Naohito Chino & Shingo Saburi
Aichi Gakuin University, Japan
March 26
The 6th International Conference on Multiple Comparison Procedures, Tokyo, Japan, March 24-27, 2009
Organization of my talk
(1) What is “asymmetric MDS”?(2) extant asymmetric MDS models(3) What is “ASYMMAXSCAL”?(4) advantages of ASYMMAXSCAL(5) shortfalls of ASYMMAXSCAL(6) our proposals to overcome the
defects
(1) What is “asymmetric MDS”?
The asymmetric MDS is a method which is specifically designed to analyze asymmetric relationships among members and display them graphically by plotting each member in a certain dimensional space, given asymmetric data.
Examples of symmetric and asym-metric relational dataSymmetric data flight mileages among 10 cities => We can recover the map of these cities.Asymmetric data degrees of sentiment relationships among 17 members measured by a 7-point rating scale => We can estimate the configuration of members in a certain dimensional space.
extant asymmetric MDS models
Chino (1978, 1990), Chino and Shiraiwa (1993), Constantine and Gower (1978), Escoufier and Grorud (1980), Gower (1977), Harshman (1978), Harshman et al. (1982), Kiers and Takane (1994), Krumhansl (1978), Okada and Imaizumi (1987, 1997), Rocci and Bove (2002), Saburi and Chino (2008), Saito (1991), Saito and Takeda(1990), Sato (1988), ten Berge (1997), Tobler (1976-77), Trendafilov (2002),Weeks and Bentler (1982), Young (1975), and Zielman and Heiser (1996).
Examples of the extant modelsO-I model (Okada & Imaizumi, 1987) -- an augmented distance
model
GIPSCAL (Chino, 1990) -- a non-distance model
,*kjjkjk rrdd
,cbas kqtjk
tjjk xLxxx
What is “ASYMMAXSCAL”Although various asymmetric MDS methods
have been proposed, these methods have remained to be descriptive until recently.
By contrast, Saburi & Chino (2008) have proposed a maximum likelihood method for asymmetric MDS called ASYMMAXSCAL, which extends MAXSCAL by Takane (1981) to asymmetric relatio-nal data.
MAXSCAL is the name for a multidimensional successive categories scaling.
ASYMMAXSCAL revisited (1)
Parameters in ASYMMAXSCAL As with MAXSCAL by Takane
(1981), it has three kinds of parameters pertaining to
(1) the representation model (2) the error model (3) the response model
ASYMMAXSCAL revisited (2)As for the representation model, the
proximity model of Oi to Oj, say, gij, can generally be written as
),,,( cxx jiij fg
where f(•) is any asymmetric MDS model,xi and xj, respectively, are coordinate vectors of Oi and Oj, and c is the remain- ing parameter vector.
ASYMMAXSCAL revisited (3)As regards the error model, the error-
perturbed proximities are written as
),0(~, 2 Neeg ijijijij
Error-perturbedproximities
proximities
ASYMMAXSCAL revisited (4)As for the response model, we
assume that subjects place error-perturbed pro-ximities in one of the M rating scale categories, C1, …, CM. Thus, these categories are represented by a set of ordered intervals with upper and lower boundaries on a psychological contin-uum.
MMm bbbbb 110 boundaries
ASYMMAXSCAL revisited (5)Accordingly, the probability that the
error-perturbed proximity of Oi to Oj falls in Cm is given by
).( 1 mijmijm bbprobp
We assume that
ASYMMAXSCAL revisited (6)
,)(1
ij
b
b ijijm dpm
m
based on Torgerson’s law of categorical judgment. Here, Ф(τij) denotes the density of the standard normal distribution.
(For computational convenience, we app-roximate it by the logistic distribution.)
ASYMMAXSCAL revisited (7) We estimate all the parameters
pertaining to ASYMMAXSCAL by maximizing the following joint likelihood of the total observations
,111
ijmYijm
M
m
n
j
n
ipL
where Yijm denotes the frequency in category Cm, in which subjects placed the error-perturbed proximity of Oi to Oj.
