University of St Andrews Full metadata for this thesis is available in St Andrews Research Repository at: http://researchrepository.standrews.ac.uk/ This thesis is protected by original copyright
University of St Andrews
Full metadata for this thesis is available in
St Andrews Research Repository at:
http://researchrepository.standrews.ac.uk/
This thesis is protected by original copyright
%1.1«
CONTENTS
Page
Summary 1
PART I 2
1*1) Introduction , 2
'I *2) Development of an optimal procedure 3
1 *3) Discrimination when the two populations are multivariate
normal 7
1'4) The sample discriminant function 10
1 *5) Distributions of D (x) and Dr_1(x), and misclassificationO 1
probabilities 11
1*6) The estimation of error rates 13
1*7) The analogy of discriminant analysis with regression 22
1 *8) Hypoth sis testing in discriminant analysis 26
1 *9) Size and shape factors 31
PART II 33
2*1) The population discriminant function for paired
observations, and its distribution 33
2*2) The sample discriminant function for paired observations,
and its distribution 36
23) The probability of misclassification 38
24) The likelihood ratio criterion for the classification of
one new pair (univariate) 46
(2*5) The L.R. criterion for the classification of two new
pairs (univariate) 30
(2*6) The allocation of a single observation by the L.R.
criterion (univariate) 52
(2*7) The procedure for populations with equal means and
unequal variances
(k'8) The L.R. criterion for the classification of one new
pair (bivariate)
(2*9) The allocat ion of a single observation "by the
L.R. criterion (bivariate)
(2*10) The geometric interpretation of the discriminant
functions of (2*8) and (2'9)
(2 "11) An example of the use of DrLo
(2*12) Two algorithms for the joint classification of k
new pairs
(2*13) Conclusion
References
A
DISCRIMINANT ANALYSIS WITH RESPECT TO PAIRED OBSERVATIONS
SUMMARY
The thesis is divided into two sections. In PART I the
theory of discrim^rit analysis is presented in a standard manner.
Particular attention is paid to those aspects of the subject which
also have relevance tothe allocation of pairs. PART II mainly concerns
the author's work on the theory of the assignment of paired observations.
It is concluded by an example which illustrates the advantages of
allocation by the discriminator developed.
c
PART I
(1*1) INTRODUCTION
In general, the problem may be stated as follows:
We are given the existence of k groups, in a population, and possess
a sample of measurements on individuals from each of the groups.
Using these measurements, we must set up a procedure whereby a new
individual can be assigned, on the basis of its own measurements,
to one of the k groups. To assess the performance of a procedure,
we usually calculate the proportion of new. individuals in each group
which we will expect to have been wrongly classified.
From an historical point of view, the first published
applications of discriminant analysis seem to have been in the papers
of Barnard (1535) and Martin (1936), both at the suggestion of R.A.Fisher.
Fisher (1936) gave another example of the methods and pointed out that
discriminant analysis was analogous to regression analysis.
P.C.Mahalanobis (1927,1930) in India and H.Retelling (193^) inthe U.S.
were concerned with similar problems. Hotelling derived the distribution. 2
of the generalised Student's ratio (T ) to test the significance of
the difference between two mean vectors. On the other hand, Mahalanobis
introduced a measure of distance between two populations (D2) as an
improvement over K.Pearson's coefficient of racial likeness, c2 (1926).Fisher (1938) showed there was a close connection between his work
3
and that'of Hotelling and Mshalanobis, while distinguishing between
the different objects for which they were developed, and derived (l9e0)
a measure of the precision with which his discriminant function had
been calculated from the data. Until recently, the nonavailability
of computers to carry out the required calculations (including the
inversion of large matrices) has deterred most people from practicing
the methods.
Discriminant Analysis should not be confused with Classification
Analysis, where we are given a sample of individuals and have to divide
them into unknown distinct groups, such that the.individuals in any
one group possess a mutual similarity.
For the remainder of the thesis we shall consider the
case of just two groups.
(1 2) THE DEVELOPMENT OF AN OPTIMAL PROCEDURE
Let us suppose we have a large or infinite population 0,
subdivided into two populations ni and II2, and each member of II has
associated with it a pdimensional vector of variables x. In other
words, n1 and n2 are two clusters of points in pdimensional space,
which are separate but overlap slightly (otherwise there would be
no difficulty in distinguishing between them). A new individual has
to be assigned to I11 or n2 depending on its scores on the response
vector x.
The diagram below represents the type of situation that
arises in two dimensions. Some boundary such as A3 must be constructed,
so as to separate X's from O's in an optimal way.
/,
Three examples where Discriminant Analysis has been
successfully applied are as follows:
Variables x Populations Hi and H2
1. Anthropological measurements
on skulls
2. Sepal and Petal, length and
width
3. External symptoms in suspected
sufferers from a certain disease
Male and Female
Two species of flower e.g.
Iris Setosa and Iris Versicolor
Presence and Absence of the
disease
Now, let x have probability density function (p.d.f.)
ff 1 (x) if x belongs to I11
hf2(x) if x belongs to II2 ,
and <£1, y>2 = 10i be known prior probabilities
of a member of II belonging to Hi and H2 ,respectively. It would be
desirable to split the whole pdimensional space R into two regions
5
Ri aad Ra such that
if x falls in Ri, assign new individual to n1
cr if x falls in R2, assign new individual to n2.
By using the above procedure, we will inevitably make some wrong
decisions, and so we must choose our optimum procedure to be the
one with the smallest probability of doing so. There are two types
of error which may arise. Firstly, a new individual which is actually
from n1 may be assigned to 02 and secondly, an individual from H2 may
be classified as FIi.
Prob. (a n, individual, chosen at random, is assigned to H2)= J f1 (x)dx = di
R.2
Similarly, a2 = ff2(x)dxRi
So that, M = Prob. (a randomly chosen member of H is misallocated)= 01«1 + $2®2
= / <f>2f2(x)dx + I 01f1(x)dxRi Ra
= / 0 2 f 2 (x)dx^ + 01  / 0ifi(x)dxRi Ri
Thus to minimise M, we minimise
I [$2f2(x)  <£ifi(x)]dxRi
This is achieved when we take into Ri all points for which
</>2f2(x)  <t>ifi(x) <
and exclude from Ri all points for which 02fa(x)  0ifi(x) > 0.
i.e. the boundary of Ri is given by
f2 (x) 01 ,—
=  constant,fi(x) 02
f 2 (x)where is the likelihood ratio for the observation x . This
f i (x)
was first proved by Welch (1939).
Looking at the problem from a more empirical angle,
we see that by a simple application of Bayes' Theorem, the conditional
probability that an observation x emanates from Hi is
01f1(g) (Anderson, 195°)f1(x) + ^2f2(x)
Now it is evident that if we assign a given observation x to that
population with the higher conditional probability, no other rule
iui smailtr misclassification probability. So if
<t> ifi(x) 4>zf 2 (x) „— > we choose IIi
0lfl(x) + 02f2(x) 0lfl(x) + 02f2(x)
If not, choose n2. Again the boundary of R1 is easily seen to reduce to
fs(x) 01
f 1 (x) 02
In the case where it is worse to misclassify members
of one population than the other, it is useful to introduce weights
ci and c 2, where
ci is the 'cost' of misclassifying an individual from nt as n2
c2 is the cost of the second type of misclsssification.
These costs may be in any units and might be introduced in circumstances
such as Example 3 , where the consequences of a wrong decision may
be drastically different viz. it is less dangerous to diagnose
presence of the disease in a healthy person, than to let the disease
go undetected in a sufferer.
We now have to minimise the expected cost 0iciai + 02C2a2
7
leading 'by similar analysis as before to the boundary of Ri being
fz(x) <£lCl
f 1 (x) ^202
We have thus the following likelihood ratio rule £, if
fi (x)> c allocate x to 311
/ f2(x)f 1 (*)
< c allocate x to n2 [1
= c allocate x to I11 with probability yf2(2£)
or to n2 with probability 1y
When f 1 (x) an(l Pz(x) are continuous, ^ f 1 (x) 1Prob. , = c = 0, and
^ f2(x) Jthe third case does not arise. If  fi(x)
Prob. ! = c } > 0, weL f2(x) J
usually take y = §■. In other words, if the new observation lies on
the discriminating boundary, we just spin a coin to decide which
population to assign it to.
Henceforth, we shallassume equal costs and equal prior
probabilities i.e. C1 = c2 and <£1 = $2 = so that c = 1.
(13) DISCRIMINATION WHEN THE TWO POPULATIONS ARE MULTIVARIATE NORMAL
We now assume Hi and PJ2 are N (Hi,£) and N (p2,S)P ~ P ~
respectively, where
£. is the vector of means in the i^*1 population (i = 1 ,2)
E is the (pxp) dispersion matrix of each population.
8
i.s. f. (x) = (2m) 12 J 2 exp 4(x ~ O x 1 (x  £. ) (i  1,2).L
Then log f 1 ^ 1 ,■ v ' V1 , % 1 /  ' „1 ,six  £i ) (x  £1) + o(x  £2) a (x  £f2(x)
I
= [*  i(£1 + £2)] 2 (£1  £2) = D (x)T
(by addition and subtraction of the same vector product and the property
that transpose (. $cc\ I &. r ) = (<x r )
Taking c=1 => log c =0 we see that, from [1], £ is given by
R1 is Dt(x)>0R2 is D.p(x)<0 J
[2]
and the boundary is x'2' (A£i  £2) = constant. h^(x) is usuallyknown as the Population Discriminant Function. Anderson (1958) credits
this approach to Wald(l944). Fisher(l936) arrived at the same result
by suggesting that an individual should be classified into one of the
two populations, using that linear function of its p measurements
which maximised the between group distance, relative to the within
group standard deviation. We no?/ give a matrix derivation of Fisher's
method.
Denote this linear function of x by a'x. According to
Fisher we want to maximise
la'Cifi  £2) !(a'2a)2
or equivalently, find the vector a which maximises1
y = [a'(pi  b2)(bi  b2)'a][a'Ea]
•Cowith respectra,
= 0 => 2(£i  £2)(£i  £2) a[a'2a]1
a«(£1  £2)(£i  £2) a[22a][a'2a] 2 = 0— 1 ' '
i.e. 2 (£1  ££2) (j£i  £2) a = a'(£i  £a)(£i  £2) a
a' 2a[31
The ojily values of a that satisfy this equation are the eigenvectors1 '
of 2 (y,  ZfsKifi ~ £2) £• The maximum of y is therefore the first
eigenvalue, with a its corresponding eigenvector.t
But Rank[ (£1  £2) (£1 ■ £2) ] = 11 1
So Rank[ 2 (£1  £e)(£i  £2) J = 1, being the product
of two matrices one of which has rank one. Therefore, there exists
only one nonzero eigenvalue.1 1
But, sum of eigenvalues = trace (2 56 )1 „
= ^ 2. > where 6 = £1  £2
Substituting this value for y in [3],
ZT'S 6'a = 5'2"16 a
1Therefore, a = mE 5 _____ [£]
6'a.where the scalar m = z—
S'2 5
As we are only attempting to separate EU and n2 (without measuring
the distance between them), m can be replaced by any ether scalar.
In other words, the coefficients of the linear discriminant function
are not unique, although their sizes are. Notice that the
values of y when a = 2 ^ (£1  £2) is substituted reduces to1 2
(£1  £2) 2 (£1  £2) = D ,
2where D is Mahalanobis's Generalised Distance, i.e. the distance
2between the means on the population discriminant function is D .
Comparing [4] with D (x) , we see that Fisher's linear
discriminant function differs only by the constant
(£1 + £2) 2" (jja ~ £2) from D (x).Thus the linear discriminant function is the best discriminator when
IIi and n2 are multivariate normal with the same dispersion matrix.
10
(1 if) THE SAMPLE DISCRIMINANT FUNCTION"
In practice, the values of £1 ,£2 and E (the parameters
of fi and f2) are rarely known; so they are usually replaced by xi,
X2 and S, the sample means and pooled sample variance/covariance
matrix, based on samples of size m from fli and n2 from n2.
Let X1 and X2 be the (mxp) and (n2xp) matrices of sample
observations from hi and n2, respectively. Define1 1 1 «
xi = — X1 e , x2 ~ X2 eni ~ni — n2 —n2
A = Xi'(l   E )X1 , A = X2'(I   E )X2Ai nt m A2 r\2 riz
and S = ^"(Av + A ) whereni+n22v Xi X2
e (i=1 ,2) is a column vector of 1's of length n..—n. 1
x
B (i=1,2) is a (n.xn ) matrix of 1's. From now on, the 'n. 'XI. X 1 X
x
subscripts will not be included, when the dimension of the vector
or matrix of 1 's is clear.
