2. Kyngdon and Richards
Attitudes can be measured, so proclaimed
L.L. Thurstone (1928). He did not mention,
however, that the semantic structure of attitude
statements is not explicitly quantitative. Hence
the claim of attitude measurement cannot be
uncritically accepted as true (Michell 1994).
Sherman (1994), Michell (1994) and Johnson
(2001) used unidimensional unfolding theory
(Coombs 1964) to investigate attitudes towards
the assimilation of immigrants, homosexuality
and nuclear conflict. These studies found, re-
spectively, that 97%, 95% and 97% of observed
paired comparison judgments were as predicted
by unidimensional unfolding. Moreover, Michell
(1994) and Johnson (2001) found support for the
double cancellation axiom ofACM (KLST 1971);
suggesting Thurstone’s (1928) claim may have
some credence.
Unidimensional unfolding theory posits the
simple argument that:
a prefers xj
to xk
iff |ia
–xj
||ia
–xk
|, (1)
where xj
and xk
are any two stimuli in a set X of
stimuli of the same kind (such as attitude state-
ments); and ia
is person a’s point of maximum
preference (or ideal point) (see Kyngdon 2006a
and b).The theory contains both ordinal and quan-
titative components. In any empirical instance
both must be true, because if one is not true then
neither is the theory (Michell 1994). Given it
is plausible that attitude statements possess no
more than ordinal structure, the investigation of
both components is behoved. Consider the pair
(xp
, xq
) formed by two attitude statements xp
and
xq
in X. This pair is unilateral to the ideal point ia
of person a if and only if either (i) xp
# ia
and xq
# ia
; or (ii) ia
# xq
and ia
# xp
. Hence,
a prefers xp
to xq
iff either ia
# xp
xq
or xq
xp
# ia
. (2)
Expression (2) is the ordinal component of un-
folding theory (Michell 1994). Let (2) be called
the Ordinal Pairwise Axiom (OPA) of unidimen-
sional unfolding.
Judgements of the pair (xp
, xq
) do not follow
the OPA if the statements straddle an ideal point.
Hence xp
and xq
are bilateral to ia
if and only if
either (i) xp
ia
xq
or (ii) xq
ia
xp
. As reduc-
tion to merely ordinal relations is not possible
with bilateral pairs (Michell 1994), reference to
distance must be made:
a prefers xp
to xq
iff |ia
– xp
| |ia
– xq
|. (3)
A stimulus pair is thus either unilateral or bilateral
relative to a specific ia and never both. Expression
(3) is the quantitative component of unfolding the-
ory; and let it be called the Quantitative Pairwise
Axiom (QPA) of unidimensional unfolding.
Michell (1994) found 93.5% of unilateral
paired comparison judgments accorded with the
OPA. Michell (1994) and Kyngdon (2006b) found
94.3% and 96% of bilateral paired comparison
judgments accorded with the QPA, respectively.
Moreover, “between – subjects” support of the
QPA was found in both studies as double can-
cellation axiom was supported. However, the
probability in each study of double cancellation
being supported at random was .5874 as only
six statements were used (Michell 1994). This
is not an unlikely event. Furthermore, Kyngdon
(2006b) and Michell (1994) counted the number
of intransitive bilateral judgments to assess sup-
port of the QPA. Counting errors is relatively
crude as it cannot indicate the degree of axiom
violation nor does it account for sample size
(Karabatsos 2001).
Unlike Johnson (2001), Michell (1994)
or Sherman (1994), Kyngdon (2006b) tested a
stochastic IRT unfolding model (the SHCMpp,
Andrich 1995). An order upon the statement
estimates was found matching the predicted
quantitative J scale statement order. Transforming
these estimates revealed linear and exponential
relationships against the transformed Goode’s
algorithm (Goode, cited in Coombs, 1964) scale.
These findings, however, were limited by the
use of only six statements and the poor fit of the
model to the data.
Hence the aim of the present study was to
expand upon the research of Johnson (2001),
Kyngdon (2006b), Michell (1994) and Sherman
(1994). The methodology was developed from
previous research in two ways. One was in the
construction of sets of eight statements using
3. Attitudes, Order and Quantity
the theory of the ordinal determinable (Michell
1994). The other was in testing the OPA and
QPA with the probabilistic multinomial Dirich-
let model (Karabatsos 2005). Hence the paper
contains the first ever direct probabilistic tests of
Coombs’s theory in its original algebraic form
(viz., Expression 1).
Study One: Attitudes
Towards Interpersonal Distance
to Homosexual People
Stimuli
Richards (2002) used the theory of the ordi-
nal determinable (Michell 1994) to construct the
following set of eight statements:
A. Not only would I be friends with a homo-
sexual and have sex with one, I would be
involved in a life partner relationship with
a homosexual.
B. I’d be friends with a homosexual and I
would have sex with one, but I would not be
involved in a life partner relationship with
a homosexual.
C. I’d be friends with a homosexual, and while
I would not have sex with a homosexual, I
would be physically intimate with one.
D. I’d be friends with a homosexual, but I
wouldn’t be physically intimate with one.
E. I’d speak to a homosexual in passing and mix
with one socially, but I wouldn’t be friends
with one.
F. I’d speak to a homosexual in passing, but I
wouldn’t be friends with one or mix with one
socially.
G. I wouldn’t be friends with a homosexual or
even speak to one in passing, but I would not
wish harm upon a homosexual.
H. I wouldn’t be friends with a homosexual or
even speak to one in passing, and in fact, I
would wish harm upon a homosexual.
The binary tree diagram of statement predicates is
contained inAppendix One.An interesting feature
of the binary tree structure is that it is symmetric
around the kernel concept, unlike the structures
used by Kyngdon (2006b) and Michell (1994).
Participants
Participants were 204 first year psychology
students at the University of Sydney. They re-
ceived course credit for participating.
Materials
The computer program RUMMFOLDpp™
Version 2.1 (Andrich and Luo, 1998) was used
to estimate the SHCMpp (Andrich, 1995).
The program uses a joint maximum likelihood
(JML) iterative algorithm for the estimation of
model parameters (see Luo, Andrich and Styles,
1998). The package S – Plus®
6.1 for Windows
(Insightful, 2002) was used conduct multinomial
Dirichlet model analyses based on the S – Plus
program created by Karabatsos (2005). Regres-
sion analyses were conducted using program
SPSS for Windows™ (Version 12.0.1) (SPSS,
Inc., 2003).
Procedure
Test administration was conducted by the
second author. A booklet containing the state-
ments was administered to the participants in
the psychology laboratories at the University of
Sydney in the year 2002. Participants completed
a rank order task and then completed 28 paired
comparison tasks. Participants were instructed for
each pair to indicate which statement they agreed
with most strongly or disagreed with the least.
Ross’s (1934) balanced optimal presentation
order for paired comparisons was used.
Results
Identifying bilateral and unilateral stimulus pairs
For n number of unfolding statements form-
ing the qualitative J scale, there are 1/2n(n–1)
number of paired comparison tasks. Let these
responses be coded “0” or “1” by defining a vari-
able Y such that:
Y
a x
a x x
p
p q, , =
ì
í
ïï
îïï
1 if chooses
0 otherwise
, (4)
4. Kyngdon and Richards
where a is the participant and (xp
, xq
) is the
stimulus pair (yet not necessarily presented in
that order). If Ya x xp q, , =1 then a orders xp
and
xq
as they are ordered in the qualitative J scale.
Thus each participant’s response pattern of paired
comparison judgments can be represented as a
vector, Za
, of 1/2n(n–1) number of “0”s and “1”s.
The total number of possible response patterns
is 2n
, however, the OPA reduces this to 2n-1
and
the QPA reduces this further to 1/2n(n–1) +1
(Michell 1994).
The QPA and the folding condition together
require transitivity to hold over all stimulus pairs,
however, the OPA only predicts the unilateral
pairs within each Za
are ordered, not that each Ya
is ordered. Hence paired comparison judgments
must be identified as either unilateral or bilateral.
This cannot be done, however, for pairs contain-
ing a’s most preferred statement (xa
). Therefore
xa
must first be identified (Michell 1994). Given
transitivity in paired comparison judgments, par-
ticipant a’s preference score for each statement xp
is determined as follows (Michell 1994):
Y Y Ya x a x x a x x
q p
n
q
p
p q p p q, , , , ,= −( )+
= +=
−
∑∑∑ 1
11
1
. (5)
Then xp
is a’s most preferred stimulus, xa
, if and
only if for all xj
0 X (xp
xj
):
Y Ya x a xp j, , ∑∑ . (6)
With intransitive judgments it may not be possible
to determine xa
. In such cases there may be more
than one statement with a maximal preference
score; so resort must be made to a’s rank order
data. If, however, a participant gives many intran-
sitive judgments it may not be possible to identify
xa
. Data from such participants is deleted.
When xa
is identified, a’s position on the
dimension, ia, is still undetermined. With the
exception of those stimuli at either “end” of the
qualitative J scale (e.g., statements A and F in
the qualitative J scale ABCDEF), each xa
has a
unique set of statement pairs of which each pair is
bilateral to xa
. Hence if transitivity holds over the
relevant set of bilateral paired comparison judg-
ments then either ia
xa
or ia
ia but not both.
Table 1 contains the number of “within
– subject” departures (intransitive bilateral paired
comparison judgments) from the QPA. The QPA
accounted for 98% of identifiable bilateral judg-
ments (Table 1).
For each xa
a set of unilateral judgments can
be identified as xp
xq
xa
iff xp
xq
ia
; and xa
xp
xq
iff ia
xp
xq
(Michell 1994). Table 2 con-
tains the number of intransitive unilateral paired
comparison judgments. The OPA accounted for
97% of identifiable unilateral judgments.
Testing the QPA and OPA
with the Multinomial Dirichlet model
Consider any preferred statement xb
and a
statement pair (xp
, xq
) that is either unilateral or bi-
lateral to xb
. A pattern of two binary preferences
R r x x r x xx x x p q q pb p q,
,( )
= = =( )1 2
exists where r1
= (xp
xq
) is the pattern supporting
(~V) either the OPA or QPA; r2
= (xp
xq
) is the
pattern violating (V); with “” representing the
relation “is preferred to”. Hence V Rx x xb p q
⊂ ( ),
and
~ ,
V R Vx x xb p q
= −( ) . The corresponding parameter
vector Q of a multinomial distribution is Q = (q1
,
q2
= 1 – q1
) with q1
= q and q2
= 1 – q where “q” is
the probability that the relevant axiom is satisfied
(Karabatsos 2005).
Let N be the total number responses to each
pair (xp
, xq
) which is calculated by summing the
relevant entries in the column labelled “#” (Tables
1 and 2). For example, N = 11 + 38 = 49 for theAD
bilateral pair (Table 1). For each pair with N ob-
servations it is assumed the sequence {x1
,x2
,...,xN
}
is exchangeable (de Finetti 1937/1964) where xn
= (xn1
, xn2
) is a 0 – 1 vector such that xn1
= 1 if the
response is pattern r1
and xn2
= 0 otherwise; with
xnjj
==∑ 11
2
(Karabatsos 2005). The number of
responses that violate the QPA is given for each
bilateral pair in the row labelled “T” (Table 1).
