
+kJ~ ,
(3.1)
,
E
G;). ;;
,
(3.2)
where ko will eCllal nr in the particular case where the same number of
samples are observed in each grade.
Now, as stated in Chapter I, the goal of such an eJq>eriment is to
detemi.ne mich method does the best job of detecting the e:x:Lstence of
a between-grades canponEilt of variance. Hence, the hypothesis :to be
tested is:
Ho·
cib
",2
w
w
JIIb
,
-;;2-"
0
•
T.h:is alternative will be one-sided when
a new procedure with a
c~pard.ng
standard or accepted procedure. Since the to IS are the same, Ho is
equivalent to
H'o
,
•
0
0
,
or
H'.
o·
cl.-
k0 b +
k
rJ2
cijci
w
w
+,2
o b
w
1-w
_ :L.
0bT.l.0usl;r, tor either of the torms, the appropriate stetiet:l.c is
(3,7)
~~
•
17
Except for the correlatioo between Xa ani ~, this s~tistic calld
be expressed as a ratio of Sne.decor's UFw,. or a difference ofF1sher's
~
ItzUt! . Since the correlation can not be ignored, the test of significance will be dependent on the cts as well as the a's and bas.
-
3.2,2__ Adjusiment for the cbrrelation.
-
The "between-grades" and
"within-grades" lines of tis ana4"sis being independent, it is possible,
from Piiman (1939), to find the distribution of the ratio~lof Ireall
squares for each line, ani then ofthe:tr ratio.
Thus, for either line
of the ana1.ysis we have from Pitman
2
t i ..
(ai/bi - <:.)i)2(~-1)
2
4(1-I'i)(ai /bi) CNi
'
(i .. 1,2);
(3.8)
where t i is distributed as Student's "tIt with (mi-l) degrees of freedom,
2
2r i .. ci/ai b i , and tis 4)i's are:
kO"~ +
.-0
Go,)
0"2
,
w
1/
k,2 +
o b
w
1
and
•
T1:e null hypothesis
m~
now be written
H~: c.>l .. c.>2
Now, tq- obtain tie distriblltion of
for
~l'
which yields
2
2
a.i {
2ti (l-%'i)
~i .. b
1 + ...,;;-------i
(mi-l)
(3 _,JJ! )
•
~b2
b:I.
a2
we must first solve (3.8)
18
Next, let
t (1-r2)1/2
i
d
i
I!\I
i
(m _1)1/2
,
(3.13)
i
then (3.11) can be written
{.>i
(~) =
1 + 2di ± 2di (1 + di>1/2
•
It is :important to note that these limits are reciprocal; i:'e.
1 + 2di + 2di (1 + di) 1/2
III
1/ [1 + 2di - 2d i (1 + di) 1/2) •
Finally, let
Fill + 2d 2 + 2d..(1 + d?)1/2 == [d + (1 + d 2)1/2
1
1
-~
-.1
1
(1
)2
'
(3.16)
,
(3.11)
and
2
F
"1 + d 22)1/2 • rd
+ (1 + d 2 )1/2 ]2,
2 i l l + 2d2 + 2d2A
~ 2
2
than
~b2
:b1a2
' , ' \ _Ii'2 (4J_,
1)
:iS diatribUted' a~ F ~w2
1
,.
(3.lEn
and, if we let W III Ii'2/F1'
:t-la2
b2
under
H~.
3.2.2.1
I
is distributed as W ,
Tests of h;ypothes:is.
An ..<-1evel test of
H~
against the
two-sided alternative (3.3) will be, ttReject H~ ifW ~ ";1"</2 or
...
W ~:W1-42' otherwise do not reject H~." To determiI)e which method is
-
most sensitive we need only observe that i f W ~ W..</2' the method
associated with
x"
is the more sensitive method; i f W ~ W1 _"</2' the
19
method associated with Xa is tb:l more sensitive method.
reject
H~
If we fail to
the results of the imve stigattion indicate that there is no
difference in sensitivity between IIlethods X and ~o
a
.An: alternative approach to'~ the tes,t is to look at the two ratios
(~/a2) and (b /b ) and to place the larger value in the numerator of W.
l
2
Hence, for the two-sided alternative, the test of significance will then
be, ltReject H~ if' W ~Wl-42' otherwise do not reject H~i~ll In this case,
provided
H~
is rej.ected by the test, the most sensitive method will be
that method associated with the nunsrator of Wo
3.2.2.2: Oonfidence mnds. Confidence bounds may be obtained for
,
the sen~itiv:t.'tyratio,GJl /Co)2' if' we first note that from the relation
in (3.15) W42
Probe
III
1/Wl-'l2.
Thus
~ '~92
b a
Wl -42 ~ &r; , ~a2 • ~-.</2 ] :: 1 - .,( •
[ ~b2/
l 2
302.) Distribution of the test statistic.
(3.20)
In order to perform the
statistical test described in the previous section we must first obtain
the distribution of W0 Rem~beriiJg tbat W = F 2/Fl ' we may set up the
following relationg
U ::
m
==
Ne1Jtt, let d1
(1/2) In(F 2/Fl)
,
In!d + (1 + d~)1/21 - In[d + (1 + di)1/2]
2
l
-1
sinh
= sinh
d 2 - s:inh
-1
~
, (3.21)
0
zl and d 2 = sinh z2' tl:en
where, since they are related to the ltbetween-grades ll and ''within-grades''
lines of the analysis, respectively, Zland z2 are independent.
20
From (3.13) we see that
~
is merely a transformation on Student t s
t-distribution; hence, zi is aJ.so a transformation on Student's t-distri-
i)
bution; i.e.,
sinh zi
where t
i
l-r 1/2 '
ti
( mi-l
==
,
is distributed with (~-1) degrees of freedom.
r(~)
'.
r( "1> - (l-ri>lf2 r<~ >f(mr1)fl+
.
1
.
cosh zi
'Sinh2
\
"i
2
)mi /2
' -
Thus,
00
< zi < CD
,
1-r
i
(3.24)
Since.
z~
and z2 are 'independent, their joint distribution is:
2 (~-1)/2
2 (~-1)/2
(1-r1 )
(1-r 2 )
r{i!)r(?)
-r(+)r(+)
x
.
(cosh
Now, put p
l!ll
2
2 m1 /2 '.' . 2
2 ~/2
zl - r 1 )
(cosh z2 - r 2 )
•
(zl + z2)/2 and q == (z2 - ~~/2,and substitute in (3.25) to
get
l(f.)r~)
r(+)r(+)
2 cosh (p-q)
cosh (p+q)
21
Thence, by integrating p out of this expression, we can obtain the distribution of q; and, since u == 2q, we would then be able to obtain the
distribution of u.
