essay on testing order hypotheses in ordered probit

Running head: HYPOTHESIS TESTING IN ORDERED PROBIT MODELS 1

Essay on Testing Order Hypotheses in Ordered Probit Regression Models - A Bayes

Factor Approach

Zhengguo Gu

Research Master in Social and Behavioral Sciences (minor: MTO)

HYPOTHESIS TESTING IN ORDERED PROBIT MODELS 2

Abstract

In this paper, we present a Bayes factor approach to testing order hypotheses in the

setting of ordered probit regression. Ordered probit models are often applied in social

sciences, yet no statistical tests are readily available for examining order hypotheses - a

type of hypotheses that depict the relation among the parameters of interest in terms of

their relative rank orders in expected magnitude. The Bayes factor approach proposed

in this paper is able to provide a holistic picture of the relationships among parameters

of interest and therefore is able to directly test the relative rank orders of expected

e�ects summarized by order hypotheses. A simulation study is presented to show the

basic features and behavior of this Bayes factor approach. The applicability of this

approach is further illustrated in an empirical example.

Keywords: Bayes factor, order hypothesis, ordered probit regression, hypothesis

testing


Essay on Testing Order Hypotheses in Ordered Probit Regression Models - A Bayes

Factor Approach

Introduction

Consider the following scenario: As a marketing analyst, you are asked to report

to the CEO of Procter & Gamble on the research question "which aspects amongst price,

product-related feature (e.g. perfume free, extra stain removal, and odor elimination

etc.), brand loyalty, and advertisement should Procter & Gamble further invest in so as

to secure the leading position of the brand Tide in the industry and to distinguish their

brand from similar brands, such as Wisk, Era, and Surf in terms of popularity among

consumers?" You have a large penal dataset regarding price, product-related feature,

brand loyalty, and advertisement, as well as the choice data of Tide, Wisk, Era, and Surf

over thousands of regular consumers in the chain stores across the United States within

a certain period of time. Specifically, the CEO of Procter & Gamble wants to know

whether their brand positioning is in line with the data: The CEO believes that the

brand Tide should primarily focus on product-related feature and on advertisement with

the latter relatively less important than the former. Focusing on these two aspects is

expected to boost consumers’ preference for Tide. In such a situation, as the analyst,

you might want to try ordered probit regression and test the following hypothesis:

—product-related feature

> —advertisement

> 0,

—product-related feature

> —advertisement

> {—price

, —brand loyalty

}.(1)

That is, both product-related feature and advertisement are positively associated with

consumer brand preference, yet product-related feature, compared to advertisement, has

a higher impact. Further, the impact of product-related feature and advertisement is

higher than that of price and brand loyalty.

However, readers should soon notice that there is no statistical tests available yet

in the setting of ordered probit regression for testing order hypotheses (such as the

inequalities above) in a direct manner. Marketing analysts therefore cannot give a

concrete answer to research questions that contain order hypotheses, if ordered probit

models are employed.


Marketing research is not the only field that often faces order hypotheses similar

to the one discussed above. This issue has been noticed by many a few researchers in

statistics and management science. For example, Klugkist, Laudy, and Hoijtink (2005)

and Klugkist and Mulder (2008) discussed how Bayesian statistics can be employed for

inequality constrained ANOVA. Mulder et al. (2009) extended the Bayesian framework

of testing order hypotheses to repeated measures. Braeken, Mulder, and Wood (2014)

provides a nice overview on testing order hypotheses in management area and presents

a Bayes factor approach to testing order hypotheses in linear regression models. Kato

and Hoijtink (2006) and Kato and Peeters (2008) extended the Bayesian approach to

testing order hypotheses in multilevel models. Hoijtink and Boom (2008) showcased

Bayesian estimation for inequality constrained latent class models. Despite that

researches in testing order hypotheses have seen fruitful results, much work remains to

be done. For instance, there is no statistical tests available for examining order

hypotheses in ordered/nominal probit/logit models, yet those models are regularly used

by researchers and practitioners, especially in social and behavioral sciences. This paper

takes the first step: we focus on ordered probit models and illustrate how to test order

hypotheses via the Bayes factor approach in such models.

This paper is organized as follows: In the next section, we present a brief overview

on the prevalence of order hypotheses in social sciences. Afterwards, we introduce the

basic idea of the Bayesian statistics and the concept of Bayes factor. We then discuss

ordered probit regression in detail and illustrate how the Bayes factor approach can be

applied to testing order hypotheses in ordered probit regression, which is followed by a

simulation study where we examine the behavior of this Bayes factor approach in

ordered probit regression and by an empirical example where we showcase its

applicability. The rest of the paper discusses some general features of this Bayes factor

approach and o�ers suggestions for future research.

Order Hypotheses in Social Sciences

Many theories in social sciences explicitly or implicitly incorporate order

hypotheses - the hypotheses that involve "relative rank order(s) for predicted e�ects in


terms of their expected magnitude" (Braeken et al., 2014). For example, ecological

studies in anthropological research assume that energy is the most significant aspect of

food resources, and nutritional factors are also important but not as important as

energy (Winterhalder, 1981). In economics, researchers have long acknowledged the

important impact of R&D (e.g., Coe & Helpman, 1995; Gerschenkron et al., 1962),

trade (e.g., Krishna & Mitra, 1998) and human capital (e.g., Benhabib & Spiegel, 1994)

on productivity growth, but theories suggest that trade, compared to the rest factors, is

the least influential factor (e.g., Gri�th, Redding, & Van Reenen, 2004). In marketing

research, marketing capabilities development is assumed to be influenced by business

strategy, organizational structure, market information processing capabilities,

organizational structure, and task routinization(e.g., Vorhies, 1998), and furthermore,

theories suggest that business strategy, compared to the rest, exerts the strongest

influence on marketing capabilities development (e.g., Vorhies, 1998; Day, 1994).

Similar examples can also be found in other social sciences such as management

research (see e.g. Braeken et al., 2014).

The examples above illustrate one commonality that is often seen in research:

Many theories involve comparing concepts/parties/factors in terms of their relative

(expected) magnitude. Therefore, researchers are often in need of statistical tests that

are able to examine the hypothesized rank orders so as to test the soundness of a

theory. A few such statistical techniques are available for models such as linear

regression models. For example, to test a hypothesis

H : —1

> —2

> —3

> 0 (2)

in a multiple regression model, researchers may follow a traditional way of null

hypothesis significance testing (NHST). However, NHST cannot test —1

> —2

> —3

> 0

directly: we must first test whether —1

> 0, —2

> 0, and —3

> 0 then test whether

—1

> —2

, —2

> —3

, and —1

> —3

. Hence, multiple statistical tests need to be performed

before we can even reach a conclusion. Further, even if multiple statistical tests are

performed, a researcher might find it di�cult to combine all the results of the multiple

tests into a single, coherent conclusion. For example, consider the following situation:


the one-sided test suggests that there is no enough evidence to reject —2

= —3

and also

no enough evidence to reject —1

= —3

at a certain significance level, but there is enough

evidence to reject —1

= —2

, which apparently contradicts the results of the former two

tests. In this situation, a coherent conclusion is not possible.

Furthermore, performing multiple testing can be problematic because of the

inflated type I error, and correction methods such as Bonferroni correction, Holm

method (Holm, 1979), and Hochberg method (Hochberg, 1988) need to be employed. In

addition, as has been pointed out by Braeken et al. (2014), it is not clear what the

alternative hypothesis is for an order hypothesis like —1

> —2

> —3

> 0 in NHST.

An alternative approach to testing order hypotheses is the Bayes factor approach.

Previous research has touched upon this new method in cases of ANOVA (e.g., Klugkist

& Mulder, 2008), linear regression models (e.g., Braeken et al., 2014), latent class

models (e.g., Hoijtink & Boom, 2008), and multilevel models (Kato & Peeters, 2008)

etc. In this paper, we will extend previous research on regression models to ordered

probit regression models. As far as we know, there is no statistical test readily available

for testing order hypothesis in ordered probit models, yet ordered probit models are

routinely applied in research and practices such as the scenario in the Introduction

section. Before delving into ordered probit models, we first present a brief introduction

to Bayesian statistics in the next section.

A Brief Introduction to the Bayes Factor Approach

To understand the foundation of the Bayes factor approach, we must start with

Bayes’ theorem, which gives the relationships among the probability of event A (i.e.

p(A)), that of event B (i.e. p(B)), the conditional probability of A given B (i.e.

p(A|B)), and that of B given A (i.e. p(B|A)):

p(A|B) = p(B|A)p(A)P (B) . (3)

In the Bayesian framework, given that we have observed some data (yyy) and that we

want to make statistical inference about the model parameter ◊, Equation 3 becomes

p(◊|yyy) = p(yyy|◊)p(◊)p(yyy) , (4)


where p(◊) denotes the prior distribution for ◊, p(yyy|◊) denotes the data distribution,

p(yyy) is often refered to as marginal likelihood and can be summarized as

p(yyy) = q◊

p(◊)p(yyy|◊) (where ◊ is discrete) or p(yyy) =s

p(◊)p(yyy|◊) d◊ (where ◊ is

continuous), and p(◊|yyy) is called the posterior distribution. Equation 4 reflects how

Bayesian data analysis works: we have some information with regard to the model

parameter ◊ before observing the data (yyy), and this information is summarized in the

prior distribution p(◊). We then update the prior information regarding ◊ by

incorporating the collected sample data, which is the likelihood distribution p(yyy|◊).