We knew that In the multivariate normal case
x. ~ N (fi.E) independently of A^ ~ W(2,n.1,p) (i=1,2)~~~i p in. a . 11 x
Since the samples from Hi and n2 are independent,
(m+n22)S ~ W(2,m+n22,p)
and, xi, X2 and S are unbiased estimators of £1, £2 and £. Substitution
of these estimates in D,p(x) yields the estimated likelihood ratio*
rule £ :
Dq(x) > 0 assign individual to I11If < S
D (x) < 0 assign individual to n2
where D (x) = [x  2"(xi+£2) ] S (xix2) is the Sample Discriminant
Function, sometimes known as Anderson's W Statistic.
With known population parameters we argued that the
[5]
procedure was optimal since we had minimised the expected loss.
Although the above procedure [5] cannot be justified in the same
way, and will inevitably introduce further misclassification errors,
Hoel and Peterson (1049) have shown that the substitution of these
estimators produces asymptotically optimal statistics.
It is worth noting that we would have obtained D (x),O
apart from a constant term, by maximising
ja'(x,  x2)T
(a'SaY
This justifies to some extent the use of the Sample Discriminant
Function when the populations are not normal with the same covariance
matrix. Another criterion (Likelihood Ratio) is discussed, in detail,
in PART II.
(1*5) DISTRIBUTIONS OF D (x) AND D (x), AND MISCLASSIFICATION PROBABILITIESO 1
Let E_^ denote expectation with respect to f^(x) (i=1,2) .
When x belongs to IIi , D (x) is univariate normally distributed with
mean
E.tBJD! = (£1 . £o2
' 1 .2= 1  £2) £ (Pi  £2) = 2^ , an(i variance
V£r [D^Cx)] = (£1 £2) 2~1 ££~1(£1 £2) = D2.p 2
Thus, if x belongs to Tin, D^,(x) ~ N(gD _,D )2 2
Similarly, if x belongs to n2, (x) ~ N(gD ,D ).
The notation we adopt for misclassification probabilities
is due to Hills(1966). He defines a (0 (i=1,2) to be the probability
of misallocating a randomly chosen member of ht when C is used.
12
l h en «,(£) = Prob. ( Dt(x) < 0 I x belongs to n, )
72~ exp
1
277"
1 i^2n2 ,^2(y2D ) dy
2exp gu du
= $(D/2) where $ is the cumulative normal distribution
Similarly a2(0 = $(D/2)
FIG. 2
In FIG. 2 , ^(0 and ^(0 are represented by shaded regions cutoff at the tails of the distributions.
The unconditional distribution of D (x) is extremely
complicated, as can be observed in the works of, amongst others,
Sitgreaves(1952) and Wald(l944). Okamoto(l9^3) has worked cut an
asymptotic expansion for the distribution, up to terms of order
(Vni)2, (1/112)2, (1/n m2), 1/m (m+n22) , l/n2 (ni fn22) , l/(m+H22)2.This will be considered briefly in the next chapter.
For large samples, it has been shown using limiting
13
distribution theory that, as m , 112 * *>
xi ■* £1, "* £2 and S » 2 , in probability.
Hence, the limiting distribution of Dq(x) is that of D (x), and forsufficiently large m , ri2 we can use the criterion as if the population
parameters were known.
However, if x belongs to n. (i=1 ,2) , D„(x) is conditionally1 O """ ""
(on xi,x2 and S) normally distributed and has mean
E.[D0(x)[xi,x2 arid S] = [p.  1+2] s \rxiX2~, and variance1
2
— ' —1 — _ —
Var[D (x)[x,,x2 and S] = [x, x2 ] S 2S [xix2].
Thus_1 _ _ _ ' _i _ _
a (£*)  f ^ (i1_X2) + 1"(X1+X2)S (xi X2)v V(xiX2) S 'ss \xixz)
It is best at this stage, to make a clear distinction between*
1) a(C ) j the conditional probability that a randomly*
chosen member of H. will be misallocated by £ ,
*
and 2) E[a (£ )] , the unconditional probability of misclassificatioA.
E denotes expectation over all possible samples of size m from ni
and n2 from n2 .
(1 *6) THE ESTIMATION OF ERROR RATES
The methods currently used to estimate the error rates
of sample discriminant functions can be divided into two categories;
in the first category, we have those assuming properties of normality
and in the second, those methods which employ sample data. The methods
dependent on normality are biased and if the degrees of freedom(n^+n 2)of the Wishart matrix S are small, may considerably underestimate the
true error rate. Their performance with nonnormal distributions
is difficult to judge and, in such cases, the empirical methods of
the second category are probably more suitable. The techniques are
summarised below :
The Holdout Method (or K Method)
If the original samples X^ and are very large, wemay select a subset of observations from each, compute a discriminant
function from them and use the remainder to estimate error rates.
The number of misallocations for Hi and 02 are binomially distributed,ir £ £
with probabilities ai(£ ) and oc2(r ). When the estimates of ai(£ )*
and a2(£ ) have been calculated, we now use the entire data set to
form the discriminant function. Since the numbers of misclassification
are binomially distributed we may also calculate confidence intervals*
/ *for a1(£ ) and a2(£ ).
The method suffers from the following disadvantages.
Firstly, large samples are not normally available in practice, for
various reasons such as cost. Secondly, the discriminant function
whose performance we have judged is not the one eventually used ;
there may be quite a difference between the two. Thirdly, different
users will obtain different estimates, since they will not select
the same subsets. Finally, we could have constructed the discriminant
function from a much smaller sample than that required for the
application of the H Method.
The Resubstitution Method (or R Method)
The method was first suggested by C.A.E.Smith (l%7).
Quite simply, the sample used to compute the discriminant function
is reused directly, to estimate the error rates. Each observation
15
is classified, ss either I11 or H2 and we calculate nn/ni and E2/112 ,
the proportions of ITi and n2 misclassified. Again,
mi ~ Bin. (m , a<(£ )) and m2 ~ Bin. (n2 , a2(^ )).
The method has. often been found to 3^ield misleading
results  even when sample sizes are fairly large, the performance
of the discriminant function is overoptimised. Obviously, seriousJ.
bias must arise when £ is judged on the sample it was calculated
from. The advantages of the method are the simplicity of the calculations
involved and the fact that we make no assumptions about the population
distributions.
ZEE ^ and fis Methods
We have already shown that, in the case of known
population parameters, a 1 (£) = a2(0 = ^("gb). Now since (x} was
obtained by substituting sample estimates in D (x)5 it appears
reasonable that replacement of D by its sample estimator f) will providej. wuV T*
us with another method of estimating «i(C ) and a2(£ ).2   ' 1  
8 = (X1X2) S (X1X2) is Mahalanobis' Generalised
Sample Distance, and our new estimator of both «i(£ ) and. a2(£ ) is
s,(c*) = az(c*) =
This method relies on the assumption of normality, and was first
proposed by Fisher(l936) ; it was known as the D method in Lachenbruch
and Mickey's(1 968) notation. For large sample sizes, it works fairly2 2
well (since 8 is consistent for D ), but for small n^,n^ bias becomesconsiderable.
^2 2We now show that D is biased for D . By definition,
if u ~ N (m,£) and B ~ W(2,f,p) , independently of u , thenp 
21 2T = fu'B u is Hotelling's T distribution, based on f degrees of
16
1 1freedom, with noncentrality parameter Xi = £ 2 p. Also it has
been proved that
f  2 t
—^ T has the noncentral Fdistribution F (p,fr>+1 , (>.1 }). [a]From Johnson and Kotz(p.190) , we know
Efy,v„v>>] = trfely1 <>2> wNow x. ~ IT (p. , 1/n. Z) (i=1,2)
~l p ~x 1
and +n22)s ~ W(Z,n1+n22,p)2 n1n2
So letting f = n,+n 2 , c  —— and taking, in the notation+n2
of the above definition,
B = fS ~ 7/(2,f,p)
and u = c(xixs) ~ N ( c(p1p2), 2) we see thatP
2,  x ' •1 ,  . 2^2 2c "(xix2) S (xi_2S2) = c fi has the 1 distribution, with f d.f.
' 1 2and Xi = (i£i ~i£2) 2 (P1P2) = D •
Therefore, from [a] and fb]
■r.r fp+I 2^2 , , 2 2 f» r ,iE[ f~ CD ] = (c D + p)  t+1fP " p(fpl)
i.e. E[D2] = (D2 + p/c2) ffp1
In practice therefore, the unbiased estimate~2 20 is usually used instead of fi where
ft2 fp1 2 ,2D = —— D  p/c .
This is the 6s Method. Lachenbruch and Mickey point out that in some
~2 2extreme cases 0 is negative and suggest that when this happens, D
f—D—1 2should be estimated by —— 0 . This estimator is obviously biased
2but less so than 5 ; it is also consistent.
17
The 0 Method
Okamoto's asymptotic expansion for the unconditional
probability of misclassification is given by
Prob. (D (x) < 0 [xclli) = + ar/ni + a2/n22 9
+a3/ni+nj2 + bii/m + b22/n2 + bi2/nui2 + bi 3/m (m +n22) +
2b23/n2(ni+n22) + b33/(m+n22) + 0Z ,
where the a's and b's are functions of the number of
variables p, and terms, where
dj = UD/2 (1=2,4,6,8)*
He also gives a similar expression for R[a2(£ )]. Lachenbruch and* *
Mickey postulate that we can estimate ai (£ ) and a2(£ ) by substituting
f) for D in the asymptotic expansion. This is the 0 Method and enables
us to obtain different estimates for ai(£ ) and a2(£ ), which was not
possible with the previous methods. Obviously, we can.improve thisA
method by using the unbiased estimator £  the OS Method.
Note that, even though Okamoto's expansions are for the* *
expected values of <*i(£ ) and a2(£ )5 the 0 and OS methods still provide£
(in practice) good estimates of a. (C ) (i=1,2).Lachenbruch's U Method (19&5)
Here we make use of all the observations, as in the
R Method, but develop estimators which are approximately unbiased.
The sample discriminant function is not estimated from all the
(ni+n2) observations, but from (m+112l) of them, and then used to
classify the sample vector omitted ; the process continues by missing
out all the ni+n2 observations, in turn.
18
Suppose x. is omitted from X. and let X, , x be the~J 1 !(j)
remaining (/yi)*J\ matrix. Then, using the same notation as in (1*4)> I
A = X.,.(l vE)X , ..1(i) ni_l 1(j)(j)1 nl  _ «
= X, (I E)X r(x.x )(x.x )1 n, 1 n, 1 —1 —.i —i
n,
= Ax, r u.u. where u. = x.x_^ n^ 1 jj j ~j 1
Hence, the pooled sample covariance matrix omitting x. is given by
(n,+n 3)S, . = A + A 12 yj X1 X2 n^. u .u.1 JJ
"l+n2~2 .
, "l1 (j) ~ VV3 ~ (nr1)(n1+n23) jj
f1 (nrl)(fl) jj *
We now need to calculate S^\ . Since each application of the U Methodwould require n^+n2 such calculations, a technique has to be introducedto reduce the number of these matrix inversions. An identity due to
Bartlett (1 95"!) enables us to avoid a lot of timeconsuming computations.
Bartlett showed that if A and B are square ncnsinguiar matrices and
u and v are column vectors such that
»
B = A + uv ,
then B = A ^  [A ''u.v A ^/(1+v A ^u) ]1 • 1
c S u.u. S1 f1 r„ — 1 J*jL
S(j) = f [S + 1  c u.'s"1u. ]1 J J
where ci 1 (^7?The U Method now requires only one specific matrix inversion that
of S. We now have to adjust the difference of sample means,
d = x, xo . to allow for the omission of x.— ~
i ~Z — i ifrom X, . Let the
mean vector of the remaining n^~A> observations be Xj ^ ^ • r"hen~1 (j) ~ n11 Xl(j) ~
ni  1
n^1 1 1
and so d/.\ = x. x„~{j) ~HjJ ~2
1d r u , v.Tith a little algebra.
n1 1 j
Also xx./.v + x = x. + x  t u. , and by substitution in1(j) 2 1 2 n.,1 j
D (x) , we arrive at D.(x), the sample discriminant function basedS J
on the n^+n^1 vectors and
D.(x) = [£«s,0) +i2)]'s(:1)a(.)f—1 r 1 / — — 1 \ 1 '
= — U j(x1+x2̂.)]X,1 ' 1
c, S u.u . S
x[s 1 + — V—j— ] [ £,  x  r~f(xx  £ )]1  c.u. s u. 1
1J "J
Now, as D.(x) was constructed from the sample dataJ
excluding x. , it may be used to classify x., which we already knov,rJ J
comes from Hi . Thus x. will be ccrrectlv allocated to I7i if
D.(x.) >0 and misclassified if D.(x.) < 0 , 'whereJ J JJ
— — 1 1 — 1D.(x.) = [x. ~ i(x, + io T h.) J "S/ dx .v
J J J 1 2 n.j1 j (j) —CJ )This process is repeated for all objects from IIi i.e. we calculate
D.(x.) for all j=1,....,n , and the proportion misallocated isJ J
20
noted bo be ni^/n^ , say. Then, rnbj/r^ is an unbiased estimate of*
ai(b. _1 ) > where the rule £ . is based on samples of size•»r1>n2 ' ❖
n.1 from n 1 and n from n2 . If n. is not small, ai(£ . ) is' n. ~ ' 5 n
* ij' '
almost the same as «i(£ ) , in which £ is constructed from samples
of size n^ and n2 . By a similar process of omitting columns of ,
we arrive at the estimate mj/n^ . This is the U Method.a.v
Define the random variable £ . byn.,1,n2
*
£ . =1 if x. is misclassifiedbni1,n2 j
=0 if x. is classified correctly."J
ni
Then m.'/n. = — } C * / \11 ni / • ni~1 'no^Jj=i J
"0W> E[Vl,„2%>J = (£n11 ,n2' = A) (0=1, n,)/ ^
and therefore, mj/n^ is approximately unbiased for ai(£ ).nl
2„ \ n *
Var. ml/n. = 1/n Var.) £ 1 „ (x.)11 1 / . n.1 ,n jU 1 2
ni A [ £ Var S1,n2%) + YZ C"(S1'"2(Sl)'S1'"S
j =1 i/j
4. A 4.
Var. £ 1 _ (O = «i(£ )[1  «i(£ )]1 9 2
* ^n 1 n 1 n f a^(^)[1 " «i (C*) ] x P1 2 1 2
*
Cov
where p is the correlation between £ . (x.) and £ . (x.)1,n2 —a Vj1,n2vj
2"!
iates
In a sampling experiment Lacnenbruch found p to be very small (<0*Cl),
in must cases, and so treatment of mjj/n^ and ml/n2 as Bernoulli variiis justified, in the calculation of confidence intervals and the standard
error.