As an alternative approach, calculate the chararnristic function
of zi
Then.,- 1£ we note that cosh z == cosh (-z), the characteristic
0
function of
111:
is the product of the characteristic functions of the
z!~so
Thus, the inversion procedures presented by Cramer (1957 sec.
J.
10.3) COllld be applied in order to obtain the distribltion of u. Let
it suffice to
say that this approach requires an equally intricate
integration to that required for the approach using equation (3.26)
0
The cll:J.racteristic function of zi' ~z)&), will be obtained here,
J.
however 0 By rewriting equation (3024) in the forM
f(zi)
2
r.)
t
=:
I-m.
K(cosh z:t)
J.,l -~
cosh zi
and expanding the negative binomial, we get
.K(¢oah
_ z.)
_lam. [.M
(-rJ. ...
)2
1 + -i
Mi
O
fez )
i
==
==
-
K
J.
;J..
·t' M2'i (Mi2' +1)
s'
2
...
sL
I.· ..
cosh zi: :,
J
~ +s-.Lj
(Mi
j
of
""rrJ.i
111i
2'(2 +
t
2
1)
/2
,
r.
( cos~ zi
2
I-mi -2s
rJ.0s (cosh zi)'
•
i&z
e
i
---~~2::----':'''- . dzJ..
zi
-z.s + M.-.J.,
(e;- + e J.)
J.
r(1 +s}ri
S
r(s+l)
(3.28)
,
22
Equation (3.29) mq then be used to gEl'lerate the moments of zi e
However, there :is a more direct approache From (3.23) we have that
"-1
Zi == sinh
2
2
Ait i , where Ai
==
l-r.
mi-~ , and t i follows the t-distribltion
with (m -1) degrees of freedom. Obviously tmn, zi is distributed
i
s;ymmetrically a1:x>ut zero; therefore,
(For tba remainder of the development of the moments of zi the subscript, i, will be omitted.)
EJlpand:ing
in a Taylorts series about
Z
zero gi.ves:
z: == At _ 1. (+t)3 + 1erl (Ag)5 ...
,
2 3
2.
~
'2":Ii":O
(+t)
7
7
+
000
0
'1'llIls,
z
2
468
==
(A.t)2 - (+;)
(At)4 - 2~t)
+
6
~ - ~+
+
¥ _
000
,
. 8
0 ••
,
(A t) 6 ... (A t) 8 + ... ,
z8
III
(At) 8 - ••• ,
and from the moments of the t-distribution [Cramer (1957)J,
2
2
2
E 2
11-r 1 f!!!-I)
1 (1-r ) 2 3 (m-l) 2 + 8 (1-r ) 3
1~(m-113
(z ) == '\'m:ll \m:J -- 3" .
(m-3) (m-55
45 m:r (m-3) (m-5) (m-7)
m:r
. 2
4
4
~ . ~ (;:1') (m-3;(~mtl-n (m-9)
l-r 2
==
iil=3 -
+ ... ,
(1-r2) 2
. 8fl-r~3
12 (1-r2) 4
(m-3)(m-5) + 3(m-3 (m- (m-7) - (m-3)(m-S)(m-7)(m-9) +
.oo
•
23
and
2
E(z8) _
l05(1-r )4
- (m-3)(m-5)(m-7)(m-9)
- ... .
These moments may now be simplified by expanding the denominators
in a binomial series, as follows:
(m-3r,l = m- l + 3m- 2 + 9m- 3 + 27m- 4 +
·.. ,
(m-3)-1(m-5rl = m-2 + 8m- 3 + 49m-4 +
·.. ,
·..
.. .. .
(m-3)-1(m-5)-1(m-7)-1 =m- 3 +15m- 4 :.(m-3)-1(m-5)-1(m-7)-1(m-9)-1 = m- 4 +
~
Letting A == l_r 2, the moments of z can be written as:
',,' 2
A, 3A-A2 9A-8A 2 + 8/3A3 27A-49A 2 + 40A3-12A4
E( z ) = - +
2 +
3
+
4
+ ••• ,
m
m
m
m
':142
2~2_10A3
E(z 4 ) == ~ + ~ 3
m
m
147A2 - J.50A3 + 49A4
15,A
E(z8)
lOtA4 +
4
-
3
3
4
+ 225A ;:lO$A +
m3
in
E( z6) ==
==
+
m
-
+ ••• ,
... ,
t •••
m
Due to the independence of zl and z2 the moments of u may be obtained
by taking the expectation of the expansion 'of un = (z2-z1)n, n == 1,2, ••••
Since zl and z2 are symmetrically distributed aboutzero,u' is also symmetr.ically distributed about zero.
Hence,
24
E. ( u2S+l)
Since
= 0,
= 0,1,2, •••
s
ro:t. will in general be larger than
,
~,
it can be seen by
inspection,: and by noting that 0' ~ '1, that all terms of third order
or higher in mil will be negligible, even for values of ~ as small as
ten. Thus, dropping terms of third order and higher, the moments'of u
become
.. 2
CTU
.A:t
= lII:t
2
2
A2 ~-Ai 3A2-A2
+- +
2
+
2
lII:t
m2
A:!.
lJ.4 (u) = 3 (iFi1 +
'
(3.30)
m2
~)2
m;
•
(3.31)
Now, since u is symmetrically distributed about zero, the skewness
coe£:fie:tent of uis
~ero;'.- aild,!.f
the-third order terms are' omitted~
25
the kurtosis coefficient of u is three.
These are the skewness and
kurtosis coefficients of the ·normal distributlon, thus for values of'
IrI:t.
and ma for which it is reasonable iD drop the higher order terms :in
mi-1 ' uI au a.pproximates the standard normal distriblltion.
For values of
IrI:t.
between five and ten, the third order terms of the
moments must be retained; thus,
whUe, for values of
~
equal to or greater than thirty-five, second and
-1
2
third order terms :in mi of au may be dropped, giving
2
~ A2
a a - + C3 .3,)
u IrI:t. m2 •
'When
l!1.
is less than five tls normal approx:iJna.tion should not be used.
Note iha t the higher the correla.tion between Xa and
v~es
of
~
~
the smaller the
and A2 will be; and, hence, the better the normal approxi-
mation will fit.
j Since the objective of this section was to obtain t:tS distribltion
of W, note that u • (1/2) In(W) and that, as has bean shown above, u is
appro:x:imately norma.lly distribl ted with mean zero and variance ~.
W has approximately a. lognormal distribution with mean
4a2
4a2
e2~
Then
and variance
e u(e u - 1).
If
t
III
u/a~, th6n tD calculate
t l _ 1/2 eo~~$pong].ng.todthear·eEl·~der
"V
2au~1' '2
Thus, Wl _-'/2 a e
-4. For example,
first obtain t:t:e critical value
the normal curve of 1--'/2.
the critical value Wl -J.!2 we need
26
:i..f .,(..
