Literature

1
Literature

• Rijmen, F., Tuerlinckx, F. De Boeck, P.,
  Kuppens, P. (2003). A nonlinear mixed model
  framework for item response theory. Psychological
  Methods, 8, 185-205.
• De Boeck, P., Wilson, M. (Eds) (2004).
  Explanatory Item Response Models A Generalized
  Linear and Nonlinear Approach. New York Springer.

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Dataset 1Verbal aggression

• 316 persons filled out a behavioral questionnaire
  on verbal aggression (Vansteelandt, 2000 Smits,
  De Boeck, Vansteelandt, 2004 )
• 24 items
• E.g. A bus fails to stop for me. I would curse.
• No 0 0
• Perhaps 1
• Yes 2

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Item design

• Item were a combination of three design factors
• situation type other-to blame versus
  self-to-blame
• A bus fails to stop for me
• I miss a train because a clerk gave me faulty
  information
• The grocery store closes just as I am about to
  enter
• The operator disconnects me when I had used up my
  last 10 cents for a call

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• Behavior
• A bus fails to stop for me. I would curse.
• A bus fails to stop for me. I would scold.
• A bus fails to stop for me. I would shout.
• Behavior mode Want vs. Do
• A bus stop fails to stop for me. I would want to
  curse
• A bus stop fails to stop for me. I would curse
• Full factorial item design 43224
• situations nested within Situation Type
  2(2)32

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Person p
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Person covariates

• Gender (243 females, 73 males)
• Trait Anger score (M 20 SD 4.85)

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Leading questions

• What is the effect of the item design factors?
• Self vs. Other-to-blamemore verbal aggression if
  other-to-blame?
• Do vs. Wantverbal aggression inhibited?
• Behaviorpopularity of behaviors?
• What is effect of person variables?
• Men vs. Women are men more verbally
  aggressive?
• Degree of Trait Anger is the tendency of a
  person to react in a verbally aggressive way
  related to trait anger?

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Leading questions

• What is the effect of the item design factors?
• Self vs. Other-to-blamemore verbal aggression if
  other-to-blame?Do people differ in sensitivity
  to others faults?
• Do vs. Wantverbal aggression inhibited?Do
  people differ in inhibition?
• Behaviorpopularity of behaviors?Do people have
  verbally aggressive styles?
• In other words,are individual differences in
  verbal aggression adequately captured by a single
  underlying dimension, a tendency to react in a
  verbally aggressive way? Or are more dimensions
  needed?

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Dataset 2 self-report study on anger

• 510 persons filled out a behavioral questionnaire
  on anger feelings (Kuppens, 2002)
• 24 aversive situations. For example
• Someone broke your bike
• A member of your family is ill
• You are home and alone
• Your beloved shows more interest for someone else
  
• To which degree would you become angry?
• 4-point scale 0 1 2 3

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Item covariates

• A second group of 25 persons rated the 24
  situations with respect to a set of situational
  characteristics
• Mean ratings (over persons) can be used as item
  covariates
• (some) situational characteristics
• Amount of control over the situation
• Predictability of situation
• Consequences for oneself
• Consequences for a third person
• Loss experience

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Person covariates

• Trait Anger score (M 1.3 SD 0.6)
• Self-esteem score (M 1.8 SD 0.6)
• Gender (179 females, 331 males)

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Leading questions

• What is the effect of the item covariates?
• E.g. does one show more anger when there are
  important consequences for oneself?
• Is the effect of an item covariate constant over
  persons?
• Are individual differences adequately captured by
  a single underlying dimension for anger? Or are
  more dimensions needed?
• What is the effect of person covariates?
• E.g. Do persons with a low self-esteem become
  more easily angry?

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Multiple person dimensions
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overview

• Multidimensional extension of the Rasch model
• Between- and within-item multidimensionality
• Item factor analysis
• Multidimensional extension of the LLTM

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Multidimensional extension of the Rasch model

• Rasch model mixed logistic regression model
  with
• Random intercept
• Item indicators as predictors (separate
  regression weight for each item)
• Linear predictor
• ?p is a latent variable, an underlying dimension
  that reflects quantitative individual differences
• Tendency to react in a verbally aggressive way
• Tendency to become angry