The TJ Method
Lachenbruch and Mickey have suggested using §(D./S )I \j.'1
£__ #
as an estimate of ) , and 1(D^/S^ ) as an estimate of «2(£ ), where
"l\—'
B1 = 1/n1 \ Dj(2j) x,J=1
n2
D2 = l/n2 V h(£o' if X2J=1
2S = sample variance of the quantities D.(x.), X£
J J J '
2and similarly for S .
2
' —1Now, D^(x) = [x  §(Afi +££2) ] 2 (P1M2) and, from (1*5),
we see that «(C) = §( —§T>) can be written
(l)k E[Dt(x)] n r k=1 if xe n,§ "
Vvar. (D.^(x)) ' k k=2 if xc" n2
By considering D.(x.), (j=1 , j^), as n1 observations on DT(x),>1
J "J
we observe that their sample mean may be taken as an estimate o
2
Hence §(D./S ) can be used to estimate a 1 (f ). Similarly a2(£ )1 U1will be estimated by §(D,/S ). This is known as the U Method.
2
E[Dm(x)1 , and their sample variance as an estimate of Var. fD^(x)],
22
The Relative Merits of the above methods
Based on a series of Monte Carlo experiments,Lachenbruch
and Mickey reached the following conclusions about the above methods.
First, the D and R Methods, which have been the most commonly used in
the past, give relatively poor results compared with the other methods.
Second, in general, the 0 Method provides cuite good estimates of cti(£ )# t t st
and a2(£ ). Third, the^three estimators, in order of perfoimance,
seem to be the 0S,U and U Methods. If normality can be assumed, the
OS and U Methods work very well ; if it cannot be assumed, the U Method
should be employed. Yrhen either f) or n^ or n^ is small, due todifficulties with Okamoto's expansion, the U and U Methods should be
used in preference to the 0 Method.
(17) THE ANALOGY OF DISCRIMINANT ANALYSIS YITH REGRESSION
Fisher(1936) showed that the coefficients of a regression
analysis using a dummy variate X , where
X = I
nr
n1 +n2n.
n1+n2
X1 if x f lit
X2 if x € n2 , were proportional
to the coefficients of the sample discriminant function. By writing
our observation matrix in the partitioned form
X =1
LX2J
n
, it can be shown that the overall sum of
squares(ssq) about the mean is
T = X (I  E )X* vi _Lr> '
n1 +n2 n.j +n2
23
"^Y + ^YX, a„ n.nn
n1no _ _
+ (x,  x )(x.  X )2 n1 ^2 V1 2/v1 —2
2 '= Sy/ + c
Since S... = fS , our vector of discriminant coefficients isV*
I = 1But the vector of regression coefficients, when X =
X1
X?n2
is
b = (Sw + c2d.d') 1X*(I  —— E)X W ' K nl+n?
r„ 1 ^SW C •* ^ t .2,~~
■ ^V" ~ 2 1 1 C —1 + c d Sp d
°V1*1 + c2fi2/f
(using Bartlett's matrix
inversion"!
Hence the regression and discriminant coefficients are equivalent2
except for the multiplicative constant fc ; this matrix2ft2f + c D
proof was first given by Healy(l9£>5) , but he seems to be mistaken
when (in his notation) he says the discriminant coefficients are
1given by Sn d . Cramer(l9^7) reduced Healy's calculations considerably.
The mean of the dummy variables is zero and their sum of2 2
squares isn1 n2
(n1+n2)'n2ni
= c
Also, the regression ssq = bed
2 ' 1C ^ 2
55 x c di+c r/f
">/
c2n2 c2 xx
f+c¥So we can form the following ANOVA table
Source d.f. ssq
222 C
Regression ssq p c x • 22f+c D
2 **Residual ssq n^+n p1 c x
f+c D
2Total ssq +n 1 c
It should be stressed that we do not have the standard
.regression analysis situation since the dependent variable \ is not
normal but a pseudovariable taking only two values, and the independent
variables are not fixed but normally distributed. Nevertheless
2/s2Regression sso/d.f.
_ £lE±l x £_JL „ f . /n2uResidual ssq/d.f. p f ^ ^ '
2 . ,If D =0 this distribution reduces to F(p,fp+1) and hence we can
apply an Ftest of significance to the regression coefficients, to
test whether the population means are equal. It is also theoreticallyfor
possible to develop significance tests^a subset of the discriminant
coefficients (say pk of them), from regression theory. The validity
of these tests will be demonstrated (in the next chapter), when we
see that they are equivalent to tests developed empirically. Also from
the ANOVA table, the square of the multiple correlation coefficient22
o T>  CD2_ Regression ssqR =
Total ssq " f+c2fi2
2o
Hence we'have the relationship, pointed out "by Fisher(l938) ,
fR22~2
T  c 1) = '
1R2
We also know that in the true regression situation, the variance
covariance matrix of the regression coefficients would be
2 1 1 2 2V(b) = (Syr + c d.d ) cr , where cr is V'ar. (X)
c2f^2 Residual ssq
ana o" = ~~d,f* " (fp+i) (f+c2f)p2)
test
We could^the significance of a single regression•coefficient b. usingi
b.t = — 1 — with fp+1 d.f.
"/Estimate of V(b.)x
Das &upta(l968) worked out the actual variancecovariance
matrix of the discriminant coefficients to be
f22 1 f_1 1 f~P+1 1 1
V(a) [D 2 + —xZ + 2 5.52 ](fp)(fp1)(fp3) P c fp1
—1 —1He also proved that as n^ ,n^ * 00 , Vf(fS d  2 5) is asymptoticallynormally distributed with variancecovariance matrix
1 ' 1 2 1 12 8.5 2 + D 2 + 2
Notice that we may choose any values such as
X =
r 1 if x in n t
i. 0 if x in n2 for the dummy variable,
and we will still achieve the proportionality of regression and discriminant
coefficients. It should be emphasised that whereas regression coefficients
are unique, only the ratios of discriminant coefficients are unique.
26
( S 8) HYPOTHESIS TESTING IN DISCEIMINANT ANALYSIS
For this section we need to introduce the following
partitioned vectors and matrices
d =
x(l) kft
5(1) k
x77)> _ 
pk 8(2) pk
d( 1 )" k
, 2 =E11 E12
d(2) pk E21 E22
1, a = 2 '5 =
a(l)
a(2)
k
pk
pk
S =S11 S12 k
S21 S22 pk
i. E11 2 ~E11 E12E2211 1
v v y22 "21 112 E221
where, 2,. = 2,,  2 21.
J11 2 11 12 22 21
J221 = S22 " Z21S11 S12 ' (See Morrison,p.68}
Now, from Kshirsagar(p.21),
1 1(I + LK) = I  L(l + ML) M  [6]
Thus '112 ~ E11 + E11 212E221221S11
= 2^ 1 + p 2,22.^/3 , where p = 2.,J21 11
Also 222 2^2^ >2 7 2. f I  2 ^2 £ S ) ^2 ^_ 222 i21U ^ 12 22 2V 11
 2 1Z 2 "1~
221 21 11
So we can write
,1 Z11 "1 +",Z22>' 1
V1~S22P
1
E221
2Then if D is the population generalised distance based on ail p variable
r'2
and P is the population generalised distance based, on the first kK
variables x(l),
8(1)2
PF  6(1)
1 • 1
E11 V222"Z22> Z 1
221
"6(1)'6(2)
= [6(2)  /56(1)!,222:J[6(2) /?6(1)]Similarly for sample distances, with analogous notation
Cp =  [d(2) ^d(l)]'s22;J[d(2)   
Thus an estimate of the contribution of the last pk variables x(2)
to the distance between FW and U2 is
[7]
1,[d(2)  /?d(l)j S ^ [d(2)  /3d(1 ) ]22
This sample estimate was given by Kshirsagar(p.200) and the above
derivation is a summary of work from several chapters of the same
book. Notice that by suitable ordering and partitioning of x , we
can find the contribution of any single variable or combination of
variables to the distance.
There are several hypotheses of interest in Discriminant
Analysis. The first is whether there is a significant difference
between the means of Hi and n2 . The null hypothesis is
Hj : if1=£2 > which is equivalent toH2 : Dp=° *
2 2 2Since T = c f) , we know that in the null case
P 22. c D
—x P ~ F(p,fp+l) , which can be tested for
28
significance in tables.
Secondly, we might ask the question whether the addition
of more variables to our discriminant function would significantly
increase its ability to separate n1 and Il2 . Our null hypothesis
here is
H3 • d,. . —Q.. —• • •
k+1 k+2..=a =0
P
i.e. H3 : a(2) = 0
Buta(l)
' 1"E221 ~5(1)"
a(2)_ ~S221/9
A
V 1221
_
6(2)
Therefore, a(2) = 0 => S22 ^(5(2) /SS(D) 0
i.e. 6(2) = ^6(1)
From [7], it is easy to see that a(2)=£ > ^ 2D = D,
P kSo the hypothesis is equivalent to
2 2H, : D = D.4 p k
Rao(l965, p.482) gives the test for this hypothesis as
= F(pk,fp+l) frEtl x °2(6'  $pk „ 22 [8]
f + c D,
If this varianceratio is not significant, the variables x(l) are
sufficient for discrimination between n1 and n2 . Obviously, by!
putting x(2)=x^ , x(1)= [x^ ,. .. ,x_. ^ »xp+j»* • ,Xp an<^ k=p1 in [8],we can test to see if omission of x^ from the analysis significantlydecreases the effectiveness of the discriminant function. Incidentally,
Rao first gave this test in 1946 (see the list of references).
Thirdly, certain variates (known as concomitant or ancillary
variables) possibly have the same mean in both populations, thereby
possessing no discriminating power on their own. However, if they
are correlated with variables which do differ with respect to I11 and n2 ,
their inclusion may actually improve the discriminant function. Cochran
and Bliss(1 948) have pointed out that the analysis of concomitant
variables is analogous to Covariance Analysis. The null hypothesis of
interest when ancillary variates are present is
: 6(2) =0 given that 6_(1) = 02 2
or : D =0 given that D, = 0bp. k
2 2= D . = 0 given that D, = 0
pk _ k
Since implies H. under the condition that 6(.1) = 0b 4 _
the Ftest [8] is often used to test or IIg as well as or .
However, Rao(l949) suggested a test based on
v = 1(6I t I).V
The statistic actually used is W = — , with density
f(w) = [r(f+2kp+i)r(£±l) / r(£^±l)r(k/2)r(£±Ebl)]k_1 fP+1 1
X w2 (1  w) 2
x F ( f'P+k+1 . f+k+1 w )2 1 2 3 2 9 2 9
where ^F^ is hypergeometric function of the second kind. Raopostulated that V was a better statistic than F[8] for testing ,
/s2since the variance of estimates of D he obtained based on F[8"l and V
P^2
given that D, = 0 , were smaller in the latter case. He also suggestedK
an approximate varianceratio
f_p+1r f_k+1 ° (Du " V n
p_—[ ~f7T x t ] = Fpk,fP+i *
Fourthly, we want to test whether the linear discriminant
function we is sufficient to discriminate between Hi and n2 .
Our null hypothesis is!
: a given function y=h x is good enough to discriminate
between IIt and n2 .
~2Tc test this hypothesis, we apply the Ftest [8] with replaced by
 — 2 '2" y2) (h d)
J) = = ;
Estimated Var. (y) h Sh
Thus if the varianceratio
2,^2 ~2.r_p+1 C ~ 1^
— x no— is significant, the linearP1 f + c
t
discriminant h x is not the best discriminator.
