**\'Optimal Designs for Binomial and Multinomial Regressions**

**with Applications in the Social Sciences\'**

**Ben Torsney, University of Glasgow**

**Abstract:**\
The presentation will briefly describe results on optimal designs\
for binary regression designs when there is a single design\
or control variable and the design space is a finite or infinite
interval.\
Results hinge on a parameter dependent transformation to a weighted\
linear design problem and can be illustrated pictorially.\
Applications in the social sciences will be cited, including\
contingent valuation studies, which aim to assess a population\'s\
willingness to pay for some service or amenity, and in educational\
testing. These lead naturally to consideration of multinomial\
regression models. Extensions to problems with more than one\
design variable will be indicated too.

# **1 Introduction**

Suppose that a survey or investigation is to be conducted in which some
variable,

on a continuous scale, denoted by X, is of interest; but that we cannot
measure it very precisely on the sample members. We record only to which
of a finite number of categories they belong,

possibly determining this by a process of elimination.

Examples arise in Adaptive Testing in Education, in Market Research
Studies and in Contingent Valuation Studies. One might record such
categorical information in a market research investigation if
respondents are likely to be reluctant to be very specific or to have
poor memory recall; for example in surveying general practitioners in
respect of what percentage of patients they assign to a specific drug.
In contingent valuation studies the primary aim is to assess a
population's willingness to pay for some non-market good or service or
towards an increase in charges e.g. for a fishing permit, or for access
to a country park, or for new medical facilities.

Since respondents may never have considered such questions it is
unrealistic to expect them to state a specific 'willingness to pay
value'. In a simple dichotomous choice question they are offered a
single 'bid' question; e.g. 'are you willing to pay \$20.?' In a double
bounded approach they would then be offered a second bid, lower, e.g.
\$10, if their response to the first 'bid' is NO

and higher, e.g. \$30, otherwise. We would then know into which of the
four ranges, below \$10, between \$10 and \$20, between \$20 and \$30,
above \$30, a respondent's willingness to pay falls.

See Alberini (1995), Kaninen (1993) and Torsney & Gunduz (1999).

A similar approach is used in adaptive testing with a view to assessing
academic attainment. Students are given, say a moderate test initially,
followed by an easier or harder test depending on their performance in
the moderate test. A fundamental question is: what bid values should be
offered to respondents?

**2 The Formal problem**

Suppose that we know that X є *X* = \[C,D\], so that this is a sample
space (which might be the real line). Suppose that we wish to place
responses into one of k categories determined by cut-points x~1~,
x~2~,........x~k-1~ , chosen in advance, satisfying C = x~0~ \< x~1~ \<
x~2~ \< .........\< x~k-1~ \< x~k~ = D.

What sets of values should be chosen for these cut-points? This defines
a non-linear regression design problem, in which the design variable is
the vector [x]{.underline} = (x~1~,x~2~,......,x~k[-]{.underline}1~).
The solution should depend on the underlying distribution of X in the
population of interest.

We make the simple but widely used assumption that X (or it could be
some function h(X), e.g. ln(X) when X is positive, as in the case of
'Willingness to Pay') has distribution function:

P(X ≤ x) = F((x - μ)/σ) , x ε *X ,*

where μ and σ are unknown location and scale parameters respectively,
and F(z) is a standardised distribution function. Equivalently

P(X ≤ x) = F(α + βx) , x ε *X ,*

where α = -(μ/σ), β = 1/σ. This is a Generalised linear model in the
parameters α, β. Let [λ]{.underline} = (α,β)^T^.

We have a two parameter model and our objective is good estimation of
some aspects of these parameters. Often μ is of particular interest.

**3 Some Design Objectives**

We wish to choose a design which will ensure good estimation of some
aspects of our model.

We could be interested in efficient estimation of either both
parameters, or , in this context, possibly only of μ. For the latter we
then wish to minimise Var($\hat{\mu}$). Since μ = -α/β, $\hat{\mu}$ =
-$\hat{\alpha}$/$\hat{\beta}$ and Var($\hat{\mu}$) ≅
Var([c]{.underline}^T^$\hat{\lambda}$) for [c]{.underline} =
∂μ/∂[λ]{.underline} ∝ -(1,μ)^T^/β. This is an example of the c-optimal
criterion. Alternatively good estimation of σ = 1/β corresponds to c =
-(0,1)^T^/β^2^.

If we want good estimation of both parameters then we wish to make C =
Cov($\hat{\lambda}$) 'small'. Possible targets are to minimise either:
det(C ) (D-optimality); tr(C ) (A-optimality); or the maximum eigenvalue
of C (E-optimality). We will return to construction of these.