As for the details of ASYMMAXSCAL, seeSaburi, S. and Chino, N. (2008). A maximum likelihood method for an asymmetric MDS model.
Computational Statistics and Data Analysis,
52, 4673-4684.
Advantages of ASYMMAXSCAL over the extant descriptive models (1)
(1) Determination of the appropriate scal-
ing level of the data by AIC, that is, ordinal, interval, or ratio level.(2) Determination of the appropriate dimensionality of the model under study by AIC.
Advantages of ASYMMAXSCAL over the extant descriptive models (2)
(3) Examination that the data are sufficiently asymmetric or not, i) prior to the scaling of objects, by applying some tests for symmetry, ii) on the way to the scaling, by selecting a model among several candidates including some symmetry models, using AIC.
Tests for symmetry, prior to the scaling of objects
The data obtained by the above method is as follows. We call it the Type A
design data, or the Type A data.One of them is the test for a special con-
ditional symmetry hypothesis for this design data.
.
nnMnnnn
M
M
YYY
YYY
YYY
,,,
..........................
,,,
,,,
21
12122121
11112111
C
1
C2 … CM
from O1 to O1
from O1 to O2
…
from On to On
rating categoriesproximity
judgments
frequencies
n11
Number of samples
n12
…
nnn
Type B (design) data in ASMMAXSCAL
Type B (design) data is obtained by rearranging the Type A data per rat-ing category.
(It is a bit different from traditional designs for the n×n×M table (Ag- resti, 2002, Bishop et al., 1975)).
Y111 … Y1n1. .Yn11 …
.Ynn1
.
Y11M… Y1nM
Yn1M
. .… YnnM
. .
C1
C2
CM
∶∶
∶
n11
nn1 nnn…. . total
The Type B design data
Tests for symmetry in ASYMMAXSCAL (2)
MmjippH jimijmcs ,,1,,:)(0
According to Saburi and Chino (2008), under the null hypothesis,
the likelihood ratio test statistic,
M
m
n
i
n
j jiij
jimijmijijmijm nn
YYnYYG
1 1 1
2 )(lnln2
asymptotically follows the central χ2-distribution with (M-1)n(n-1)/2 degrees of freedom.
,2
)(lnln2
1 1 1
2
M
m
n
i
n
j
jimijmijmijm
YYYYG
with Mn(n-1)/2 degrees of freedom under the null hypothesis.
By the way, the traditional conditional symmetry test with the special n×n×M contingency table, of which test statistic is given by
shortfalls of ASYMMAXSCAL (1)(1) Shortfall of the model selection method At present, ASYMMAXSCAL enables us to
select the most appropriate model among several candidate models which include some variants of symmetry models using AIC. However, such a model selection method by some information criterion does not consider the nature of the data.
It will be necessary to select the representation model which reflects the nature of the data most.
shortfalls of ASYMMAXSCAL (2) To do this job, it might be necessary to utilize
various smmetry related tests which have been developed in the branch of mathematical statistics.
Chino and Saburi (2006) attempted to administer these tests prior to the scaling step of ASYMMAXS-CAL. Figures 1 and 2 show this. However, relation of inclusion of these tests shown in Figure 1 is very complicated.
=> Chino, N. & Saburi, S. (2006). Tests of symmetry in asymmetric MDS. Paper presented at the 2nd German Japanese Symposium on Classification, Berlin, Germany.
shortfalls of ASYMMAXSCAL (3)(2) Lack of taking overall statistical errors into account In performing such sequential tests shown in
Figure 2, we did not take overall statistical errors into account.
It will be interesting and useful to examine whether these tests are mutually statistically independent or not. For simplicity, we shall, at
present, exclusively consider a two-dimensional square contingency table.
We have recently conjectured that, at least, two of these tests are statistically independent.