Lastly, since discriminant function coefficients are
not unique, we cannot test an hypothesis of the form Pp~k * However,their ratios are unique and thus we can construct a null hypothesis
of the typea.
Hp : — = k .o a.
n2 ^If D .is the generalised sample distance based on all variables
p1
except xj \ > we can if k is the true value of the ratio usingX."
^ (jut + t<"X^
31
(fp+1) X
2 / ^2 ^2 \
° <dp  °Pi)
pi
i,fp+i
This test is again due to Rao(l96p).
(19) SIZE AND SHAPE FACTORS
We conclude PART I , with a brief look at the special<na.tr i\
case, first considered by Penrose(1947) , when the dispersion*has the
form
2 . =
1 P.. .
P 1 P. •
P..
P
P
PP 1
= (1  p)l + e.<
Bartlett(1 951) showed that
1e.e
1p 1p 1+P(p1) , and consequently
' —1 ' *(Pi  P2) Z X oC[ —7  £ ]x + [ p1 ] £ x
e_ 6 1 +p(p1)
Penrose called the two sets of coefficients, h and £ , the Shape
and Size Factors, respectively. These factors are uncorrelated and
hence independently distributed. It can be shown that the discriminant
function can be expressed in the form
—2 h x + —2 g x [9]
where 5^ is the difference in the means of the shape factor innt and n2 ,
and o; is the variance of the shape component. 6 and c~ areh g g
defined similarly. According to Bartlett, Penrose has shown that the
discriminant function [9] gives good results even when S is not exactly
of the required form. As such [9] has been successfully applied to
the analysis of biological organs, where size and shape are particularly
relevant.
3?.
PART II
(21) THE POPULATION DISCRIMINANT FUNCTION FOR PAIRED OBSERVATIONS
AND ITS DISTRIBUTION
Suppose our overall population n can be split into
Males(M) and Females(F), with equal prior probabilities of an individual
coming from either and equal costs of misclassification. 7/e introduce
the restriction that a new pair of observations (z^,z^) must be ofdifferent sexes i.e. only two cases can occur :
1) z, is Male, z„ is Female !~z,,<fM,z eF]' \ '2 12
2) z^ is Female, z^ is Male [z^eF,z^eM]It Is also assumed that z^,z^ are independent. No?/, if in the univariate
2 2case Males ~ N(hi,cr ) and Females ~ N(^2,cr ),
Prob. (z^eMl there exist two pops. M and F , &/>JL no oi z*)1 (
exp —~2 (Z1 ~ Pi)=
12 1 2exp —p (zi""^0 + exp —r (z(/i2)
2cr 2cr
with similar formulae for the conditional probabilities that z^eF,
z^eMjZ^eF . Thus, using Bayes Theorem,
Prob. (z.j eM,Z2eF [ either case 1) or 2) must hold)
3 A
1 U \2 f ,2,exp r !(z,pi) +(z p2) ]
2a"
exp [(z Pi)%(z p2)2] + exp ~ [(?..p2) + (s0l'1)2'2a" * 2o *
1
1 + exp  ~2(Z2Z1 )a
We will assign z.j>M,z2>F ^ >
i.e. if exp  ~2 (z2~z1)(^2~Pi) < 1cr
i.e. if (z1z2)(pip2) > 0
Thus if we are given §=p1P2 > 0 , we have a very simple assignment
procedure :
r If z, > z„ , allocate z .»M,z »FC j 2 2 [10]
If z, < z0 , allocate z.>F,z~>M12' 1 ' 2
Similarly, in the multivariate case, when Males ~ N^(pi,S), andFemales ~N^(£2,l) and we have a new pair of vector observations (z,j ., z,2.) »
Prob. .fM,z_2>eF one is Male, the other is Female)
f(£1l £1) x f C^21 ££2)• f (z^ 1^1 )x f(z2p2) + fCzJps) x f(z2pO
1 1 1 f= 1 / (1 + exp gtr. [2 (^£2) (^.£2) + 2 (z.2£i ) (z.2""E1 )
~2 1 (£] ~E1 ) (±.5 e0  2 1 (z2"E2 ) ( z2~E2 ) ^ )
= 1 / (1 + exp (^£2) 2 ' (E2E1 ) ) = P >
the obvious generalisation of the onedimensional case,
35
Now let _z,j ~x , and ,z_^ have vector of midpoints X i.e. z?=2XxThen, xcM , and the other member of the pair eF , if p > r? ,
_ ' _ 1 V*i.e. if (x  X) 2 (p,  £2) > 0     [11]
Hence we seethat as the discriminating boundary for pairs,
^PXT(—) ~ ~ 2 1 ~~2^ = 0 ' differs from Dr,,(x)
only by a constant, D  (x) represents a A in pdimensional space,I AI
parallel to D^Cx), and passing through the midpoint of the pair.Thus for a new pair (z^ ,z_0) , we first calculate the midpoint
vector X = (z,+.z9)/2 and then D  (x) , where x is either z_ or z .I C. * i.A.i i rd
The assignment procedure £ is as follows :
^ If D  (x) > 0, assign x to M, and the other observation to Fr Ai
If D—^x) < 0, assign x to F, and the other observation to Iff.rAl
The twodimensional situation is represented in the diagram below.
Next we consider the distribution of D  (x). When xcM ,Jta l
x  X ~ N ( (D1P2)/2 , S/2) , so that_ _
p _ 
E[DpxT (—) ] = ~~z) S" '(H1£2)/2 = 152/2Also Var.[DpT(x) ] = (£i£z) 2~1SS 1(£t£a) = D2/2
Thus if xeM , Dpp(x) ~N(D2/2,DZ/2)Similarly if xeF , DpT(x) ~ N(D2/2,dV2)
(2*2) THE SAMPLE DISCRIMINANT FUNCTION FOR PAIRED OBSERVATIONS
AND ITS DISTRIBUTION
We shall discuss two criteria for dealing with the situation
when the parameters of the Male and Female distributions are unknown.
The first is analogous to the criterion of (1*4), and the second is
the likelihood ratio criterion. This chapter is only concerned with
the first of these.
Taking samples of size m from the Male population and n from
the Females, we calculate Xj > —p an<^ samP^e nieans and pooledsample dispersion matrix, as defined in PART I. Then substitution in
D^YTiCi) giyes us (x), the sample discriminant function for pairs,rAl x.Ao
where
dpxs(~) = ~ 2.)  E?) .
The unconditional distribution of D  (x) is not normal. However,x ao
if , we see that from the independence of S and (x^  x ),
E[Dp^s(x)]  E[x  X]'e[S~1 jEt^  £2]
37
Now, Das Gupta(l968) has shown that if S ~ V<r(2,n,p) ,
E[S_1] = ~~r S"1np1
= J5±Sz^ b2/2m+npp
= m+n~2 E[D  (x) ]ra+np3 L PXTV/J
Hence, even under normality, the coefficients of x in D  (x) are1 AO
biased for the coefficients of x in D_^^,(x). But, we have seen inX" Jv_L
PAKF I that discriminant functions, in general, are unaffected by
scale changes, and so the factor m+n is of no importance.m+np3
If x^M , D™ (x) is conditionally normally distributedIT.A.O
with mean
E[Bps(x) ] =!(£,  £2) S"1^  x2)and variance
Var. [Ep^g(i) ] =^(i1  i2) SV1^ ~ i2)where the expectations are conditional on x.] > —2 an<^ canfind similar expressions when xfF . Our estimated allocation rule is
* r If D_r: (x) > 0, assign x>M , other individual >FC j FXS
If Dp^g(x) < 0? assign x>F , other individual >M .
Notice that since D  (x) is parallel to the sampleIAO
discriminant function for classifying single observations, the regression
analogy (and consequently all the work of (1*7)) still holds for pairs.
We may also apply the hypothesis tests of (1 8) to the coefficients
of Dp^ (x) , and the theor3r of size and shape factors can be easilyIAD
38
modified. Replace x "by in [9] to get
2ex cr
This simplified discriminant function may have particular relevance
to the sexing of pairs when we discriminate on the hasis of certain
organs e.g. wing, beak. Since the above function reduces to the sample
discriminant function in the bivariate case, we were not able to apply
it to to the fulmar data.
(23) THE PROBABILITY OF MISCLASSIFICATION
Define to be the probability of misallocating
a randomly chosen pair, eM , , using £ . Letting x=z< , wes60 "fchsi/fc
= f(D//2)
Similarly, a2(£) = §(D/"/2)*
We also define ai(£ ) to be the conditional probability of misallocating. *
the pair (z^ =xeM , z^F) when £ is used. Then
°i(c) = Prob. (Dp^T(2S) < °l 2£eM and
«,(C') = Prob. (Pp^gCx) < °f 2£eM^2eF'1'2 and S'x ' 1 ,
*
Similarly for a2(£ ) .
It is suggested that ai(£ ) may be estimated "by most of the methods
outlined in (1 *6) . Okamoto's expansion will have to be adapted but
the 6 method, for instance, will yield the estimate
«,(£*) = §(fi//2)
We now describe the modified U Method in more detail. Having omitted
one Male and one Female sample vector, we classify the pair they form
using the paired sample discriminant function estimated from the
remaining m+n2 observations. The procedure^by successive omission
of each possible pair of Male and Female from the sample matrices
X1 (Males) and X^Females) .
It can easily be seen that the pooled sample covariance
matrix when x. is omitted from X, and x. from X„ , is given byx 1 j 2
_ m+n2 „ m ' n u ,Qu(ij) m+n4 (m1 ) (fi.) —i1—i1 (nl)(fl) J
where u.. = x.x , u = x.x~x1 x 1 jA j ~2
I mNow if
(i) ' = Jl2 S ~ (m1 ) (fX)m
iixi }
S(ij) S(i)' (ni)(ft) "J2~j2
Hence, using Bartlett's matrix inversion
S/ "J _ s "I + °2S(ibai2i2SfiVs(u)  s(i)' M . c u's,f 2~j2 (i) j2
where
Z,0
5(i) ^[s1 ci3 iiiis C u 3 u.,1—1.1 x1
m, n
c„ — / . \ r4 ana c — / , \ _ a1 (m1)f 2 (ni)f
Thus in the paired case, we still only require one specific matrix
inversion  that of S. If d,.. N is the difference between the sample~Uj)means of the remaining m+n2 observations, it can be shown that
u.u ~
d, , = a  tVxj/ — m1 n1
The paired sample discriminant function computed without x. and x. isJ
1
h/xW Cil iz) 8(J)i(ij) .
which can be legitimately used to classify the pair (x.,x.) • Thus,J
we see that the pair (x.,x.) will be misclassified ifJ
f 1D. . (x.,x.) = (x. x.) S/..N d/..x < 0
_ j) (ij)ij "x ~J
The process is repeated for all the mn possible pairs from M and F ,
and the proportion misallocated is recorded as 6.£
Defining C . ,,x ) bym1 .n1 x ,1
rA (x.,x.m1,n1 i j
"J
r = 1 if (x.,x.) is misallocated1 J
^ 0 otherwise ,
_m pi
e E)M
mxn
, * V
Thus 6 is exactly unbiased for a, (£ A ) and approximatelyJ m1,n1 ^ Junbiased for at) , where the notation is the same as in (l'6) .
The method is <xn /*«.«t" upc.\ the U Method for single observation*
since we average over inn £ values as opposed to m+n in the standard
method. Consequently, the modified U Method should provide very good
estimates of paired misclassification probabilities.
Notice that since D/V2 < D/.2 , §(D/vr2) < §(D/ 2) ,
the probability of misclassification 1'or adding a single observation
i.e. we are less likely to make a misclassification error when we
allocate a pair of observations, under the conditions of (2*1), than
than when a single observation is sexed. Thus if two individuals are
known to be a pair, the extra information imparted by this knowledge
leads to a distinct improvement in their chances of being classified
correctly. . , , ,• ,it/itk fteyaa LThe univariate case^is now considered in more detail
the assignment procedure £ has already been given in [10]. Suppose
z1 ~ N(pi ,<x2) >2  independently ,
z2 ~ N(p2,cr ) J2
then z  z^ ~ N(pip2 ,2cr ) ,
and a,(£) = Prob.(z z <o z^eM , z eF) = §(5/72),
where 5=pip2/cr . We now take samples (x^ ,... ,x^ ) from M and\ ^
(x10,... 5xno) f>ron an<3 estimate £ by £ which says
• if (x.x )(z z ) > 0 assign z„>M , z >F_1 _2 1 2 1 2 [12]t if (xrx2)(zrz2) < 0 , assign z^F , z^M .
r Prob. (z >z ) if x <xr«i(0 = . J 12 _1 _2I Prob. (z1<z2) if x1>x2
1  $(^2Pi/cr/2) if x <x
1  $(p1 P2/0V2) if x1 >X,
12 r i[a J
— 2 — 2Now x.. ~ N(^i,cr /n) independently of x2 ~ N(/i2,o" /n) .