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• The Rasch model is a unidimensional model
• Assumption all items are located on the same
  dimension, they measure the same construct
• Verbal aggression data if person 1 has a higher
  probability than person 2 to shout in one
  situation, person 1 has a higher probability to
  shout in all other situations as well
• Anger data if person 1 has a higher probability
  than person 2 to become angry in one situation,
  person 1 has a higher probability to become angry
  in all other situations as well

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• Extension vector of
  latent variables
• Linear predictor
• Xri (Xri1,,XriKr) specifies to which extent
  item i is measuring each of the Kr dimensions

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Between- and within-item multidimensionality

• Between-item multidimensionality
• test consists of several subtests, each measuring
  a separate dimension
• An item loads on one dimension only Xri is
  indicator vector
• Subtests are unidimensional
• E.g. Verbal aggression data separate dimensions
  for do-items and want-items. Person 1 might want
  to shout to a higher degree than person 2, but
  shouts less (individual differences in
  inhibition).

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• Within-item multidimensionality
• The test measures multiple dimensions
• An item can load on more than one dimension
• E.g. anger study whether one becomes angry in an
  aversive situation might depend on two
  dimensions
• The tendency to become angry
• Tendency to inhibit socially undesirable behavior
• can be thought of as a weighted sum of the
  locations of the item on the individual
  dimensions

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• Verbal-aggression data
• Between-item twodimensional model
• Separate dimensions for do and want items

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• Wanting is more likely than doing
• Cursing is the most likely behavior, shouting the
  least likely
• Correlation between dimensions .78
• Rasch model is nested within two-dimensional
  model, LR(12) 92, p lt .001

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Item factor analysis

• Often, one cannot specify apriori to which extent
  an item is loading on each of the dimensions
• Item factor analysis elements of Xri are
  unknown.
• Estimate loadings from the data

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Multidimensional extension of the LLTM

• Linear logistic test model mixed logistic
  regression model with
• Random intercept accounts for person main
  effects
• item properties as (item) predictors account
  for item main effects
• Linear predictor

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• Interactions between item property and persons
  are not taken into account by the LLTM (parallel
  slopes)
• But often of central interest
• E.g. anger study
• Consequences for oneself is an item property
• Its effect on anger might depend on the person
• Interaction between an item property and persons

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• Random weights LLTM
• Multidimensional extension of the LLTM
• Allows for person-specific regression weights for
  (some of) the item properties
• Random intercept and slope(s)
• Linear predictor

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• Anger data (dichotomised)
• Random weights LLTM
• Fixed effects
• Amount of control over the situation
• Predictability of situation
• Consequences for oneself
• Consequences for a third person
• Random effects
• Intercept
• Consequences for oneself

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Results
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• Tendency to show anger increases when
• Less control over the situation
• More predictability of situation
• More consequences for oneself
• More consequences for a third person
• Need for a random slope for consequences for
  oneself?
• LLTM is not rejected when tested against RWLLTM

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Polytomous data

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overview

• Common use of polytomous data
• Recoding the data
• Multivariate extension of the generalized linear
  mixed model
• Distribution
• Link function
• Linear predictor
• Specifying the link function
• Applications

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Common use of polytomous data

• data are often polytomous or multicategorical
• A bus fails to stop for me. I would scold.
• No
• Perhaps
• Yes
• Someone broke your bike. To which degree would
  you become angry?
• Not angry at all
• Slightly angry
• Angry
• Very angry

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• Solve for x x2 4
• 16 (wrong)
• 2 (partially correct)
• 2, -2 (correct)
• I consider myself to be a
• Liberal
• Socialist
• Conservative
• Categories can be ordered or nominal
• Only one response category can be chosen

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How to handle polytomous data?

• Dichotomize the data
• Loss of information
• Treat data as if continuous
• Not possible for nominal data
• Data are treated as if on a metrical scale (equal
  distances between categories)
• Data are often far from normal (inflated
  zero-category)

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You are home alone and bored (anger data)
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How to handle polytomous data?