These are (1) test of the quasi-symmetry hypothesis,
and (2) test of the symmetry hypothesis under the condition that the quasi-symmetry hypo- thesis holds
Our proposal to overcome the shortfalls
Parameter spaces for these tests (1)
(1) total parameter space of the log-linear model
(2) parameter space for the quasi-
symmetry hypothesis
)0()2()1()12()0()2()1()12( ,,,,,,, jiijjiij
)0()2()1(
)12()12()0()2()1()12(
,,
,,,,
ji
jiijjiijQS
Parameter spaces for these tests (2)
(3) parameter space for the symmetry hypothesis
(4) parameter space for the equality of the row and
column effects
)0()2()1(
)12()12()0()2()1()12(
,
,,,,,
ii
jiijjiijS
)0()12(
)2()1()0()2()1()12(
,
,,,,
ij
iijiijERC
Null and nonnull hypotheses of these tests
(1) the quasi-symmetry hypothesis,
(2) the symmetry hypothesis under QSH0
QSQS
QSQS HagainstH θθ :: 10
SQSS
SS HagainstH θθ :: *
1*0
Ω=ω0
ωQS ωS
ωERC
Θij(12)=Θji
(12) Θ i (1) =Θi (2)
.0 SQS It should be noticed that
Dissolution of a likelihood ratio statistic (1)
The likelihood ratio λS for testing the usual sym-metry hypothesis can be written, for example, as
Therefore, we have
or
.)ˆ(
)ˆ(
)ˆ(
)ˆ(
)ˆ(
)ˆ(*
00SQS
QS
SQSSS L
L
L
L
L
L
),ln2(ln2ln2 * QSSS
Dissolution of a likelihood ratio statistic (2)
This statistic is due to Caussinus (1965), and follows asymptotically χ2 distribution with r-1 degrees of freedom.
.222* QSSS GGG
Test statistic for the quasi-symmetry hypothesisIt is well known that under the
hypothesis,
follows asymptotically the χ2 distribution with
(r-1)(r-2)/2 degrees of freedom. Here, satisfies the following equations:
QSH0
,ˆlnln21 1
2ijij
r
i
r
jijQS ffG
ij
jiijjiijjjii ffff ˆˆ,ˆ,ˆ
Joint sufficient statistic for the nuisance parametersThe statistics
corresponding to the nuisance parameters
under the quasi-symmetry hypothesis, give their joint sufficient statistics.
QSH0
)(,),(,,1 1
ijji
r
i
r
jijji ffffnff
)(,,,, )12()12(0
)0()2()1(ijijji
Ancillary statistic for these nuisance parameters
Moreover, under the hypothesis of quasi-symmetry, the statistic, , is free of these nuisance parameters, because the term, , are estimated as functions of the data,
.In other words, the statistic, , is
ancillary for these nuisance parameters.
2QSG
ij
jiijji ffff ,,
2QSG
Independence of the two statistics
As a result, due to the theorem by Hogg & Craig (1956), is statistically independent of
Hogg, R. V. & Craig, A. T. (1956). Sufficient sta-
tistics in elementary distribution theory. Sankhyā, 17, 209-216.
2QSG
).,,,,(2222* jiijjiijQSSS ffnfffGGGG
Control of the error rates of the two kinds
Since the two statistics are mutually stochastically independent, we may set the error rate of each of the test statistics to 1-(1-α)1/2 if we want to control the error rate of the first kind at α.
Furthermore, we can construct a more powerful test than Dunn’s and Holm’s, if we set the error rate of the quasi-symmetry test to α and set that of the symmetry test under the quasi-symmetry hypothesis to α/2, according to Hochberg (1988).
Quasi-symmetry ?
Multiple CP
Non-QS-familyof MDS models
rejected
Symmetry Under QS?
accepted
Multiple CP
SymmetricMDS
QS-family ofMDS models
accepted
rejected