—  2So x]X2 ~ ~^2 >2°" /n) « and averaging over all possible
samples (x.^ ,.. . 'xn) from M and (x , ... >xn2^ from P, we obtain
p[ai(£ )] = $(6/vr2) §(6Vn/vr2) + §(6//2)$(8Vn/V2)
= the unconditional probability of misallocation
If we now substitute the estimates x^ , x2 and s for p'1 , Ne and crin [a], we get
°i(s*) =
VxisV"2
r1 _ §( 21 ) < x2
x x
1  1 ) x1 > x2s/2
= 1  §( [81/V2)n""i
whe re 6 =irs2 2 ) + KrV2
, s =
2(nl)
To calculate Efai(^ )] , let cr  A . Then*
VX2 „ „ V*2E[a,(C )]  Prob. ( ~f2~ < 0 and X < )
x.x X.X
+ Prob. ( J2 >0 and x> )2 ^ '
43
where X'~ N(0,1), independently of x,,x and the expectation is everI eL
all possible samples size n from I,'and F .
* X1"X? , X"X'xTherefore, E[ai(£ )] = Prob. ( —7^— < 0 ) + Prcb. ( X + —77, ~~ < 0)
_ 2xProb. ( V!i < 0 and _x + Vf2 < 0}
But ^ „ N( Pi^ ^ l/n }
o ^ V X1 "X2 ^ TcrC Pi Pa „ 1 \ru ^ + y*2 ^ v~2 ' ' + n
Therefore, E[Si(£ )] = §(5vn/vr2) + 5(5/n/V2(n+1 )) 2$(5vrn/vr2 , ~6Vn/V2(n+1) ; p)
where $(a,b;p) is the bivariate cumulative normal distribution with
correlation p .
Now E[(X + ^)(^)] = iE[(5rx2)2]1i 2
= — + 'g'Cpipa) , and after a little
algebra it is easily seen that
p = 1/Vn+1
The table below shows
1) The true probability of misclassifieation using C, °i(t)
2) The unconditional prob. of misclassification using £ , E[ai(£ )]
3) The expected value of the estimated prob. of misclassification,
E^CC*)!.Assuming M ~ N(pi,1) , F ~ N(p2,1) , l)>2) and 3) have been calculated
a
for various values of SpijU2 and n, the common sample size. The figures
for E[ai(£ )] and E[Si(£ )] were obtained by interpolation in tables
of the univariate normal and cumulative bivariate normal distributions
(National Bureau of Standards, 1959).
n 5 10 15 20 30 50 100 200
6=0 *5
<*i(0 •3618 •3618 •3618 •3618 •3618 •3618 •3618 •3618
E[«i(C )] •4211 •3982 •3854 •3775 •3691 •3622 ■3619 •3618
E[Si(c')] •3323 •3517 •3622 •3650 •3595 •3608 •3625 •3622
6=1 0
«i(0 •2399 •2399 •2399 •2399 •2399 •2399 •2399 •2399
E[«I(C )] •2695 •2465 T—4"C\J •2404 •2400 •2399 •2399 •2399
E[ai(C )] •2543 •2480 •2462 •2453 •2436 •2420 •2409 •2404
6=1 "5
«1 (c)
E[«i(£*)]E[Si(OJ
•1444 •1444 •1444 •1444 •1144 •1444 •144+ •144
•1507 •1447 •1444 • 1444 •1444 •1444 •U44 ■ 1444
•1666 •1553 •1522 •1503 •1485 •1469 •1457 • 1450
A similar table giving misclassification probabilities
for the addition of a single observation can be found in Hills(l966).
By comparison of the two tables, it is clear that the true, unconditional
and estimated misallocation probabilities, for the same sample size
and distance between the populations, are always smaller when we assign
paired observations. There are two further points of interest. From$
the above table, it seems that the inequality «i(C) < E(ai(£ )],
/> 5
hypothesised by Hills for single observations, still holds for pairs.*
/s *Secondly, the approximation )«i(C)]=0, which was good
in Hills' paper, no longer seems appropriate for pairs, even when
sample sizes are large. This might indicate that the D Method does
not yield very good estimates of paired misclassification probabilities.
Further research needs to be carried out into the relative merits of
all the estimation techniques when they are applied to pairs ; sampling
experiments should be performed on the lines of Lachenbruch and Micke\r's
(1968). ^ *
Incidentally, it appears that Hills' formula for E_Qi(£ )J
is incorrect and should be, in his notation,
E[Si(£*)] = G(a) + G(b) '2G(a,b;p ) .
It was then thought that an improvement in the magnitude
of the misclassification probabilitiesAif the allocated pair (z^,z9)were used in the formation of a new discriminant rule , where
 if (z z) ) (x x ))o assign z »M , z >F=> 
if (z^z^) (x^ x^)co assign Zy>F , z^>M
where x^ is the new estimate of Pi from the n+1 sample values(x. x „) together with either z. or z„ . Obviously,v 11 ' ' n1 1 2
Bl(0 = 1  §(8//2) , still.
r if x'< x*„ fr*\ f 11 X1 2
L 1  *("57T) ■ X1> X2
But Prob. (x < x ) = Prob. (ri^+z^ rn^+z^O^ zjlx^ xjjz^ > 593)
+ Prob. (mq+2^ nx2+zj^\[z^< zjlx^ x^z^ > zo0x1 > x^)
£6
= Prob. (z„> z/lx < x„) i Prob. (z < z fix > x )i £ i 2 i 2 i 2
= Prob. (x,< x2)3{l r
Thus E[ai(£2)] ~ )] 5 an(t there vri.ll be no gain in informationif the unknown pair (z^,z ) is included with the original sample data,to form a new allocation procedure for the sexing of subsequent unknown
pairs.
(24) THE LIKELIHOOD RATIO CRITERION EOR THE CLASSIFICATION OF ONE
NEW PAIR (UNIVARIATE)
In this chapter, we consider the application of the
likelihood ratio criterion in the formation of population and sample
discriminant functions for classifying paired observations. Firstly,
in the univariate normal case with known parameters,1 1 ,2
the Male p.d.f. is ^(x) = V2ircr ex^ ~ —22cr
1 12and the Female p.d.f. is f„(x) = fn exp  — (x^z)F
2a
The likelihood of the new pair (z^,z^) being (M,F) is
2 1 2 2f/2rrcr x exp  —^ [ (z Pi) + (z^Pz) ] ,
2o
and the likelihood of it being (F,M) is
2 1 2 21/27TCT X exp  —2 [ (z,^2) + (ZpUi) ] .
2o"
But, from PART I , we know that the discriminating boundary is given
by the likelihood ratio (L.R.) = 1 and
12 22 22 22L.R. = exp—p[z 2z +z/"2z p2+p2 ~z +2z./j2/j2 z
2a ^ ^ ii 2+2z ]
log(L.R.) = (z1~z2) (pip2)/cr2 .
Thus our discriminating boundary is (z^z )(hi^2) = 0 which leadsto the same rule, £, as [10] . £ will now be referred to as the L.R.
❖
rule. Obviously, £ is the estimated L.R. rule.
When the population parameters are unknown, we take samples
Cj 1 j • • •11 mi
2xn'* >x^ ~K(pi,o'") from M
2and x. ,x ~ N(/u2,a ) from F12 n2
Denote the hypothesis that (zj,z ) is (M,F) by , andthe hypothesis that (2 ,z ) is (F,M) by H .
I C. \J
Then, under , the likelihood of obtaining the samples
x^ j. .. , x ^ from Mand x_j2,... >xn2 *>rom proportional to
11 = l/oJn+n 2 exp—^[2(xjVi)2 + £(x.2h2)2 + (z.^i)2 + (z IJZ)2]2a
1 m+n+2 _2 1 rs/ \2 «, \2 , x2L1 = log 11 g— loS a ~ —2^ (xil"A'1) + ^^xi2~^2 +
2a"
+ (z2p2) "]
The maximum likelihood estimates of Pi,p2 and cr2 , under , are
given by setting their partial derivatives with respect to L^ equalto zero. Thus
mx,+z. nx +zQ — —J L u —  —
11 ' 21,m+1 n+1
L'6
ana °  m+n+2 f + ^(xj_2~^z 1 ^ + (zjhii) + (Z2"'J21) 1
/\ /\ /n 2 2where Pm, h'ai and <x^ " are the M.L.E.'s of hi,/J2 and cr under H.
Substitution of these M.L.E.'s in =>
m+n+2 ~ 2 m+n+2L log cr  —
Similarly, under the alternative hypothesis
mx^ +z2 nx„+z,h 1 O = , h 20 
m+1 n+1
/s2 1 r ^ \2 ^A 12 f ,2 . >2 ,
°0 = M+2 ^ ^xi1^1°) + 2^xi2"^20^ + (ZT^20) + vz2Pio) ]
... c m+n+2 a 2 m+n+2Also Lq =  y log oQ  T
*• c .m+n+2 , a 2 2Thus L1 > Lq if —— log crQ /o^ > 0
 2  2i.e. if crQ >
i.e. if 2(x^£io)^ + Z(x^p2o)^ + {z^Pzo)^ + (zpPio) >
SCx^Pll)2 + 2(*i2h2l)2 + (zr/Jll)2 + (z2Pzi )2which, on substitution of the M.L.E.'s, gives us
2 2—2 — 2(mS^+Zg) (nx2+z1) (mx^z^ (nx2+z2) _ (z2~zJ! + _ _ 2mx^— :—
m+1 n+1 m+1 n+1 m+1
1*9
_ (zozi) (nx +z ) (ax,+zj (mx +z )+ 2nx2 2 7^ 2z? + 2z1
nh1 n+1 " mt 1 m+1
„ (nVZ2)+ 2z > 0
n+1
This inequality simplifies to
(z1z2)[(nm)(zJj+z2) + 2m(n+1 )x.;  2n(m+l)x2] > 0 ,
Thus, in the univariate situation, we have the following assignment
procedure, rj, for the pair (z^,z )
r If > 0 assign zf>M , z^FIf DpLS^zi'Z2^ < 0 assign z^F , z2>"
win 're DPLS^Z1 'Z2^ = (zi~z2^^n~m^zl+z2^ + 2=l(n+1)x1 " 2n(m+1 )x2]
Notice that if the sample sizes are the same i.e. n=m , y) reduces to
the estimated L.R. rule for paired observations, £ (see [12]). The
unconditional distribution of D (z., ,z ) is complicated and noriLiS ' 2
attempt has been made to evaluate it. However, we can look at its
expected value.2 2 2
Suppose z^ eM , z^F. Then, E[z^ J = Pi + <r and2, 2 2
E[z2 ] = p2 + cr . Then, taking .. expected values over allsamples (x^,... »xm1Jz,) fnom M and (x^,... >xn2'z2^ from F>
2 2
E[DpLs(V22)] = (nm)(pi p2 ) + (pi p2) [2m(n+1 ^  2n(m2
= [m(n+l) + n(m+1)](Pip2)^
= [m(n+1) + n(m+l)]S[DpT(z1,z2)] ,
50
where Dpp(zrz2)  Oj^KUi ~Vz)
Therefore, the coefficients of z, ,2„ in D^„ „(z. ,zn) are biased.1* 2 PnS 1 2'
for the coefficients of zpz2 zj >once more usingthe fact that scale changes do not affect allocation procedures, we
see that ^ppg(z]'z2} ^"s rea^l 'unbiased' for D^(z^ , z^).
(25) THE LIKELIHOOD RATIO CRITERION FOR THE ADDITION 0? TWO NEW PAIRS
(UNIVARIATE)
In practice.we may observe several new pairs and require
to allocate them as a whole. Ideally, for the the purposes of practical
discrimination, when all the new pairs have been introduced, the rule
would break down into the independent assignment of each pair.
Suppose we have two new pairs (z^z^) and (w^,w2), andthe usual samples size m from M , n from F . Then, since there are
four different ways of allocating the two pairs, we have to consider
four hypotheses. They are :
1) H1 : (z^MjZ^F) and (w^MjW^F)By the obvious extension of the previous case, the M.L.E.'s under H^ are
mx.+z.+w. nx +z +wn 111 ^ 2 2 2P11 = , /i21 ,
m+2 n+2
^ 2 1 r Q f ~ v 2 H / A * 2 t ,2 f avv2 / ^
a1 " rn+n+4 ^ + 2(xi2U2i) + (ZlPn) + (z2/j£i) + (w^Cti( A \ ^ i+ \.w2/i2i) J .
<> m+n+4 t ~ 2 m+n+4Also, o1 =  —J— logecr  5
51
2) H2 : (z^MjZ2e?) and (w^eF^eM)3) : (z1eF,z2eM) and (w^M^eF)4) : (z1eF,z2eM) ana (w^F^eM)
The M.L.E.'s and log likelihoods under H2,H^ and are similar tothose under . We will have independent assignment of the pairs
if £„L2 leads to the same procedure as L L. , and LL_ leadsI j 2 4 12
A A
to the same procedure as L^L^ .
A A 2 A 2Now £^ >0 if o_ > 6^ . After a little algebra,
it can be shown that
> ° lf (zi~z2^^n_in)(zi+z2^ + 2(n+2)(w1+mx1) ~ 2(m+2)(w2+nx2) J> 0 .
