The paired comparison method for latent variables An Application of - - PowerPoint PPT Presentation

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

The paired comparison method for latent variables An Application of the Bradley Terry Model

Almut Thomas Michaela Gareiß Regina Dittrich Reinhold Hatzinger

nd Workshop on Psychometric Computing February th - February th  LMU, Department for Statistics

Introduction

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Instrument

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Freizeit-Interessen-Test (FIT), Stangl  based on Holland’s () RIASEC-model . . . . Realistic: practical, physical

Instrument

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Freizeit-Interessen-Test (FIT), Stangl  based on Holland’s () RIASEC-model . . . . Realistic: practical, physical Investigative: intellectual, scientific

Instrument

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Freizeit-Interessen-Test (FIT), Stangl  based on Holland’s () RIASEC-model . . . . Realistic: practical, physical Investigative: intellectual, scientific Artistic: creative, independent

Instrument

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Freizeit-Interessen-Test (FIT), Stangl  based on Holland’s () RIASEC-model . . . . Realistic: practical, physical Investigative: intellectual, scientific Artistic: creative, independent Social: supporting, helping

Instrument

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Freizeit-Interessen-Test (FIT), Stangl  based on Holland’s () RIASEC-model . . . . Realistic: practical, physical Investigative: intellectual, scientific Artistic: creative, independent Social: supporting, helping Enterprising: competitive, persuading

Instrument

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Freizeit-Interessen-Test (FIT), Stangl  based on Holland’s () RIASEC-model . . . . Realistic: practical, physical Investigative: intellectual, scientific Artistic: creative, independent Social: supporting, helping Enterprising: competitive, persuading Conventional: detail-oriented, organizing

Example

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Which of the two alternatives would you prefer?

Example

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Which of the two alternatives would you prefer?

Build a greenhouse (R)

• grow and maintain

rare plants (I) in your own garden.

Example

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Which of the two alternatives would you prefer?

Build a greenhouse (R)

• grow and maintain

rare plants (I) in your own garden. Play as a musician (A)

• be a conductor (E)

in a folk group.

Example

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Which of the two alternatives would you prefer?

Build a greenhouse (R)

• grow and maintain

rare plants (I) in your own garden. Play as a musician (A)

• be a conductor (E)

in a folk group. Produce (R)

• sale (E)

christmas decoration.

Standard Procedure

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Sum the selected items of each scale Compare the means of the sub-scales

Standard Procedure

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Sum the selected items of each scale Compare the means of the sub-scales Each item has a different attractivity

Standard Procedure

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Sum the selected items of each scale Compare the means of the sub-scales Each item has a different attractivity The selection of an item depends on the offered alternative Comparison of sub-scale means is not appropriate

Way out

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Use of methods for Paired Comparisons

Way out

Introduction ❍ Instrument ❍ Example ❍ Standard Procedure ❍ Way out Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Use of methods for Paired Comparisons FIT:  different items  different items for each sub-scale  comparisons e.g. R : I, I : A,

Analysing FIT with Paired Comparisons Methods

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Problem

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

[R] [I] [I] [A] [A] [S] ... [R] 

• 

    ... [I]

• 

     [I]   

• 

  [A]  

• 

   [A]     

• 

[S]    

• 

 ... ...

Linear dependencies in the design matrix

Object Covariate

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Each sub-scale is treated as an object: R I A S E C

Object Covariate

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Each sub-scale is treated as an object: R I A S E C Each item is assigned to a sub-scale

Object Covariate

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Each sub-scale is treated as an object: R I A S E C Each item is assigned to a sub-scale

Categorical Object Covariates

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

The fact that an item belongs to one scale is treated as categorical object covariate

Categorical Object Covariates

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

The fact that an item belongs to one scale is treated as categorical object covariate e.g. item R has the attribute Realistic ⇒  on the object covariate R

Categorical Object Covariates

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

The fact that an item belongs to one scale is treated as categorical object covariate e.g. item R has the attribute Realistic ⇒  on the object covariate R ln m(jk)j = µ(jk)j + λO

j − λO k

Categorical Object Covariates

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

The fact that an item belongs to one scale is treated as categorical object covariate e.g. item R has the attribute Realistic ⇒  on the object covariate R ln m(jk)j = µ(jk)j + λO

j − λO k

linear reparameterization λO

j = xj · R + xj · I + xj · A + xj · S + xj · E + xj · C

Categorical Object Covariates

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

The fact that an item belongs to one scale is treated as categorical object covariate e.g. item R has the attribute Realistic ⇒  on the object covariate R ln m(jk)j = µ(jk)j + λO

j − λO k

linear reparameterization λO

j = xj · R + xj · I + xj · A + xj · S + xj · E + xj · C

λO

R =  · R

Reparameterization Matrix

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

R I A S E C R       I       I       A       A       S       . . .