This update - the multiplication of prior distribution p(◊) and likelihood distribution

p(yyy|◊) - leads to the posterior distribution p(◊|yyy). Further, as it is not a function of ◊,

the marginal likelihood p(yyy) is often left out, resulting in the following formula

p(◊|yyy) Ã p(yyy|◊)p(◊), (5)

indicating that the posterior distribution p(◊|yyy) is proportional to the likelihood times

the prior. Figure 1 shows the relationship among a prior distribution, a likelihood

distribution, and their posterior distribution. The posterior distribution, combining the

information in the prior and that in the likelihood, stands in the middle between the

prior and the likelihood. If the prior bears little information (e.g. a vague prior), then

the likelihood (i.e. the data) dominates the posterior distribution - the posterior

distribution is close to the likelihood distribution. If the prior brings in a large amount

of information, then the posterior distribution is "dragged" towards the prior

distribution.

Leaving out the marginal likelihood is not a problem if we focus on estimating ◊ in

the posterior distribution; however, the marginal likelihood must be taken into account

if we are to compare models by calculating Bayes factors, because it is very informative

in this respect (Braeken et al., 2014), which will be explained in the following

subsection.

Bayes Factors and Model Comparison

In the Bayesian framework, testing one hypothesis against another hypothesis is

treated as comparing one model to another model. As a criterion for comparing


Bayesian models, the Bayes factor is defined as

BFmodel1, model2

= p(yyymodel1

)p(yyy

model2

) , (6)

where p(yyymodel1

) and p(yyymodel2

) denote the marginal likelihood of model 1 and model 2,

respectively. In this paper, we will use Bayes factors to compare hypotheses, because

Bayes factors are able to quantify the relative evidence between two scientific theories

or hypotheses (Braeken et al., 2014; Kass & Raftery, 1995). To be specific, if, say,

BFmodel1, model2

= 5, then we can conclude that there is 5 times more evidence in favor of

model 1 than model 2. If BFmodel1, model2

= .2, then we can conclude that, compared to

model 2, model 1 receives 5 times less evidence. Di�erent than traditional null

hypothesis significance testing, where a significance level (e.g. – = .5) serves as a

reference point, the Bayes factor is interpreted without such references. However,

tentative guidelines with regard to the interpretation of Bayes factors are also available:

Table 1, adopted from Kass and Raftery (1995), presents some guidelines for

interpreting Bayes factors.

Testing Order Hypotheses by the Bayes Factors Approach

As mentioned in the previous subsection, marginal likelihoods are an

indispensable component of Bayes factors; this however brings about a new challenge -

it is in general very complex or even impossible to calculate (i.e. integrate) marginal

likelihoods. A shortcut, namely, the encompassing prior approach (Klugkist, 2008), is

available to avoid complex problems of integrating marginal likelihoods under certain

circumstances. The encompassing prior approach states that, given two models - an

unconstrained model (Mu

) and a constrained model (M1

) with the latter nested in the

former - the Bayes factor for M1

and Mu

is

BFM

1

, Mu = f1

c1

, (7)

where f1

and c1

denote posterior fit and prior complexity, respectively. For interested

readers, Appendix A provides a brief illustration on how to derive Equation 7. Before

discussing this encompassing prior approach in detail, we would like to remind readers

that this shortcut may not be applicable in every situation. For example, one vital


prerequisite is that an unconstrained model must exist so that constrained models are

nested within the unconstrained model. However, we expect that, after reading the

following sections where we discuss the methodology of this encompassing prior

approach and where we showcase its usage in the empirical example, readers will

recognize its superior applicability in a wide range of situations.

Although readers can find detailed information on posterior fit f1

and prior

complexity c1

in e.g. Klugkist, Kato, and Hoijtink (2005), Hoijtink, Klugkist, and

Boelen (2008), and Braeken et al. (2014), we still feel it necessary to discuss these two

concepts in this paper so that readers may find it easier to understand the core idea of

the encompassing prior approach. We focus on the following order hypotheses (M1

and

M2

):

M1

: —1

> 0, —2

> 0 (8)

M2

: —1

> —2

(9)

Mu

: —1

, —2

, (10)

as an example. M1

hypothesizes that —1

> 0 and —2

> 0. M2

hypothesizes that —1

> —2

.

Mu

states that —1

and —2

are left unconstrained.

Prior complexity. We borrow the proportionality idea from Braeken et al.

(2014) to illustrate the concept of prior complexity. Figure 2 presents the concept of

prior complexity on a two-dimensional space. The shaded areas indicate the values that

satisfy the constraints of each hypothesis (M1

and M2

) mentioned previously. When —1

and —2

are unconstrained, the complexity of this unconstrained model is one; that is, —1

and —2

can take any values, and therefore the shaded area covers the entire space (see

the plot on the top left corner). The plot on the right hand side of Figure 2 represents

the prior complexity under M1

, where —1

and —2

are allowed for only positive values.

The shaded area hence shows that the prior complexity is 1/4. As for M2

, where

—1

> —2

, the shaded area shows that the prior complexity in this case is 1/2. To sum

up, M1

is less complex than M2

, which is less complex than the unconstrained Mu

.

Posterior fit. Again, we borrow the proportionality idea from Braeken et al.

(2014) to illustrate the concept of posterior fit. Suppose we observe two hypothetical


datasets A and B (see Figure 3). Under the unconstrained model (Mu

), dataset A is

situated in the vicinity of (.2, .7) - that is, maximum likelihood estimates are —̂1

= .7

and —̂2

= .2 under the unconstrained model, and dataset B is situated in the vicinity of

(.7, .2) - that is, maximum likelihood estimates are —̂1

= .2 and —̂2

= .7 under the

unconstrained model. Posterior fit indicates the extent to which the posterior

distributions are covered within the shaded areas under the constrained models

compared to the situation under the unconstrained model. To give an example, the fit

measure for dataset B in M2

(—1

> —2

) is zero, as it is not covered by the shaded area,

whereas the fit measure for dataset A in M2

is one - the maximum fit level. As for M1

,

it is not di�cult to see that the fit measures for data set A and B are both one.

Bayes factor and hypothesis testing. Recall that Bayes factor can be

defined as f1

/c1

in the encompassing prior approach where an order-constrained

hypothesis is tested against the unconstrained model. Now it is clear that Bayes factor

o�ers a balanced view of posterior fit and prior complexity. For data set A under M1

,

fA

= 1 and cA

= 1/4 , and hence the Bayes factor for model 1 (M1

vs. Mu

) is

fA

/cA

= 4, meaning that there is 4 times more evidence in favor of M1

over Mu

. For

dataset A under M2

, fA

= 1 and cA

= 1/2 , and hence the Bayes factor for model 2 (M2

vs. Mu

) is fA

/cA

= 2, meaning that there is 2 times more evidence in favor of M2

over

Mu

. In the similar fashion, for dataset B, the Bayes factor is fB

/cB

= 4 under model 1

and fB

/cB

= 0 under model 2.

Comparing a constrained model to the unconstrained model might not be of

interest to researchers. For example, suppose a researcher has found that the Bayes

factor is in favor of the model —1

> —2

> 0 over the unconstrained model {—1

, —2

}. This

outcome does not necessarily mean that —1

> —2

> 0 is the most salient relationship

among —1

and —2

in the data: it is possible that other relationships, say, —2

> —1

> 0,

are more frequently seen in the data than —1

> —2

> 0. However, because the Bayes

factor comparing —1

> —2

> 0 with the unconstrained model {—1

, —2

} does not take into

account other models such as —2

> —1

> 0, researchers might overlook other potentially

more important relations between —1

and —2

. Hence, it is more interesting and


informative if we test the model —1

> —2

> 0 against the complement model where —1

and —2

can have any values that do not satisfy the constraint of —1

> —2

> 0 (denoted as

—1

> —2

> 0). In other words, instead of comparing a constrained model to its

unconstrained counterpart, we compare a constrained model to another constrained

model. Comparing the constrained model (e.g. —1

> —2

> 0) to another constrained

model (e.g. —1

> —2

> 0) can be processed as follows: Let M1

denote the model

—1

> —2

> 0 and M2

denote the model —1

> —2

> 0 and their unconstrained model

(—1

, —2

) be denoted as Mu

. Then the Bayes factor is defined as:

BFM

1

,M

2

= BFM

1

,Mu

BFM

2

,Mu

(11)

The aforementioned approach is particularly useful when two models (i.e. hypotheses)

are compared. However, researchers might encounter situations where more than two

hypotheses are compared at a time. In such situations, researchers may resort to

"posterior model probabilities" (PMP). Given a set of hypotheses, a PMP for a

hypothesis reflects the relative support of this hypothesis among the hypotheses set,

based on the Bayes factors of all the hypotheses and the unconstrained model

(Klugkist, 2008). For example, suppose there are three hypotheses, M1

, M2

, and M3

,

and the unconstrained model Mu

, then the PMP for M1

is

PMP (M1

) = BFM

1

, Mu

BFM

1

, Mu + BFM

2

, Mu + BFM

3

, Mu + BFMu, Mu

, (12)

where BFMu, Mu = 1. The fifth column of Table 2 presents the PMP’s for dataset A.

Model 1 (— > 0 and — > 0) receives the highest support (PMP= .57). Alternatively, we

may calculate PMP without the unconstrained model (Klugkist, 2008)

PMP (M1

) = BFM

1

, Mu

BFM

1

, Mu + BFM

2

, Mu + BFM

3

, Mu

, (13)

and as is shown in Table 2 Model 1 receives relatively higher support (PMP= .67).

So far we have presented a brief introduction to the Bayes factor approach. In the

following section, we will present a brief introduction to ordered probit regression and

how to perform order hypothesis testing in ordered probit regression models.


The Bayesian Ordered Probit Regression

A Brief Introduction to Ordered Probit Regression

Many research questions in reality involve ordinal, non-numeric variables as the

dependent variables. For example, in marketing research, the size of soda (e.g. small,

medium, and large) in a restaurant is assumed to be associated with food

consumptions. Health status (e.g. excellent, very good, good, fair, and poor) might be

related to age, gender, and hours of exercise etc. In cases where the dependent variables

are ordinal, ordered probit regression is often used in research.