■By symmetry, f^l, > 0 if
(z^z2)[(nm)(z^+z2) + 2(n+2)(w2+mx,j)  2(m+2)(v^ +nx2)] > 0 .
Therefore, we cannot reduce the allocation procedure for two new pairs
into their independent assignment (except in the trivial case of w^=w2 ,
or z^=z2). It is worth noting that the situation might arise in practicewhereby we know (z^,z2),(w^,w2) are either (M,F),(M,F) or (F,M),(F,M),For instance, a biologist may have classified some observations using
a common coa^Aa. that the first individual in a bracket belongs to
one group, and the second belongs to the other group ; a second biologist
looking at the data may not be able to tell which way round the pairs
are. For this problem, we need to compare with H, . After moreA A
algebra, it can be shown that if
(z1z2)[(nm)(z1+z2) + 2(n+2)mx1  2(m+2)nx2]  2(nm)(z1w1 + z^)+ (w1w2)[(nm)'(w1 +w2) + 2.(n+2)mx> 2(m+2)nx2"' > 0
COi
Hence the second biologist will be able to come to a decision as to
whether the notation used was (M,F),(M,F) or (F,M),(F,M) .
We can extend the analysis to the addition of k new pairs
(z.„,z ) (i=1,... ,k), when we vri.ll have to choose between 2^ differentv i1 i2
hypotheses. A search procedure should be introduced to shorten trie
number of computations required (maximum of 2 likelihoods). Two
algorithms are suggested for the bivariate case in (212), where their
application to the Fulmar data is discussed.
(26) THE ALLOCATION OF A SINGLE OBSERVATION BY THE LIKELIHOOD RATIO
CRITERION (UNIVARIATE)
Suppose the new individual is w and we have the usual
samples from M and F. From Anderson's (15§$) multivariate result,
a little algebra leads us to the conclusion, that we assign w>M if
2 — — — 2 — 9
DLg(w) = (nm)w  2w[(m+l)nx2  (n+l)mx^] + (m+l)nx2  (n+l)mXj > 0 [ 13]
Putting n=m in [13], we see that w is assigned to M if
^i+x2(x x ) (w  — ) >0 , which is the same rule as f5 I.
2
If n/m , there will be cases of reversal. The diagrams belcw illustrate
how the discrimination rules depend on the sample sizes.
In FIG, k we see that the sample means x^ ,x2 have thesame distribution (with different means), and the cutoff point is
simply (x^+x2)/2 . FIG. 5 shows how, when n>m and consequently2 2
cr /m > cr /n , reversals will occur to the left of the point A .
— 2 — 2Females x^ ~N(p2,cr /n) Males x^ ~K(jUi,cr /m)
FIG. 4
FIG. 5
If n>m , the parabola D (w) has its vertex downwardsLS
and we will assign w to F if w lies between
_ _ — —— 2 — 2 ^~~r 2~~[(m+l)nx2  (n+1 )mx^ ] + v [ (m+1 )nx,_,(n+1 )mx^ ]  (nrn) [ (m+1 )nx,, (n+1 )mx^ ]
n  m
i.e. weF if w lies between
[ (m+1 )nxp(n+1 )mx^ ] ± (x^x )V mn(m+1)(n+1)nm
5^
If we are given weM ,
O
E[D (w) ] = (nm)hi  2 I" (m+1 )nH2(n+1 )mhi ]Lb
+ (ra+1 )n(^2^+o"^/n)  (n+1 )m(pi ^+0"2/m) + (nm)cr22
= (m+1 )n(hi p2)
= 2(m+l)nE[DT(w)]2
Similarly, if weF , E[D (w) ] = (n+1)m(hip2)iJO
(2*7) THE PROCEDURE FOR POPULATIONS WITH EQUAL MEANS AND UNEQUAL
VARIANCES
2 2Suppose M ~ N(p,o~i ) and F ~ N(r,ct2 ) , and we have
to classify the new pair (z^,z^) . The probability of (z.j,z ) being(M,F) is
1 / n2 1 , v
—(Zlh)  (z^p)1 1 / n2 1 , n2
~2—2 eXp27ro1 cr2 2Cm 2ct2
and of being (F,M) is
L— exp _ ~(z n)2 _ —!_( p)22770102 2o, 2O2
Putting the logarithm of the L.R. equal to zero, we find our discriminating
boundary to be2 2 Z1+Z2
(v5^) (°"1 °~2 )( 2 " H) = o •
When the population parameters are unknown, we take the usual samples
from M and F, and form two hypotheses :
1) H. : {Zj[ ,z2)  (M,F) . Under ^
<?2?(mx.j +2 . ) + a,f(nx +zQ)^1   ~  —
(ra+l)5"2i + (n+1 )*i?
<Ti? = [ Z(xi1/^i)2 + (z^Pi)2 ]/(m+1) , 5"22 = [2(xi2/Ji)2 + (s2£i)2]/(n+l)
, a. m+1 , ~ 2 n+1 , ^2 1,and L1 =  log^cru  —— log^cr^  ^{m+n+2)
2) Hg : (z.j ,£?) = (F,M) . The M.L.E.'s and Hog L.R. under are
similar to those under .
We choose in preference to , if L^L^ > 0
• j> m+1 a 2 /<. 2 n+1 ^ 2 2i.e. if ——log CTio/CTn + —jlog O2o/O"2 1 > 0
i.e. if
a 2 m+1 a 2 n+1
*T©T • • 2
Similarly, it can be shown that we classify a single observation w as M ifav 2 m a 2 n
2 f \ 2 52o^ •*:— > 1 , where tne M.L.E.'s are2 / , '1 /
given by similar expressions to the M.L.E.'s under the two hypotheses,
in the paired case.
(28) THE LIKELIHOOD RATIO CRITERION FOR THE CLASSIFICATION OF ONE NEW
PAIR (BIVARIATE)
We now extend the univariate techniques into two dimensional
ones. The bivariate case was studied, in preference to the multivariate
problem, since it was felt that a geometric interpretation would be
simpler to carry out ; also, there was some bivariate bird data for
pairs readily available, to which the discriminant functions could be
applied.
Suppose the males are distributed bivariate normally with
mean (pi 1,^12) , the females are distributed bivariate normally with
mean (^21,^22) , and they have common dispersion matrix
Z =
2cr 1 po"1 cr2
P<J~1 0~2 0*2
We now have one pair of new observations (zi>zi2^ an(^
^Z21'Z22^* ^en we know from (2*1) that the population discriminantfunction is
2 = 0
i.e.Z11"Z21
Z12_Z22
cy2 ~po~1 o 2
2po, cr2 cr1
U1121
N12"P22
= 0
When we have no knowledge of population parameters we take the samples
(Sil^i,) ,m from Mand (xi2'^i2^ i=1 > • • • >n from F
Under : (z] > zj 2^ (Z21'Z22^ as ^ ' we ^ave ^he Following
M.L.E.'s ana log likelihood
P11  (mx1+z11)/(m+l) , pi 2 = (my1+z12)/(m+l) , p2 1 = (nx2+z21 )/'(n+1) ,
P?2 = (ny2+z22)/(n+1) ,
^12 = [ 2(xi1Pii)2 + Z(xi2p2i)2 + (z^Pn)2 + (z21P2i)2 ]T (m+n+2)
a 2 .
0b 1 = similar expression
Pi = [ 2(xi1Pn)(yi1Pi2) + S(x.2P2i)(yi2P22) + (z11 Pi 1) ( z1 2~Pi 2)+ (z21p2i)(z22p22) ] r (m+n+2)a16L2
= (ni+n+2)log o"i i<x2 i/TpT2
Under Hq : (z^jZ^) is F, (Z2i»Z22^ ^"S ^ M.L.E.'s can be expressedin a similar fashion to those under . The log likelihood is
yv / 2Ln = (m+n+2)log <Xi ocr2oV1 p oU G
A A
We choose H. in preference to H„ if L. > L„1 c 0 10
• * r* A 2/. A 2\ a 2 2/. A, 2.i.e. if oi o O"2o (1Po ) > cTi i cr2i (1 pi )
where So =
i.e. if So > SiA 2 A A Acr1 o Pocr i ocr2o
A A  2P0CX1 ocr2o cr2o
After expanding the above inequality and a great deal of tedious
algebra, we have the final result that (z>z)2^ ^~s ass^Sne(^ "to
(Z2i'z22) to F> if DPLS^1'~2^ > ° Wh6re
b8
2  __
(m+n2)(z12z22)s1 [ (nin) (2+z?2) + 2(n+l)my1  2(m+l)ny9l +
2  
(m+n2)(z^z2j)s2 [(nm)(z^1+z91) + 2(n+l)mx1  2(m+l)nx2] 
2(m+n2)rs1 s2[ (nm) (z^ z1 2~z21 z22) + (z^z^ ) ( (n+1 )my.) (m+1 )ny2)
+ (z12z22)((n+l)mx1(ra+i)nx2)j +
2<m(zu',2z2^22)(x^2x^) *
2mn(z11z22z12z21)[3(x2y1x1y2) + (*,*2)(z12+z22) ~ (yny2)(211+22V( J +
mn^z11_Z21^y1+y2^('Z11+Z21^yry2^ + 2^ly2"x2y1^ +
mn(z12z22)(x1+x2)[(z12+z22)(x1x2) + 2(x^x^)] , and
s.j2 — [ ^(x^x^2 + E(xi2x2)2 ]/(m+n2) ,
s 2 = [ + 2(yi2y2^2 ]/(m+n2) ,
and r = [S(x±1 x1)(y±1y.,) + 2(x±2x2)(y±2y2)]/(m+n2)s1s22 2
are unbiased estimators for o~i , cr2 and P, respectively.
Let the inequality Dw (z ,z„) > 0 be denoted by [14].rIjo 12
Then, by putting m=n in [14] , we see that for equal sample sizes
(z11}z12) is allocated to M, (z21>z22) to F if2   2  4(n1) (n+1 )sJ (z12z22)(y1y2) + 4(n1 ) (n+1 )s2 (Z11z21 ) (x1x2) 
4(nl)(n+l)rs1s2[(z11z21)(y1y2) + (z^2~z22)(£,x2)] +
59
2n(z11Z12"221ii22)(;^2i^1) +
2n(z11Z22"Z12Z21^3(X2y1~X1 y2> + (X1"^2^Z12+Z22^ " ^y1 ~y2^Z11+Z21 ^
n^Z1l"Z21^y1+y2^^Z11+Z21^y1~y2^ 2(^y2~x2y1^ +
n(z12~z22)(x1+x2)[(z12+z22)(xx2) + 2(x2y1 ~X1y2' ^ > [15]
L, > L„ if1 0
Letting m,n><» in [14] , it is clear that in the limit
2 2
^(Z1 2_z22^a". (^12~^22) + ^"(Z11 Z21 ^CT'2 (^11~^21)
 4pcricra[(z11221)(^i2/i22) + (z12a22)(/in#i2i)l > 0
i.e. if2 2, 2.
cri or2 (1p )
Z1TZ21
Z12~Z22
2o~2 —per1 o 2
2—peri o~2 o~i
Pi 1P21
P1 2 ~P2 2
which is the bivariate population discriminant rule for pairs (see Ml])
We now show how the L.R. sample discriminant function
through the midpoint of a pair can be written in matrix notation. Let
the midpoint be X = (X,Y) , and (z^z ) be (x,y) ; then (z2>z22)is (2Xx,2Yy) and from [14] , it follows that (x,y) is M if
DPXlS(x'y) =
2(m+n2)(yY)s12[(nm)2Y + 2(n+1)myl  2(m+1 hy^J +
2 
2(m+n2)(xX)S2 [(nm)2X + 2(n+1)mx^  2(m+l)nx2] 
2(m+n2)rs1s2[(nm)2(xY+Xy2XY) + 2(xX)((n+1)my1(m+1)ny2)
2(yY)((n+1)mx (m+1)nx ) ]
6U
4mn(xY+Xy2XY) (x^x^ ) +
4(xYXy)[3mn(x2y1x1y2) + 2mnY(x1~x2)  2mnX(y,]y2) ] +
2mn(xX)(y1+y2)[2X(y1y2) + 2'^y^x^^) ] +
2mn(,yY)(x1+x2)[2Y(x1x2) + 2(x^x^)] > 0 , which eventually
simplifies to DpxLS^X'y^ =
rs1S2
+mn
xX"i
(m+n2)/Y_
xX y
x1+x22X y1+y22Y
rS1 S2
y.
Yc
(nm)X + (n+1)mx^(nm)Y + (n+1)my
X Y
V*2 yry2
 (m+1)nxr<L
 (m+1 )ny_
It was hoped that, from this matrix equation, a generalisation
to p dimensions would be forthcoming. However, it is obvious from the
above equation and the univariate case of (1*4) that we can only go
so far as to say that in p dimensions, the L.R. sample discriminant
function through the midpoint of a pair will be of the form
DPXLS^ = (m+n2) OS"!.) s 1[(nm)X + (n+l)mx1  (m+1 )nx2 ]1
+ constant x x product of determinants , which is not very
helpful.