For the moment we note that for non-linear models optimal designs
typically depend on the unknown parameters of such models. They are
called locally optimal designs. Provisional estimates of parameters are
needed for these to be of practical value. We will focus on the
construction of such designs. We can characterise this parameter
dependence through a parameter dependent transformation to a
standardised problem.

Let Z = (X - μ)/σ = α + βX, z = (x - μ)/σ = α + βx,

A = (C - μ)/σ = α + βC, B = (D - μ)/σ = α + βD,

Then

P(X ≤ x) = P(Z ≤ z) = F(z) , z ε *Z* = \[A,B\].

We have in Z a transformed standardised version of X. We can focus on
determining cut-points

z~1~, z~2~,.........,z~k-1~ satisfying A = z~0~ \< z~1~ \< z~2~
\<.........\< z~k-1~ \< z~k~ = B. We have a design problem

with design vector [z]{.underline} = (z~1~, z~2~,.........,z~k-1~). Of
course z~j~ = (x~j~ - μ)/σ = α + βx~j~ , j = 0,1,2...,k.

Ford, Torsney and Wu (1992) used this approach for the two-category case
which we now review.

# 4 **Case** of **k = 2 Categories**

This is simple in that [x]{.underline} = x~1~ is scalar. Let x~1~ = x ε
*X* = \[C,D\].

We focus on construction of design measures ξ. That is we seek a
distribution ξ~X~ on *X* which will identify the optimal proportions of
observations to take at each point in *X*.

(Note that we are assuming that we are free to take x to be any value in
*X* = \[C,D\].

This can be permissible. However we could be restricted to a subset of
*X*, say \[c,d\].

We will return to this later.)

Then Cov($\hat{\lambda}$[) ∝ M^-1^(ξ~X~),]{.smallcaps} where
[M(ξ~X~)]{.smallcaps} is the expected information matrix per
observation.

If ξ~X~ assigns weight ξ~i~ to a discrete set of values x~1~,
x~2~,.....ξ~i~ \> 0, ∑ξ~i~ = 1, then

M(ξ~X~) = E~ξx~{I~x~} = ∑ξ~i~I~x~${}_{i}$

where I~x~ is the expected information matrix of a single observation at
x or a one point design at x. Using properties of generalised linear
models Ford, Torsney and Wu (1992) show that

I~x~ = w(z)(1,x)^T^(1,x) ,

where w(z) = {f(z)}^2^/{F(z)(1-F(z))}, f(z) = F′(z) and z = α + βx.

Now under this z-transformation (1,z)^T^ = B(1,x)^T^ where B is the
matrix

B = $\begin{pmatrix}
1 & 0 \\
\alpha & \beta
\end{pmatrix}$

Hence

I~x~ = B^-1^I~z~(B^-1^)^T^ ,

where

I~z~ = w(z)(1 z)^T^(1 z).

Further

M(ξ~X~) = B^-1^M(ξ~Z~)(B^-1^)^T^ ,

where ξ~Z~ is the distribution induced on *Z* = \[A,B\] by ξ~X~ on *X* =
\[C,D\] so that

M(ξ~Z~) = E~ξz~{I~z~} = ∑ξ~i~I~z~${}_{i}$ , z~i~ = α + βx~i~ .

Also

det(M(ξ~X~)) ∝ det(M(ξ~Z~)) ,

[c]{.underline}^T^M^-1^(ξ~x~)[c]{.underline} = [c]{.underline}~B~^T^
M^-1^(ξ~x~)[c]{.underline}~B~ , [c]{.underline}~B~ = B[c]{.underline} .

Thus D-optimal and c-optimal criteria, as functions of ξ~X~, transform
to other D-optimal and c-optimal criteria as functions of ξ~Z~.

Hence we focus on finding the design ξ~Z~ which either maximises
det(M(ξ~Z~) (D-optimality) or minimises [c]{.underline}~B~^T^
M^-1^(ξ~Z~)[c]{.underline}~B~ (c-optimality). Note that if
[c]{.underline} = -(1,μ)^T^/β,

[c]{.underline}~B~ ∝ (1,0)^T^, and if [c]{.underline} = -(0,1)^T^/β^2^,
[c]{.underline}~B~ ∝ (0,1)^T^. Denote these by
[e]{.underline}~1~-optimality and

[e]{.underline}~2~-optimality respectively.