.05 and
~
is lar goa,
•.
W. 975 = e
3.3 Case II:
3.29
(~+ ~) 1/2
Ill:!.
m2
•
Each Measurement Taken on Separate Subsamples
from the Same Experimental Unit
Suppose now that two test procedures,
3.3.1 !vpothesisand notation.
or methods of ITeasurement, are being compared which are both of a destructive nature.
In this case it is not possible to apply both tests to the
same e:x;perimental units.
that
e
.
sa:rrq>les be taken from the
~
These
The experimental design here 'WOuld reCjl:ire
~
l'
th
grade of the material under s'bldy.
samples 'WOuld tmn be divided equally between the two test
procedures so that each procedure is applied to nr sampleso
If
m
~
. . . (~)
r~
II
N, the analysis of variance and covariance will be the same
.
as :in Section 30201, Tabla 301, except that the sums of sql1ares from
the nnthin-grades u line
or
the ana.lysis are now independent; i.eo, the
sum of products term c 2 will not appear 0
The hypothe sis to be tested is exactly the same as for Case IJ
However, for this case t:te right-hand ratio is a ratio of independent
var:iances, and its estimate a /b is distriblted as Fo((l/,,2),
2 2
wmre FQiollows an F-elistrilntion with ~ degrees of freedom for both
the :Dl1IIElrator and denominator
0
27
It should be noted that the approach for Case II is
~
restricted
to the situation where both methods are of a destructive nature.
Even
though both methods of measurement may be applied to the same sample, it
may be desirable to use separate samples for each method.
This might
occur i f a time effect exists between successive measurements on the same
e:x;perimental unit and we wish to eliminate it in this way.
It will be
shown in Section 3.5, however, that the test for Case II is not as
statistically powerful as the test for CaseL
3.3.2
Distribution of the test statistic.
The test of hypothesis
and confidence bounds discussed in Sections 3.2.2.1 and 3.2.2.2 are also
applicable to Case II; however, the test statistic
l 2
F0
is now distributed as F
la2
1
a b
~
((...,)G.> 1
2)
;
where F0 is distributed as Snedecor t s F with m2 degrees of freedom for
both the numerator and denominator, and Fl is as in Section 3.2.2.
W1
= Fo1F1,
Let
then under H~
ab
l 2
b la
2
is distributed as W1 •
(3.35)
, Next, to obtain the distribution of WI ~ let
u'
=
(1/2) In(Fo/Fl ) ,
=
(1/2) In(F ) - (1/2) In(F )
o
l
=
Zo - zl
,
(J .36)
,
where Zo is distributed as Fisher's z-distribution with (m2,m ) degrees
2
of freedom, zl is as in equation (3.24), and Zo and zl are indep e7;ldent •
28
The moments of zl have been presented in Section 3.2.3; thus, to
obtain the moments of u' we need the moments of zO.
First, note that
when the numerator and denominator degrees of freedom are equal, Fisher's
z-distribution is symmetrically distributed about zero; hence,
2s+1E( Zo ) . 0 , s • 0,1,2,...
•
The even moments of Zo are [Cornish and Fisher (1937)]:
E(Z~)
•
,
E(Z~ =
Therefore, since Zo and zl are independent, u' • Zo - zl is also
sjlmmetrically distributed about zero, and its moments are
6
E(u' )
Referring to the discussion at the end of Section 3.2.3 we see that
it is reasonable to drop higher order terms of the moments.
TJ:ms, far
29
and
1'4(ul) • 3
(~+ ~) 2
(3.38)
•
Again a normal approximation ~ be used in that
ul is approx:lmately
U I /O'
distributed as a standard normal.
Hence, WI is approximately distriblted
?
2
2
20- I
40' I 40' I
as a lognormal with mean e u and varianoe e u (e u - 1) •
For values of 1T1. between five a:rxl ten, use
2
O'u I ..
(~.. + -1) + (3~-Ai
1) + (9~-8Ai
2 + 2
3+ 8/3
-
~
~
and, for values of
,
2
O'u t
~
~
m2
Ai + -3
2.) ;
3m
IIlJ.
(3.39)
2
greater than thirty-five, use
Al
1
= m;+ iii;
•
(3.40)
The proposed test should again not be used for values of m less than
l
five.
To calculate the critical value
Wl-42 ..
where
.1-42
Wi-.42 '
2O'U I ! 1-42
e
is the normal deviate corresponding to an area under the
normal curve of 1-420 For example if.,( ...05 and ml is large,
3.4 Determination of relative sample size
0
In conjunction with
any comparison of sensitivities of measurement methods, we may also be
interested in comparing the cost of obtaining a given sensitivi"Gy-o
Suppose, for example, in an acceptance-~sting situation it has been
30
determined that we should test n samples from a
particular shipment of
material, using a standard test procedure.. Further suppose that an
eJq>eriment designed to compare sensitivities has shown this accepted
method, s8\Y' XCI' to be more sensitive than some al ternativa method, say
~.
Now, the question arises as to how many samples 'WOuld be required with
method
~
to attain the same sensitivitY' that we have using n samples and
method X ? And, using these sample sizes, which method has the min:i.mwn
a
cost? By .r.ef:errihg' to thehyp:o.the:sis :actual:ty\re~ ,tested:» l1ee.ahp1W.~
vide anansWertothe:s'equ.estions.
To obtain an answer to this problem, let us first consider what
this difference in sensitivi ty implies.. When we say that method X is
a
more sensitive than method ~j we are saying that
a~ ,J~
->2
D~~
0
a rJ.2
w w
Now, if n samples are observed in each grade, (3.42) may be writ!ben
2 ,/2
nab Dl"b
->- 0
(3043)
a2
w
,2
w
Hence, in order 'Ix> make these two sensitivities equal we must obtain a
multiplier for the right-hand s;tde of (3043); 1oeo,
2
nab
2=
a
w
where hba, is
,/2
~anpb
p.2
a.c~ua.lly
'
w
the relative sample size of method
~
given the
sample size to be used with method Xa 0 The sample size to be used with
method ~ to attain a sensitivity equal to that given by method Xa when
n samples are used will than be hoa.n.
31
first step in obtaining the value of hba is to obtain est:iJnates
2
of O:b, a 2, ,pb2 , and t),2 from the analysis of variance and covariance. These
The
w
w·
will be:
,
(3.45)
Thus, substituting these est:imates in (3.44) and solving for
~
aA2/~2.
Pb
b
==
~
w
~
w
~,
we get
(3.46)
•
To determine the total cost, i f C is the cost per sample using
.
a
method Xa , and Cb is the cost per sample using method~, then
Ta
==
Can
, and Tb == (Cbhoon)
(3.47)
If' Tb is less than T , method ~ is
are the respective total costs •
a
the most economical for a given sensitivitYJ and, if Ta is less than
~,
method Xa is the most economical.