For theAD bilateral pair this figure is 1. Hence the
number of responses which conform to the QPA,
s, for the AD bilateral pair is s = N – 1 = 48.
An objective or “non-informative” Bayesian
analysis (Berger 1985) is obtained with non-
informative priors which result in the posterior
6. Kyngdon and Richards
distribution p(Q|x1
,...,xi
,xN
) being mostly a prod-
uct of the data obtained (Karabatsos 2005). One
such prior is the improper uniform, for which
every possible rj
is expected to occur with a
probability of 1 / J (Berger 1985). Another is the
proper reference prior which leads the posterior
distribution being most influenced by the data
obtained (Karabatsos 2005). The present study
used both these priors, the values of which were
~V = V = 1 and ½ ~V = ½ V = .5, respectively.
Thus the reference prior used was equivalent to
Jeffreys’ prior (Jeffreys 1961) for multinomial
distributions (Karabatsos 2005).
Several null hypotheses can be tested upon
the same set of data within a Bayesian framework
(Berger 1985). The null hypothesis H0
: q Î[cmin
,
cmax
] Í [0,1] that an axiom (either QPA or OPA)
was true was tested at four levels (H0
: q $ .50,
H0
: q $ .75, H0
: q $ .95 and H0
: q $ .99) against
the alternative hypothesis H1
: q Ï [cmin
, cmax
] that
the axiom was violated (Karabatsos 2005). If
the null hypothesis H0
: q $ .50 was rejected, the
axiom was interpreted as being violated; if either
H0
: q $ .50 or H0
: q $ .75 was not rejected then
the axiom was “weakly” or “moderately” sup-
ported, respectively; and if either H0
: q $ .95 or
H0
: q $ .99 was not rejected then the axiom was
“strongly” supported (Karabatsos 2005).
These null hypotheses were tested by calcu-
lating the Bayes factor, which is an index of the
evidence in the data in support of H0
against H1
(Berger 1985). The Bayes factor (B) is (Karabat-
sos 2005, Eqn 9):
B H c c
p H s p H s
p H p H
0
0 1
0 1
: ,
| |
min maxq ∈[ ]( )=
( ) ( )
( ) ( )
, (7)
where the numerator is the post-analysis evi-
dence in support of H0
having observed the data
s (posterior odds ratio); and the denominator
being the pre-analysis evidence in support of H0
(prior odds ratio) (Berger 1985, Definition 6). If
B H c c0 1 10∈[ ]( )min max, then it is considered
there is “decisive” evidence against the null hy-
pothesis (Berger 1985; Jeffreys 1961; Karabatsos
2005).
Table 3 contains the results of the multino-
mial Dirichlet model analyses of the QPA and
OPA upon the data contained in Tables 1 and 2, re-
spectively. Table 3 shows the statement pairs with
the corresponding number of axiom supporting
judgments (s), the total number of observations
for that pair (N), and the null hypothesis sup-
ported with the Bayes factor under the reference
(Jeffreys) and uniform (Uniform) prior distribu-
tions. In the bottom left corner of Table 3 appear
the Bayes Factor Global (BFG) statistics which
indicate support of the dataset overall under a
given null hypothesis and prior.
Under both the reference and uniform priors,
each bilateral and unilateral pair was “strongly”
supported (either H0
: q $ .99 or H0
: q $ .95) by
the multinomial Dirichlet model (Table 3). More-
over, the BFG statistics suggested the bilateral and
unilateral data overall “strongly” supported (H0
:
q $ .99) the QPA and the OPA, respectively.
Statistical independence over both bilateral
and unilateral pairs was tested by the calculation
of Bayes factors (under the data – sensitive Jef-
freys prior) aggregated over pairs that shared one
common statement (Karabatsos 2005). The set of
statement B common stimulus bilateral pairs did
not reject the H0
: q $ .95 hypothesis whilst all
other sets did not reject H0
: q $ .99. The sets of
unilateral pairs that had B, F and G as common
stimuli did not reject the H0
: q $ .95 hypothesis
whilst all other sets did not reject H0
: q $ .99.
Hence the QPA and OPA were “strongly” sup-
ported in the relevant sets of bilateral and unilat-
eral common stimulus paired comparisons.
Coherent Tests of Triple Cancellation
upon I Scale Dominant Path
Midpoint Orders
The bilateral paired comparison judgments
were used to derive participants’ I scales (see
Kyngdon 2006b). Figure 1 is an I scale proximity
graph for eight stimuli ABCDEFGH. Seventy-
seven percent of I scales were located on the
dominant path depicted in Figure 1.
All 140 tests of the double cancellation
axiom of ACM were satisfied by the obtained
interstimulus midpoint order. The use of eight
statements, however, leads to the creation of 4
7. Attitudes, Order and Quantity
x 4 conjoint matrices upon which tests of triple
cancellation (KLST 1971) must be conducted to
verify the presence of additive structure. Whilst
coherent tests of double cancellation were iden-
tified by Michell (1988), coherent tests of triple
cancellation were only recently identified by
Richards (2002).
Consider a relation $ on AHX, where A and X
are finite sets containing elements a1
, a2
, a3
, a4
and
x1
, x2
, x3
, x4
respectively. Michell (1990) provided
a general statement of n-th order cancellation in
terms of a trivially true instance based on an (n +
1) termed sequence of values. In the case of triple
cancellation, this may be written as follows:
if a1
x1
$a2
x2
and
a3
x3
$a3
x3
and
a4
x4
$a4
x4
then
a1
x1
$a2
x2
.
Setting out the antecedent terms of this de-
generate condition in a 3H4 matrix (A* below), if
the elements within any column of the array are
permuted, then a different matrix will be gener-
ated (e.g. B* below, produced by swapping the
order of elements in column four of A*):
A* =
a x a x
a x a x
a x a x
1 1 2 2
3 3 3 3
4 4 4 4
B* =
a x a x
a x a x
a x a x
1 1 2 4
3 3 3 2
4 4 4 3
and so produce a corresponding distinct set of 3
inequalities, the antecedent of a triple cancella-
tion condition having the same consequent as the
trivial instance given above:
if a1
x1
$a2
x4
and
a3
x3
$a3
x2
and
a4
x4
$a4
x3
then
a1
x1
$a2
x2
.
If $ on A1
HA2
is an additive structure, then if
each of these three antecedent inequalities is true,
it follows that a1
x1
$a2
x2
. All permutations within
the column terms in A* produce a new matrix,
Table 3
Bayesian analysis results, interpersonal distance eight statements, psychology student sample.
Bilateral H0
not rejected (Bayes Factor) Unilateral H0
not rejected (Bayes Factor)
s N Jeffreys Uniform s N Jeffreys Uniform
AC 11 11 $.99 (8.46) $.99 (12.69) AB 179 191 $.95 (1.52) $.95 (3.8)
AD 48 49 $.99 (3.53) $.99 (9.72) AC 153 153 $.99 (170.61) $.99 (336.39)
AE 173 174 $.99 (30.95) $.99 (108.66) AD 28 28 $.99 (17.86) $.99 (33.50)
AF 186 188 $.99 (10.46) $.99 (41.11) AE 14 14 $.99 (10.1) $.99 (16.11)
AG 190 194 $.99 (1.27) $.99 (4.92) AF 8 8 $.99 (6.79) $.99 (9.37)
AH 200 201 $.99 (40.29) $.99 (148.97) AG 1 1 $.99 (2.14) $.99 (2)
BD 36 38 $.99 (.29) $.99 (.70) BC 144 155 $.95 (.8) $.95 (1.94)
BE 159 163 $.99 (.71) $.99 (2.57) BD 30 30 $.99 (19.03) $.99 (36.19)
BF 172 177 $.99 (.27) $.99 (.97) BE 15 16 $.99 (.66) $.99 (1.23)
BG 190 197 $.95 (27.77) $.95 (66.4) BF 10 10 $.99 (7.91) $.99 (11.57)
BH 200 204 $.99 (1.5) $.99 (5.91) BG 3 3 $.99 (3.72) $.99 (4.06)
CE 123 125 $.99 (4.21) $.99 (15.18) BH 2 2 $.99 (2.98) $.99 (3.03)
CF 132 139 $.95 (5.26) $.95 (12.73) CD 41 41 $.99 (25.82) $.99 (51.99)
CG 142 145 $.99 (1.72) $.99 (6.3) CE 27 27 $.99 (17.29) $.99 (32.17)
CH 151 152 $.99 (23.55) $.99 (82) CF 21 21 $.99 (13.92) $.99 (24.50)
DF 14 14 $.99 (10.1) $.99 (16.11) CG 15 15 $.99 (10.1) $.99 (16.11)
DG 20 20 $.99 (13.37) $.99 (23.26) CH 13 13 $.99 (9.55) $.99 (14.96)
DH 27 27 $.99 (17.29) $.99 (32.17) DE 64 65 $.99 (5.46) $.99 (16.31)
EG 6 6 $.99 (5.63) $.99 (7.22) DF 59 59 $.99 (38.68) $.99 (81.94)
EH 13 13 $.99 (9.55) $.99 (14.96) DG 50 52 $.99 (.61) $.99 (1.63)
FH 7 7 $.99 (6.22) $.99 (8.29) DH 51 51 $.99 (32.66) $.99 (67.96)
EF 176 184 $.95 (10.66) $.95 (26.34)
Bayes Factor G. Jeffreys Uniform EG 169 177 $.95 (8.82) $.95 (21.79)
Bilateral $.95 3.57 x 1042
1.54 x 1046
EH 176 176 $.99 (230.81) $.99 (487.42)
Bilateral $.99 547163676 2.95 x 1017
FG 178 191 $.95 (.88) $.95 (2.17)
Unilateral $.95 5.07 x 1044
2.29 x 1047
FH 188 190 $.99 (10.72) $.99 (42.2)
Unilateral $.99 2.095108 287994339 GH 194 196 $.99 (11.52) $.99 (45.56)
8. Kyngdon and Richards
Figure 1. Proximity graph depicting the dominant I scale path, interpersonal distance eight statement set psy-
chology student sample
ABCDEFGH 2
ab
BACDEFGH 2
ac
BCADEFGH 3
bc ad
CBADEFGH 2 BCDAEFGH 1
ad
bc
ae
CBDAEFGH 5 BCDEAFGH 1
bd ae bc af
CDBAEFGH CBDEAFGH 7 BCDEFAGH 1
cd ae bd af bc ag
DCBAEFGH 6 CDBEAFGH CBDEFAGH 4 BCDEFGAH 2
ae cd be af bd ag bc ah
DCBEAFGH 2 CDEBAFGH CDBEFAGH CBDEFGAH 4 BCDEFGHA 1
be cd af cd af be
ag
bd
ah
bc
DCEBAFGH DCBEFAGH CDEBFAGH CDBEFGAH 4 CBDEFGHA 1
ce af be cd
ag
cd
bf
ag
be
ah bd
DECBAFGH 1 DCEBFAGH DCBEFGAH 6 CDEFBAGH CDEBFGAH CDBEFGHA