The ordered probit regression model can be expressed as follows

zi

= ———Õxxxi

+ ‘i

(‘1

, ‘2

, · · · , ‘n

≥ i.i.d.N(0, 1)) (14a)

yi

= g(zi

), (14b)

where ——— is the unknown parameter in the regression model (Equation 14a) and g(·) is

the link function that connects the zi

’s and the ordinal variable yi

’s. Thus, an ordered

probit model consists of two parts: 1) a regression model where the regression

coe�cients ——— describe the relation among the independent variables xxx and the latent

variable zzz, and 2) the link function g(·) that associates the latent variable zzz with the

ordinal variable yyy. g(·) is defined as a non-decreasing function, and therefore the

positive/negative signs of ——— indicate the increase/decrease in yyy. Note that here we set

‘1

, ‘2

, · · · , ‘n

≥ i.i.d.N(0, 1) and the intercept has been omitted, because the link

function g(·) can represent the scale and the location of the distribution of yyy (Ho�,

2009).

The link function g(·) "transfers" the continuous latent variable zzz into the ordered

variable yyy, which is done in the following way: assume yyy contains C values, for example,

{1, 2, ..., C}, then the link function g(·) consists of (C ≠ 1) "thresholds", say,


{t1

, t2

, ..., tC≠1

} (t1

< t2

< ... < tC≠1

). Then yyy and zzz are linked as follows:

y = g(z) =

Y_____________]

_____________[

1, if ≠ Œ < z < t1

2, if t1

< z < t2

...

C, if tC≠1

< z < Œ.

(15)

Prior Specification

Prior distributions for —’s ( and ttt1) need to be specified before Bayes factors can

be calculated. Although it seems to involve subjective judgment, the elicitation of priors

in practice abides by certain formal methods and recommendations (Liang, Paulo,

Molina, Clyde, & Berger, 2008; Kass & Wasserman, 1995; Berger et al., 2001). In this

paper, we let the priors be symmetrically centered around zero in a (multivariate)

normal form. Such priors are viewed as "objective" and "impartial" (Mulder, Hoijtink, &

Klugkist, 2010; Braeken et al., 2014) because they do not show prior preference for any

hypothesis (i.e. no prior preference for positive/negative e�ect of independent variables

on the ordinal dependent variable). Since the mean has been set at zero, much

attention should be paid to the specification of the variances of the priors. In this

paper, we discuss two types of priors that di�er in (co)variance structure: 1) the same

conjugate prior for each —i

: —i

≥ N(0, ‡2

—

) where we primarily focus on the case in which

‡—

= 1 and compare its results with situations in which ‡—

= 100 and 10000, and 2) a

unit-information prior ——— ≥ N(000, n(XXX ÕXXX)≠1) for all —’s together. We will first explain the

elicitation of the second prior, as it is more complex, and afterwards we will discuss the

similarities between the first prior and the second prior.

When it comes to normal linear regression models, researchers conventionally

resort to the conjugate Normal-Gamma family for model-specific parameters (e.g.

George & McCulloch, 1993; Berger et al., 2001). Zellner’s g prior (Zellner, 1986)

——— ≥ N(000, g‡2(XXX ÕXXX)≠1) for — coe�cients are widely welcomed because it is

1Note that the prior for ttt is not of primary concern, and we will follow common practices when setting

a prior for ttt, which will be briefly mentioned later.


computationally e�cient in terms of evaluating marginal likelihoods and because it

o�ers simple, understandable interpretation (Liang et al., 2008). Furthermore, this prior

is also invariant for linear transformation of the covariates in XXX (Ho�, 2009); that is, it

gives rise to the result that the parameter estimation does not depend on the scale of

the regressors. This is an attractive feature in empirical research where an independent

variable might be coded in terms of di�erent units (e.g. hour/minute/second).

If we let g = n in Zellner’s g prior ——— ≥ N(000, g‡2(XXX ÕXXX)≠1), then the prior becomes

a unit information prior (Kass & Wasserman, 1995). This is because, for a linear

regression model, (XXX ÕXXX)/‡2 can be regarded as the amount of information in n

observations, and therefore (XXX ÕXXX)/(n‡2) is viewed as the amount of information in one

observation. Note that the information is actually the inverse variance for —̂——ols

, and

hence ——— ≥ N(000, n‡2(XXX ÕXXX)≠1) is a unit information prior. Finally, recall that we set

‡ = 1 in ordered probit models to identify the model, we thus get ——— ≥ N(000, n(XXX ÕXXX)≠1)

in this paper as a unit information prior. Choosing a unit information prior is in line

with the recommendation from Kass and Wasserman (1995), and the Bayes factors

correspondingly behave like the BIC (Liang et al., 2008).

As for the first prior —i

≥ N(0, 1), now it is not di�cult to see that this prior is

very close to the unit information prior ——— ≥ N(000, n(XXX ÕXXX)≠1), because the latent variable

zzz is assumed to be a standard normal distribution N(0, 1), and therefore assigning a

prior of —i

≥ N(0, 1) to the model is analogous to adding a single observation to the

data. Thus the two types of priors share the same theoretical vintage point.

Full Conditional Distributions for ———, zzz, and ttt under the Prior —i

≥ N(0, ‡2

—

)

Full conditional distribution of ———. In a general form, given the same prior

— ≥ N(—0

, ‡2

—

) for each —, the corresponding posterior distribution is also a normal

distribution conditional on zzz, with mean and variance

E(—i

|zzz) = (—0

‡2

—

+ xxxÕi

zzz—i)(

1‡2

—

+ xxxÕi

xxxi

)≠1 (16a)

V ar(—i

|zzz) = ( 1‡2

—

+ xxxÕi

xxxi

)≠1, (16b)


where zzz—i = —

i

xxxi

+ ‘‘‘. Appendix B explains how Equation 16a and Equation 16b are

derived.

Full conditional distribution of zzz. The distribution of zi

is a normal

distribution zi

≥ N(—1

xi1

+ —2

xi2

+ ... + —n

xin

, 1). If yi

is observed, then zi

œ (tyi≠1

, tyi).

Therefore the full conditional distribution of zi

given ———, yyy, and ttt is

p(zi

|———, yyy, ttt) Ã dnorm(zi

, µi

, 1) ◊ ”(tyi≠1

,tyi )

(zi

), where µi

= —1

xi1

+ —2

xi2

+ ... + —n

xin

. The

full conditional distribution of zi

is therefore a constrained normal distribution.

Full conditional distribution of ttt. Given that we have observed yyy and

obtained zzz, how to settle ttt is quite straightforward. It is clear that the threshold tc

meets the following criterion:

tc

œ (max{zi

: yi

= c}, min{zi

: yi

= c + 1}); (17)

that is, the threshold tc

should be smaller than all the z values for y = c + 1 and be

larger than all the z values for y = c. In this paper, in line with previous research (e.g.

Ho�, 2009), we let the full conditional density of ttt to be a normal density N(µc

, ‡2

c

)2

with the constrain tc

œ (max{zi

: yi

= c}, min{zi

: yi

= c + 1}).

Full Conditional Distributions for ———, zzz, and ttt under the Prior

——— ≥ N(000, n(XXX ÕXXX)≠1)

Full conditional distribution of ———. Given the unit information prior

——— ≥ N(000, n(XXX ÕXXX)≠1), the posterior distribution is a multivariate normal distribution

with

E(———|zzz) = n

n + 1(XXX ÕXXX)≠1XXX Õzzz (18a)

V ar(———|zzz) = n

n + 1(XXX ÕXXX)≠1. (18b)

Appendix C illustrates how to derive the above equations.

Full conditional distribution of zzz. The distribution of zi

is a normal

distribution zi

≥ N(———Õxi

, 1). If yi

is observed, then zi

œ (tyi≠1

, tyi). Therefore the full

conditional distribution of zi

given ———, yyy, and ttt is2We set µc = 0 and ‡ = 100 in this paper.


p(zi

|———, yyy, ttt) Ã dnorm(zi

,———Õxi

, 1) ◊ ”(tyi≠1

,tyi )

(zi

). The full conditional distribution of zi

is

a constrained normal distribution.

Full conditional distribution of ttt. The full conditional distribution of ttt in

this case is not di�erent from what we have discussed previously: we let the full

conditional density of ttt be a normal density N(µc

, ‡2

c

) with the constrain

tc

œ (max{zi

: yi

= c}, min{zi

: yi

= c + 1}).

Gibbs Sampling

As a special case of Metropolis-Hastings sampling, the gibbs sampler (Geman &

Geman, 1984) is often applied when it is di�cult to directly sample from a multivariate

probability distribution. The posterior distribution of the ordered probit model we

discussed previously involves zzz, ——— matrix, and ttt (from the g(·)), and therefore we will

not sample directly from the poster distribution but will employ Gibbs sampling.

Instead of considering the joint probability distribution as a whole, the Gibbs sampler

focuses on univariate conditional distribution; that is, one random variable is considered

at a time while other random variables are fixed. When using the same prior

—i

≥ N(—0

, ‡2

—

) for —’s (e.g. —1

, —2

, and —3

), the Gibbs sampler in this paper will proceed

as follows:

zzz(i)

≥ p(zzz|—1

= —1(i≠1)

, —2

= —2(i≠1)

, —3

= —3(i≠1)

, ttt = ttt(i≠1)

) (19a)

—1(i)

≥ p(—1

|zzz = zzz(i)

, —2

= —2(i≠1)

, —3

= —3(i≠1)

, ttt = ttt(i≠1)

) (19b)

—2(i)

≥ p(—2

|zzz = zzz(i)

, —1

= —1(i)

, —3

= —3(i≠1)

, ttt = ttt(i≠1)

) (19c)

—3(i)

≥ p(—3

|zzz = zzz(i)

, —1

= —1(i)

, —2

= —2(i)

, ttt = ttt(i≠1)

) (19d)

ttt(i)

≥ p(ttt|zzz = zzz(i)

, —1

= —1(i)

, —2

= —2(i)

, —3

= —3(i)

), (19e)

where (i) indicates the ith round of iteration. Take Equation 19a for example, zzz in ith

round is drawn from the full conditional distribution of zzz conditional on —’s and ttt in

(i ≠ 1)th round. Once we have obtained zzz in ith round, we can update —1

conditional on

zzz in ith round and —2

, —3

, and ttt in (i ≠ 1)th round. In such as way, zzz, —’s and ttt are

updated sequentially3.3Note that it is also necessary to set the initial values for —’s; in this paper —1(0) = —2(0) = —3(0) = 0.