Ford, Torsney and Wu (1992) exploited the fact that these (non-linear)
design problems are equivalent to corresponding weighted linear design
problems with weight function w(z). Tools for constructing designs for
linear models can be invoked. For example there are geometrical
characterisations of D-optimality and c-optimality relating to the
design locus:

G = {(g~1~,g~2~) : g~1~ = √w(z) , g~2~ = z√w(z) , z ∈ *Z* = \[A,B\] } .

They established, that, for several choices of F(z), D-optimal designs
need to take observations at only two distinct points (support points)
in *Z*, in which case optimal weights are ½, ½ . (Note that two distinct
points are needed to ensure estimation of both parameters.) Optimal
designs are then of the form:

z z~1~ z~2~

ξ~z~ ½ ½

For the logistic and normal/probit choices of F(z) (for which *Z* = ℜ),
z~1~ = - z, z~2~ = z, with z = 1.543 and 1.138 respectively. These two
values are well established in the literature.

For two others (for which *Z* = ℜ), three support points are needed, in
which case optimal weights are not uniform. Torsney & Musrati (1993)
report the following optimal designs:

Double Exponential Double Reciprocal

z -1.594 0 1.594 z -√2 0 √2

ξ~z~ 0.282 0.436 0.282 ξ~z~ 0.262 0.476 0.262

In the case of c-optimality either one or two support points are needed.
If only one is needed it is the value z such that [c]{.underline}~B~ ∝
√w(z)(1,z)^T^. If two are needed these are fixed for all
[c]{.underline}~B~ and there is an explicit solution for the optimal
weights which do vary with [c]{.underline}~B~. Thus optimal designs are
of the form:

z z~1~ z~2~

ξ~z~ ξ~z~${}_{1}$ ξ~z~${}_{2}$

For the logistic and normal/probit choices of F(z) (for which *Z* = ℜ),
z~1~ = - z, z~2~ = z, with z = 2.399 and 1.157 respectively. These are
the support points for e~2~-optimality

each having equal weighting. For e~1~-optimality there is one support
point; namely z = 0.

We now turn to the k-category case. First we note that I~z~ has the
expansion

I~z~ = (1,z)^T^f(z)(1,-1)D~θ~^-1^(1,-1)^T^f(z)(1,z) ,

where D~θ~ = diag(θ~1~,θ~2~) , θ~1~ = F(z), θ~2~ = 1 -- F(z). Similarly

I~x~ = (1,x)^T^f(z)(1,-1)D~θ~^-1^(1,-1)^T^f(z)(1,x) , z = α + βx .

# 5 **One Point Designs : k Categories**

One key difference from the two-category case is that with 3 or more
categories it is possible to estimate all parameters using the same
cut-points for all respondents; i.e. the same design point. This is a
one point design.

Let the cut-points be x~1~, x~2~,.........x~k-1~ and x~0~ = C, x~k~ = D
.

Let

θ~1~ = P(X ≤ x~1~) = F(α + βx~1~) = F(z~1~) , θ~k~ = 1 -- F(z~k-1~) ,

θ~i~ = P(x~i-1~ ≤ X ≤ x~i~) = F(z~i~) - F(z~i-1~) , i = 2,3,....,k-1 .

Then the Fisher Information matrices at [x]{.underline} =
(x~1~,x~2~,......,x~k-1~)^T^ and at

[z]{.underline} = (z~1~,z~2~,......,z~k-1~)^T^, z~i~ = α + βx~i~ are:

I~X~ = XQX^T^ , I~Z~ = ZQZ^T^

where

X^T^ = ([1]{.underline}~k-1~[x]{.underline}) , Z^T^ =
([1]{.underline}~k-1~[z]{.underline})

[1]{.underline}~n~ = (1,1,..........,1)^T^ ∈ ℜ^n^

Q = D~f~HD~θ~^-1^H^T^D~f~

D~f~ = diag{f(z~1~),f(z~2~),........... f(z~k-1~)}

D~θ~ = diag(θ~1~,θ~2~,............,θ~k~)

H = (I~k-1~[0~k-1~]{.underline}) -- ([0~k-1~]{.underline}I~k-1~)

[0]{.underline}~n~ = (0,0,..........,0)^T^ ∈ ℜ^n^

I~n~ = identity matrix of order n .

Clearly

Z = BX , I~X~ = B^-1^I~Z~(B^-1^)^T^

We have the same relationship as in the two-category case. We can focus
on determining an optimal [z]{.underline}^\*^. Below are results for the
logistic and normal/probit choices of F(z) for k = 3,4. In view of the
symmetry of these distributions we must have

[z]{.underline}^\*^ = (-z^\*^,z^\*^) for k=3, and [z]{.underline}^\*^ =
(-z^\*^,0,z^\*^) for k=4. We then have an univariate optimisation to
perform.