de af ce bf
cd
ag
be cd
ah
cd
ag
bf ah
be
EDCBAFGH DECBFAGH DCEFBAGH DCEBFGAH 2 DCBEFGHA 1 CDEFBGAH 4 CDEBFGHA
af
de
bf
ce ag ce ag bf cd
ah
be cd bg ah bf
EDCBFAGH DECFBAGH DECBFGAH DCEFBGAH 5 DCEBFGHA CDEFGBAH 1 CDEFBGHA
bf
de ag de cf ag bf ah ce bg
ah bf cd
ah
cd
bg
EDCFBAGH EDCBFGAH DEFCBAGH DECFBGAH 1 DECBFGHA DCEFGBAH 9 DCEFBGHA CDEFGBHA 3
cf de
ag bf
de ah de ag cf bg
ah ce
bf ce ah bg
bh
EDFCBAGH EDCFBGAH EDCBFGHA DEFCBGAH DECFGBAH 6 DECFBGHA DCEFGBHA 3 CDEFGHBA 3
df
ag cf de
ah
bg de bf de bg cf
ah
ah cf bg ce
bh cd
EFDCBAGH EDFCBGAH EDCFGBAH EDCFBGHA DEFCGBAH 4 DEFCBGHA DECFGBHA 1 DCEFGHBA 4
ef
ag
df bg ah bgcf ah cf bg de cg de ah
bg
de cf
bh ce
FEDCBAGH EFDCBGAH EDFCGBAH EDFCBGHA EDCFGBHA DEFGCBAH 20 DEFCGBHA 1 DECFGHBA 1
ag ef bg df ah df cg
ah
bg cf de
bh de
de
ah cg bh cf
FEDCBGAH EFDCGBAH EFDCBGHA EDFGCBAH 2 EDFCGBHA EDCFGHBA DEFGCBHA 10 DEFCGHBA 2
bg
ef
ah ef
ah
cg
df ah
bg df
cg bh cf de bh de
cg
FEDCGBAH FEDCBGHA EFDGCBAH EFDCGBHA EDFGCBHA EDFCGHBA DEFGCHBA 18
cg ef
ah bg
ef
dg
ah
cg bh df bh df cg
de ch
FEDGCBAH FEDCGBHA EFGDCBAH EFDGCBHA 1 EFDCGHBA EDFGCHBA 1 DEFGHCBA 22
dg ah
cg
ef
ef ah
bh ef dg bh
cg
df ch de
FEGDCBAH FEDGCBHA FEDCGHBA EFGDCBHA EFDGCHBA EDFGHCBA 5
eg
ah dg
bh
cg ef bh ef
dg ch df
FGEDCBAH FEGDCBHA FEDGCHBA EFGDCHBA EFDGHCBA 3
fg ah eg bh dg
ch ef ch ef
dg
GFEDCBAH 1 FGEDCBHA FEGDCHBA FEDGHCBA EFGDHCBA 2
ah fg bh eg ch
dg
ef
dh
GFEDCBHA FGEDCHBA FEGDHCBA 5 EFGHDCBA
bh fg ch eg dh
ef
GFEDCHBA FGEDHCBA 1 FEGHDCBA
ch fg dh eg
GFEDHCBA 2 FGEHDCBA
dh fg eh
GFEHDCBA 2 FGHEDCBA
eh fg
GFHEDCBA 1
fh
GHFEDCBA 1
gh
HGFEDCBA 1
Figure 1. Proximity graph depicting the dominant I scale path, interpersonal distance eight statement set psychology student sample
9. Attitudes, Order and Quantity
and thus a new set of inequalities. Given that $ on
A1
HA2
is an additive structure, then each of these
triples of inequalities also implies a1
x1
$a2
x2
.
The three elements in each column may be
permuted in 3! ways. Thus there are (3!)4
= 1296
matrices realisable with this procedure. Permuting
the elements in the first column does not alter the
composition of the inequalities, only the order
in which they are stated. This leaves (3!)3
= 216
triples of inequalities produced by permuting the
elements in columns two, three and four. Empiri-
cal tests for each of these 216 different versions
of triple cancellation could be developed.A triple
cancellation condition, however, is redundant if its
conclusion follows directly from the assumption
of a lower order condition; that is, at least any
one of independence or double cancellation or
transitivity. Consider the following three anteced-
ent inequality sets:
a1
x3
$a2
x2
a1
x3
$a3
x4
a1
x3
$a3
x2
a3
x1
$a4
x3
a3
x4
$a4
x3
a3
x4
$a4
x3
a4
x4
$a3
x4
a4
x1
$a2
x2
a4
x1
$a2
x4
It is clear that in the first triple, if indepen-
dence is taken to be true, then this version could
never be false, as the third inequality (a4
x4
$a3
x4
)
restates independence. In the second set, the first
and second inequalities hold the transitivity of
the levels on A1
HA2
and thus the conclusion of
this triple follows directly from the assumption of
transitivity. In the third set, both a4
and x4
may be
cancelled from the second and third inequalities,
producing the new statement a3
x1
$a2
x3
.Thus these
two inequalities combine by double cancellation
and as such, if double cancellation is assumed,
then this version of triple cancellation could not
be false. If single cancellation, transitivity and
double cancellation are assumed, then only twelve
of the 216 possible triple cancellation conditions
need to be tested in a 4H4 data matrix. The an-
tecedents are:
1. a1
x3
$a2
x4
2. a1
x3
$a2
x4
3. a1
x4
$a2
x3
4. a1
x4
$a2
x3
a3
x4
$a4
x3
a3
x1
$a4
x2
a3
x3
$a4
x4
a3
x1
$a4
x2
a4
x1
$a3
x2
a4
x4
$a3
x3
a4
x1
$a3
x2
a4
x3
$a3
x4
5. a1
x3
$a2
x4
6. a1
x3
$a2
x4
7. a1
x4
$a2
x3
8. a1
x4
$a2
x3
a3
x4
$a4
x2
a3
x1
$a4
x3
a3
x3
$a4
x2
a3
x1
$a4
x4
a4
x1
$a3
x3
a4
x4
$a3
x2
a4
x1
$a3
x4
a4
x3
$a3
x2
9. a1
x3
$a4
x4
10. a1
x3
$a3
x4
11. a1
x4
$a4
x3
12. a1
x4
$a3
x3
a3
x4
$a2
x3
a3
x1
$a4
x2
a3
x3
$a2
x4
a3
x1
$a4
x2
a4
x1
$a3
x2
a4
x4
$a2
x3
a4
x1
$a3
x2
a4
x3
$a2
x4
These twelve triples of antecedent inequali-
ties fall into three different ‘forms’ of triple can-
cellation, which differ with respect to the position
of the elements of the conclusion (i.e., a1
x1
$a2
x2
)
within the three inequalities (KLST 1971). In the
first form, a1
and a2
both appear in one inequality
and x1
and x2
both appear in another. In the second
form a1
and a2
appear in two different inequalities
and x1
and x2
both appear in the third. The final
form occurs when x1
and x2
appear in two different
inequalities and a1
and a2
are both appear in the
third. Of the 12 versions of triple cancellation four
cases each fall into these three different forms.
In testing the 12 versions of triple cancella-
tion in a 4H4 matrix, the variables a1
, a2
, a3
, a4
and
x1
, x2
, x3
, x4
may be substituted as the four row
elements( say a, b, c, d) and the four column ele-
ments (say w, x, y, z) respectively. Therefore there
are 4!H4! = 576 different substitution instances of
each version of triple cancellation in a 4H4 matrix.
Within each of the three distinct forms of triple
cancellation, the substitutions produced are iden-
tical; and within each form the 576 substitution
instances are not all logically independent of one
another. Rather, the substitution instances fall into
different clusters. Within each cluster the relevant
order relations are between the same four pairs
of cells of the matrix; and if triple cancellation
is supported or rejected in one instance, then the
results of all other instances are known. Between
clusters, the instances of triple cancellation are
logically independent of one another.
As before with the inequality sets, these clus-
ters are not all independent of the independence
axiom. Independence can be graphically repre-
sented in a data matrix if and only if there exists
some permutation of the rows and columns such
that the cells in each row are ordered from least to
greatest from left to right, and the cells within each
column are ordered from least to greatest from top
to bottom. Consider the substitution instance in
Figure 2. The arrows in this matrix point from the
greater to the lesser of the two cells in the order
relation. Satisfaction of this test therefore follows
directly from independence.
10. 10 Kyngdon and Richards
Generally, for the first form of triple can-
cellation, all but 9 clusters of tests are satisfied
whenever independence is satisfied. For both the
second and third forms, all but 6 clusters are satis-
fied whenever independence is satisfied. These 21
different cluster patterns are represented in Figure
3 (graphs a through i for the first form, j through
o and p through u for the second and third forms
respectively). Note that double cancellation does
not define any set order relations on a matrix and
so does not reduce the number of tests further.
Not all 21 patterns in Figure 3, however,
represent coherent empirical tests. Consider those
substitutions where lines intersect. Each such
instance involves contradictory partial sets of
inequalities (Michell, personal communication,
2003). Consider the following example:
In Figure 4, the pair of arrows on the right
hold that ay$bz and cz$dy.Also, by independence
we know that bz$cz and dy$ay. Thus we have
ay$bz$cz$dy and dy$ay, a contradictory set of
inequalities. At the same time, note that the pair
of arrows on the left is not contradictory. Rather,
the contradiction occurs in a substitution instance
whenever arrows of opposite direction intersect,
and arrows of opposite direction intersect once,
and only once, in every instance in Figure 3 where
lines cross. There are seven such substitution in-
stances in Figure 3 (tests c, f, g, h, i, n, r). So, if
a weak order satisfies independence and double
cancellation in a 4H4 matrix, it is sufficient to
perform 14 tests of triple cancellation to test the
hypothesis of additivity relative to that matrix (i.e.
those remaining tests in Figure 3).
Logically, a test of triple cancellation is
violated only if (1) all the antecedent inequalities
are true and the consequent is false, or (2) all the
antecedents are false and the consequent is true.
The possible outcomes for a test of triple cancel-
lation are given in Table 4.
This represents the order relations among
four pairs of cells: ay/bz, cz/dy, dw/cx, and aw/bx.
Whilst not all of the tests of triple cancellation
involve these pairs of cells, the data patterns that
Figure 2. This substitution instance is satisfied when-
ever independence is satisfied.
Figure 4. A contradictory cancellation test.