When using the prior ——— ≥ N(000, n(XXX ÕXXX)≠1), the Gibbs sampler is simpler:

zzz(i)

≥ p(zzz|——— = ———(i≠1)

, ttt = ttt(i≠1)

) (20a)

———(i)

≥ p(———|zzz = zzz(i)

, ttt = ttt(i≠1)

) (20b)

ttt(i)

≥ p(ttt|zzz = zzz(i)

,——— = ———(i)

). (20c)

After a su�cient number of iterations, the Gibbs sequence should reach to a

stationary distribution. This stationary distribution is not dependent on the initial

values, and this distribution is used for us to make inference about the characteristics of

the joint distribution that we do not sample directly from. Note that in practice, it is

necessary to discard the first few numbers of iterations (i.e. burn-in period); that is, we

need to remove the influence of the initial values.

In the last part of this section, we discuss a few issues that are interrelated and

are important when it comes to testing order hypotheses in ordered probit models.

Power, Prior Information, and Prior Shrinkage

Power. Power is generally not of primary interest in Bayesian hypothesis

testing, because, as mentioned in previous sections, Bayes factor reflects the extent to

which one hypothesis is favored over another, and therefore there is no need to consider

type I and type II errors. However, we believe that it is helpful to discuss power in this

paper, because on the one hand readers who are familiar with frequentist statistics

might find the Bayes factor approach more accessible via the idea of power and on the

other it facilitates illustrating a few features of Bayesian probit models to readers.

The "power" in this paper is conceptually di�erent than the power in frequentist

statistics. In this paper, power can be viewed as the extent to which the evidence

derived from the posterior is capable of reflecting the true hypothesis (in other words,

the true relation among parameters of interest) in population. To give an example,

suppose there are two samples randomly selected from the population with one sample

of five observations (n = 5) and the other 100 observations (n = 100) and suppose a

non-informative prior is assigned. Then it is not di�cult to imagine that the former

sample has lower power than the latter sample, because the former sample is less likely


to be a representative sample, compared to the latter sample; in other words, the

evidence based on the former sample is less likely to be able to reflect the true relations

among parameters of interest, compared to the evidence based on the latter sample.

Another factor that may influence power is the true —’s, which can be viewed as

e�ect sizes. To give an example, suppose sample A contains 100 observations drawn

from the population where the true —1

= .9 and —2

= .3, whereas sample B contains 100

observations drawn from the population where the true —1

= .9 and —2

= .85, and

further suppose a researcher plans to test the hypothesis —1

> —2

> 0 in both two

samples. Then it is not di�cult to imagine that sample A has more power than sample

B, as the di�erence between the true —1

and true —2

is much larger in sample A than

that in sample B.

Sample size and e�ect size are not the only factors that influence power: we will

discuss in the remaining section two interrelated factors that are related to power and

also to sample size and e�ect size, namely prior information and prior shrinkage.

Prior information. Prior information is directly related to the power we

discussed previously. Recall that the relative position of the posterior distribution is

somewhere between the prior distribution and the likelihood distribution. To give an

example, suppose that the order hypothesis to be tested is —1

> —2

> 0, that the prior is

—i

≥ N(0, ‡2

—

), and that the true relationship between —1

and —2

is —1

= .7 and —2

= .3

(and therefore is in line with the hypothesis), then the vaguer the prior (i.e. ‡ æ Œ),

the closer the posterior distribution is to the likelihood distribution, and therefore the

larger the area of the posterior is within the order-constrained space (—1

> —2

> 0),

meaning that the posterior fit f is larger. Further recall that BF = f/c in the

encompassing prior approach, the Bayes factor in this case is larger. On the other hand,

if the true relationship between —1

and —2

is —1

= ≠.7 and —2

= ≠.3 (and therefore is

not in line with the hypothesis), and a vaguer prior will let the posterior distribution be

located farther away from the order-constrained space (—1

> —2

> 0), which in turn

gives rise to a smaller value of Bayes factor. In general, the example above illustrates

that when the prior is located on the boundary of the order-constrained space (i.e. the


mean of the prior is 0), a vaguer prior leads to higher power. Similar results have been

found by Mulder (2014).

Coming back to the two priors to be examined in this paper. The information in

the prior ——— ≥ N(000, n(XXX ÕXXX)≠1) is relatively fixed in the sense that it contains one unit of

information, which is defined by the data (i.e. XXX and n) itself. This prior is relatively

vague when the sample size n is large. For this prior, the sample size plays the major

role in terms of how much power we obtain. On the other hand, priors like — ≥ N(0, ‡2

—

)

are a bit complex because, as illustrated above, ‡ decides how vague the prior is, which

in turn a�ects the power. However, we expect the impact of ‡—

would be salient only

when the sample size n is small. When n is su�ciently large, the impact of prior

information is negligible because the posterior in such cases is dominated by the data.

In general, it is of necessity to examine the impact of prior information on power

when the sample size is small, because this impact is in fact directly determined by

researchers: the type of priors specified by a researcher will decide how the priors

influence the power. We will test in this paper how di�erent ‡—

’s in — ≥ N(0, ‡2

—

) would

influence hypothesis testing through power in ordered probit models.

Prior shrinkage. The relation between prior information and the relative

position of the posterior distribution between the prior and the likelihood reflects the

reason why Bayesian estimators are often called shrinkage estimators. The e�ect of

prior shrinkage depicts the influence of the prior on the posterior. In addition to the

prior information, sample size n is a major concern when it comes to the shrinkage

e�ect. Recall that in the regression part of the ordered probit regression model

E(———|zzz) = n/(n + 1)(XXX ÕXXX)≠1XXX Õzzz, V ar(———|zzz) = n/(n + 1)(XXX ÕXXX)≠1, when the prior

——— ≥ N(000, n(XXX ÕXXX)≠1) is assigned. Readers might have noticed that the expect values

and variances depend on the number of observations n: When n æ Œ, —̂—— æ —̂——ols

.

Furthermore, the number of observations n is also implicitly included in XXX and zzz. This

means that the posterior summaries - E(———|zzz) and V ar(———|zzz) - are partially influenced by

n. If the prior —i

≥ N(—0

, ‡2

—

) is assigned, then we have

E(—i

|zzz) = (—0

/‡2

—

+ xxxÕi

zzz—i)(1/‡2

—

+ xxxÕi

xxxi

)≠1 and V ar(—i

|zzz) = (1/‡2

—

+ xxxÕi

xxxi

)≠1. Again, the


expect values and variances implicitly include the number of observations n (in xxxi

and

zzz—i), and they are therefore influenced by n. Now it is clear that the impact of

shrinkage e�ect depends on the sample size n. The larger n is, the smaller impact the

shrinkage e�ect will exert, which is analogous to having a vaguer prior; this relation in

turn should lead to greater power when the prior is located on the boundary of the

order-constrained space.

Similar to the e�ect of prior information, the e�ect of shrinkage on power, on a

macro level, is directly decided by researchers, as it depends on what types of priors are

specified by researchers. In this paper, therefore, it is of interest to see how the two

types of priors (i.e. —i

≥ N(—0

, ‡2

—

) and ——— ≥ N(000, n(XXX ÕXXX)≠1)) will experience the

shrinkage e�ect, when the sample size n is small.

To sum up, this section presents a general framework of testing order hypothesis

in ordered probit regression models via the Bayes factor approach. We have listed a few

components that are required to employ this Bayes factor approach, namely, prior

specification, full conditional distributions for ———, zzz, and ttt, and Gibbs sampling. Further,

we have stressed a few issues that this paper will examine, including power, prior

information, and prior shrinkage. In the next section, we provide a simulation study to

showcase the general behavior of the Bayes factor approach in testing order hypotheses

in ordered probit models.

Numerical Simulation

This section presents a simulation study on testing the order hypothesis -

—1

> —2

> —3

, given di�erent sample sizes and true — values.

Simulation Design

We apply the Bayes factor approach to 40 simulated datasets that di�er in sample

sizes and true — values. Specifically, the datasets are organized in 8 groups in terms of

the true — values, and they can be further grouped into 4 categories in terms of their

e�ect sizes (that is, the distances between the — values and zero): Small e�ect sizes (S),

including {—1

= .2, —2

= .1, and —3

= 0} and {—1

= ≠.2, —2

= ≠.1, —3

= 0},

Small-medium e�ect sizes (S-M), including {—1

= .4, —2

= .2, —3

= 0} and {—1

= ≠.4,


—2

= ≠.2, —3

= 0}, Medium-large e�ect sizes (M-L), including {—1

= .6, —2

= .3, —3

= 0}

and {—1

= ≠.6, —2

= ≠.3, —3

= 0}, and Large e�ect sizes (L), including {—1

= .8,

—2

= .4, —3

= 0} and {—1

= ≠.8, —2

= ≠.4, —3

= 0}. Further, each of the 8 groups

contains 5 datasets, and each dataset consists of 100 samples; the five datasets in the

same group di�er in terms of sample sizes; that is, in the first dataset each sample

contains n = 10 observations, in the second dataset each sample contains n = 50

observations, in the third dataset each sample contains n = 100 observations, in the

fourth dataset each sample contains n = 500 observations, and in the fifth dataset each

sample contains n = 1000 observations. For each dataset, we test whether the data

support —1

> —2

> —3

. Note that, instead of testing —1

> —2

> —3

against the

unconstrained model {—1

, —2

, —3

}, we test —1

> —2

> —3

against —1

> —2

> —3

(that is, —1

,

—2

, and —3

can take any values that do not satisfy —1

> —2

> —3

); In other words, we

calculate the Bayes factor for each dataset and see whether the Bayes factor is higher

than one. By varying sample sizes and — values, we aim to understand how e�ective the

Bayes factor is when testing order hypotheses under two priors - 1) the same conjugate

prior for each —i

: —i

≥ N(0, ‡2

—

) (where ‡—

= 1, 100, and 10000), and 2) a

unit-information prior ——— ≥ N(000, n(XXX ÕXXX)≠1). Further, we examine how power is

influenced by prior information and prior shrinkage.