The criteria considered are :

D-optimality : maximise {log det (I~Z~)}

A-optimality : maximise {-tr(I~Z~^-1^)}

e~1~-optimality : maximise {-e~1~^T^ I~Z~^-1^e~1~}

e~2~-optimality : maximise {-e~2~^T^ I~Z~^-1^e~2~}

E-optimality : maximise {-maximum eigenvalue of (I~Z~^-1^)}

We wish to to choose z to maximise φ(z) = ψ(I~Z~) for Z^T^ =
([1]{.underline}~k-1~[z]{.underline}), z = (-z,z) or

(-z,0,z), where ψ(.) is one of the above criteria.

RESULTS:

LOGISTIC k = 3 k = 4

CRITERION z^\*^ φ(z^\*^) z^\*^ φ(z^\*^)

D-optimality 1.5 -1.5574 2.0 -1.2485

A-optimality 1.2 -5.2012 1.7 -4.3790

e~1~-optimality 0.7 -3.3751 1.1 -3.2000

e~2~-optimality 2.2 -1.0229 2.2 -1.0229

E-optimality 0.7 -3.3751 1.1 -3.2000

NORMAL/PROBIT k = 3 k = 4

CRITERION z^\*^ φ(z^\*^) z^\*^ φ(z^\*^)

D-optimality 1.1 -0.2072 1.4 -0.0999

A-optimality 1.0 -2.2800 1.3 -1.9443

e~1~-optimality 0.6 -1.2350 1.0 -1.1332

e~2~-optimality 1.5 -0.7668 1.5 -0.7668

E-optimality 0.8 -1.2620 1.0 -1.1332

## 6 Future Work

We list a number of topics to be pursued.

**6.1 Arbitrary Design Intervals**.

We assumed that the design interval was the sample space of X or Z.

In practice we may be restricted to a subset of *Z* = \[A,B\], say
\[a,b\].; e.g. a positive subset when the variable X is really positive
(as in the case of Willingness To Pay in Contingent Valuation Studies)
although we might have assumed X to have a logistic or normal
distribution.

For the two-category case Ford Torsney and Wu (1992) derived c-optimal
designs for all

possible choices of \[a,b\]. Again either one or two support points are
needed. If two are needed

either one or both of a,b can be support points.

In the case of D-optimality two or three support points are needed. They
conjectured that for

many choices of F(z) only two would be needed and that the structure
would be similar to the c-optimal solution. Gunduz and Torsney (2002a)
went on to confirm these conjectures. Also Sebastiani and Settimi (1997)
focussed on the logistic case. It is likely that there will be similar
solutions in the k-category case.

**6.2 Determining Cut-points**[.]{.underline}

# In practice the cut-points z~1~, z~2~,.........,z~k-1~ or equivalently the cell probabilities θ~1~, θ~2~,....θ~k~ must be determined numerically. One possible algorithm is the multiplicative iteration:

θ~j~^(r+1)^ = θ~j~^(r)^m(d~j~^(r)^)/∑ θ~i~^(r)^m(d~i~^(r)^)

where m(.) is a positive increasing function and d~j~^(r)^ = ∂ψ/∂θ~j~
[θ]{.underline} = [θ]{.underline}^(r)^.

This kind of algorithm was developed originally for finding optimal
design weights;

see Torsney (1983, 1988), Mandal & Torsney (2000), Torsney & Mandal
(2000, 2002).

**6.3 Multiple Design Points**

We assumed the same set of cut-points for all respondents; i.e. a
one-point design.

In practice several sets may be needed. Each is defined by a set of cell
probabilities. These distributions and also the distribution defined by
the set of design weights (the proportion of observations to be taken at
each set) can be determined by a natural extension of the above
algorithm. This has been explored in another context by Mandal & Torsney
(2002)

**6.4 Choice of k**

How many cut-points are needed? The same number for each design point?
These numbers could be determined by starting with large values of k. If
zero cell probabilities emerge k has been reduced and arguably found.
Alternatively we could search through increasing values of k.