(29) THE ALLOCATION OF A SINGLS OBSERVATION BY THE L.R. CRITERION
(bivariate)
We have the same distributions and samples as in (2*8),
and we must classify the new individual (z^,z^) . Again by reducing
6 a
o be inserted at the end of (2.9)
It has been pointed out to the author that the generalisation
to p dimensions is in fact straightforward. f A is the pooled
sample covariance matrix, and =2;  x j for i,j = 1,2 , thenthe allocation of (jg^tz^)depends on whether
Iff#A + Aii + ^x" i.22 J22 is 6reater or loss than
A + n+T ^12 12 + m^l 21 *^21
Using Bartlett's (1951) matrix Inversion result and the fact that• 1 * .1I* * SS, I • A  I 1 + £ A a J we see that
 A + ss' + bb'l a  A  (X + a' A.""1 a + b' A""^b + a' Arl£ b' A"Xb  (a'/."1^)2)\ \
If the^ are substituted for a and b and we simplify
to the twodimensional case, the formula for I)..,VT c on page 60 is1 Xl.D
verified.
61
Anderson's multivariate result, we assign (z z ) to M if D (z >0 ,i ii S> ! c
where V (zj>z2) =
2 2 —  / •— 2 — 2s^ [(nm)z2  2z2((m+1 )ny2~(n+1 )my1 ) + (m+1)ny2  (n+ljmy^J +
2 2   2 —2s2'[(nm)z1  2z ((m+1)nx2~(n+1)mx1) + (m+l)nx2  (n+1 jmx^ ] 
2rs1s2[(nm)z1z2  z2((m+1)nx2~(n+1)mx1)  z ((m+1)ny2~(n+1)my,) 
(n+l)mx1y1 + (m+1 )nx^ ]
In matrix notation, ^;ls^z1,Z2^ ~
2(m+1)nx 1
LS
2
"rs1S2
rs1 s2
(nm)z^ + 2(n+l)mx,j(nm)z2 + 2(n+l)my1 2(m+1)ny0 i
CJ
(m+1) n"rs1s2
"rs1S2
 ( n+1 )□
r i»
xi
yi "rsls2
Mien m=:n ' dls(Vz2> =
"rS1S2I
J.71
z1  (x1+x2)/2z2  (yA+y2)/2
1 X1 X2
y1 y2
= ^g^Z1'Z2^' samP''e discriminant function for a bivariate population(see [1 *4]) . Thus, only in the case of m=n does the likelihood ratio
discriminant reduce to the sample discriminant function. Mien m/n ,
reversals will occur and the situation is analogous to the univariate
case illustrated in FIG. 5 , except that in the bivariate case we are
not interested in the intersections of normal curves, but normal surfaces.
62
(210) THE GEOMETRIC INTERPRETATION OP THE DISCRIMINANT FUNCTIONS
OF (28) AND (2Q)From A.C.Jones(l 912), we know that the general second
degree equation
2 2S = ax + 2hxy + by + 2gx + 2fy + c = 0 , can represent
several different loci. If
a h g
A h b f =0 , S is a line pair.
g f c
If A/0 , S represents one of the following :
2101) An ellipse if abh > 0 and, A.a and A.b are both
negative.o
102) Has no points if abh'i> 0 and, A.a and A.b are both
positive i.e. the locus is imaginary.2
10*3) An hyperbola if abh < 0 .
Thus, if we first consider the shape of ^pg^zi'z2^' we see ^romthat the coefficients, in Jones' notation are :
(nm)s22(nm)s^ 2(nm)rs^ s^
2 — — _ _
[s ((m+l)nx2(n+l)mx1)  rs^s2((m+1)ny2~(n+1)my1)]2 «. — •
~[s^ ((m+On^Cn+^m^)  rs1 s2((m+1 )nx2~(n+1 )mx1 ) 1
s1 2((m+1 )ny22(n+1 Vy^2) + 2((m+1 )'nx22(n+1 )mx1 2)2rs1s2[(m+l)nx2y2(n+l)mx1,y1 ]
a 
b =
h =
g =
f =
c =
It can "be shown, with a lot of algebra, that
A = (nm)mn(m+l)(n+l)s sA2(lr2)[s12(y2y1)2  2rSl ) (*2x1)2f  v21+ s2 VX2_X1 ) I •
Thus, A=0 and we have a line pair when at least one of the following
conditions is satisfied :
1 ) m=n
2) r=12 2 2 2
3) s1 (y2y*) 2rs1s2(y2y1)(x2x1) + s2 (xgx,,) 0
2) and 3) are trivial and we have already shown that if m=n, D (z z )LS i 2
is the sample discriminant function. We now show that if A/0 9
2   2     2   2s (y2"yf ~ 2rsi s2(y2~y1 ^x2x1 ^ + s2 (X2~X1 ) is always positive.
There are two cases to consider :
a) r(y2y1)(x2x1) > 0. Then
2  — 2 — — —  2   2 2  — 2sj (y^y^  2rSl s2(y2y1)(x2x1) + s2 (x^x^ > s., (y2~y1) 
— — — — 2  — 2 — — 2 — 22s1s2(y2y1)(x2xi) + s2 (x2~xi) = [s1(y2y1)  s2(x2xp] > o
b) r(y2yA)(x2x^) < 0. Then
s^2(y2y^)2  2rSls2(y2y.,)(x2~x1) + s^^x^2 >
s *(y2yi)2 + s22(x2x^)2 > o
Thus if none of the conditions 1),2) or 3) are satisfied, A is negative
if n>m and positive if n<m. But a is positive if n>m and negative if
n<m and so A.a and A.b are alwrays negative.
XI: k Wa fcr bk~t s pr« ^ Sw.r&U*u« ; tu t beW—fcU
64
2Since ab  h is positive, it is clear from 10*1) that
Lf A/o , D_ (z z ) is an ellipse. This ellipse has centre (>..£/) whereLb \ tL
hf  bg (m+1)nx  (n+1)axx = _ = 2 a
ab  h n  in
gh  af (m+l)ny  (n+l)myH = = L
ab  h n  m
< n>m (A,p) is nearer (x2,y )We now prove that if •I n<m (W,p) is nearer (x^,y.)The distance between (W,p) and (x ,y2) in the Xdirection is
(m+l)nx2  (n+1)mx^ _ (n+l)m(x x^)n — m n  m
and between (^,^0 and (x^,y^) is(m+1 )n(x x )
z i
n  m
It follows easily that the centre of the ellipse is closer to the mean
which has been estimated from the larger of the two samples.
Vle now analyse D„T _(z ,z ) , taking it to be a functionjtIJO I c.
of , with z_0 constant. Then in the general case with m/n , in
Jones' notation
2 — — — 2 — 2a = (nm)(m+n2)s2  2mnz22(y1y2) + mn(y1 y2 )
2    2  2b = (nm)(m+n2)s^  2mnz2^(x^x2) + mn(x^ x2 )
h = (nm)(m+n2)rs1s2 + mn(x2y2x.)y1 )' + maz^^x^
+ mnz21(y1y2)
65
2It can "be shown that abh 
/ \ / rt\2/. 2n 2 2 2 2r /  (~ ~~ \ (" \ 12(nm) (ra+n2) (1r )Sl s2  inn L (x^ ) +z^ (y1 y2) x2) J +
2 — 2 — 2 2 — — 2 — — 2 — 2 — 2
mn(nm)(m+n2)[s2 (^ "x2 )2s2 (x1~x2)z212s1 z22(y1y2)"l's1 ~^2 "> +
2rs1s2(x2J2x1y1)+2rs1 ^2Z22^x2)+2rs1 s2z21y2)]
If n=m , abh2 < 02
If n/m , abh can be of any sign since in the trivial2
case of m=1 => s^ =0 ,
2 22——*—— 2 — — — — —
abh = n(nl) s2 (x1x2)(x1+x2~2z21)  n [ (x^x^ )+z21 ^y2)
"222{Sr;2)1?Thus, if n^m and A/o (it has not yet been proved that such cases exist),D (z,,z ) may be of any conical shape. We now restrict our attentionPLS 12
to the case where n=m. Then after more tedious algebra, we find that
A can be expressed as
9 — — — — — — — 2 — — 2n [x1y2x2y1+z21(y1y2)z22(x1x2)] x [4(n1 )(n+1)[(2z21(x1+x2))(s2 x
— _ — _ _ 2 — — — —
(x1x2)rs1 Sg^yg)) + (2z22(y1+y2))(s1 (y.,^)^ s^Xg)) ] 
— — _ _ —  —  2  — — —
2n(x1y2x2y1)(z21(y1+y2)z22(x1+x2))  nz21 (x^Xg) (x1x2)2  ~  —

nz22 (y1+y2)(y1y2)]2 2 — — — — 2 — — — — 2
4n(nl) (n+1) [x2)(x1+x2~2z21)(s2 (x1x2)rs1s2(y1y2))    2 2
+ (y1y2)(yi+y2_2z22^si (yiy2)rsis2(xrx2^ ^
Thus for n>1 , A=0 if one of the following situations exist
a ) s21 = (x1+x2)/2 , z22 = (y.,+y2)/2 i.e. (z21'z22) is the EamPlecentre of gravity*
2    
b) s2 (x1x2)rs1s2(y1y2) = 0
2 _ _ __
S1 (y1y2)rs1s2(x1x2) = 0
=>
r=±1 and the gradient of the line
joining the sample means is rs^/sJ
and either x^x^+z^ (^y^z^^x2) = 0
i.e. (Z2]'Z22^ lies on line joining the sample means
222 222     —  _ _
or z21 (x1 x2 )+z22 (y1 y2 )+2(x1y2x2y1)(z21(y1+y2)z22(x1+x2
= 0 [16]
2    
c) s2 (x1x2)rs1s2(y1y2) = 0
z22 = (yi+y2V2 are all satisfied
[16]
2 — — — —
i) s1 (y1y2)rs1s2(x1x2) = 0
Z21 = (*1+^2^/2
[16]
are all satisfied
e) (v^) = (x2,y2)
The cases b),c),d) and e) 'are trivial and give rise to
line pairs, but if we put z ^ = (x^+x^)/2 and z22 = (y.j.+y2)/2 inD (z^zj , it can be shown that the discriminant reduces to
PLS 1 2 '
67
(m+n2) (z12 ^l2^)s12[(niii)(s12f + 2(n+l)iayi  2(m+1 )ny2 ] +
(ra+n2)(z11 ~is)s22[(nm)(211+ + 2(n+l)mxi  2(m+l)nx2] 
2(m+n2)rs1s2[(nm)(z11z12 X1 1^2) + (z^ ~) ( (n+1 )myi(m+1 )ny2 )
+ (z12~ ^L^LS") ((n+1 )mx1(m+1 )nx2) ] .
Hence for case a) , when m=n ,
D ,(zvz2) = (z12~ + s22(X1X2) pl;
rs1s2[(z11 ^L~a)(yiy2) + (z12 iLl^)(x152) ]
= h (^z. ) , the sample discriminant function for allocat 1O •
on
of a single observation. Incidentally, we have proved that in general
(when m/^n), the fact that one of the pair is positioned at the sample
centre of gravity does not reduce the problem to the classification of
the other member of the pair by the sample discriminant function,
Summing up, we have the following nontrivial cases :
21) (z21'Z22^ "^es a" sample centre of gravity => A=0 , abh =0and DDTO(£i,z ) is the the sample discriminant function (B ).xLS ' S
2) (Z21,Z22^ '^es on Hne joining the sample means, but not at2
the centre of gravity => A/o , abh =0
and „(z„ ,z„) represents a parabola.PLS 1 2
3) (Z21'Z22^ ^oes no^ ^e on line joining the sample means => A/0 ,2
abh <0 and D (.z, ,£„) represents an hyperbola with centre (^,p) wherePli S 1 c
68
2(n1)(n+1)[Sl"(y1 y2)"rsiS2(*1 x2)]X = x +  z  3— ■' — 3———:—21 n[x1y2x2y1+z21(y1y2)z22(x1x2)]
r— — "11r 2
X1 ~x2 s2 rs1s2 X1+X2~2Z21
2(nl)(n+l) (xix2) y1 y22
rs.Sg s1 /1+y22z22    _ _ _ 2
[x1y2x2y1 +z21 (y1y2)z22(x1~x2^ ^
sample discriminant function
2(n1) (n+1) [ s22(5c1 Xg)rs, s2(y1 y2) 1n[x1y2x2y1+z21 (y1y2)z22(x1*2)]
X1 x2 "rS1S2T'r. ^
x1x2
Ly1 ~y2J
[x1y2x2y1 +z21 (yiy2)z22(x^x2) T
centre of gravity  (z , z ) *PXSV1'2y
*1
FIG. 6
We observe from the above diagram that when lies on—2
the line joining the sample means, the situation may arise whereby
D and D  differ in their sexing of the pair. Obviously, similari L o i X S
conflicting results will occur when m/n and neither of the pair lies
on the line joining the sample means. The only exception to this
general rule is when m=n and one of the pair lies at the centre of
gravity (since D and D  are then equivalent).r.Lo rXo
(211) AN EXAMPLE OF THE USE OF DOT_jLl)
The male and female of the fulmar are almost indistinguishable
in external appearance ; only by dissection can wre be certain about the
sex of a bird. However, there are slight differences in billlength
and billdepth between the male and female of the species. As there
is some overlap of these measurements and they are often available in
pairs, it looks as if we can apply to good effect. Note also
that since it is known that there are equal numbers of male and female
fulmars in the whole population, our assumption of equal prior probabilities
is valid.