Results

The results are generated by the Gibbs sampling procedure with 50000 iterations,

and the first 5000 iterations, as the burn-in period, have been discarded.4

The impact of sample sizes and e�ect sizes when the prior —i

≥ N(0, 1)

and the prior ——— ≥ N(000, n(XXX ÕXXX)≠1) are assigned respectively. By altering sample

sizes and e�ect sizes, we find that the Bayes factor approach works better on testing

—1

> —2

> —3

when the sample size and the e�ect size are larger. Table 3 provides a

summary for the results under the prior —i

≥ N(0, 1), where we list the percentages of4We have found that 100000 or more iterations would be better in terms of convergence according

to convergence diagnostics such as the Geweke method (Geweke et al., 1991). However, we conclude

that (due to the time constraint and limited computing power) 50000 iterations is enough, because the

convergence does not noticeably influence the results of this study.


Bayes factors that fall into categories that are in correspondent to the recommendation

from Kass and Raftery (1995) on how to interpret Bayes factors.

Specifically, when the relation among —’s is in line with the hypothesis, the S

e�ect size requires large sample sizes so that the hypothesis —1

> —2

> —3

can be tested

accurately: When sample size n = 10, 54% out of the 100 Bayes factors are in favor of

—1

> —2

> —3

over —1

> —2

> —3

(i.e. BF > 1); when n = 100, 78% of the Bayes factors

support —1

> —2

> —3

; when n = 500, 95% of the Bayes factors support the hypothesis

(i.e. BF > 1). As the e�ect sizes are getting larger (from S-M to L), smaller sample

sizes are enough to test the hypothesis —1

> —2

> —3

. For example, starting from the

S-M e�ect size, when n Ø 50, over 90% of Bayes factors support the hypothesis

—1

> —2

> —3

over —1

> —2

> —3

(i.e. BF > 1). Furthermore, as the sample size is

getting larger, higher amount of Bayes factors fall into the "BF > 20" category, which

according to Kass and Raftery (1995) indicates "strong (or very strong)" evidence in

favor of —1

> —2

> —3

.

On the other hand, if the hypothesis is not in line with the true relationship

among —’s, then a sample size as large as n = 50 is enough no matter the e�ects sizes

are large or small. As is shown in Table 3, when the sample size n Ø 50, at least 90% of

the Bayes factors are in favor of —1

> —2

> —3

over —1

> —2

> —3

(i.e. BF < 1).

In general, we can conclude that, when the prior —i

≥ N(0, 1) is assigned, the

Bayes factor approach works very well: Firstly, if the hypothesis being tested is in line

with the true relations of parameters of interest in population, a sample size of at least

100 is preferred. (If researchers prefer more stringent criteria such as BF > 3 instead of

BF > 1, then larger sample sizes are preferred.) Further, given the same e�ect size, the

larger the sample size is, the easier it is for the Bayes factor to test the hypothesis

accurately. In addition, given the same sample size, the the larger the e�ect size, the

easier it is for the Bayes factor to test the hypothesis accurately. Secondly, if the

hypothesis being tested is in not line with the true relations of parameters of interest in

population, then a rather small sample size (e.g. n = 50) is able to detect it.

As for the impact of sample sizes and e�ect sizes when the prior


——— ≥ N(000, n(XXX ÕXXX)≠1) is assigned, in line with what we have found when the prior

—i

≥ N(0, 1) is assigned, the Bayes factor approach works very well (see Table 4): given

the same e�ect size, the larger the sample size is, the easier it is for the Bayes factor to

test the hypothesis accurately; given the same sample size, the the larger the e�ect size,

the easier it is for the Bayes factor to test the hypothesis accurately. In general, the

results based on the prior ——— ≥ N(000, n(XXX ÕXXX)≠1) and those based on the prior

—i

≥ N(0, 1) are very close, and therefore we will not discuss the results here in detail.

The impact of selection of ‡—

in —i

≥ N(0, ‡2

—

). We further test the same

order hypothesis under the priors p(0, ‡—

) = N(0, 1), N(0, 100), and N(0, 10000) and

find that, given the same e�ect size, di�erent ‡—

’s do not influence the results noticeably

when the sample size is large enough (n Ø 50). For the sake of simplicity, we provide

the results based on the simulated datasets with S e�ect size and M-L e�ect size in

Table 5. The results are generally in line with our expectation: the influence of prior

information on the posterior depends on the sample size.

Prior shrinkage. The results that have been discussed so far also demonstrate

the behavior of prior shrinkage in testing order hypotheses under the two types of

priors. We will not repeat reporting the results again. The takeaway for readers is that

the e�ect of shrinkage is salient when the sample size is very small (n = 10) and that it

becomes less and less salient when the sample size becomes larger.

An Empirical Example

In the beginning of this paper, we asked readers to consider a scenario in

marketing research. In fact, that scenario is based on a real case of the company ABB

Electric, which later on became a well known case study for MBA students. In this

section, we revisit the scenario and show readers how to examine order hypotheses from

the perspective of the Bayes factor approach.

The Background of the ABB Electric Case Study

ABB Electric was founded by ASEA-AB Sweden and RTE Corporation in 1970

with the purpose of designing and manufacturing medium power transformers, breakers,

and relays and the aim of entering the North American Market. The market was


dominated by big companies such as General Electric, Westinghouse, and

McGraw-Edison, and therefore ABB Electric had a tough time breaking through the

market. Furthermore, the market was experiencing a severe industry-environment

change in 1974 - industry sales dropped by around 50%. In order to face the challenges,

ABB Electric invited Dennis Gensch, an outside consultant, to help develop a type of

management science models, namely multi-attribute disaggregate choice models.

Thanks to the management sciences models, ABB Electric survived the attacks from

major competitors and the harsh industry-wide environmental challenges and became

the dominant firm in the industry.5

In this paper, we will not analyze the entire case study but focus on how to

identify the positioning of ABB Electric via ordered probit models in the frequentist

framework and via the Bayes factor approach.

Data and Hypothetical Research Question

The dataset used in this paper is adapted from Lattin, Carroll, and Green (2003).

88 customers evaluated four companies (General Electric, Westinghouse,

McGraw-Edison, and ABB Electric) with regard to the following 4 attributes: price,

maintenance, warranty, and quality. The four companies are treated as the ordinal

dependent variable in terms of market share of each company6, with General Electric

being coded as 1, Westinghourse as 2, McGraw-Edison as 3, and ABB Electric as 4.

The data of the 4 attributes came from 4 corresponding items (9-point scale) in a

questionnaire with the highest score (9) indicating that a company performed the best

on that particular item.

Suppose that the CEO of ABB Electric believed that the positioning of ABB

5Note that we adopt the background information from Gensch, Aversa, and Moore (1990).6Note that we do not have accurate information regarding the exact market share of each company;

however, available information gives rise to the following tentative conclusion: General Electric accounted

for the highest amount of the market share, followed by Westinghouse, McGraw-Edison, and ABB

Electric.


Electric can be summarized in the following hypothesis7.

—price

< 0

|—quality

| > |—maintenance

| > |—warranty

|8,

(21)

that is, the higher the price, the less likely that customers would choose ABB Electronic

products. Furthermore, the CEO believed that, among the attributes quality, warranty

and maintenance level, quality was ranked the most important in terms of the absolute

value of the expected e�ect, followed by maintenance level, and warranty. That is, in

general, the CEO conjectured that the positioning of ABB Electronic was such that its

products and services should charge relatively lower prices compared to the rest three

companies and that the company should primarily focus on quality followed by

maintenance and warranty.

Results

We first analyze this ordered probit model in the frequentist approach, as this

approach is often employed when such types of data are encountered in business

research. Then we reanalyze the data by applying the Bayes factor approach.

Ordered probit regression analysis with the frequentist approach. We

notice that in practice, researchers who apply ordered probit models might follow

slightly di�erent procedures. In this paper, we follow the procedure recommended by

Lattin et al. (2003), which is often practiced in marketing research. Compared to the

null model where only the intercept is included, the final model with parameters price,

quality, warranty, and maintenance shows significant improvement at – = .1 level7Note that this hypothesis is di�erent from the original ones; the original research questions and

hypotheses are more complex. We use this hypothesis to illustrate the Bayes factor approach to testing

hypotheses in ordered probit models. Readers should not confuse this hypothesis with original ones.8Some readers might wonder whether there is any relation between price and other product attributes.

It is indeed possible. However, in theory and practice, price is often regarded as a unique attribute in

the sense that consumers may view price as an overarching attribute that combine many aspects (e.g.

cost (Lichtenstein, Ridgway, & Netemeyer, 1993), reputation (Liu & Soman, 2008), and fairness (Bolton,

Warlop, & Alba, 2003) etc.), and we therefore follow the convention and do not examine the relation

between price and other attributes of ABB products.