If',instead,the relative sample size of method Xa , given''l:.be sample
size to be used with method~, is desired, (3044) will be written
habnO"~ n~~
2
T ·
0:
aw
(3.48)
s&w
Substituting the estimates of the components of variance in (3.48) and
solving for hab, we get
~~/~~
hab=~ ~
w
w
•
(3.49)
32
. Note that hab
II
l/~a.
Ta
The total costs :in this instance will be
(0 aa
h bn)
II
,
and
Tb
II
0bn ,
(3.;b)
respectively •
It must be pointed out that the development in this section is not
a procedure for detenn.ining the
sampling problem.
s~le
size required for a particular
It is assumed that the optinn.un sample size has been
determined previously; and, hence, the number of samples required for at
\
mast one of the methods is known.
3.2
Power of the tests.
In order to assess the value of the tests
described:in Sections 3.2 and 3.3, it is necessary to look at the power
of the tests.
The power of a test is defined as the probability of
n:aking the correct decision when, in fact, the a1 ternativa hypothesis
is tt-ue. Since the main goal of the test of sensitivities is to select
the best measuring procedure for a particular type of problem, it is
:important to know how capable the test is of detecting differences when
they exist.
In both Case I and II when the normal approxiJrJation is appropriate
it has been pointed out that the test statistic, W, follows the lognormal distribution.
Tlms, the first step in calculating the power is
"Ix> obtain the critical values of tha test under
H~.
As shown previously, the critical vaJ.ues under
test are
,
and
,.
H~
for the two-tailed
33
where ~ is distriblted N(O,l).
The critical values under the alternative
will be
and
Hence, for a parti.cular simple alternative
value of Gl /
l
Cr)
i .•e., a parti.cular
2' the power of the test may be calculated from the relation
Power = prob.(W~wi
prob.(W~~1~1/Q2).
/t:Jl /G.>2)+
= (1/2)
Remembering that u
Power
~othesis;
In(W), (3.53) can be written
= Prob. (U$ u~l~l/""2)
+
Prob.(u~ u~ICJ1/~2)
,
where
~
==
(1/2) In(~)
,
u~
= (1/2)
In (~~
,
and
=-[ O'~~ 1-4'/' (1/2) In (~l/GJ 2) J
To obtain the desired probabilities for equation
0.56)
•
(3.~)
it is only
necessary to consult a table of the CUIlltllative Normal Distribution,
2
since u /O'u is distributed N(O,l).
Figure 3.1 presents power curves of tl:e tests of sensitivity for
..( = .05, rn:t = 10,
r
l
and r
2
~
= 55 and
three values of r l and r 2 • For simplicity,
have been set equal to each other; however, this would not
generalJ.y be the case.
on the same figure.
Curves for 'both Case I and Case II are presented
Note that except for Figure 3.1 (a), where the 1NTo
34
Power
1.0
0.6
Case .I
0.2
case I
Case I
. . --- -ca.;e-----II
Figure 3.1. Power curves of the tests for sensitivity
e
e
e
Power
i~o
1
------------ -2
3
0.9
0.8
.,. ...
.",,......
.,,'"
".,.,~-"..-­
'"
0.7
0.6
1
r l =·8
,
--"2 r l a .8 ,
r 2=·8
-----).
r 2=·8
,
r =.5 ,
4 l
o .2.4 .6 .8
1.0
1.,
2.0
2.5
CJ
Figure' 3.2. Rower
3.0
l /CJ.)2
3.
4.0
r l =·5
4.
r 2=.5
r 2=.5
.0
~
curve. of the test for sensitivity; indicating the effects of changes in correlation
36
cases have the same power, the test for Case I is uniformly more powerful
than the test for Case II.
Note also tm t the power increases as the
correlation between X and 11, increases. In fact, the power is inversely
a
proportional to au; and, hence, an increase in the correlation or in the
degrees of freedom, which will in tum bring about a decrease in Q'u' will
result in an increase in power.
Figure 3.2 again presents power curves of the tests of sensitivity
~
for ..<
-.05 and ml • 10,
of r
and r 2 • The canparison of curve 1 with curve 2, and curve 3 with
4, illustrates the effect of changes in r 2, while the comparison
l
curve
• 55, but with various combinations of values
of curve 1 with curve 3, and curve 2 with curve 4, illustrates the effect
of changes in r • Note that changes in r 1 have a marked effect on power,
l
'Whereas changes in r have a negligible effect on power. Also, note that
2
any increase in correlation increases the power.
It has been pointed out previously tht t even :if both methods of
measu:rement can be applied to the same sample, it is still valtLd to use
an e:x;perimental design which would be classified under Case II.
How-
ever, the reduction in power from Case I to Case II, illustrated in
Figure 3.1, would indicate that a design suitable to Case I should be
used.
3.6 Case III:
Approximation for the Schumann-Bradley Test
3.6.1 Notation and Ji'pothesis. As a means of demonstrating the
closeness of the· appro:x:imations developed in Sections 3.2 and 3.3 we
mll develop an approximation for the Schu.rna:nn-Bradley test, and
compare the critical values obtained by the approximation to the
37
exact values obtained by SchUIna.nn and Bradley (1959).
situation cons idered by Schuma.rm and Bradley
i~
The experimental
the case in which two
independent experinents are conducted and a different method of measuremnt is applied to each experiment.
They have developed an exact test
for the comparison of the sensitivities of the two measursn.ent methods
mich involves a ratio of the F-ra:tios for treatments from the analyses
of variance for the two independent experiments. .As explained previously,
the use of their tables requires that the experiments be identical; i.e.,
that the degrees of freedom for the F-ratios for treatments and the
number of observations for a given treatment are the same in both e:xperiments.
Using the approach of Sections 3.2 and 303, it is possible to
develop an appro:x::i.ma.te test for this case.
Suppose m treaments are selected which will produce differences in
tlle characteristic of interest..
th
Let nr be the number of samples observed
m
treaiInent, and let ~. (n )= N.. Next, suppose that two experir=l r
ments are conducted in which the same m treatments are used, and n
in the r
r
samples are observed in the r th treatment in each experiment. Further,
suppose 'that measurement method X is applied to one experiment, and
a
measurement method ~ is applied to the other experiment. Then the sums
of squares from the analysis of each method would be the same as in
Table 3.1. For this situation, however, the sums of products terms, cl
and c , would not appear.
2
The hypothesis to be tested is exactly the same as for tm previous
two cases; i.e.,
o
H(Wfever, these are now ratios of independent variances, and their est:tmates
~/bl and a2/b2 are each distri~uted as Fi~' i = 1,2; where the Fits
follow
an F-distribution
in which the degrees of freedom are the same for
both the numerator and the denominator, arrlthe Wi's are as in eqaations
(30'/) and (3;1.0)••
It should be pointed out that for the case Of independent experiments
it is not necessary to write the null hypothesis in this form.