Figure 3. The 21 clusters of triple cancellation tests in a 4H4 data matrix.
12. 12 Kyngdon and Richards
may obtain between the cells of interest in any test
will have an identical form. Separating $ into its
components and =, it is apparent that any of the
three relations, , or =, may hold between any of
these pairs of cells. There are then 33
distinct data
patterns possible. Those patterns marked by “*”
violate triple cancellation. For each of them the
substitution instances are such that whenever all
of the antecedents are true, the consequent is false,
or vice versa. They are rejections. The remainder
are acceptances. For each of them the substitution
instances are such that whenever all antecedents
are true, the consequent is always true.
As Michell (1988) noted, when employing
the cancellation conditions to test for additivity
in a data matrix, if a test at any level of the hi-
erarchy fails, then the hypothesis of additivity is
contradicted by the data. If independence is not
satisfied in a given dataset, then the hypothesis of
an additive representation must be rejected. In the
present case of 4H4 matrices, the 14 tests of triple
cancellation are necessary, but not sufficient, to
satisfy an additive representation. Rather, the
conjunction of the satisfaction of independence,
the possible tests of double cancellation and the 14
tests of triple cancellation are required to confirm
the hypothesis of additivity on a complete 4H4
data matrix.
Accordingly, given the satisfaction of double
cancellation by the interstimulus midpoint order
entailed by Figure 1, the 14 coherent instances of
triple cancellation were tested upon this midpoint
order. All 14 tests were satisfied (Figure 5).
Figure 6 is the Goode’s algorithm scaling
solution (see Kyngdon 2006b) derived from the
interstimulus midpoint order obtained.
Analyzing the Complete Set
of Paired Comparison Data
with the IRT SHCMpp
The SHCMpp (Andrich 1995) is:
Pr
cosh
cosh cosh
xnij
n j
n j n i
={ }=
−( )
−( )+ −( )
1
b d
b d b d
, (8)
where xnij
0[0,1] is the response of person n to
the pair of items i and j, bn
is estimate of person
n, di
and dj
are the estimates of items i and j, re-
spectively; and cosh(x) = (exp(x)+exp(–x))/2. For
each set of statements participants’ entire sets of
1/2 n(n–1) paired comparisons were subject to
analysis using the SHCMpp.
Table 5 contains the SHCMpp item estimates,
standard errors and fit statistics for the interper-
sonal distance eight statement set. The order upon
the item estimates matched the quantitative J scale
order predicted by the ordinal determinable. The
model displayed an excellent fit to the data.
The Goode’s algorithm (Figure 6) and SHC-
Mpp scaling solutions were linearly transformed
onto 1 – 101 scales (statement F = 1 and A = 101).
A simple linear regression was fitted to the data
using the program SPSS for Windows™ (SPSS,
Inc., 2003). The dependent variable was the
transformed SHCMpp scale. The fit of the model
was good (R2
= .996; F = 679.23, p .001; b0
=
1.03, p .001; b1
= –2.4, n.s) (Figure 7) with the
exception of a non significant constant (b1
).
Discussion
As suggested by Kyngdon (2006b), Study
One employed a set of eight statements rather
than six; and collected complete sets of paired
comparison judgments such that both bilateral
and unilateral judgments could be identified for
each participant.
The OPA was strongly supported (Table 4)
by the multinomial Dirichlet model (Karabat-
sos 2005). A “within-subjects” test of the QPA
using this model found strong support for the
axiom (Table 4); thus lending support to the
“error – count” results of Michell (1994) and
Kyngdon (2006b). The “between – subjects” test
of the QPA comprised all 14 coherent instances
of triple cancellation. These were satisfied; thus
supporting the less stringent results of Michell
(1994), Johnson (2001) and Kyngdon (2006b).
No evidence was found of dependencies within
sets of bilateral and unilateral pairs that shared
one common statement. This may have been con-
trolled by the use of balanced paired comparison
orders (Ross 1934).
Kyngdon (2006b) argued that multi-predicate
statements are not a source of confounding error
13. Attitudes, Order and Quantity 13
Table 5
SHCMpp item locations, standard errors and fit statistics, interpersonal distance eight statement,
psychology student sample.
Statement di
S.E. A B C D E F G H
A 9.018 0.122 • .1 .59 .55 .03 3.89 1.24 .98
B 6.579 0.1 • .28 .01 .8 .51 .48 .15
C 4.416 0.103 • .68 .2 .16 .24 .02
D –0.672 0.109 • .82 .25 .08 .37
E –2.547 0.096 • .46 .09 .48
F –3.782 0.087 • .48 .17
G –4.845 0.092 • .05
H –8.166 0.130 •
Overall SCHMpp model chi-square = 14.17, d.f., = 63, p . 1
† All chi-square fit values for item pairs have 3 degrees of freedom.
Figure 5. Satisfactory tests of the triple cancellation condition. Solid arrows indicate the antecedent relations.
Open arrows indicate the consequent relation.
14. 14 Kyngdon and Richards
in the context of unidimensional unfolding theory,
provided that such statements were constructed
with the ordinal determinable (Michell 1994).
The results of the present study supported this
argument.
A linear relationship was found between
the transformed SHCMpp (Andrich 1995) and
Goode’s algorithm (Figure 7) scales. Kyngdon
(2006b) found both linear and exponential re-
lationships. Given the fit of the SHCMpp to the
data was considerably better than that found by
Kyngdon (2006b), the findings of Study One
support a linear relationship. Such a relationship
is indicative of the SHCMpp and Goode’s scales
providing measurements in different units, rather
than using different conceptions of additivity
(Luce 2001) as speculated by Kyngdon (2006b).
However, Kyngdon (2006b) developed an asym-
metric (with respect to the kernel concept) binary
tree structure to construct statements. Study One
used a symmetric structure. It is possible the
asymmetric structure of Kyngdon’s (2006) study
may have caused the exponential relationship he
discovered.An experimental investigation into the
effect of binary tree structure is thus warranted,
given Johnson (2001) and Michell (1994) also
created asymmetric binary tree structures.
Furthermore, both Study One, Johnson
(2001), Michell (1994) and Kyngdon (2006b)
used convenience samples of participants. Such
samples are not representative of the adult popula-
tion and are therefore biased.
Study Two: Symmetric and Asymmetric
Binary Tree Structures
Method
Stimuli
A set of eight statements expressing attitudes
towards illegal immigrants in Australia was cre-
ated using Michell’s (1994) theory of the ordinal
determinable (see Appendix One). These were
as follows:
A. Not only should boat people be allowed to
land on Australian shores and not be de-
tained, they should automatically be entitled
to become full Australian citizens.
Figure6.Goode’sminimumintegerscalingsolutionfortheinterstimulusmidpointorderentailedbyFigure1.
15. Attitudes, Order and Quantity 15
B. Boat people should be allowed to land in
Australia without being detained and they
should be entitled to become Australian
citizens, but not automatically.
C. Although boat people should be allowed to
land in Australia, they should be detained
but should automatically be entitled for
residency.
D. After arriving inAustralia and being detained,
boat people should not automatically be en-
titled for residency but only a temporary visa.
E. After arriving in Australia and being de-
tained, boat people should not be automati-
cally entitled even to a temporary visa.
F. Boat people should not be allowed to land in
Australia and they must be detained in some
offshore country, but they should be entitled
to apply for a temporary visa.
G. Boat people should not be allowed to land
in Australia and they must be detained in
some offshore country and they should not be
entitled even to apply for a temporary visa.
H. Boat people should not be allowed to land in
Australia, nor should they be detained in an
offshore country, but rather should be sent
directly home.
As the interpersonal distance statements were
constructed with a symmetric ordinal determin-
able, the illegal immigrant statements were con-
structed from an asymmetric one. The symmetric
interpersonal distance determinable was modified
to produce an asymmetric determinable with the
following set of six statements:
A. Not only would I be friends with a homo-
sexual and have sex with one, I would be
involved in a life partner relationship with a
homosexual.
B. I’d be friends with a homosexual and have sex
with one, but I would not be involved in a life
partner relationship with a homosexual.
C. I’d speak to a homosexual in passing and mix
with one socially, but I wouldn’t be friends
with one.
Figure 7. The linear relationship between the transformed Goode’s and SHCMpp scales, interpersonal distance
eight statement set, psychology student sample.
16. 16 Kyngdon and Richards
D. I’d speak to a homosexual in passing, but I
wouldn’t be friends with one or mix with one
socially.
E. I wouldn’t be friends with a homosexual or
even speak to one in passing, but I would not
wish harm upon a homosexual.
F. I wouldn’t be friends with a homosexual or
even speak to one in passing, and in fact, I
would wish harm upon a homosexual.
Theasymmetricillegalimmigrantdeterminable
was modified to produce a symmetric determinable
with the following set of six statements:
A. Not only should boat people be allowed to
land on Australian shores and not be de-
tained, they should automatically be entitled
to become full Australian citizens.
B. Boat people should be allowed to land in
Australia without being detained and they
should be entitled to become Australian
citizens, but not automatically.
C. Although boat people should be allowed to
land in Australia, they should be detained
but should automatically be entitled for
residency.
D. Boat people should not be allowed to land in
Australia and they must be detained in some
offshore country, but they should be entitled
to apply for a temporary visa.
E. Boat people should not be allowed to land
in Australia and they must be detained in
some offshore country and they should not be
entitled even to apply for a temporary visa.
F. Boat people should not be allowed to land in
Australia, nor should they be detained in an
offshore country, but rather should be sent
directly home.
Design
The study employed an Internet based experi-
mental survey design consisting of 8 conditions.
Participants were randomly assigned into one of
these conditions. Participants in each condition
completed one set of the interpersonal distance
statements and one set of the illegal immigrant
statements. Hence the conditions were:
1. Interpersonal distance 8 statements + illegal
immigrant 8 statements.
2. Interpersonal distance 8 statements + illegal
immigrant 6 statements.
3. Interpersonal distance 6 statements + illegal
immigrant 8 statements.
4. Interpersonal distance 6 statements + illegal
immigrant 6 statements.
The remaining four conditions were simply
the above conditions reversed to control for the
order of presentation (e.g., illegal immigrant 8
statement + interpersonal distance 8 statement).
For each set in each condition participants com-
pleted three different tasks. The first task was to
rank the statements from either 1 to 6 or 1 to 8.
The second task was to rate each statement on a
7-point Likert (1932) type rating scale where “0”
corresponded to “Strongly Disagree” and “6” to
“Strongly Agree”. The third task was to respond
to a full set of either 15 or 28 paired comparisons.
Ross’s (1934) balanced optimal presentation order
for paired comparisons was used.
Materials
Materials used in Study Two were the same
as those employed in Study One.
Participants and data collection procedure
The setup and administration of the survey
via the Internet was conducted by the private com-
pany SurveyEngine Pty Ltd. The company also
recruited participants such that a sample repre-
sentative of theAustralian adult population (with
respect to age and gender) was obtained. Each
participant was invited via email to participate
by the company. Those choosing to participate
simply logged onto the relevant Internet website
and followed the instructions. The participants
were free to discontinue their participation at any
time simply by logging off.