(‰2(4) = 8.037, p = .09). The goodness-of-fit AIC criterion reduced by 8.037, indicating

a better fit of the full model. The estimated parameters are presented in Table 6.

Quality is significantly di�erent from zero at – = .05 level (—quality

= .469, p < .05),

whereas price, warranty, and maintenance are not significantly di�erent from zero at

– = .1 level (—price

= ≠.199, p = .197; —warranty

= .045, p = .611; —maintenance

= ≠.179,

p = .235). The — coe�cients seem to suggest that, in line with what the CEO has

conjectured, the higher the price, the less likely that customers would opt for products

from ABB Electric9 and that quality is indeed more important than maintenance in

terms of absolute values, which is in turn more important than warranty. However, note

that the statistical tests above only tell us whether the regression coe�cients are

significantly di�erent from zero, and therefore we do still not know, based on the

frequentist approach, whether the proposed the order hypothesis {—price

< 0,

|—quality

| > |—maintenance

| > |—warranty

|} receives more evidence from the data than its

complement.

Ordered probit regression analysis with the Bayes factor approach.

Calculating the Bayes factor is rather straightforward, as is shown in previous sections.

Let M1

denotes the hypothesis {—price

< 0, |—quality

| > |—maintenance

| > |—warranty

|}, and M2

{—price

< 0, |—quality

| > |—maintenance

| > |—warranty

|} covers all the possible relationships

among —’s that are not in line with M1

. Then, based on the unit information prior

——— ≥ N(000, n(XXX ÕXXX)≠1), the Bayes factor of M1

against M2

is 30.8, indicating that the

Bayes factor approach is strongly in favor of M1

, suggesting that the CEO’s conjecture

{—price

< 0, |—quality

| > |—maintenance

| > |—warranty

|} is supported. On the other hand, one

may also calculate posterior model probabilities for M1

and M2

. The PMP for M1

is

.97, and that for M2

is .03, which indicates that the evidence from the data is in favor

of {—price

< 0, |—quality

| > |—maintenance

| > |—warranty

|} over

{—price

< 0, |—quality

| > |—maintenance

| > |—warranty

|}. Compared to the frequentist approach

above, our Bayes factor approach speaks to the order hypothesis directly and therefore9Some researchers would discard nonsignificant parameters and redo the analysis. Considering the

purpose of this section is not to explain how to conduct regression analysis but to compare the frequentist

approach with the Bayes factor approach, we do not discard nonsignificant parameters in this paper.


enables decision-makers to draw confident conclusions with regard to the positioning of

ABB Electric.

Discussion

In this paper, we present a Bayes factor approach to testing order hypotheses in

ordered probit regression models. This Bayes factor approach is, as far as we know, the

only method available for testing order hypotheses in ordered probit models in a direct

manner. Compared to the frequentist way of hypothesis testing, this Bayes factor

approach has its own advantage: First of all, the Bayes factor approach enables us to

test order hypotheses directly in only one step, whereas traditional NHST usually

involves a general omnibus testing step, followed by a few more tests where the original

order hypothesis is decomposed. When it comes to ordered probit models, the

aforementioned way of NHST is even not applicable. Therefore, hypothesis testing via

the Bayes factor approach, compared to that via the frequentist approach, is much

easier and enjoys higher statistical power. Moreover, this Bayes factor approach, unlike

the frequentist approach, does not rely on large sample theory, which is also an

advantage. Secondly, because it directly tests order hypotheses, the Bayes factor

approach o�ers a better conceptual clarity. Theories built upon relative rank orders of

expected e�ects can be examined precisely.

Most importantly, we believe that the Bayes factor approach helps researchers to

make better decisions in theory development. This approach not only tells researchers

whether there is evidence to confirm an order hypothesis (say, —1

> —2

> —3

) but also

tells researchers to what extent the hypothesis —1

> —2

> —3

is supported against other

possible relationships among the parameters (e.g. —1

> —2

> —3

). This is of utmost

importance because it prevents researchers from making partially true conclusions.

Furthermore, our paper has shown that the Bayes factor approach generally does not

require a very large sample size, especially when the hypothesis is not in line with the

true relationship in population. This feature also benefits scientific research and

practices: As the Bayes factor is quite "sensitive" to false hypothesis, researchers can

halt their research projects timely to re-examine their theories.


In this paper, we present a simulation study on testing order hypothesis

—1

> —2

> —3

based on two types of priors, namely, p(—) ≥ N(0, ‡—

) (where ‡—

= 1, 100,

and 10000) and ——— ≥ N(000, n(XXX ÕXXX)≠1), respectively. The results show that the Bayes

factor approach works well when the sample size is reasonably large (at least larger than

50). This finding is expectable: First, intuitively the larger the sample size, the more

information there is to be used to test the hypothesis, and therefore more power10.

Second, the e�ect of prior shrinkage is smaller as the sample size is getting larger, which

in turn also increase power. E�ect size is another factor that influences order hypothesis

testing: the results suggest that the larger the e�ect sizes, the easier it is for the Bayes

factor approach to test the hypothesis, which reflects the increase in power.

Both two priors work well and generate very similar results, which is in line with

our intuition as well: the standard normal prior p(—) ≥ N(0, 1) can be viewed as adding

one observation to the data, because the distribution of the latent variable zzz is assumed

to be a standard normal distribution N(0, 1); On the other hand, we have explained

previously that ——— ≥ N(000, n(XXX ÕXXX)≠1) is a unit information prior - that is, this prior

contains the same amount of information as one observation would contain. Therefore,

both two priors can be regarded as adding one observation to the data, and yet the two

priors are di�erent: correlations among —’s are not taken into account by the prior

p(—) ≥ N(0, 1) but by the prior ——— ≥ N(000, n(XXX ÕXXX)≠1). This simulation study also

illustrates that results can be noticeably influenced by prior information and prior

shrinkage when the sample size is very small (e.g. n = 10). Therefore, readers are

recommended to pay particular attention to situations with small sample sizes.

The empirical study of ABB Electric further showcases the applicability of our

Bayes factor approach. The key takeaway is that this Bayes factor approach

substantially facilitates managerial decision-making by testing order hypotheses in a

direct manner, whereas the conventional frequentist approach fails to provide adequate

10We would like to remind readers again that the concept of power in this paper is conceptually di�erent

from that in the frequentist approach. In this paper, power can be viewed as the extent to which the

evidence derived from the posterior is capable of reflecting the true hypothesis (in other words, the true

relation among parameters of interest) in population.


solutions.

We are confident that, due to its theoretical clarity and practical applicability, the

Bayes factor approach discussed in this paper o�ers a promising line of research. We

propose that the following directions could be of interest to future research. First of all,

we have examined two types of priors p(—) ≥ N(0, ‡i

) and ——— ≥ N(000, n(XXX ÕXXX)≠1) in this

paper. Yet, there might exist other priors that work well (or even better) for testing

order hypotheses in ordered probit models. Two candidates have drawn our attention in

this respect: One candidate, proposed by Gelman, Jakulin, Pittau, and Su (2008), is a

weakly informative prior with a Cauchy distribution with center zero and scale 2.5 for

logistic regressions. This prior is said to be able to provide stable estimates while

remaining vague enough as a default in applied work (Gelman et al., 2008). A second

type of priors contain mixtures of Zellner’s g priors proposed by Liang et al. (2008),

who have shown that mixtures of g priors are able to resolve consistency problems that

traditional fixed g priors are faced with, and they in the meanwhile are able to maintain

computational e�ciency. We believe that, given their merits, the aforementioned two

types of priors might be useful for hypothesis testing in ordered probit models. Further

research could focus on understanding the behavior of the Bayes factor approach in

situations where the priors proposed by Gelman et al. and Liang et al. are involved.

The second direction for future research is of practical importance: We suggest a

thorough examination on the sample size requirement for testing order hypotheses via

the Bayes factor approach. The simulation study in this paper suggests that a sample

size of 100 or larger should su�ce to test an order hypothesis in an ordered probit

model with three independent variables. We expect that a larger sample size will be

needed when more independent variables are included in the model. Future research

could focus on understanding what the minimum sample sizes are in complex situations

with more independent variables.

Finally, this paper focuses on one type of order hypotheses that are often seen in

social sciences: —1

> —2

> —3

. The Bayes factor approach can also be extended to other


types of order hypotheses, for example:

—1

> —2

= —3

> —4

(22)

—1

> —2

± ‘ > —3

11. (23)

Hence, future research could also focus on this direction.

Conclusion

This paper presents a Bayes factor approach to testing order hypotheses in ordered

probit models. This Bayes factor approach is, as far as we know, the only method

available for testing order hypothesis in the setting of ordered probit models. This

approach is conceptually clear and straightforward - it enables researchers to test rank

ordered relationships among expected e�ects such as —1

> —2

> —3

in a direct manner,

which is particularly useful when researchers are developing and verifying complex

theories. To illustrate how to conduct hypothesis testing via this Bayes factor approach,

we provide a simulation study and an empirical example where the advantage of this

approach over the common practices based on the frequentist approach is showcased.

This paper serves as a stepping stone for research on testing order hypotheses in

ordered/nominal probit/logit models. We are confident that our Bayes factor approach

is very promising in this line of research, which will eventually facilitate theory

development and research practices in the near future.

11Here, ‘ denotes any very small number. Hence —1 > —2 ± ‘ > —3 indicates that —1 is not strictly

larger than the number —2 but larger than the region [—2 ≠ ‘, —2 + ‘], and this region is larger than —3.


