The Bayes Factor for Inequality and About Equality Constrained Models

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The Bayes Factor for Inequality and About
Equality Constrained Models

• Irene Klugkist
• Faculty of Social and Behavioral Sciences
• Utrecht University, The Netherlands
• i.klugkist_at_uu.nl

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Psychological illustration (1.1)

• Experimental research by Van Well and others
  about gender effects on stress levels.
• Stressor Cold Pressor Test (hand in icecold
  water)
• Outcome Responsivity (stress level, a.o. blood
  pressure)
• Factors Sex (male / female)
• Gender Role identification (masculine /
  feminine)
• Gender manipulation (masc. / fem. / neutral)
• Covariate Baseline stress level (i.e., BP for BP
  outcome)
• Traditional analysis 2 ? 2 ? 3 ANCOVA

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Psychological illustration (1.2)
Match Effects Sex M1 (µ1, µ4) gt (µ2, µ3, µ5,
µ6) and (µ8, µ11) gt (µ7, µ9, µ10, µ12) GRI M2
(µ1, µ5) gt (µ2, µ3, µ4, µ6) and (µ7, µ11) gt (µ8,
µ9, µ10, µ12) Mismatch Effects Sex M3 (µ2,
µ5) gt (µ1, µ3, µ4, µ6) and (µ7, µ10) gt (µ8, µ9,
µ11, µ12) GRI M4 (µ2, µ4) gt (µ1, µ3, µ5, µ6)
and (µ8, µ10) gt (µ7, µ9, µ11, µ12)
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Psychological illustration (1.3)
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Psychological illustration (1.4)
Sex match?? Sex mismatch?? GRI match?? GRI
mismatch??
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Psychological illustration (2.1)

• Topic Gender differences in route learning (from
  the EU Wayfinding project (Fp6grant) De Goede,
  2009)
• Outcome travelled distance
• Two strategies landmark based (moreoften used by
  females) and based on geometric cues (moreoften
  used by males)

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Psychological illustration (2.2)

• Question are the gender differences in the
  strategy used based on ability or preference?
• Experiment with encoding and retrieval phase and
  3 conditions
• Encoding with landmarks, Retrieval with
  landmarks
• Encoding no landmarks, Retrieval no landmarks
• Encoding with landmarks, Retrieval landmarks
  removed
• Hypotheses (M1 is male in cond.1, F2 female in
  cond.2, etc.)
• Ability M1lt (M2M3) F1lt (F2F3) with (M2-M1)
  lt (F2-F1)
• Preference M1lt (M2M3) F1lt F2 lt F3 with
  (M2-M1) (F2-F1)

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Psychological illustration (2.3)
females
females
males
males
1 2 3
1 2 3
Ability Preference
M1lt (M2M3) M1lt (M2M3) F1lt (F2F3)
F1lt F2 lt F3 (M2-M1) lt (F2-F1)
(M2-M1) (F2-F1)
H1
H2
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Classical Inequality Constrained Hypotheses

• Hypothesis testing can handle both
• H0 µ1 µ2 µ3 µ4 vs. HA µ1 lt
  µ2 lt µ3 lt µ4
• and
• H0 µ1 lt µ2 lt µ3 lt µ4 vs. HA µ1 ,
  µ2 , µ3 , µ4
• References Barlow, Bartholomew, Bremner, Brunk
  (1972)
• Robertson, Wright, Dijkstra (1988)
• Silvapulle, Sen (2005)
• But not
• H1 µ1 lt µ2 lt µ3 lt µ4 vs. H2 µ1 gt
  µ2 gt µ3 gt µ4
• Reference for non-Bayesian model selection for
  H1 versus H2
• Anraku (Biometrika, 1999)

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Bayesian Model Selection

• Bayes factors for a constrained model Mt versus
  the corresponding unconstrained model M0 (e.g.
  M1 ?1lt ?2 lt ?3 versus M0 ?1, ?2, ?3)
• Bayes factor is ratio of two marginal
  likelihoods
• Using the expression
• we get

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Encompassing prior approach

• All models imposing inequality constraints on
  parameters are nested in the unconstrained model
  M0
• One prior distribution is specified for M0 and
  the constraints of Mt are incorporated by
  truncation of this encompassing prior
• Similarly, the posterior of a unconstrained model
  Mt is

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Estimation of Bto
ct-1 proportion of unconstrained prior in
agreement with constraints of Mt dt-1 proportion
of unconstrained posterior in agreement with
constraints of Mt
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Specification of encompassing prior

• Specification of the encompassing prior in the
  absence of subjective prior knowledge
• Goal objective w.r.t. the model comparisons
• Definition of complexity (Mulder, Hoijtink,
  Klugkist submitted)
• Loosely each ordering of parameters is a priori
  equally likely

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Specification of encompassing prior

• In practice when inequality constraints are
  imposed on ?j (j1, ,J) then , that is, the
  prior is symmetric around ?1 ?J
• As a consequence complexity/size does not depend
  on the exact specification of the encompassing
  prior.
• E.g. Same uniform prior for each ? on (0,25),
  (-10,10), (-100,100), or,
• same normal prior N(0,1), N(5,10), N(-3,25) all
  provide a complexity
• measure that only depends on the constraints of
  the hypothesis
• Furthermore, using vague priors, also the
  posterior (fit) is hardly
• affected by the encompassing prior.