Two hundred and one participants completed
all tasks for the set of eight interpersonal distance
statements, with 50 only completing either or
both of the rank order and rating tasks only. Two
hundred and four participants completed all tasks
for the set of six interpersonal distance statements,
17. Attitudes, Order and Quantity 17
with 41 only completing either or both of the rank
order and rating tasks. One hundred and eighty
six participants completed all tasks for the set of
eight illegal immigrant statements, with 50 only
completing either or both of the rank order and
rating tasks. Two hundred and eight participants
completed all tasks for the set of six interpersonal
distance statements, with 47 only completing
either or both of the rank order and rating tasks.
Results
Interpersonal distance eight statement set
Out of the 201 participants who completed
all rank order, ratings and paired comparisons
tasks for this set of statements, only 179 gave suf-
ficiently transitive paired comparison judgments
for the most preferred statement to be identified.
Table 6 contains the multinomial Dirichlet model
Bayesian analyses of the QPA and OPA.
The DE unilateral pair violated the OPA. The
BFG statistic suggested “weak” overall support
for the OPA (H0
: q $.50 not rejected). As the H0
: q $.95 hypothesis was not rejected, the bilateral
data overall “strongly” supported the QPA.
There was “strong” support for the QPA
(H0
: q $.95 not rejected) in all sets of common
stimulus bilateral paired comparisons.A “lateral-
ity” (Coombs 1964) dependence effect between
the sets of common stimulus unilateral paired
comparisons was found. Within sets where the
common stimulus was either A or H (the most
“extreme” attitude statements) the OPA was
“strongly” supported (H0
: q $.95 not rejected).
Sets pertaining to statements B, C, F and G
“moderately” supported the OPA (H0
: q $.75 not
rejected); whilst sets pertaining to statements D
and E “weakly” supported the axiom (H0
: q $.50
not rejected).
Table 6
Bayesian analysis results, interpersonal distance eight statement set, Internet sample.
Bilateral H0
not rejected (Bayes Factor) Unilateral H0
not rejected (Bayes Factor)
s N Jeffreys Uniform s N Jeffreys Uniform
AC 12 17 $.75 (.97) $.75 (1.18) AB 136 153 $.75 (231812) $.75 (250195)
AD 39 41 $.99 (.35) $.99 (.86) AC 122 129 $.95 (3.81) $.95 (9.16)
AE 70 76 $.95 (.86) $.95 (10.28) AD 91 94 $.99 (.5) $.99 (.156)
AF 137 145 $.95 (3.47) $.95 (1.89) AE 23 25 $.95 (1.7) $.95 (3.06)
AG 150 155 $.99 (.15) $.99 (.51) AF 12 15 $.75 (3.82) $.95 (.13)
AH 160 166 $.95 (22.28) $.99 (.16) AG 4 4 $.99 (4.39) $.99 (5.1)
BD 23 24 $.99 (1.21) $.99 (2.62) BC 119 138 $.75 (3202.5) $.75 (3772.7)
BE 56 59 $.99 (.12) $.99 (.31) BD 100 103 $.99 (.65) $.99 (2.11)
BF 122 128 $.95 (7) $.95 (16.67) BE 30 34 $.95 (.31) $.95 (.57)
BG 129 138 $.95 (1.49) $.95 (3.6) BF 23 24 $.99 (1.21) $.99 (2.62)
BH 141 149 $.95 (3.93) $.95 (9.62) BG 13 13 $.99 (9.55) $.99 (14.96)
CE 32 35 $.95 (1.14) $.95 (2.2) BH 8 9 $.99 (.28) $.99 (.42)
CF 99 104 $.95 (6.17) $.95 (14.27) CD 111 120 $.95 (.73) $.99 (1.7)
CG 107 114 $.95 (2.24) $.95 (5.3) CE 44 51 $.75 (76) $.75 (87.45)
CH 124 125 $.99 (16.3) $.99 (55.54) CF 39 41 $.99 (.35) $.99 (.86)
DF 69 69 $.99 (46.99) $.99 (101.07) CG 26 30 $.95 (.2) $.95 (.35)
DG 77 79 $.99 (1.55) $.99 (4.83) CH 22 26 $.95 (.11) $.95 (.19)
DH 87 90 $.99 (.44) $.99 (1.35) DE 13 75 Axiom violated Axiom violated
EG 9 10 $.99 (.33) $.99 (.52) DF 58 65 $.95 (.16) $.95 (.33)
EH 18 21 $.95 (.26) $.99 (.43) DG 49 54 $.95 (.55) $.95 (1.13)
FH 10 11 $.99 (.38) $.95 (.62) DH 50 50 $.99 (31.94) $.99 (66.29)
EF 84 100 $.75 (127.64) $.75 (158.26)
Bayes Factor G. Jeffreys Uniform EG 74 89 $.75 (57.36) $.75 (72.13)
Bilateral $.95 1.85 x 1014
1.36 x 1020
EH 79 85 $.95 (1.36) $.95 (3.07)
Bilateral $.99 1.32 x 10–32
1.45 x 10–24
FG 133 158 $.75 (736.77) $.75 (917.88)
Unilateral $.50 +¥ +¥ FH 146 154 $.95 (4.57) $.95 (11.24)
Unilateral $.75 0 0 GH 157 164 $.95 (11.06) $.95 (26.86)
Unilateral $.95 0 0
Unilateral $.99 0 0
18. 18 Kyngdon and Richards
Figure 8 is the I scale proximity graph depict-
ing the I scales obtained from the bilateral paired
comparison data. As the order of the BG and CD
midpoints to each other was undetermined, and
similarly for the DG and EF midpoints, the domi-
nant path condition was not satisfied.
Thirty six of the 140 tests of the double
cancellation axiom of ACM (KLST 1971) were
rejected by the midpoint order entailed by Figure
8. These tests appear in Appendix Two.
Table 7 contains the SHCMpp item esti-
mates, standard errors and fit statistics for the
interpersonal distance eight statement set. The
order upon the item estimates did not match the
order predicted by the ordinal determinable. The
model displayed an excellent fit to the data with
the exception of the CH statement pair.
Illegal immigrant eight statement set
Ofthe186participantswhocompletedalltasks,
158 gave sufficiently transitive paired comparison
judgments for the most preferred statement to be
identified.Table8containsthemultinomialDirichlet
model analyses of the QPA and OPA.
ABCDEFGH 9
ab
BACDEFGH 2
ac
BCADEFGH 1
bc ad
CBADEFGH 2 BCDAEFGH 4
ad
bc
ae
CBDAEFGH BCDEAFGH 2
bd ae bc af
CDBAEFGH 6 CBDEAFGH BCDEFAGH 2
cd ae bd af bc ag
DCBAEFGH 7 CDBEAFGH CBDEFAGH BCDEFGAH 5
ae cd be af bd ag bc ah
DCBEAFGH CDEBAFGH 2 CDBEFAGH 1 CBDEFGAH 1 BCDEFGHA 1
be cd af cd af be
ag
bd
ah
bc
DCEBAFGH DCBEFAGH CDEBFAGH CDBEFGAH 0 CBDEFGHA
ce af be cd
ag
cd
bf
ag
be
ah bd
DECBAFGH DCEBFAGH DCBEFGAH CDEFBAGH 2 CDEBFGAH 1 CDBEFGHA
de af ce bf
cd
ag
be cd
ah
cd
ag
bf ah
be
EDCBAFGH DECBFAGH DCEFBAGH DCEBFGAH DCBEFGHA CDEFBGAH 4 CDEBFGHA
af
de
bf
ce ag ce ag bf cd
ah
be cd bg ah bf
EDCBFAGH DECFBAGH DECBFGAH DCEFBGAH 2 DCEBFGHA CDEFGBAH 2 CDEFBGHA
bf
de ag de cf ag bf ah ce bg
ah bf cd
ah
cd
bg
EDCFBAGH EDCBFGAH DEFCBAGH DECFBGAH DECBFGHA DCEFGBAH 2 DCEFBGHA CDEFGBHA 2
cf de
ag bf
de ah de ag cf bg
ah ce
bf ce ah bg
bh
EDFCBAGH EDCFBGAH EDCBFGHA DEFCBGAH DECFGBAH 2 DECFBGHA DCEFGBHA 1 CDEFGHBA 1
df
ag cf de
ah
bg de bf de bg cf
ah
ah cf bg ce
bh cd
EFDCBAGH EDFCBGAH 2 EDCFGBAH 4 EDCFBGHA DEFCGBAH 0 DEFCBGHA DECFGBHA DCEFGHBA 4
ef
ag
df bg ah bgcf ah cf bg de cg de ah
bg
de cf
bh ce
FEDCBAGH EFDCBGAH EDFCGBAH EDFCBGHA EDCFGBHA DEFGCBAH 6 DEFCGBHA 1 DECFGHBA 1
ag ef bg df ah df cg
ah
bg cf de
bh de
de
ah cg bh cf
FEDCBGAH EFDCGBAH EFDCBGHA EDFGCBAH 28 EDFCGBHA EDCFGHBA 2 DEFGCBHA 1 DEFCGHBA
bg
ef
ah ef
ah
cg
df ah
bg df
cg bh cf de bh de
cg
FEDCGBAH FEDCBGHA EFDGCBAH EFDCGBHA EDFGCBHA 3 EDFCGHBA DEFGCHBA 1
cg ef
ah bg
ef
dg
ah
cg bh df bh df cg
de ch
FEDGCBAH FEDCGBHA EFGDCBAH 1 EFDGCBHA EFDCGHBA EDFGCHBA 5 DEFGHCBA 7
dg ah
cg
ef
ef ah
bh ef dg bh
cg
df ch de
FEGDCBAH 1 FEDGCBHA FEDCGHBA EFGDCBHA EFDGCHBA 1 EDFGHCBA 19
eg
ah dg
bh
cg ef bh ef
dg ch df
FGEDCBAH 2 FEGDCBHA FEDGCHBA EFGDCHBA EFDGHCBA 2
fg ah eg bh dg
ch ef ch ef
dg
GFEDCBAH 5 FGEDCBHA FEGDCHBA FEDGHCBA 1 EFGDHCBA 1
ah fg bh eg ch
dg
ef
dh
GFEDCBHA FGEDCHBA 1 FEGDHCBA 1 EFGHDCBA
bh fg ch eg dh
ef
GFEDCHBA 1 FGEDHCBA 2 FEGHDCBA
ch fg dh eg
GFEDHCBA 4 FGEHDCBA
dh fg eh
GFEHDCBA 1 FGHEDCBA 2
eh fg
GFHEDCBA 0
fh
GHFEDCBA 0
gh
HGFEDCBA 4
Figure 8. Proximity graph depicting the obtained I scales, interpersonal distance eight statement set, Internet sample
Figure 8. Proximity graph depicting the obtained I scales, interpersonal distance eight statement set; Internet
sample.