Table 1

Table for Interpreting Bayes Factors Adopted from Kass & Raftery (1995)

BFmodel1, model2

Evidence in Support of Model 1

< 1 Negative

1 - 3 Anecdotal

3 - 20 Positive

20 - 150 Strong

> 150 Very strong


Table 2

Posterior Model Probabilities for Dataset A

c f BF PMP (Full Set) PMP (M1

and M2

)

M1

(—1

> 0 and —2

> 0) 1/4 1 4 .57 .67

M2

(—1

> —2

) 1/2 1 2 .29 .33

Mu

(—1

, —2

) 1 1 1 .14 –


Table 3

The Percentages of Bayes Factors under Various Sample Sizes and E�ect Sizes When

the Prior —i

≥ N(0, 1) Is Assigned

1 < BFM1,M2 Æ 3 3 < BFM1,M2 Æ 20 BFM1,M2 > 20

Small e�ect size (S)

—1 = .2, —2 = .1, —3 = 0

n=10 23% 29% 2%

n=50 39% 34% 1%

n=100 33% 37% 8%

n=500 2% 50% 43%

n=1000 1% 20% 76%

—1 = ≠.2, —2 = ≠.1, —3 = 0

n=10 15% 12% 1%

n=50 9% 1% 0

n=100 7% 1% 0

n=500 0 0 0

n=1000 0 0 0

Small-medium e�ect size (S-M)

—1 = .4, —2 = .2, —3 = 0

n=10 28% 24% 4%

n=50 23% 50% 17%

n=100 14% 54% 28%

n=500 0 6% 94%

n=1000 0 0 100%

—1 = ≠.4, —2 = ≠.2, —3 = 0

n=10 19% 4% 0

n=50 3% 1% 0

n=100 0 0 0

n=500 0 0 0

n=1000 0 0 0

Medium-large e�ect size (M-L)

—1 = .6, —2 = .3, —3 = 0

n=10 33% 25% 6%

n=50 10% 46% 39%

n=100 3% 29% 68%

n=500 0 0 100%

n=1000 0 0 100%

—1 = ≠.6, —2 = ≠.3, —3 = 0

n=10 10% 4% 0

n=50 0 0 0

n=100 0 0 0

n=500 0 0 0

n=1000 0 0 0

Large e�ect size (L)

—1 = .8, —2 = .4, —3 = 0

n=10 36% 32% 5%

n=50 5% 37% 58%

n=100 1% 11% 88%

n=500 0 0 100%

n=1000 0 0 100%

—1 = ≠.8, —2 = ≠.4, —3 = 0

n=10 8% 3% 0

n=50 0 0 0

n=100 0 0 0

n=500 0 0 0

n=1000 0 0 0

Note. M1 stands for the hypothesis —1 > —2 > —3, and M2 denotes the complement model —1 > —2 > —3. The three categories are defined

based on Kass & Raftery (1995) (see Table 1).


Table 4

The Percentages of Bayes Factors under Various Sample Sizes and E�ect Sizes When

the Prior ——— ≥ N(000, n(XXX ÕXXX)≠1) Is Assigned

1 < BFM1,M2 Æ 3 3 < BFM1,M2 Æ 20 BFM1,M2 > 20

Small e�ect size (S)

—1 = .2, —2 = .1, —3 = 0

n=10 23% 26% 3%

n=50 39% 33% 2%

n=100 33% 38% 7%

n=500 4% 51% 40%

n=1000 1% 18% 78%

—1 = ≠.2, —2 = ≠.1, —3 = 0

n=10 13% 8% 4%

n=50 8% 1% 0

n=100 6% 1% 0

n=500 0 0 0

n=1000 0 0 0

Small-medium e�ect size (S-M)

—1 = .4, —2 = .2, —3 = 0

n=10 31% 21% 4%

n=50 22% 50% 16%

n=100 15% 31% 30%

n=500 0 6% 94%

n=1000 0 0 100%

—1 = ≠.4, —2 = ≠.2, —3 = 0

n=10 18% 5% 0

n=50 4% 1% 0

n=100 0 0 0

n=500 0 0 0

n=1000 0 0 0


—1 = .6, —2 = .3, —3 = 0

n=10 29% 27% 7%

n=50 9% 49% 36%

n=100 2% 30% 67%

n=500 0 0 100%

n=1000 0 0 100%

—1 = ≠.6, —2 = ≠.3, —3 = 0

n=10 10% 5% 0

n=50 0 0 0

n=100 0 0 0

n=500 0 0 0

n=1000 0 0 0

Large e�ect size (L)

—1 = .8, —2 = .4, —3 = 0

n=10 25% 31% 8%

n=50 5% 37% 58%

n=100 1% 12% 87%

n=500 0 0 100%

n=1000 0 0 100%

—1 = ≠.8, —2 = ≠.4, —3 = 0

n=10 11% 1% 1%

n=50 0 0 0

n=100 0 0 0

n=500 0 0 0

n=1000 0 0 0

Note. M1 stands for the hypothesis —1 > —2 > —3, and M2 denotes the complement model —1 > —2 > —3. The three categories are defined

based on Kass & Raftery (1995) (see Table 1).

HY

POT

HESIS

TEST

ING

INO

RD

ERED

PROBIT

MO

DELS

35

Table 5

The Percentages of Bayes Factors When Priors p(0, ‡—

) = N(0, 1), N(0, 100), and N(0, 10000) Are Assigned

‡ = 1 ‡ = 100 ‡ = 10000

1 < BF Æ 3 3 < BF Æ 20 BF > 20 1 < BF Æ 3 3 < BF Æ 20 BF > 20 1 < BF Æ 3 3 < BF Æ 20 BF > 20

Small e�ct size (S)

—1 = .2, —2 = .1, —3 = 0

n=10 23% 29% 2% 19% 25% 6% 17% 26% 6%

n=50 39% 34% 1% 42% 31% 2% 41% 32% 2%

n=100 33% 37% 8% 32% 37% 8% 33% 37% 8%

n=500 2% 50% 43% 3% 53% 39% 3% 53% 39%

—1 = ≠.2, —2 = ≠.1, —3 = 0

n=10 15% 12% 1% 12% 8% 5% 9% 7% 5%

n=50 9% 1% 0 9% 1% 0% 9% 1% 0%

n=100 7% 1% 0 8% 1% 0% 7% 1% 0%

n=500 0 0 0 0% 0% 0% 0% 0% 0%


—1 = .6, —2 = .3, —3 = 0

n=10 33% 25% 6% 23% 23% 12% 23% 25% 12%

n=50 10% 46% 39% 9% 46% 39% 11% 44% 39%

n=100 2% 29% 68% 1% 30% 68% 2% 29% 68%

n=500 0 0 100% 0 0 100% 0 0 100%

—1 = ≠.6, —2 = ≠.3, —3 = 0

n=10 10% 4% 0 8% 5% 1% 6% 4% 2%

n=50 0 0 0 0 0 0 0 0 0

n=100 0 0 0 0 0 0 0 0 0

n=500 0 0 0 0 0 0 0 0 0

Note. BF denotes BFM1,M2 , where M1 (—1 > —2 > —3) is compared against M2 (—1 > —2 > —3). The three categories are defined based on Kass & Raftery (1995) (see Table 1).


Table 6

Estimated Parameters of the Ordered Probit Model via the Frequentist Approach

Estimated Value p

Price - .199 .197

Warranty .045 .611

Quality .469 .006

Maintenance - .179 .235


Figure 1 . The posterior distribution, combining the information in the prior and that in

the likelihood, stands in the middle between the prior and the likelihood. (Note: the pic

is retrieved from Lynch (2007).)


Figure 2 . An illustration of prior complexity. The prior complexity for Mu

is one, that

for M1

is 1/4, and that for M2

is 1/2.


Figure 3 . An illustration of posterior fit. The posterior fit for dataset A is one under

M1

and M2

. The posterior fit for dataset B is one under M1

and zero under M2

.


Appendix B

Deriving the Mean and Variance of the Posterior Distribution for —

Firstly, we rewrite multiple linear regression with latent variable zzz

zzz = ———ÕXXX + ‘‘‘ (‘1

, ‘2

, · · · , ‘n

≥ i.i.d.N(0, 1)) (29)

in the following form:

zzz = —1

xxx1

+ —2

xxx2

+ ... + —i

xxxi

+ ... + —n

xxxn

+ ‘‘‘. (30)

Further, let

zzz— i

= zzz ≠ —1

xxx1

≠ —2

xxx2

≠ ... ≠ —i≠1

xxxi≠1

≠ —i+1

xxxi+1

≠ ... ≠ —n

xxxn

+ ‘‘‘

= —i

xxxi

+ ‘‘‘.

(31)

The likelihood distribution is therefore

p(z—i1

, z—i2

, ..., z—in

|xxxi

, —i

) = (2fi)≠ n2 exp{≠1

2(zzz— i

≠ —i

xxxi

)Õ(zzz— i

≠ —i

xxxi

)}. (32)

Furthermore, the prior for —i

is —i

≥ N(—0

, ‡2

—

), where —0

and ‡—

are known, then the

posterior distribution is

p(z—i1

, z—i2

, ..., z—in

|xxxi

, —i

) ◊ p(—i

) =(2fi)≠ n2 exp{≠1

2(zzz— i

≠ —i

xxxi

)Õ(zzz— i

≠ —i

xxxi

)}

(2fi‡2

—

)≠ 1

2 exp{≠ 12‡2

—

(—i

≠ —0

)2}

Ãexp{≠12[(zzz

— i

≠ —i

xxxi

)Õ(zzz— i

≠ —i

xxxi

) + 1‡2

—

(—i

≠ —0

)2]}.