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Prior sensitivity

• Some simulation results for an ANOVA model, H1
  unconstrained, and
• H2 µ1 gt µ3 gt µ2 gt µ4
• H3 µ1 gt µ2 gt µ3 gt µ4
• Note the nuisance parameters are more or less
  ignored in this presentation.
• For the specification guidelines of encompassing
  priors (including all parameters) based on a
  training data approach for the multivariate
  normal linear model, see Mulder et al.
  (submitted).

Nj10 sample means (sd) 100, 79, 86, 84
(15 1)
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Intermezzo Static priors

• Symmetric prior based on minimal training sample
  for comparing
• H1 µ1 gt µ2 gt µ3 gt µ4 gt µ5 gt µ6 gt µ7 , s2
• H0 µ1 , µ2 , µ3 , µ4 , µ5 , µ6 , µ7 , s2

• Sample means are (s210)
• 0, 3, 6, 9, 12, 15, 14
• Complexity c-1 is fixed
• (7!)-1 1/5200

Jeroen Ooms, 2009, master thesis
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Models stating equality of parameters

• Encompassing prior approach and corresponding
  Bayes factor estimate are tailored to nested
  models that are specified by truncation of the
  parameter space
• Does H1 ?1 ?2 (versus H0 ?1, ?2) fit in this
  framework?
• NO. Parameter space has a different dimension
  (less parameters)
• YES. The support for this model can be
  approximated via
• ?1 - ?2 lt e for e?0
• However, sampling the unconstrained prior and
  posterior to estimate proportions ct-1 and dt-1
  is very inefficient for small e

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Stepwise sampling procedure
Sample from grey area count proportion of
iterations in yellow area With some
modification this procedure can also be used for
models containing both and lt, gt constraints
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Prior sensitivity

• The estimate Bt0ct/dt clearly illustrates the
  prior sensitivity for models of different
  dimensions, or, with constraints
  (Bartlett-Lindley paradox).

For a diffuse encompassing prior, the value for
dt-1 is hardly affected by the prior. But, ct-1
becomes smaller and smaller for increasing
diffuseness.
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Prior sensitivity

• Some simulation results for same ANOVA model as
  before, H1
• unconstrained, and H4 µ1 µ3 µ2 µ4

Nj10 sample means (sd) 100, 79, 86, 84
(15 1)
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The CPP-approach

• To obtain reasonable priors (objective but not
  too diffuse for the data at hand) we use training
  data
• Non-informative (improper) priors are updated
  with small part of the observed data ?
  posterior priors
• To maintain the prior insensitivity for
  inequality constraints, we restrict the posterior
  priors to be symmetric
• Non-informative (improper) priors are updated
  with small part of the observed data under the
  restriction that equal priors are obtained for
  constrained parameters ? constrained posterior
  priors (CPP)
• For an introduction of the CPP for multivariate
  normal linear models, see
• Mulder, Hoijtink, Klugkist (submitted)

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Motivation for constrained posterior prior
Evaluate H1 µ1 lt µ2 (grey area)
Ref Van Wesel, Hoijtink, Klugkist (submitted)

• Results for three possible samples 1, 2, 3
• Small circles represent posteriors (based on
  minimal TS)
• Larger circles/ellipses denote the priors with
  (plot II on the right) and without (plot I
• on the left) the symmetry requirement

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Intermezzo Bounded Bayes factors

• Bayes factors comparing a constrained with the
  unconstrained model
• are bounded by the size of the model
• Consider an unconstrained posterior that falls
  completely in area of Mt
• This is due to overlap in the models. Bounded
  Bayes factors can
• be avoided by comparing exclusive models only.
• E.g. H0 ?1 , ?2
• H1 ?1 gt ?2
• H2 ?1 lt ?2

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Results for the route finding example

• Route learning based on two strategies landmark
  or geometric cue based
• Hypotheses (M1 is male in cond.1, F2 female in
  cond.2, etc.)
• H1 M1lt (M2M3) F1lt (F2F3) with (M2-M1) lt
  (F2-F1)
• H2 M1lt (M2M3) F1lt F2 lt F3 with (M2-M1)
  (F2-F1)
• Results based on the CPP approach (Mulder)
• B1,unc 1.1
• B2,unc 41.7
• B21 37.9 in favor of the
  preference theory

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References

• Klugkist, I., Hoijtink, H. (2007). The Bayes
  Factor for Inequality and About Equality
  Constrained Models. Computational Statistics and
  Data Analysis, 51, 6367-6379.
• Mulder, J., Hoijtink, H., Klugkist, I.
  (submittted). Equality and Inequality Constrained
  Multivariate Linear Models Objective Model
  Selection Using Constrained Posterior Priors
• Van Wesel, F., Hoijtink, H., Klugkist, I.
  (submitted). Choosing priors for constrained
  analysis of variance methods based on training
  data
• Ooms, J. (master thesis, 2009). The Highest
  Posterior Density Posterior Prior for Bayesian
  Model Selection
• Springer book
• Hoijtink, H., Klugkist, I. and Boelen, P.A.,
  eds. (2008). Bayesian Evaluation of Informative
  Hypotheses. New York Springer.
• Application in psychology
• Van Well, S., Kolk, A.M. and Klugkist, I.G.
  (2008). Effects of Sex, Gender Role
  Identification, and Gender relevance of Two Types
  of Stressors on Cardiovascular and Subjective
  Responses Sex and Gender Match/Mismatch Effects.
  Behavior Modification, 32, 427-449.