19. Attitudes, Order and Quantity 19
Under the Jeffrey’s prior, both the QPA and
OPA were “moderately” supported (H0
: q $.75
not rejected). However, the uniform prior did not
reject the H0
: q $.95 hypothesis for the QPA, sug-
gesting “strong” support for the axiom.
The OPA was “moderately” supported (H0
:
q $.75 not rejected) by all sets of common stimu-
lus unilateral paired comparisons except that set
pertaining to statement A where the OPA was
“strongly” supported (H0
: q $.95 not rejected).
Table 7
SHCMpp item locations, standard errors and fit statistics, interpersonal distance, eight statement,
Internet sample.
Statement di
S.E. A B C D E F G H
A 5.654 .095 • .3 .44 .77 .2 1.15 .76 2.11
B 4.450 .083 • .31 1.71 .13 .98 .62 4.25
C 3.305 .086 • 1.13 .27 .27 1.75 15.25**
D –1.305 .095 • 1.12 .24 .03 .05
E –1.127 .086 • .15 .07 1.13
F –2.461 .081 • .3 .12
G –3.506 .081 • .06
H –5.459 .107 •
Overall SCHMpp model chi-square = 35.668, d.f, = 63, p = .997
** p .01 † All chi-square fit values for item pairs have 3 degrees of freedom.
Table 8
Bayesian analysis results, illegal immigrant eight statement set, Internet sample.
Bilateral H0
not rejected (Bayes Factor) Unilateral H0
not rejected (Bayes Factor)
s N Jeffreys Uniform s N Jeffreys Uniform
AC 11 11 $.99 (8.46) $.99 (12.69) AB 139 146 $.95 (6.53) $.95 (15.84)
AD 18 21 $.95 (.26) $.95 (.43) AC 130 136 $.95 (9.03) $.95 (21.51)
AE 59 62 $.99 (.14) $.99 (.37) AD 91 95 $.95 (9.45) $.99 (.29)
AF 67 75 $.95 (.13) $.95 (.26) AE 78 82 $.95 (5.71) $.99 (.15)
AG 102 107 $.95 (6.86) $.95 (15.9) AF 49 50 $.99 (3.64) $.99 (10.09)
AH 106 109 $.99 (.77) $.99 (2.54) AG 48 48 $.99 (30.53) $.99 (63)
BD 9 10 $.99 (.33) $.99 (.52) BC 78 137 $.50 (18.2) $.50 (17.97)
BE 41 51 $.75 (8.61) $.75 (10.98) BD 92 96 $.99 (.1) $.99 (.3)
BF 58 64 $.95 (.41) $.95 (.86) BE 76 83 $.95 (.54) $.95 (1.19)
BG 85 96 $.75 (4132.6) $.75 (4498.4) BF 48 51 $.95 (3.25) $.99 (.18)
BH 88 98 $.75 (17493) $.95 (.21) BG 48 49 $.99 (3.53) $.99 (9.72)
CE 35 41 $.75 (33.57) $.75 (39.04) BH 1 1 $.99 (2.14) $.99 (2.01)
CF 51 54 $.95 (3.81) $.99 (.23) CD 97 107 $.95 (.17) $.95 (.38)
CG 81 86 $.95 (3.08) $.95 (6.99) CE 86 94 $.95 (.44) $.95 (.97)
CH 76 84 $.95 (.24) $.95 (.51) CF 61 62 $.99 (5.07) $.99 (14.96)
DF 13 13 $.99 (9.55) $.99 (14.96) CG 57 60 $.99 (.13) $.99 (.33)
DG 40 45 $.95 (.27) $.95 (.51) CH 9 12 $.75 (1.82) $.75 (2.13)
DH 42 47 $.95 (.32) $.95 (.62) DE 70 104 $.50 (5474.7) $.75 (.11)
EG 30 32 $.99 (.2) $.99 (.43) DF 53 72 $.75 (1.25) $.75 (1.69)
EH 28 34 $.75 (10.43) $.75 (12.63) DG 62 70 $.75 (822.22) $.95 (.17)
FH 2 2 $.99 (2.98) $.99 (3.03) DH 20 22 $.95 (1.27) $.99 (.15)
EF 60 113 $.50 (2.92) $.50 (2.9)
Bayes Factor G. Jeffreys Uniform EG 94 111 $.75 (288.1) $.75 (352.46)
Bilateral $.75 1.57 x 1072
1.13 x 1071
EH 57 63 $.95 (.38) $.95 (.8)
Bilateral $.95 2.46 x 10–5
4.34 FG 111 124 $.75 (82296) $.75 (86518)
Bilateral $.99 6.39 x 10–63
9.36 x 10–57
FH 69 76 $.95 (.36) $.95 (.76)
Unilateral $.75 7.69 x 10100
1.56 x 1099
GH 78 108 $.75 (.66) $.75 (.19)
Unilateral $.95 0 0
Unilateral $.99 0 0
20. 20 Kyngdon and Richards
The QPA was “strongly” supported by those sets
of common stimulus bilateral paired comparisons
pertaining to statements A, C, D, F and G (H0
: q $.95 not rejected); whilst sets pertaining to
statements B, E and H “moderately” supported
the axiom (H0
: q $.75 not rejected). Thus limited
evidence of common stimulus pair dependence
was found in the bilateral data.
Figure 9 is the I scale proximity graph depict-
ing the I scales obtained from the bilateral paired
comparison data. As the order of the BF and CE
midpoints to each other was undetermined, the
dominant path condition was not satisfied.
Nine of the 140 tests of double cancellation
were rejected by the midpoint order entailed by
Figure 9 (Figure 10).
Table 9 contains the SHCMpp results for
the illegal immigrant eight statement set. The
order upon the item estimates did not match the
order predicted by the ordinal determinable. The
model displayed an excellent fit to the data with
the exception of the AH stimulus pair.
Interpersonal distance six statement set
Of the 204 participants who completed all
tasks, 190 gave sufficiently transitive paired
Figure 9. Proximity graph depicting the obtained I scales, illegal immigrant eight statement set.
ABCDEFGH 1
ab
BACDEFGH 4
ac
BCADEFGH 2
bc ad
CBADEFGH BCDAEFGH 1
ad
bc
ae
CBDAEFGH 0 BCDEAFGH 2
bd ae bc af
CDBAEFGH 2 CBDEAFGH BCDEFAGH
cd ae bd af bc ag
DCBAEFGH 5 CDBEAFGH CBDEFAGH 1 BCDEFGAH 2
ae cd be af bd ag bc ah
DCBEAFGH 1 CDEBAFGH CDBEFAGH 1 CBDEFGAH 1 BCDEFGHA
be cd af cd af be
ag
bd
ah
bc
DCEBAFGH DCBEFAGH 1 CDEBFAGH CDBEFGAH CBDEFGHA
ce af be cd
ag
cd
bf
ag
be
ah bd
DECBAFGH 1 DCEBFAGH DCBEFGAH 2 CDEFBAGH 1 CDEBFGAH CDBEFGHA
de af ce bf
cd
ag
be cd
ah
cd
ag
bf ah
be
EDCBAFGH DECBFAGH DCEFBAGH 1 DCEBFGAH DCBEFGHA 1 CDEFBGAH CDEBFGHA
af
de
bf
ce ag ce ag bf cd
ah
be cd bg ah bf
EDCBFAGH DECFBAGH DECBFGAH 1 DCEFBGAH DCEBFGHA 1 CDEFGBAH CDEFBGHA 2
bf
de ag de cf ag bf ah ce bg
ah bf cd
ah
cd
bg
EDCFBAGH EDCBFGAH DEFCBAGH DECFBGAH DECBFGHA 0 DCEFGBAH DCEFBGHA 0 CDEFGBHA
cf de
ag bf
de ah de ag cf bg
ah ce
bf ce ah bg
bh
EDFCBAGH EDCFBGAH EDCBFGHA DEFCBGAH DECFGBAH 1 DECFBGHA 0 DCEFGBHA CDEFGHBA 2
df
ag cf de
ah
bg de bf de bg cf
ah
ah cf bg ce
bh cd
EFDCBAGH EDFCBGAH EDCFGBAH EDCFBGHA DEFCGBAH DEFCBGHA 6 DECFGBHA 1 DCEFGHBA
ef
ag
df bg ah bgcf ah cf bg de cg de ah
bg
de cf
bh ce
FEDCBAGH 1 EFDCBGAH EDFCGBAH EDFCBGHA EDCFGBHA DEFGCBAH 2 DEFCGBHA 1 DECFGHBA
ag ef bg df ah df cg
ah
bg cf de
bh de
de
ah cg bh cf
FEDCBGAH EFDCGBAH EFDCBGHA 1 EDFGCBAH 2 EDFCGBHA EDCFGHBA DEFGCBHA 3 DEFCGHBA 1
bg
ef
ah ef
ah
cg
df ah
bg df
cg bh cf de bh de
cg
FEDCGBAH 1 FEDCBGHA 3 EFDGCBAH EFDCGBHA EDFGCBHA EDFCGHBA 1 DEFGCHBA 2
cg ef
ah bg
ef
dg
ah
cg bh df bh df cg
de ch
FEDGCBAH FEDCGBHA EFGDCBAH EFDGCBHA EFDCGHBA EDFGCHBA DEFGHCBA 7
dg ah
cg
ef
ef ah
bh ef dg bh
cg
df ch de
FEGDCBAH FEDGCBHA 3 FEDCGHBA 1 EFGDCBHA EFDGCHBA EDFGHCBA 5
eg
ah dg
bh
cg ef bh ef
dg ch df
FGEDCBAH FEGDCBHA FEDGCHBA 1 EFGDCHBA EFDGHCBA 0
fg ah eg bh dg
ch ef ch ef
dg
GFEDCBAH FGEDCBHA FEGDCHBA FEDGHCBA 12 EFGDHCBA 2
ah fg bh eg ch
dg
ef
dh
GFEDCBHA FGEDCHBA 1 FEGDHCBA 2 EFGHDCBA 2
bh fg ch eg dh
ef
GFEDCHBA FGEDHCBA FEGHDCBA 3
ch fg dh eg
GFEDHCBA FGEHDCBA 1
dh fg eh
GFEHDCBA FGHEDCBA 3
eh fg
GFHEDCBA 0
fh
GHFEDCBA 2
gh
HGFEDCBA 48
Figure 9. Proximity graph depicting the obtained I scales, illegal immigrant eight statement set
21. Attitudes, Order and Quantity 21
comparison judgments for the most preferred
statement to be identified. Table 10 contains the
multinomial Dirichlet model analyses of the QPA
and OPA.