However,
this form of the hypothesis gives us the advantage that the F-ratios to
be used in determining the distributionof' the test statistic have the
same degrees of freedom in both the numerator and denominator 0 Hence,
when we transform to the z-distribution we obtain symmetry about zero.
The development in Section 3.6.4 indicates the effect on the calculation
of the critical value of not having the numerator and denominator degrees
of freedom equal 0
3.6.2 Distribution of the test statistic.
Once again the discus-
sion of the tests of hypothesis and confidence bounds in Sections 3.2.2.1
and 302.2.2 is applicable 0 However, the test statistic
~~2 is now
a2 1
distributed as
where F02 follows an F-distribution with
follows an F-distribution with
(rrt:J.'rrt:J.)
are as in Section 3.2.20 If we let
~b2
wit
Ol\G3 2
(~,~)
,
degrees of freedcn, F
degrees of freedom, and the
Ol
GJ t S
i
= F02/F01 '
- b is distributed as Wit
a2 1
under
H'.
.
0
~02 ~l)
,
(3.59)
39
To obtain the distribution of W", let
u lt
= (1/2)
= (1/2)
In(Fo2!F01)
,
In(F02 ) - (1/2) In(F01 )
where zOl and z02 are distributed as Fisher's z· with
,
(lll:L'IIl:I.)
and (~,~)
degrees of freedom, respectively.
Now, the moments of the z-distribution, when both degrees of freedom
are the
S8m3,
E(Z~rl)
=
have been presented previously in Section 3.3.2. They are:
0
,
s = 0,1,2,...
,
and
,
,
•
Hence, the moments of u it = z02 - zOl are:
E(u It2 s+1 )
and
E(u..2 )
=
0
, s
= 0,1,2,...
,
40
Again we see that it is reasonable to drop higher order terms and to
use a normal approx:i.ma.tion; i.e., UIt/CTutt is approxi.ma.tely distributed
as a standard normal.
For
tn:I. between five and ten use
(1~. = (~+~) +(~+~) +(3~ +~)
for
,
(3.61)
tn:I. between ten am thirty-five use
(1~.
and for
= (~
+~) +(~ +~) ,
tn:I. greater than thirty-five use
2
u" =
CT
(1..
.
rr:t + 1...)
m2
As for the tests developed in Sections 3.2 and 3.3, tb9 distribution
of 'Wit will be approximately a lognormal distribution with mean
2
2
e2~..
and
. ".
4CT lt 4CT l t
va.riance e U (e u - 1). Hence, the critical value is
Wi_~2
where
~ 1-~2
=
2CTU It ~ 1- "2
e~'
(3.64)
is the normal deviate corresponding to an area under the
normal curve of 1-~2t am CTu It is as above.
FiDally, it should be pointed out that the determination of relative
sample size presented in Section 3.4 is also applicable to this case.
3.6.3
Comparison of the approximate test to the exact test.
In
order to compare the approximate test to the Schumann-Bradley test,
Table 3.2 has been prepared.
This table presents the critical values
for both tests for various values of the degrees of freedom.
The upper
values are from the table presented by Schumann and Bradley (1957) and
(1959), while the lower values are the critical values calculated as
41
Table 3.2.
Comparison of critical values for the 5% level of signi:f'icance
for the sChumann-Bradle) test (upper value) and the new appro~
i.ma.te test (lower value
§
6
8
10
12
J.h
16
18
20
30
(J)
5
8.87
8.94
7,0'70
7.77
7.11
7·10
6.68
6.75
6.42
6.42
6.23
6,30
6.09
6,11
5.96
6,05
5.64
5.,70
5.05
5,10
6
7.86
7.92.
6.77
6.82
6.18
6.2~
5.78
5.87
5.54
5.58
5.38
50 42
5.25
5. 26
5.13
5.15
4.82
4.86
4.~1
5.75
,5.'76
5.21
5. 21
4.88
4090
4.65
4.66
4.48
4.48
4.35
4.~5
4.25
4 0 26
3.44
4.92
4.95
4.59
4.62
4.36
4.19
4.18
4·07
4.10
~.97
3.94
3097
3.68
3.71
4.70
4.68
4.37
3.98
3.86
3.76
4.~5
4.14
4.:14
~098
~o86
~.74
3.48
3049
2.98
'2 097
4.52
4.50
4.19
4.18
3.98
~.98
3.82
3..82
3.70
3.71
3.60
3.60
3.32
3032
2.82
2.83
4.05
4.06
3.84
3082
3.68
30 67
3.56
3.56
3.46
3.46
3.18
3.19
2.69
2069
3063
3.63
3.47
3.46
3..36
3.35
3.26
30 25
2.98
3'000
2.49
2.48
3.54
3.52
3039
3039
3.27
3.25
3.18
3.19
2.90
2.92
2.40
2.41
16
3.32
3.32
3.20
;.20
3.11
3.13
2.83
2.86
2.34
2.34
17
3.26
30 25
3.14
3.13
3.05
3.06
2.77
2.77
2.28
2027
3009
3.10
3.00
3.00
2.72
20 72
2.22
2.22
20
2..91
2.89
2.63
2.64
2.12
20 -12
21
2.87
2.86
2.59
2.59
2.08
2.08
2.36
2.36
1.85
1,84
8
.<
9
10
11
12
14
18
30
5.45
,.41
4.~5
3.98
'
4.28
~,046
3.18
~019
42
above. Since the Schumann-Bradley table presents critical values such
that 1 - G(W ) == .05, the critical value for the approximate test is
o
2
.3. 90'u"
W" .95 == e
•
A.s one might expect, far the small~st WluesQf
m:r.,
:f'ive or six,
the value for the approximate test is not as close to the value for the
e:s.acttestasit is for larger values. Howaver, even for these two
values the maxLmum discrepancfJ is only + .09, while the average
discrepance is + .046. For the remainder oftha table -the ma.xi.nmm
discrepance is + .04 and the average discrepance is + .0015.
there is no negative discrepance greater than .02.
paT'ticularly for small values of
tn:L'
Note that
In general then, and
we can say that the appro:x::i.mate
test is a slightly mare conservative test of the null hypothesis· of
equal sensitivity.
3.6.. 4 Dissimilar experiments. Oocasionally, situations might arise
wherein the experiments to be used for comparing the sensitivities of
two measuring methods do not have the same degrees of freedom.
This
might be the case i f some experimental units are accidentally destroyed
in one experiment and not in the other; or possibly, even though the
same treatments are used in each experiment, the experiments are based
on different munbers of replications or have different experimental
designs ..
In this case, neither the degrees of freedom nor the coefficient
of the between-grades component of variance will be the same from one
experiment to the other.. When this is the case, the discussion of the
hypothesis
H~,
presented in Section 3.201, will not be applicable..