(33)

For the sake of simplicity and clarity, let

M = (zzz— i

≠ —i

xxxi

)Õ(zzz— i

≠ —i

xxxi

) + 1‡2

—

(—i

≠ —0

)2. (34)

Then, reorganizing M in the following way:

M = (zzz— i

≠ —i

xxxi

)Õ(zzz— i

≠ —i

xxxi

) + 1‡2

—

(—i

≠ —0

)2

= (zzz— i

≠ —i

xxxi

)Õ(zzz— i

≠ —i

xxxi

) + 1‡2

—

(—2

i

≠ 2—0

—i

+ —2

0

)

Ã ≠2—i

xxxÕi

zzz—i + —2

i

xxxÕi

xxxi

+ 1‡2

—

—2

i

≠ 2—0

‡2

—

—i

= ( 1‡2

—

+ xxxÕi

xxxi

)[—2

i

≠ 2( —0

‡2

—

+ xxxÕi

zzz—i)(

1‡2

—

+ xxxÕi

xxxi

)≠1—i

]

(35)


Therefore, the posterior distribution is proportional to a normal distribution:

p(z—i1

, z—i2

, ..., z—in

|xxxi

, —i

) ◊ p(—i

) Ã exp{≠12[(zzz

— i

≠ —i

xxxi

)Õ(zzz— i

≠ —i

xxxi

) + 1‡2

—

(—i

≠ —0

)2]}

= exp{≠12

—2

i

≠ 2( —

0

‡

2

—+ xxxÕ

i

zzz—i)( 1

‡

2

—+ xxxÕ

i

xxxi

)≠1—i

1

‡

2

—+ xxxÕ

i

xxxi

}.

(36)

Hence, it is clear that

E(—i

) = (—0

‡2

—

+ xxxÕi

zzz—i)(

1‡2

—

+ xxxÕi

xxxi

)≠1 (37a)

V ar(—i

) = ( 1‡2

—

+ xxxÕi

xxxi

)≠1. (37b)


Appendix C

Deriving E(———|zzz) and V ar(———|zzz), Given the Prior ——— ≥ N(000, n(XXX ÕXXX)≠1)

Before deriving E(———|zzz) and V ar(———|zzz) we first need obtain the posterior distribution for

———, given prior ——— ≥ MV N(—0

—0

—0

, �0

) as a general form.

Given a multiple regression

yyy = XXX——— + ‘‘‘, ‘‘‘ ≥ N(000, ‡2III), (38)

and the prior ——— ≥ MV N(—0

—0

—0

, �0

), the posterior distribution is

p(———|yyy,XXX, ‡2) Ã p(yyy|XXX,———, ‡2) ◊ p(———)

Ã exp{2———ÕXXX Õyyy ≠ ———ÕXXX ÕXXX———

2‡2

≠ 12(≠2———Õ�≠1

0

———0

+ ———Õ�≠1

0

———)}

= exp{≠12———Õ(�≠1

0

+ XXX ÕXXX

‡2

)——— + ———Õ(�≠1

0

———0

+ XXX Õyyy

‡2

)},

(39)

which apparently is a multivariate normal distribution with

E[———|yyy,XXX, ‡2] = (�≠1

0

+ XXX ÕXXX

‡2

)≠1(�≠1

0

———0

+ XXX Õyyy

‡2

) (40a)

V ar[———|yyy,XXX, ‡2] = (�≠1

0

+ XXX ÕXXX

‡2

)≠1. (40b)

Therefore, when —0

—0

—0

= 000, �0

= n(XXX ÕXXX)≠1, and ‡ = 1, the above equations become

E(———|zzz) = (XXX ÕXXX/n + XXX ÕXXX)≠1(XXX Õzzz) = n

n + 1(XXX ÕXXX)≠1XXX Õzzz (41a)

V ar(———|zzz) = (XXX ÕXXX/n + XXX ÕXXX)≠1 = n

n + 1(XXX ÕXXX)≠1. (41b)


References

Benhabib, J., & Spiegel, M. M. (1994). The role of human capital in economic

development evidence from aggregate cross-country data. Journal of Monetary

economics, 34 (2), 143–173.

Berger, J. O., Pericchi, L. R., Ghosh, J., Samanta, T., De Santis, F., Berger, J., &

Pericchi, L. (2001). Objective bayesian methods for model selection: introduction

and comparison. Lecture Notes-Monograph Series, 135–207.

Bolton, L. E., Warlop, L., & Alba, J. W. (2003). Consumer perceptions of price (un)

fairness. Journal of consumer research, 29 (4), 474–491.

Braeken, J., Mulder, J., & Wood, S. (2014). Relative e�ects at work bayes factors for

order hypotheses. Journal of Management, 0149206314525206.

Coe, D. T., & Helpman, E. (1995). International r&d spillovers. European

Economic Review, 39 (5), 859–887.

Day, G. S. (1994). The capabilities of market-driven organizations. the Journal of

Marketing, 37–52.

Gelman, A., Jakulin, A., Pittau, M. G., & Su, Y.-S. (2008). A weakly informative

default prior distribution for logistic and other regression models. The Annals of

Applied Statistics, 1360–1383.

Geman, S., & Geman, D. (1984). Stochastic relaxation, gibbs distributions, and the

bayesian restoration of images. Pattern Analysis and Machine Intelligence, IEEE

Transactions on(6), 721–741.

Gensch, D. H., Aversa, N., & Moore, S. P. (1990). A choice-modeling market

information system that enabled abb electric to expand its market share.

Interfaces, 20 (1), 6–25.

George, E. I., & McCulloch, R. E. (1993). Variable selection via gibbs sampling.

Journal of the American Statistical Association, 88 (423), 881–889.

Gerschenkron, A., et al. (1962). Economic backwardness in historical perspective.

Economic backwardness in historical perspective..

Geweke, J., et al. (1991). Evaluating the accuracy of sampling-based approaches to the


calculation of posterior moments (Vol. 196). Federal Reserve Bank of Minneapolis,

Research Department.

Gri�th, R., Redding, S., & Van Reenen, J. (2004). Mapping the two faces of r&d:

productivity growth in a panel of oecd industries. Review of Economics and

Statistics, 86 (4), 883–895.

Hochberg, Y. (1988). A sharper bonferroni procedure for multiple tests of significance.

Biometrika, 75 (4), 800–802.

Ho�, P. D. (2009). A first course in bayesian statistical methods. Springer.

Hoijtink, H., & Boom, J. (2008). Inequality constrained latent class models. In

Bayesian evaluation of informative hypotheses (pp. 227–246). Springer.

Hoijtink, H., Klugkist, I., & Boelen, P. A. (2008). Bayesian evaluation of informative

hypotheses. Springer.

Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian

journal of statistics, 65–70.

Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the american statistical

association, 90 (430), 773–795.

Kass, R. E., & Wasserman, L. (1995). A reference bayesian test for nested hypotheses

and its relationship to the schwarz criterion. Journal of the american statistical

association, 90 (431), 928–934.

Kato, B. S., & Hoijtink, H. (2006). A bayesian approach to inequality constrained

linear mixed models: estimation and model selection. Statistical Modelling, 6 (3),

231–249.

Kato, B. S., & Peeters, C. F. (2008). Inequality constrained multilevel models. In

Bayesian evaluation of informative hypotheses (pp. 273–295). Springer.

Klugkist, I. (2008). Encompassing prior based model selection for inequality

constrained analysis of variance. In Bayesian evaluation of informative hypotheses

(pp. 53–83). Springer.

Klugkist, I., Kato, B., & Hoijtink, H. (2005). Bayesian model selection using

encompassing priors. Statistica Neerlandica, 59 (1), 57–69.


Klugkist, I., Laudy, O., & Hoijtink, H. (2005). Inequality constrained analysis of

variance: a bayesian approach. Psychological Methods, 10 (4), 477.

Klugkist, I., & Mulder, J. (2008). Bayesian estimation for inequality constrained

analysis of variance. In Bayesian evaluation of informative hypotheses (pp.

27–52). Springer.

Krishna, P., & Mitra, D. (1998). Trade liberalization, market discipline and

productivity growth: new evidence from india. Journal of Development

Economics, 56 (2), 447–462.

Lattin, J. M., Carroll, J. D., & Green, P. E. (2003). Analyzing multivariate data.

Thomson Brooks/Cole Pacific Grove, CA, USA.

Liang, F., Paulo, R., Molina, G., Clyde, M. A., & Berger, J. O. (2008). Mixtures of g

priors for bayesian variable selection. Journal of the American Statistical

Association, 103 (481).

Lichtenstein, D. R., Ridgway, N. M., & Netemeyer, R. G. (1993). Price perceptions and

consumer shopping behavior: a field study. Journal of marketing research,

234–245.

Liu, M. W., & Soman, D. (2008). Behavioral pricing. Handbook of consumer

psychology, 659–681.

Lynch, S. M. (2007). Introduction to applied bayesian statistics and estimation for

social scientists. Springer.

Mulder, J. (2014). Bayesian hypothesis testing of order constraints on correlations in

unstructured correlation matrices. (Manuscript submitted for publication)

Mulder, J., Hoijtink, H., & Klugkist, I. (2010). Equality and inequality constrained

multivariate linear models: objective model selection using constrained posterior

priors. Journal of Statistical Planning and Inference, 140 (4), 887–906.

Mulder, J., Klugkist, I., van de Schoot, R., Meeus, W. H., Selfhout, M., & Hoijtink, H.

(2009). Bayesian model selection of informative hypotheses for repeated

measurements. Journal of Mathematical Psychology, 53 (6), 530–546.

Vorhies, D. W. (1998). An investigation of the factors leading to the development of


marketing capabilities and organizational e�ectiveness. Journal of strategic

marketing, 6 (1), 3–23.

Winterhalder, B. (1981). Optimal foraging strategies and hunter-gatherer research in

anthropology: Theory and models. Hunter-gatherer foraging strategies:

Ethnographic and archaeological analyses, 13–35.

Zellner, A. (1986). On assessing prior distributions and bayesian regression analysis

with g-prior distributions. Bayesian inference and decision techniques: Essays in

Honor of Bruno De Finetti, 6 , 233–243.

essay on testing order hypotheses in ordered probit

Documents