The QPA was “strongly” supported (H0
: q
$.99 not rejected) under both priors. Moreover,
the axiom was “strongly” supported (H0
: q $.99
not rejected) by all sets of common stimulus bilat-
eral paired comparisons. For the OPA, “moderate”
(H0
: q $.75 not rejected) and “strong” support (H0
:
q $.95 not rejected) was found under the Jeffrey’s
and uniform priors, respectively. The OPA was
Table 9
SHCMpp item locations, standard errors and fit statistics, illegal immigrant eight statement, Inter-
net sample.
Statement di
S.E. A B C D E F G H
A 4.917 .116 • 0.16 .11 .19 .41 .33 .15 10.27*
B 2.943 .091 • .23 .09 .31 .11 .39 .72
C 2.836 .093 • .04 .16 .09 .34 .66
D –.278 .119 • .22 .03 .19 .12
E –1.797 .098 • .25 .13 .09
F –1.487 .091 • .04 .09
G –3.181 .090 • .37
H –3.954 .094 •
Overall SCHMpp model chi-square = 16.3, d.f., = 63, p . 1
**p .05 † All chi-square fit values for item pairs have 3 degrees of freedom.
Figure 10. Rejection tests of the double cancellation axiom, illegal immigrant eight statement dataset.
22. 22 Kyngdon and Richards
Table 10
Bayesian analysis results, interpersonal distance six statement set, Internet sample.
Bilateral H0
not rejected (Bayes Factor) Unilateral H0
not rejected (Bayes Factor)
s N Jeffreys Uniform s N Jeffreys Uniform
AC 37 38 $.99 (2.4) $.99 (6.1) AB 130 139 $.95 (1.55) $.95 (3.73)
AD 133 138 $.99 (1.5) $.99 (5.4) AC 35 39 $.95 (.5) $.95 (.96)
AE 142 146 $.99 (.48) $.99 (1.67) AD 29 31 $.99 (.18) $.99 (.4)
AF 174 176 $.99 (9) $.99 (34.97) AE 1 1 $.99 (2.14) $.99 (2)
BD 99 100 $.99 (11.01) $.99 (36.23) BC 48 52 $.95 (1.3) $.95 (2.71)
BE 105 108 $.99 (.75) $.99 (2.47) BD 41 44 $.95 (2.15) $.99 (.11)
BF 134 138 $.99 (.4) $.99 (1.34) BE 13 14 $.99 (.54) $.99 (.96)
CE 8 8 $.99 (6.79) $.99 (9.37) BF 13 13 $.99 (9.55) $.99 (14.96)
CF 37 38 $.99 (2.4) $.99 (6.1) CD 62 82 $.75 (2.37) $.75 (3.21)
DF 30 30 $.99 (19.03) $.99 (36.19) CE 44 52 $.75 (38.9) $.75 (46.53)
CF 49 51 $.99 (.58) $.99 (1.54)
Bayes Factor G. Jeffrey’s Uniform DE 134 152 $.75 (72520) $.75 (80395)
Bilateral $.95 1.26 x 1018
1.79 x 1020
DF 147 151 $.99 (.54) $.99 (1.91)
Bilateral $.99 15939.36 478604870 EF 157 159 $.99 (7.16) $.99 (27.29)
Unilateral $.75 +∞ +∞
Unilateral $.95 0.000072 0.1555453
Unilateral $.95 0 0
“strongly” supported (H0
: q $.95 not rejected) in
the common stimulus sets germane to statements
A, B and E; and in the set germane to F (H0
: q
$.99 not rejected). Sets pertaining to statements
C and D “moderately” supported the OPA (H0
:
q $.75 not rejected), suggesting limited common
statement unilateral pair dependence.
Figure 8 is the I scale proximity graph depict-
ing the I scales obtained from the bilateral paired
comparison data. The dominant path condition
was satisfied; with 85% of I scales obtained lo-
cated on this path.
Double cancellation was satisfied by the mid-
point order entailed by Figure 11 (Figure 12).
The Goode’s minimum integer algorithm was
used to derive a scaling solution for the midpoint
order entail in Figure 11 (Figure 13).
For the interpersonal distance six statement
set, the SHCMpp produced an order upon the
item estimates that matched the order predicted
by the ordinal determinable (Table 11). The fit of
the model to the data was excellent.
The Goode’s algorithm (Figure 9) and SHC-
Mpp scaling solutions for this set of statements
were linearly transformed onto 1 – 101 scales
(statement F = 1 and A = 101). A simple linear
regression was fitted to the data using the program
SPSS for Windows™ (SPSS, Inc., 2002). The
dependent variable was the transformed SHCMpp
scale. The fit of the model was good (R2
= .993;
F = 653.23, p .001; b0
= 1.048, p .001; b1
=
–2.698, n.s) (Figure 14) with the exception of a
non significant constant (b1
).
Illegal immigrant six statement set
Of the 208 participants who completed all
tasks, 178 gave sufficiently transitive paired
comparison judgments for the most preferred
statement to be identified. Table 12 contains the
multinomial Dirichlet model results for the QPA
and OPA.
The QPA was “strongly” supported (H0
: q
$.95 not rejected) and the OPA “moderately” sup-
ported (H0
: q $.75 not rejected) under both prior
distributions.All sets of common stimulus unilat-
eral paired comparisons “moderately” supported
the OPA (H0
: q $.75 not rejected) except that set
pertaining to statementA where it was “strongly”
supported (H0
: q $.99 not rejected). The sets of
common stimulus bilateral paired comparisons
germane to statements A, C and F “strongly”
supported the QPA (H0
: q $.95 not rejected); as
did the set germane to statement D (H0
: q $.99
not rejected). Sets germane to statements B and E,
however, “moderately” supported the axiom (H0
:
q $.75 not rejected), suggesting limited common
statement bilateral pair dependence.
23. Attitudes, Order and Quantity 23
Figure 15 is the I scale proximity graph
depicting the I scales obtained from the bilateral
paired comparison data.The dominant path condi-
tion was satisfied; with 88% of I scales obtained
located on this path.
Double cancellation was rejected by the mid-
point order entailed by Figure 15 (Figure 16).
For the illegal immigrant six statement set,
the SHCMpp produced an order upon the item
estimates that did not match the order predicted
by the ordinal determinable (Table 12). The fit of
the model to the data was excellent.
Discussion
The aim of this paper was to expand upon
the research of Johnson (2001), Kyngdon
(2006b), Michell (1994) and Sherman (1994).
This involved direct probabilistic tests of the
OPA and QPA in addition to testing triple can-
cellation (KLST 1971). Complete sets of paired
comparison judgments enabled more satisfac-
tory tests of the SHCMpp (Andrich 1995) than
those conducted by Kyngdon (2006b).A random
sample of the Australian adult population was
also obtained.
Figure 11. Proximity graph depicting the dominant I scale path (bold font I scales), interpersonal distance six
statement dataset. The numbers of participants who gave an I scale are in parentheses.
24. 24 Kyngdon and Richards
Figure 12. Satisfactory tests of the double cancellation axiom, interpersonal distance six statement set.
Figure 13. Goode’s scaling solution for the interstimulus midpoint order given by Figure 8.
Table 11
SHCMpp item locations, standard errors and fit statistics, interpersonal distance six statement,
Internet sample.
Statement di
S.E. A B C D E F
A 5.735 .108 • .27 .86 1.36 1.92 6.27
B 4.243 .097 • 1.27 1.06 2.72 1.4
C –.860 .100 • .44 .16 .70
D –1.749 .097 • .15 .25
E –2.434 .090 • .04
F –4.935 .118 •
Overall SCHMpp model chi-square = 18.851, d.f., = 30, p = .943
† All chi-square fit values for item pairs have 3 degrees of freedom.
25. Attitudes, Order and Quantity 25
Table 12
Bayesian analysis results, illegal immigrant six statement set, Internet sample.
Bilateral H0
not rejected (Bayes Factor) Unilateral H0
not rejected (Bayes Factor)
s N Jeffreys Uniform s N Jeffreys Uniform
AC 40 40 $.99 (25.17) $.99 (50.48) AB 137 129 $.95 (2.67) $.95 (6.48)
AD 43 45 $.99 (.44) $.99 (1.1) AC 128 132 $.99 (.34) $.99 (1.12)
AE 103 106 $.99 (.71) $.99 (2.32) AD 71 71 $.99 (48.77) $.99 (105.13)
AF 119 125 $.95 (6.36) $.95 (15.11) AE 52 52 $.99 (33.39) $.99 (69.64)
BD 5 5 $.99 (5.02) $.99 (6.15) BC 62 133 $.50 (.28) $.50 (.28)
BE 52 66 $.75 (6.25) $.75 (8.21) BD 64 72 $.75 (1188.2) $.95 (.2)
BF 73 85 $.75 (268.78) $.75 (317.2) BE 52 53 $.99 (3.98) $.99 (11.22)
CE 58 61 $.99 (.13) $.99 (.35) BF 1 1 $.99 (2.14) $.99 (2.01)
CF 73 80 $.95 (.46) $.95 (.99) CD 96 112 $.75 (687) $.75 (818.77)
DF 19 19 $.99 (12.82) $.99 (22.04) CE 83 93 $.75 (6821.7) $.95 (.14)
CF 37 41 $.95 (.59) $.95 (1.15)
Bayes Factor G. Jeffreys Uniform DE 90 98 $.95 (.54) $.95 (1.21)
Bilateral $.95 3.606215 544.2666 DF 40 46 $.75 (80.15) $.95 (.16)
Bilateral $.99 1.65 x 10–28
2.5 x 10–25
EF 82 107 $.75 (3.65) $.75 (4.98)
Unilateral $.75 1.35 x 1054
3.88 x 1052
Unilateral $.95 0 0
Unilateral $.99 0 0
Figure 14. The linear relationship between the transformed Goode’s and SHCMpp scales, interpersonal distance
six statement set, Internet sample.
26. 26 Kyngdon and Richards
The interpersonal distance eight and six state-
ment datasets (Tables 4 and 10) were the only
datasets to satisfy the ACM cancellation axioms
(Figures 5 and 12). In only these datasets did the
multinomial Dirichlet model not reject the null
hypothesis H0
: q $.99 for the QPA under both
priors. No evidence of dependence between com-
mon statement bilateral pairs was found in either
dataset. The OPA was “strongly” supported (H0
:
q $.99 not rejected) (Table 4) whilst in Table 10
the OPA was “strongly” supported (H0
: q $.95 not
rejected) under the uniform prior. These probabi-
listic results are consistent with the argument that
the OPA and QPA must both hold for Coombs’s
(1964) theory to be true (Michell 1994).
For every other dataset (Tables 6, 8 and 12)
the QPA was strongly supported (H0
: q $.95 not
rejected) under at least one prior distribution.
There was no commensurate support of the OPA
in these datasets. In the datasets of Tables 6, 8
and 12 the OPA received only “weak” (H0
: q
$.50 not rejected) or “moderate” (H0
: q $.75 not
rejected) support. Moreover, strong evidence of
dependence between common statement unilat-
eral pairs was found for the interpersonal distance
eight statement dataset of Study Two. As sets of
Figure 15. Proximity graph depicting the dominant I scale path (bold font I scales), illegal immigrant six state-
ment dataset.