In
that section it was pointed out that the null hypothesis we wished to
test was:
,
and it was shown that, since both
k~! ~
Were :the same, this could be
Oi~". '_li.<!::t=>_,
'.
written
•
If the kots are not the same, it is not possible to rewrite Ho as
H~.
Hence, for this case, Ho can not be tested merely by COOlparing the
F-ratios for treatments from the analyses of variance.
This situation has been discussed by Schumann and Bradley (1~57)
and
(1~5~), and
they suggested a procedure for testmg Ho • They
suggest thm, since Ho implies (~/cr;) ~ (~ht;) == ~o' we obtain an
2 2 ·
estimate of (Ob/O'w) from the analysis of method Xa , and an estimate
of (~/,;) from the analysis of method
JIb.
They then let q equal
the average of these two estimates, and, under Ho' q is assumed to be
an estimate of 9>
In the notation of this paper their test statistic
0
would be:
(~)(l + ~q)
(~~)(1 + ~q)
where (~,~) and
k:L
F
distributed as
F2l ;
are the degrees of freedom and coefficient Gf the
betwem-grades component of variance associated with the analysis of
measurement method Xa , (mf'~) and k2 are the degrees of freedom and
coefficient of the between-gradeseomponem of variance associated
44
with the analysis of measurement method
:JIb'
and F and F2 each follow an
l
F-distribution with (m.t'~) and (m{,~) degrees of .freedom, respectively.
Following the procedure used previously in this chapter, i f we set
Wit
= (Fl !F2)'
and let
ult
we get that u lt
(1/2) In(WIt)
==
= zl
(3.66)
,
- z2' where zl and z2 each follow a z-distribution
with (m.t'~) and (mJ.'~) degrees of .freedom, respectively.
For this
ca.se, ;. and z2 are not symmetrically distributed abOllt zero.
Hence,
using the moments of the general z-distribution as developed by Cornish
and Fisher (1937), the mean and variance of u tt will be
and
2 2'1(1- + -1 + -1 + 1)
(1
+ 1
2 2
O'u tl =
m2
~
rn:t
mI.
+ 1
~
2
+
rn:t
1. . + 71 ),
-;2'
m2
ml
Now, since ult is not symmetrically distributed about zero, we :find
that (l1u lt
-
ult)/O'u tt appro:xima.tes the standard normal distribution; and
the critical value Wi_"</2 will now be calculated .from
where
0';1t and I1U
lt
are as above, and
f l -42
is the normal deviate corres-
ponding to an area under the normal curve of 1-42.
45
As Schumann and Bradley point out, this is a somewhat crude method
of obtaining an approx:tmation to the critical value for this situation,
but the su.:Ltab:l.lity of the procech1re is not yet nbject to check. Where
one experiment :l.s c onch1cted and both methods of nsasurement are applied
to the same experimental un:Lts, this situat:Lon can not ar:Lse.
CHAPl'ER IV
APPLICA'l'ION
4.1
Test Procedures
As stated previously, experiments may be carried out specifically
tor the purpose ot comparing the sensitivities ot two methods ot measmoement. When Sl1ch experiments are conducted, it is assumed that it is
possible to select materials which are known to span the range ot values
tor the characteristic ot interest; or that it is possible to select
treatment s such that when they are applied to the ma.terial under study
they bring abCllt difterences in the characteristic ot interest. It is
not necessary to know the true difterences between experimental units,
only that difterences e:x::tst. It is important to note that it is not the
ettect'i: ot the treatments themselves that are ot interest, but rather
that the dif'ferences due to the treatments are ot SIltticient magnitUde
that they would be considered to be of practical importance. The
treatments then are used only as a means of obtaining a sample of the
population we wish to study•
.Altho'tlgh, in the tests ot sensitivity described in this paper,
the:~statistic Wis
calculated using sums of squares, W is in reality a
ratio ot the F-ratios tor treatments from the analysis for each measurement method. The most sensitive method is defined as that method which
better demonstrates the e:x::tstence of a "between-treatments lt component
ot variance (Model II of the analYsis of variance). Thus, equality of
the F-ratios implies equal sensitivities for the two methods of measurenent. The null' hypothesis representing this equality of sensitivities is
47
Ho' .•
ri
+ k .2
oOb_2
w
rJ.2
w
CT
W
+ k
J2
,
ol"b
,;,2
w
.where the terms in the ratios are the expected mean squares from the
analysis of variance and covariance as illustrated in Table 3.1 and
Section 3.2.1. In the theoretical development presented in Chapter III,
~l
and c..>2 were defined as
2
c:u
CTW
1
= rJ.2
w
2
+ koOb
+ k ",2
0'0
(4.2)
'
2
and
CT
W
CoJ
2
=
2
~w
·
The null hypothesis was then written
H~
: ('4)1
= (.J2
(4.4)
•
There are three alternatives against which
H~
may be tested.
They are
Hh : ~l > (.)2
'
(4.5)
H~2 : CUI < ~2
'
(4.6)
and
From the theoretical development we can see that the test against the
alternative
since
H~
H~2
is merely the reciprocal of the test against
implies equality of sensitivities,
H~l
H~l.
.usa,
implies that the method
associated with Xa is more sensitive than the method associated with
H~2
"
while
implies that the method associated with
H~3
lb
lb,
is the most sensitive,
merely implies that the two methods under consideration have
different sensitivities.
48
H~
An ..<-level test far
1)
against
Hh will be; "Reject
H~ ~
W>Wl _-<, otherwise do not
reject HI.1t
o
2)
H~2 will be; ''Reject H~ if W< l,!Wl_r:' otherwise do not
reject HI.tt
o
3)
H~3 will be; "Rt3jeot H~ if W< 1/Wl_~/2 or W>Wl _.y!2'
otherwise do not reject H~.1t
In the discussion above the test statistic has been referred to. as
W, the statistic for Case IJ however, it should be kept in mind that all
comments made 'in this seotion are equally applicable to WI, the statistic
for Case II, and
WI·,
the statistic for Gase III. The basic difference
in the significance tests for each case is whether or· not it is necessary
to calculate the correlation coefficients, r
apd T2 J i.e., to calculate
l
the critical value for a particular case we must first obtain the appro2 2
2.
.22
priate (iu' (iu" or ClU'• which, in turn, are dependent on r l , and r 2, only
ri, and neither ri nor
r~, respectively.
To perform the tests of significance for comparing the sensitivities
of two measurement methods the following steps are required.
1.
Do the analysis of variance and covariance for the experiment
as shown in Table 3.1, Section 3.2.1.
facilitate
(Table
3~1
is reproduced here to
reference~
Table 3.1.
Analysis of variance and covariance
d.f •
S?C~
SXa~
S~
Between Grades
tn:I.