• Dichotomize the data
• Loss of information
• Treat data as if continuous
• Not possible for nominal data
• Data are treated as if on a metrical scale
• Data are often far from normal (inflated
  zero-category)
• Treat the data as they are polytomous

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Recoding the data

• Ypi m response of person p to item i is m
• p 1,,P
• i 1,,I
• m 0,,Mi-1 Mi categories
• Without loss of generality Mi M for all i
• Ypi can be recoded into a vector Cpi of Q M-1
  dummy variables

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• Example verbal agggression data
• 3 response categories (M 3) recoded into 2
  dummies (Q 2)
• Response Ypi Cpi1 Cpi2
•  No  0 0 0
•  Perhaps  1 1 0
•  Yes  2 0 1
• For simplicity M3 for the remainder

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Multivariate extension of the generalized linear
mixed model

• Distribution
• Link function
• Linear predictor

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Multivariate extension of the generalized linear
mixed model distribution

• Dichotomous outcomes Bernoulli (binomial with
  total count equal to one)
• Bernoulli distribution belongs to the exponential
  family

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• Polytomous outcomes multivariate Bernoulli
  (multinomial with total count equal to one)

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• Multinomial distribution is member of
  multivariate exponential family
• Mean Vector of probabilities
• Variance

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Multivariate extension of the generalized linear
mixed model link function

• Dichotomous outcomes single-valued link function
  transforms the mean into the linear predictor
• Polytomous outcomes vector-valued link function
  transforms the mean vector into the linear
  predictor vector
• Several vector-valued generalizations of
  single-valued link functions possible

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Multivariate extension of the generalized linear
mixed model linear predictor

• Dichotomous outcomes
• For simplicity only item predictors

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• Polytomous outcomes

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Predictor types

• Dichotomous outcomes
• Person predictors
• Item predictors
• Person-by-item-predictors
• Polytomous outcomes
• Person predictors
• Item predictors
• Category(-contrast) predictors
• Person-by-item-predictors

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Specifying the link function

• Focus logit link
• Logit link for dichotomous outcomes can be
  generalized in different ways
• Baseline-category logits nominal data
• Adjacent-category logits
• Cumulative logits ordinal data
• Continuation-ratio logits
• Generalized logits reduce to the regular
  single-valued logit when the data have only two
  categories
• Each logit type comes with its own interpretation

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Baseline-category logits

• qth baseline-category logit is the log odds of
  category q versus category 0 (baseline-category)
• All categories can be chosen as baseline
  category, but category 0 is often a reasonable
  choice

Pr(Ypi0)
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Adjacent-category logits

• qth adjacent-category logit is the log odds of
  category q versus category q-1

Pr(Ypi0)
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• Adjacent-category logits can be expressed as
  baseline-category logits

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Cumulative logits

• qth cumulative logit is the log odds of category
  q or higher versus a category lower than q

Pr(Ypi gt1)
Pr(Ypilt1)
Pr(Ypi gt2)
Pr(Ypi lt2)
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Continuation-ratio logits

• qth continuation-ratio logit is the log odds of
  category q versus a category lower than q

Pr(Ypilt1)
Pr(Ypi lt2)
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Verbal agression data partial credit model and
rating scale model

• Partial credit model (Masters, 1982)
• Random intercept
• Adjacent-category logit
• Item-by-category-contrast interactions (Item by
  category-contrast predictors)
• Linear predictor
• value of where probabilities of
  responding in category q-1 and q are equal
  (category crossing parameter)

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Figuur 1
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• Inverting the link function gives the
  probabilities

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Figuur 3.4
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• Rating scale model (Andrich, 1978)
• Random intercept
• Adjacent-categories logit
• Item and category-contrast predictors
• Linear predictor
• Restricted PCM
• Category crossings are at the same distance apart
  from each other for all items

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• PCM or RSM ?
• LR(23) 52, plt.001

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• Inclusion of gender and trait anger as person
  predictors
• Latent regression
• GENDER 1 for men, and 0 for women
• Linear predictor

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• LR(2)29, plt.001
• Odds for responding responding  perhaps  rather
  than  no  ( yes  rather than  perhaps ) are
  exp(0.28)1.3 times higher for men
• Increase of 1 SD in Trait Anger multiplies
  adjacent-categories odds with exp(4.850.06)1.3

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• Analoguous to the linear logistic test model for
  dichotomous outcomes
• Category crossings can be modelled as a weighted
  sum of item predictors -gt Linear PCM
• Item locations of a RSM can be modelled as a
  weighted sum of item predictors -gt Linear RSM

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Conclusion

• The generalized (nonlinear) mixed model for
  dichotomous outcomes can be extended to the
  multivariate generalized (nonlinear) mixed model
  for polytomous outcomes
• Different link functions can be specified,
  depending on which sets of categories are
  contrasted with each other
• Familiar IRT models for polytomous data can be
  understood as multivariate generalized
  (nonlinear) mixed models for polytomous outcomes
• This way, extensions are easily incorporated