27. Attitudes, Order and Quantity 27
attitude statements constructed with the ordinal
determinable (Michell 1994) are strictly ordered
in terms of intrinsic favourability towards the
kernel concept, empirical support of the OPA is
not an unreasonable hypothesis. As substantial
“between – subject” violation of the QPA was
found in these datasets, the most plausible conclu-
sion that can be made was the Coombs’s (1964)
theory was not supported by these datasets.
Given the failure to support theACM cancel-
lation axioms in Study Two, it was not possible to
fully ascertain the effect of ordinal determinable
structure upon scaling solutions. Nevertheless,
a linear relationship was discovered (Figure 14)
Figure 16. Tests of the double cancellation axiom for the six statement illegal immigrant dataset. In tests 1, 3
and 4 the consequent relation (open arrow) upon the interstimulus midpoints contradicts the antecedent relations
(solid arrows). Thus double cancellation was rejected.
Table 13
SHCMpp item locations, standard errors and fit statistics, illegal immigrant six statement, Internet
sample.
Statement di
S.E. A B C D E F
A 5.735 .108 • .27 .86 1.36 1.92 6.27
A 5.034 .136 • .05 .14 .09 .97 .03
B 2.225 .102 • .20 .19 .97 .03
C 2.556 .109 • .25 .39 .10
D –1.249 .134 • .04 .28
E –3.622 .109 • .05
F –4.944 .118 •
Overall SCHMpp model chi-square = 2.939, d.f., = 30, p . 1
† All chi-square fit values for item pairs have 3 degrees of freedom.
28. 28 Kyngdon and Richards
between the transformed Goode’s scale (Figure
13) and the transformed SHCMpp (Andrich 1995)
scale (Table 11) of the interpersonal distance
six statement set. As this set of statements was
produced by an ordinal determinable forming
an asymmetric binary predicate tree, this find-
ing tentatively suggests binary tree structure
does not influence relationships between scaling
solutions.
Perhaps the most important finding of the
present study was the behaviour of the SHCMpp
(Andrich 1995). The overall fit of the SHCMpp
across all datasets (Tables 5, 7, 9, 11 and 13)
was excellent. The standard psychometric inter-
pretation of this result is to conclude the relevant
attitudes are quantitative; and the SHCMpp is a
genuine theory of psychological measurement:
…the fact that one hypothesises a latent
variable to underlie one’s observations
does not imply that the model con-
structed in this fashion cannot be tested.
Once formulated, IRT models can most
certainly be tested against empirical
data, and, in fact, this is routinely done.
Nobody working in IRT [Item Response
Theory], and we dare to make this
statement as a universal claim, accepts
the hypothesis that attributes are quan-
titative without testing the model for
its empirical adequacy. As a matter of
fact, IRT models are regularly rejected
because they do not adequately fit the
data. (Borsboom and Mellenbergh 2004,
p. 113, authors’ emphasis)
Both the results of the present study and
Kyngdon (2006b) cast strong doubt upon this
argument. If the fit of a theory to data is to be
considered meaningful, it must place restrictions
on possible outcomes (see Roberts and Pashler
2000). The implied restriction in Borsboom and
Mellenbergh’s (2004) argument is that IRT mod-
els should not fit data caused by a non-quantita-
tive structure. The present study, however, found
the SHCMpp fitted datasets (Tables 6, 8 and 12)
which exhibited “between – subject” failure of the
QPA (viz., rejection of theACM double cancella-
tion axiom). Moreover, poor fit of the SHCMpp
to two datasets which did not reject double can-
cellation was found by Kyngdon (2006b). These
findings suggest SHCMpp model fit does not
differentiate between sets of paired comparison
data that contain quantitative structure and those
which do not. These findings are consistent with
the argument that IRT model fit indices are not
indicative of the absence of quantitative structure
in psychological data (Michell 2004).
This conclusion is further evidenced in the
data of the interpersonal distance eight statement
set in Study Two. The Bayesian Dirichlet model
analysis found the unilateral pair “DE” com-
pletely rejected the OPA (Table 6). No indication
of such axiom violation is given by the SHCMpp
analysis in Table 7. Instead, the qualitative J scale
order for these statements was reversed such that
the order upon the SHCMpp statement locations
was ABCEDFGH. The standard psychometric
interpretation of the SHCMpp analysis would be
to conclude that statement E expresses a more
favourable attitude with respect to interpersonal
distance than does statement D. These statements,
however, were formed by the bifurcation of the
kernel concept by the predicate “I would be
friends with a homosexual” (Statement D) and
its logical opposite “I would not be friends with a
homosexual” (Statement E) (seeAppendix One).
By virtue of this predicate structure statement E
cannot logically express a more favourable at-
titude than statement D; therefore the SHCMpp
results contradict the logical predicate structure
of these statements. Hence it is more plausible to
conclude that there was a failure of the OPA with
respect to this unilateral pair as suggested by the
multinomial Dirichlet model.
Similar findings were made in two other
datasets. For the illegal immigrant six statement
set the SHCMpp statement location order was
ACBDEF (Table 13).The Bayesian analysis of the
“BC” unilateral pair (Table 12) found the data for
this pair only “weakly” (H0
: q $.50 not rejected)
supported the OPA. For the eight statement set the
SHCMpp statement location order was ABCD-
FEGH (Table 9). The Bayesian analysis of the
“EF” unilateral pair (Table 8) found the data for
this pair only “weakly” supported (H0
: q $.50 not
29. Attitudes, Order and Quantity 29
rejected) the OPA. Indeed, only within those sets
of data which at least double cancellation was sat-
isfied did the SHCMpp statement location order
match the quantitative J scale order predicted by
the ordinal determinable (Tables 5 and 11).
Michell (2000) proposed that measurement is
a two stage process. The first stage is the scientific
task of measurement, which is the creation of
experimental situations “…that are differentially
sensitive to the presence or absence of quantita-
tive structure” (Michell 2000, p. 649). Within the
context of unidimensional unfolding (Coombs
1964), the results of the present study suggest
the Bayesian multinomial Dirichlet model (Kara-
batsos, 2005) is suitable for the scientific task of
measurement, as it appeared differentially sensi-
tive to the presence and absence of both ordinal
and additive structure in error – contaminated
paired comparison data.
In contrast, the SHCMpp appears more suited
to the second stage instrumental task of measure-
ment which involves “…devising standardised
procedures for estimating measures of the attri-
bute involved” (Michell 2000, p.649). Using the
SHCMpp to provide a scaling solution for a set of
attitude statements satisfying double cancellation
is certainly a less laborious method than Goode’s
(Goode 1964, cited in Coombs 1964) minimum
integer algorithm.
Further research, however, is needed to
critically evaluate the study findings. The pres-
ent study employed relatively small sample sizes
(around 200 persons); and so hence it cannot be
ruled out these small sample confounded the fit of
the SHCMpp to all datasets. Increasing the sample
size (to 500 persons, for example) is not in itself
an appropriate solution, given the chi-square is a
statistic confounded by both large sample sizes
and the small expected cell frequencies yielded by
unfolding data (Roberts, Donoghue and Laughlin
2000). Furthermore, the p value of the chi square
fit statistic does not have any direct mathematical
relationship to the probability of the SHCMpp
being true given the observed data (see Jeffreys
1980; Lindley and Phillips 1976). This places an
important theoretical caveat on the conclusion the
SHCMpp is not a scientific theory of measure-
ment. Future research should perhaps empirically
test the SHCMpp within a Bayesian framework.
This would avoid the problems associated with
chi-square fit statistics as applied to IRT unfolding
models (Roberts, et al., 2000). Such a framework
could be extended to include the other IRT models
of unfolding; and may well provide insight into
the more general question of whether or not IRT
models can differentiate between quantitative and
non-quantitative psychological attributes.
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31. Attitudes, Order and Quantity 31
Appendix One
Ordinal Determinables and Binary Tree Diagrams
Let T represent the kernel concept of interpersonal distance towards homosexuals. Let
there be a set of predicates and their logical opposites P which logically conjoin T and thus
form a binary tree structure. The predicates of this set and the symbols that represent them
are as follows:
F = “I would be friends with”; S = “I would have sex with”; LP = “I would be involved in a life part-
ner relationship with”; PI = “I would be physically intimate with”; SP = “I would speak in passing
with”; MS = “I would mix socially with”; WH = “I would wish harm on”.
Let T represent the kernel concept of attitude towards illegal immigrants in Australia. Let
there be a set of predicates and their logical opposites P which logically conjoin T and thus
form a binary tree structure. The predicates of this set and the symbols that represent them
are as follows:
L = “Should be allowed to land in Australia”; D = “should be detained”; C = “should automatically
be entitled to become full Australian citizens”; R = “should automatically be entitled for residency”;
V = “should automatically be entitled for a temporary visa”; A = “should be entitled to apply for a
temporary visa”; O = “detained in an offshore country”; H = “should be sent directly home”.
Appendix One Figure 1. Binary tree diagram of ordinal determinable used to construct the set of eight interper-
sonal distance statements.
Appendix One Figure 2. Binary tree diagram of ordinal determinable used to construct the set of eight illegal
immigrant statements.
32. 32 Kyngdon and Richards
Let T represent the kernel concept of interpersonal distance towards homosexuals. Let
there be a set of predicates and their logical opposites P which logically conjoin T and thus
form a binary tree structure. The predicates of this set and the symbols that represent them
are as follows:
F = “I would be friends with”; S = “I would have sex with”; LP = “I would be involved in a life
partner relationship with”; SP = “I would speak in passing with”; MS = “I would mix socially with,
WH = “I would wish harm on”.
Let T represent the kernel concept of attitude towards illegal immigrants in Australia. Let
there be a set of predicates and their logical opposites P which logically conjoin T and thus
form a binary tree structure. The predicates of this set and the symbols that represent them
are as follows:
L = “Should be allowed to land in Australia”; D = “should be detained”; C = “should automatically
be entitled to become full Australian citizens”; R = “should automatically be entitled for residency”;
A = “should be entitled to apply for a temporary visa”; O = “detained in an offshore country”; H =
“should be sent directly home”.
Appendix One Figure 3. Binary tree diagram of ordinal determinable used to construct the set of six interpersonal
distance statements.
Appendix One Figure 4. Binary tree diagram of ordinal determinable used to construct the set of six illegal
immigrant statements.
33. Attitudes, Order and Quantity 33
Appendix Two:
Thirty Six Violating Tests of Double Cancellation
for the Interpersonal Distance 8 Statement Set
(Appendix Two continued on next page.)
34. 34 Kyngdon and Richards
Appendix Two:
Thirty Six Violating Tests of Double Cancellation
for the Interpersonal Distance 8 Statement Set
(Appendix Two continued from previous page.)