**6.5 Bivariate Approach**

Consider a one-point design with 4 categories and 3 cut-points. When
this is implemented iteratively, as in Contingent Valuation (CV) Studies
(a first bid is offered and then a lower or higher bid depending on the
response to the first bid), some approaches in the CV literature (see
Alberini (1995)) assume a change in the distribution of X (Willingness
To Pay (WTP) in CV studies) from the first to the second bid. This
points to the need for a joint distribution for X~1~ and X~2~
representing WTP at the two bids respectively. It is conventional to
assume common marginal distributions, indexed by a location and scale
parameter μ and σ as above, for some function h(.) of X~1~ and X~2~;

e.g. h(X) = ln(X). Let U~i~ = (h(X~i~) - μ)/σ, i = 1,2. A common
marginal (standardised) cdf F(u) for U~1~ and U~2~ is implied if we
assume that

P(U~1~ ≤ u~1~, U~2~ ≤ u~2~) = G{F(u~1~),F(u~2~)}

where G(r,s) is a copula; i.e. a joint cumulative distribution function
on 0 ≤ r,s ≤ 1, such that G(r,1) = r and G(1,s) = s. The approach of
section 5 should readily extend to this model.

**6.6 Conditional Approach**

Whether or not we assume a change in distribution between bids, one
approach to design construction when bids are offered iteratively,
particularly when there is a time gap between offers, is to consider
designing for the second or next stage by changing the cdf F(z) to that
corresponding to the conditional distribution of X (or X~2~) given the
response to the first or previous bid. See Gunduz and Torsney (2002b).
Design points could still be sets of cut-points.

**6.7 Extensions to Higher Dimensions**

In the two-category case we often view one of the two categories as a
'response' of interest.

If this equates to 'X ≤ x' then

P(response) = F(α + βx) .

Sitter and Torsney (1995a, 1995b) consider the extension of this model
to two and to more than two design variables respectively, so that

P(response) = F(α + [β]{.underline}^T^[u]{.underline}) , [u]{.underline}
∈ *[U]{.underline}* ⊆ ℜ^m^

Following Ford. Torsney and Wu (1992) they consider a linear
transformation from [u]{.underline} to [z]{.underline} such that z~1~ =
(α + [β]{.underline}^T^[u]{.underline}) with the remaining z~j~ to be
chosen by the experimenter. With the possible exception of z~1~,
[z]{.underline} must be bounded. This will be the case if
*[U]{.underline}* is bounded. They argue that any design space
*[Z]{.underline}* must be equivalent to a subset of

*[Z]{.underline}*~W~ = {z~1~ ∈ \[A,B\] , -1 ≤ z~j~ ≤ 1 , j = 2,3,....,m}

where \[A,B\] is the sample space of F(z). This is a widest or 'largest'
possible design space.

On *[Z]{.underline}*~W~ observations need only be taken at z~j~ = ±1, j
= 2,3,...,m, while z~1~ plays the role of z in section 4. In fact the
total weight at a value of z~1~ can be split uniformly across the 2^m-1^
combinations of z~j~ = ±1 or over subsets of these forming Hadamard
matrices. Hence we can focus on the marginal design on z~1~. Solutions
have a similar structure to the one-design variable case. This includes
the case of z~1~ restricted to a subset \[a,b\] of \[A,B\]. See Torsney
and Gunduz (2000, 2002).

We could now consider the possibility of two or more cut-points for
z~1~. In this case the matrix Z in the definition of the information
matrix I~Z~ changes. Suppose the cut-points are [z]{.underline}~1~ =
(z~11~,z~12~,...,z~1(k-1)~)^T^. Then in the case m = 3 there are four
possible forms for Z, namely:

([1]{.underline}~k-1~[z]{.underline}~1~-[1]{.underline}~k-1~-[1]{.underline}~k-1~)^T^
,
([1]{.underline}~k-1~[z]{.underline}~1~-[1]{.underline}~k-1~+[1]{.underline}~k-1~)^T^
,
([1]{.underline}~k-1~[z]{.underline}~1~+[1]{.underline}~k-1~-[1]{.underline}~k-1~)^T^
,
([1]{.underline}~k-1~[z]{.underline}~1~+[1]{.underline}~k-1~+[1]{.underline}~k-1~)^T^
.

These correspond to the four possible combinations (z~2~,z~3~) =
(±1,±1).

Suppose we use this set of cut-points for all design points (i.e. for
all respondents regardless of their z~2~ and z~3~ status). Then this
implies a one-point marginal design on [z]{.underline}~1~. If we wish to
choose [z]{.underline}~1~ to optimise any of the criteria in section 5,
then the information matrix I~Z~ must be replaced by it's sum over the
above four Z-matrices.

For the logistic and normal/probit choices of F(z) this optimisation
problem again reduces to an univariate optimisation in the case k = 3,4
as [z]{.underline}~1~ must be of the form (-z,z)^T^ or (-z,0,z)^T^
respectively.

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