~
cl
bl
Within Grades
~
a2
c2
b2
Source
49
2.
F@rmulate the null hypothesis,
hypothesis,
H~,
and the appropriate alternative
H~i.
3. Decide on the significanoe level, 0<, to be used
4. Caloulate W = w'
.
=
wit
= (~b2)/(a2bl)' seleoting the appropriate
statistio depending on whether the experiment falls into Case I, II, or
m.
5•
CompUte
A:t = (l-ri),
2
rl
using
2
2
01
O2
2
and
r
2
~bl'
=
=
-b
a2 2
•
If the test statistio to be used is WI, only
A:t
need be oomputed.
If
the statistio to be used is Wit, this entire step may be omitted.
6. Determine the critioal value for the partioular case being used
from
or
Wl -0<
=e
w:l.~..(
=e
Wit
=
1-0<
20'U~1-42
. . 20'u I
e
e1.. 42
20'uU~1_ d2
vy
where ~ 1-0< is the normal deviate oorresponding to an area under the
normal curve of 1-0<,· and O'~ is determine<;l aooording to the oase under
oonsideration as follows:
a)
For Case I
50
b) For Case II
c)
For Case III
2 =. (1- + -1) + (12 + 21) + (2
~ + 32)
~ m2
tn:L m
2 .3mi 3lT12
O"u lt
For values· of
for
~
~
·
from five to ten calculate the variances as shown above;
between ten and thirty-five the third bracketed tarriJ. on the right
may be dropped before calculating the variances; and, for nJ:t equal to or
greater than thirty-five the second and third bracketed terms may be
dropped.
7)
Compare the test statistic with the appropriate critioa1 value
and rejeot Ht in favor of Hl in acoordance with the rules set forth
o
ai
previously.
Confidenoe bounds may be obtained for Ql/G.J 2 by the relation
Frob.
{~~/Wl-42"G>1/Q 2 <; ~~. W"1-42}· 1 -
,,(. (4.13)
This relation will hold for all three cases by using the appropriate
critioal value.
Finally, the relative sample size for either of:;,the three:.cases
may be oalou!J.ated from the following relations.
a)
For the relative sample size of method
size to be used with method Xa =
Xb
given the sample
51
b)
For the relativa. sample size of method I a given the sample size
to be used with method
Xl,:
•
4.2 Example of a Case I Experiment
The following example is taken from a study by Fromm (1959).
The
ob'jeoti'Ve of the study was to compare two methods for determination of
egg shell permeability. The first method utilized a dye penetration
technique and a measure of optical density, while the second was a
measure of the egg weight loss.
It was known that the permeability of
the egg shell increases with age so that nine ages were selected as the
treatments, and eight pairs of eggs were observed far each age.
The
'Weight observations on each pair were avera.ged, while the optical density
reading was made on a pooled. sample of the shells from each pair.
Hence,
there was only one optical density reading per pair.
The test for comparison of the sensitivities of the two methods
now follows:
1.
The analysis of variance and covariance is given in Table
Table 4.1.
Analysis of variance and covariance for egg data
Source
Between Ages
Within Ages
S~' SXa~
d.f.
8
63
.7896
.8194
I a =Optical density reading
Xl, =We'ight loss determination
S~
7.2760 81.3923
.3189 6.0666
4.1.
52
2.
H~3 : <:'>1
The hypothesis to be tested is H~ : e,)1
rI
= .05.
4;
w. :~~~t x 8~:~~g~ = .0718 •
5:
ri· (.7~§g;t~:;~2j)
2
6.
.8237
,
~=
• .0205
,
A2
-,a
. ( .,2189)2
= (.8194}(6.0666)
a; " .1~63
g
+ .9 §5 + 3C.1763
= .0468337
au
against
G,)2·
3. It is specified that ..(
r2
= w 2'
= .2164
tii(
.1763 •
= .9795
.1763)2 + 3(
•
.9795§9~99795)2
,
•
•
The third term of a~ was not used here, even though
m:t < 10,
since it was
obviously le ss than .0001
W.975
= e 3.92(.2164) =
W;025 =. e -3.92(.2164) =
7.
Since W<W
,
2.3396
•
4274
•
42, HQ.is rejected in favor
of
H~3;
i.e., The
results of this experiment indicate that there is a significant difference
in the sensitivities of the two methods.
Obviously, the weiglrh loss
prooedure is the most sensitive procedure.
A 95% confidence interval for the sensitivity ratio is given by
W x W.025 'CA)1/ CN2 ~ Wx W. 975
;
that is,
(.0718)(.4274)
~ GY1/~2 ~
(.0718)(2.3396)
.0307 ~ c..>1/c.J 2' .1680
,
•
53
Or, since the method corresponding to
Xr,
was the most sensitive method,
we might prefer the confidence interval for
W.025/W ~
..
GJ 2/c.J l
G.J
2/G)1.
This is
,
~ W. 975/W·
which gives
•
4.3 Example of a Case II Experiment
The following example is taken from a study presented by Hart (1958).
The objective of the study wa.s to develop a more rapid test method for
determining the durability of Type II (water-resistant) hardwood plywood.
Since the standard test requires up to fifteen days for acceptance or
rejection of a shipment, while the proposed accelerated procedure requires
only twenty-four hours for a determination of durability, it was of
interest to determine which test does the better job of measuring durability. For the comparison of sensitivities the standard test was modified.
Normally the test is run for a maxim:um. of fifteen cycles of wetting
and drying, and the cycle at which two inches of continuous delaminatiC%1
along any glue line of the test specimen occurs is recorded..
The speci-
mens are required to average ten cycles or better to prove acceptable ..
For this study the test was continued beyond fifteen cycles, and the cycle
at which the required delamination finally occurred was recorded. The
new test, which introduced a new type of test specimen as well as an
accelerated wetting-drying cycle, was run for twenty-four cycles and
the measure of durability used was the percent of the exposed edges on
which delamination had occurred.
54
'e
For the portion of the study presented here, three glues were
seleoted, two urea resin glues and. one protein glue •. These glues will be
referred to as UF65-100, UF65';'150, and soybean.
It was of interest to
oompare the sensitivities both within the various glues, and over all .'
Ii
three glues. The treatments used to effect differenoes in durability
were oombinations of species of wooa, veneer thiokness, and oonstruotion
(ply).
In all, sixteen panels were prepared for eaoh glue mixture, and
two specimens from each panel were tested by eaoh prooedure.
'For the
comparison of sensitivities, those panels whose specimens failed on the
first oycle with the standard prooedure and also registered 100%
delamination on the aooelerated prooedure were dropped.
Hence, for the
final analysis there were less than sixteen panels for two of the glues.
For the analysis of the data, a logarithmio transformation has been
applied to the results of the standard procedure, Ial while an arcsine
transformst ion has been applied to the results of the acoelerated
procedure,
~.