The sample data used was from Figure 2 of Dunnet and
Anderson(l961). From the sample of 17 males and 17 females, the sample
discriminant function (D ) was calculated to beO
Bill depth = 0*7553Bill length + 469475
or y = 0 *7553x + 46*9475
Notice that this differs slightly from Dunnet and Anderson's discriminator
(y = 0 *5068x + 37*0958) since the data we used was only approximated
from their diagram. Since m=n17 , we know that D = D and theLo d
above function can be used to classify individual birds.
Dunnet and Anderson also give measurements on 22 pairs
of fulmars (Table IV , p.125) • We now classify these pairs using
D _ and compare with how D allocates them individually. A computerPLS S
program was written (in FORTRAN) and the results were recorded in the
table below (see p.7l)
FIG. 7 is a graphic representation of the sample discriminant
function and the pairs 1,2,5,14,16,17 From the table and the diagram,
it can be seen that the pairs 5,14,16 are allocated differently when
considered as a pair, than individually. Dunnet and Anderson were
also dubious about pair no. 17 but its allocation is straightforward
from our analysis.
In conclusion, it is the author's opinion that since the
sample discriminant function for pairs is only an approximation to the
sample likelihood ratio function, and as such may lead to an incorrect
assignment (see the end of (210)), D should be preferred, in spiteiLo
of the fact that it does not possess the easy geometric interpretation
of D • As can be observed in the above example, we also obtain anS
exact index of the certainty with which we make each paired assignment.
This index could prove to be of more importance in the classification
of paired observations than the computation of misclassification
probabilitiesj further research needs to be carried out on these lines.
71
Pair Bill Bill x 1 0"6FLS
Sex D0 x 10 6O
Sex
No. Length Depth
391 157 F 0 *0014 F1 0*1158
40 *6 190 M 0 *0022 M
382 158 F 0 *0019 F2 0*1160
43*0 176 M 0 *0025 M
38*8 16*6 F 0*0008 ' F3 0*0859
42 1 17*2 M 0*0017 M
378 176 F 0*0006 F4 0*0470
40 1 176 M o *0003 M
387 179 M 0 *0001 M
5 0 *004739*0 17*5 F 0*0000 M
363 15*8 F 0*0030 F6 0*1535
41 2 18*0 M 0 *0018 M
408 15*7 F 0*0004 F
7 0*0588401 18*2 M o *0013 M
368 17*5 F 0*0013 F8 0*1327
41 1 18*7 M 0*0023 M
400 18*1 M 0 *0011 M
9 0*105837*3 16*1 F 0*0022 F
360 16*2 F 0 *0029 F10 0*1628
41 *6 18*2 M 0 *0022 M
407 19*0 M 0*0023 M11 0*1160
382 16*5 F 0 *0013 F
41 5 18*5 M 0 *0024 M12 0*1110
385 16*5 F 1 O o o —X F
360 159 F 00031
l
F
13 01516391 187 M 00010 M
388 178 F 00001 M
14 00350405 178 M 00012 M
407 18*2 M 00016 M
15 00724390 167 F 00006 F
383 170 M 00008 F
16 00456373 160 F 00023 F
383 182 M 00001 M
17 00294382 173 F oooo6 F
397 176 M 00005 M
18 00741374 165 F 00018 F
41 5 178 M 00018 M
19 0070239*0 170 F 1 o o o o *r F
41 2 17*8 M 00016 M20 01337
368 160 F 00026 F
41 3 172 M 0 0012 M21 0 0443
39*2 171 F 00002 F
401 180 M 0 0011 M22 00916
377 163 F 00013 F
* Note that the DQ value of the second member of pair no. 5 has
been rounded off by the computer. The actual value should be a very
low positive number.
(2*12) TWO ALGORITHMS FOR THE JOINT CLASSIFICATION OF k PAIRS
It was demonstrated in [25] that when k new pairs are
observed and have to be allocated as a whole, the procedure does not
break down into the independent allocation of each pair. For large k,lr
it is not computationally feasible to find the maximum of 2^ different
likelihoods. For this reason, the following two procedures are suggested
1) In the multivariate case, suppose the k pairs are
(lll >112) ' * * * '(k1'k2^ and WS have saraPles >2^ 1 from Mand X] 2' * * ' ,x n from F, where n.,n„ > k.
7^2^ 12Suppose further that after random assignment of these
pairs, we have the combined (known and unknown) sample matrices X^ ^
and X£q j of males and females, respectively i.e.
10
11
ml
11
*k1
m +k20
12
—n2 2
12
■k2
n2+k say.
•V
Now from the likelihood ratio criterion we know that
our optimum assignment is achieved when the determinant of the pooled
sample covariance matrix, including the new pairs, is minimised. Denote
the determinant of the covariance matrix formed from X._ and X„„ by10 20
2q[. Next, we reverse the assignment of the first pair and, leavingall other rows intact, obtain the modified matrices
75
11
2£ 1V12
21
*k1
12
x ,
2C
11
22
—k2
After finding the determinant of the covariance matrix
of these modified matrices, we return to the matrices X,„,X„„ and' 10 20
repeat, the procedure for each of the remaining k1 pairs, in turn. If
the minimum of the k determinants evaluated, j [, is less than I,we now reverse the sexing of the pair (the i ^ say) corresponding to
this minimum, and repeat the whole of the above process for the second
stage combined sample matrices
11
x,.—11
11
—i1 1
—i2
i+1 1
—k1
X21
12
21n2212
i1 2
^11
i+1 2
—k2
Eventually the procedure will terminate when we reach an
overall minimum which corresponds to the optimum joint assignment of
76
the k pairs. This optimum car he checked "by starting from different
random assignments.
A modification of this algorithm is for the initial
assignment not to be random but that of the pairs considered separately.
Then the optimum should be reached in fewer iterations e.f. Beale('J 971) ,
who observes that, with reference to cluster analysis, "in general, a
good starting solution leads to a good final solution".
2) Alternatively, we make an initial random assignment
of the first k1 pairs to get the combined sample matrices ,i ,k
and X2,k
where
1,k
11
xi
n11*11
k1 1
n +k11
X2,k
n2212
k1 2
n2+k1
we use D
Then, treating X^ and X^ , as known sample matrices,I y ilC ^ j K
to classify the kth pair. Tie next omit the (k1 ) ^PLS
pair and classify it using DCT formed from the matrices X andX J_J O I y K— I
X2,k1 *'here
77
x1 ,k1
11
x i
~n1^11
—k2 1
*k1
X2,k1
x12
22
12
—k~2 2
k2
We continue in the same fashion until each of the k pairs
has been classified, and replace the initial random assignment by this
first approximation.
The procedure is repeated until we reach the optimum
assignment, when the system has settled down and will make no alterations
in the next iteration. Differentrandom starts are made to check the
result. Again, the algorithm will find the optimum quicker if we
take the initial state as that of the pairs allocated separately.
Notice that this algorithm differs from the first in that " block
moves " (Beale) are made.
The above algorithms were applied to the Fulmar data of*
(2*11). Three different initial assignments were tried out for both
algorithms, and all led to the same optimum  that of the pairs allocated
separately, by D . The corresponding minimum value of the determinantHLS
of the combined sample covariance matrix was 0*5555 .
The first initial assignment was, in fact, that of the pairs
separately and no iterations were required in either l) or 2). The
second initial assignment was as follows :
* with the aid of two FORTRAN programs, written by the author.
78
Pairs 1,5,21 the same way round as allocated by D Q in
the table of (211) ; pairs 2,3,4,6,... ,20,22 the opposite to that
of the table in (2*11).
For algorithm 1) , the values of the determinant, and
the number of the actual pair altered, at each iteration, are given
in the table on page 79* Notice that pair no. 1 is the first pair
to be altered but reverts to its initial assignment at the 15 ^ iteration.
Algorithm 2) only required two iterations but, of course, block moves
were made. The D values of the pairs, based on samples of sizePLS
n^+k1 and n +k1 , for each iteration are also given in the tableon page 79. Again pair no. 1 and also pair no. 5 this time, are altered
at the first iteration, but then revert to their initial assignment.
Note too, the suspicion with which pair no. 17 is treated before it is
finally allocated, thus confirming Dunnet and Anderson's doubts. Finally?
it is worth making a comparison between the tables on p. 7j and p. 79
of the order of the pairs, in terms of likelihood (D value). When1JJS
classified separately, the order (from the largest likelihood downwards)
of the pairs is
10 613 20 811 2 1 12 9 22 3181519 7 4 16 21 14 17 5
Similarly, when considered as a whole, the order is
10 6 13 2 20 1 11 12 9 8 22 15 18 3 19 7 16 21 4 17 14 5
Clearly, since the two orders are not equivalent, when we
take all the 22 pairs into account in our discriminant procedure, we
become more certain that we have made a correct decision with some
pairs e.g. no. 2 , but less confident about others e.g. no. 4 .
79
Algorithm 1)
Iteration
No.
i
Pair
Altered
Determinant
o 1 8532
1 1 18271
2 4 18062
3 10 17787
4 8 1 7354
5 2 1 6815
6 6 1 61 67
7 20 15401
8 3 1 4599
9 19 14001
1 0 12 1 3425
11 13 1 2788
12 9 12054
13 11 1 1261
14 22 1 0412
15 1 09466
16 18 06576
17 15 07712
18 7 07025
15 16 06372
20 17 05932
21 ' 14 0 5555
Algorithm 2)
Dtt r a"t Iteration NFLSa.
0 1 2
0 2322 13000 1 4588
01758 1 5695 1 6310
01338 09447 08748
01097 05620 04403
00299 00187 00934
01731 16777 18038
00494 02235 06981
0•1066 1 3183 13007
00706 1 0998 1 3067
02220 1 *7952 1 8494
00753 1 1319 1 4370
00818 1 1764 13673
01026 1 3376 1 6630
00753 04212 03312
00333 07457 0 9222
00121 0 4710 05931
00119 02448 03653
00599 08220 08925
00785 08043 08123
01516 1 4769 1 5594
00473 05410 04473
00579 05677 1 1383
. These are all values of
Pair
7\T r~\
2
3
4
5
b
7
8
9
1 0
11
12
13
14
15
16
17
18
19
20
21
22
N.B
80
The third initial assignment was the opposite of the second i.e., only
the pairs 1,5,21 differed from the optimum ; this was achieved after
three iterations of Algorithm 1) and one of Algorithm 2).
We conclude (212) with a. warning about the use of the
two algorithms. When the original known samples were reduced to a
randomly chosen 5 from each sex, the algorithms produced conflicting
results for the above three initial assignments. From the first and
third, the usual optimum assignment was attained ; however, from the
second initial assignment, both algorithms led to the exact opposite
of the usual optimum i.e. ail the males of the optimum were classified
as F, a.nd all the females as M.
Thus, when the original sample sizes n^,n^ < k , thenumber of pairs, the above results indicate that we either accomplish
the true optimum or an outcome close to its opposite. To counter this
effect, it is suggested that Algorithm 1) should be employed with the
following modification :
When the system has reached a local optimum, this
allocation is reversed. If the determinant of the covariance matrix
of this reversal is lower than the determinant of the covariance matrix
of the local optimum, the algorithm should continue with this new
initial assignment until it reaches the true optimum. If the determinant
corresponding to the local optimum is lower than that of its opposite,
it is in fact the true optimum. Algorithm 2) may now be used to
calculate D„ values, whereby we can tell which pairs are morePLS
likely to have been correctly classified.
(2*13) CONCLUSION
By means of the likelihood ratio criterion, we have
developed and analysed a theoretically exact procedure for the assignment
of paired observations. The usefulness of this procedure is evident
from the results, that we obtained, of its application to some Fulmar
data. When the new pairs have to be assigned a.s a whole, the two
algorithms (with modifications for small sample sizes) also seemed
to work well with respect to the same data. Notice that throughout
the thesis, the theory is always simplified if we consider the sample
sizes, of the original known data, to be equal ; obviously, this
condition will always be satisfied if the original samples are, in
fact, pairs.
Although further research needs to be conducted, such as
into the best method for estimating paired misclassification probabilities
the thesis serves as an introduction to this particular branch of
discriminant analysis.
The author wishes to thank Prof. R.M. Cormack for suggesting
the problem and for many helpful discussions during the course of this
research. The work was supported by a St. .Andrews University research
grant.
82
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