The test for the comparison of the sensitivities of the standard
test and the accelerated test is given below.
1. Since it is of interest to campare the sensitivities for the
three glues separately, Table 4.2 presents the analysis for eaoh of
the glues •
..
55
Table 4.2. Analysis of varianoe and oovarianoe for
the three glue mixtures.
Souroe
2
Sx.2
SXaJlb.o
Sxa
d.f •
-7.4312
Within Panels
UF65..100
,
.
13 4;0879
14
.3872
16.5562
3.6788
Between Panels
Within Panels
UF65-150
15 4.5834
16
.6377
-6.1242
17.7663
2.9139
-2.4373
5.3969
4.1896
Between Panels
SoYbean
10
11
Between Panels
Within Panels
2.
The null hypothesis is
hypothe sis is H~3 : cu1
=I
CAJ 2 •
1.6985
.8705
H~
: c;.)1
==
2' and the alternative .
Go)
The two-sided alternative is used here
since the new prooedure is twelve to fifteen times faster than the
standard prooedure; and, henoe, might be used even i f it is less sensitive,
depend:ing on how much less sensitive it is.
3. The signifioanoe level to be used is
For
..< =
.05.
UF65-100:
4 • St
= 1 ~.0879
.5562
2-
5•
T1 -
6.·
CUI
2
x
3.6788
.3872
(~7.g312)
2
(4.0579)(16.5562)
= .1 841+
13
= 2 •358··
=
,.
85
•19 ,
A.:I.
2
==
.1841 •
1 + 3(.1841)~.1841) + 1
14
.
1
. 196
= .09378
•
56
Thus,
O"u' = .329
•
Thus,
W• .975 -- e 3.92( .329)
= 3;6328
· .-3.92(.329)
W
.025 = e
=
5
.27 3
,
•
•
6
..•
2
O"u'
= .352
+ 1 +
10
'i'i
O"u'
= .379
2
3(.352)-(.352) + 1
100
121
•
Thus,
.-3 .92( .379)
W• .975 -- e
=
w· .025 = e- 3 •92 (.379)
=
4371•
.,
.2254
•
1..
/4..
= .14368
,
57
.7• For all three glues we do not reject
H~.
Hence, the results of
this experiment indicate that there is no significant difference between
the sensitivities of the new accelerated test and the standard test.
Since the sensitivities of the two methods are essentially the same
for all three glues, we will now pool the data to obtain an estimate of
the sensitivity ratio for all three glues.
The analysis of the pooled
data is as follows:
Table 4.3. Analysis of var~ance and covariance for
the pooled plywood data
1.
2
•
3.
Source
d.f.
,Sx2
Between Panels
Within Panels
40
41
11.9338
1.8954
WI - 11.9338 x 10·h723 - 1 6511
- 41.0982
1. 954 - •
2
rl
(j
ul
-17.3624
,
(-17.3624)2
= (11.9388)(41.0982) =.6416
= .184
2
s~
a
,
~
...3854 •
•
Tlms,
W~975 .'. e 3 •92 ( .184)
i=,2~Q5.'44
..92(.leL), •
WI.025 -- e-3".
"
.4868
•
Then confidence limits for the overall sensitivity ratio are
41.0982
10.7723
58
Even i f the results of an experiment to compare the sensitivities of
two methods of measurement indicate that there is no difference in sensitivity between the two methods, we may still wish to calculate relative
sample size.
For this partioular example one would not need to determine
relative sample size.
However, in order to demonstrate
th~
computations
involved, the determination of the relative sample size of method
~,
given that two samples are to be observed if method Xa is used, is as
follows:
Hence, the sample size to be used with the new method to insure that the
same degree of sensitivity is attained will be (1.135)x(2)
This is as expected for this example.
= 2.27
or 2.
CHAPl'ER V
SUMMARY
5.1
Summary
In this disserta.tion statistical tests of significance are developed
for comparing the sensitivities of two methods of measuring some characteristic or response of interest, where both measuring methods are applied
to the same experiment.
Two experimental situations were of primary
interest, namely:
(i)
Case· I: Both measurement methods applied to the same
experimental units,
(ii)
Case IT: Each measurement method applied to separate
sub-samples from the same e:xperimental units.
A technique is presented for determining the relative sensitivity
of the two measurement methods.
The relative sensitivity is then used
to determine the number of additional samples reqnired with the least
sensitive method to provide the same sensitivity as would be obtained
with the most sensitive method.
The power of the tests developed for Cases I and II is discussed,
and power curves are presented for several values of the parameters.
As an outgrowth of the study for Case I and Case II, a test was
developed for a third experimental situationJ viz.,
(iii)
case III:
Each measurement method applied to independent
experiments.
Two situations are dealt wi. th; the first is the case where both e:xperiments have the same degrees of freedom, and the second is the case where
'.
the degrees of freedom are different in each experiment.
A table is pre-
sented canparing critical values for the 5% significance level of the test
developed in this study, and the corresponding critical values of a test
developed previously by Schumann and Bradley.
IDustrations are given in Chapter IV to present the computational
procedures for use of the tests for Case I and Case
n.
Confidence
bounds are calculated for the sensititivy ratios, and the use of the
relative sample' size for determining required sample size is demonstrated.
5.2
Suggestions for Further Research
The development in this dissertation has been restricted to Model II
(the random effects model) of the analysis of variance.
•
There is a need
,for the development of a statistical test for the comparison of sensitivities of two measurement methods applied to the same experiment when
Model I (the fixed effects model) of the analysis of variance is the
applicable model.
An extension of the procedures presented here for comparing more than
two measurement methods applied to the same experiment is also needed.
For this case some form of ranking procedure will be necessary.
Finally, the criterion used here selects:as the most sensitive
method that method which emibits the optimum combination of maximum
discrimination of differences in the characteristic oj interest and
maximum repeatability.
In some situations it seems reasonable to assume
that what may be required is some weighting of these two factors, either
•
in favor of more precision or in favor of greater ability to discriminate.
Hence, a critical study of the criterion for the most sensitive measurement method would be a worthwhile investigation.
61
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496-510.
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Table. Ann. Math. Stat .. .J.Q:581-583.
Carlin, A. F., Kempthorne, 0., and Gordon, J. 1956. Some aspects of
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-
Cochran, W. G. 1943. The comparison of different scales of measurement
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Gramer,
H. 1957. Mathematical Methods of Statistics.
University Press.. Princeton, N. J ..
Princeton
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Jour • .!mer. Stat. Assoc. ~: 243-264 ..
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Of
Lashof, T. W., Mandel, John, and Worthington, V.. 1956. Use of the
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62
lL
Mandel, J. and Lashof, T. W. 1959 • The interlaboratory evaluation of
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