Including Subject Covariates

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

account for subject covariates: sex (G)

Including Subject Covariates

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

account for subject covariates: sex (G) ln m(jk)j|g = µ(jk)jg + λO

j − λO k + λOS jg − λOS kg

Including Subject Covariates

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

account for subject covariates: sex (G) ln m(jk)j|g = µ(jk)jg + λO

j − λO k + λOS jg − λOS kg

Sample

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Students of the University of Klagenfurt F Cultural Technical G sciences sciences total female    male      

Solution

Introduction Analysing FIT with Paired Comparisons Methods ❍ Problem ❍ Object Covariate ❍ Categorical Object Covariates ❍ Reparameterization Matrix ❍ Including Subject Covariates ❍ Sample ❍ Solution Results  Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Design matrix y µ R I A S E C G F R y  

• 

      I y 

• 

       I y   

• 

     A y  

• 

      A y    

• 

    . . .

Results 

Introduction Analysing FIT with Paired Comparisons Methods Results  ❍ Model Selection ❍ Worthplot Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Model Selection

Introduction Analysing FIT with Paired Comparisons Methods Results  ❍ Model Selection ❍ Worthplot Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

anova(G+F,G,F)

• Resid. Df Resid. Dev

Df Deviance G+F  . G  .

• 
• .

F  . 

• .

Worthplot

Introduction Analysing FIT with Paired Comparisons Methods Results  ❍ Model Selection ❍ Worthplot Possible Refinement Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

estimated worth

Preferred Interests

0.10 0.15 0.20 female male R C I S E A A E C I S R

Possible Refinement

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement ❍ Account for the Differences ❍ Account for the Differences ❍ Item Difficulties ❍ Reparameterization Matrix ❍ Design Matrix ❍ Account for Item Difficulties Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Account for the Differences

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement ❍ Account for the Differences ❍ Account for the Differences ❍ Item Difficulties ❍ Reparameterization Matrix ❍ Design Matrix ❍ Account for Item Difficulties Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Whom would you choose as your statistical consultant?

Account for the Differences

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement ❍ Account for the Differences ❍ Account for the Differences ❍ Item Difficulties ❍ Reparameterization Matrix ❍ Design Matrix ❍ Account for Item Difficulties Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

easy items difficult items low weights high weights

Item Difficulties

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement ❍ Account for the Differences ❍ Account for the Differences ❍ Item Difficulties ❍ Reparameterization Matrix ❍ Design Matrix ❍ Account for Item Difficulties Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Where do we get these weights from?

Item Difficulties

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement ❍ Account for the Differences ❍ Account for the Differences ❍ Item Difficulties ❍ Reparameterization Matrix ❍ Design Matrix ❍ Account for Item Difficulties Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Where do we get these weights from? item difficulties from Rasch Model

Reparameterization Matrix

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement ❍ Account for the Differences ❍ Account for the Differences ❍ Item Difficulties ❍ Reparameterization Matrix ❍ Design Matrix ❍ Account for Item Difficulties Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

item difficulties are continous object covariates choosing item R wants an certain amount of the attribute ’Realistic’

[R] [I] [A] [S] [E] [C] R . . . . . . I . . . . . . I . . . . . . A . . . . . . A . . . . . . S . . . . . .

Design Matrix

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement ❍ Account for the Differences ❍ Account for the Differences ❍ Item Difficulties ❍ Reparameterization Matrix ❍ Design Matrix ❍ Account for Item Difficulties Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

y µ R I A S E C G F R y  .

• .

      I y 

• .

.       I y   .

• .

     A y  

• .

.      A y    .

• .

    . . .

Account for Item Difficulties

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement ❍ Account for the Differences ❍ Account for the Differences ❍ Item Difficulties ❍ Reparameterization Matrix ❍ Design Matrix ❍ Account for Item Difficulties Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

consider different item difficulties

Account for Item Difficulties

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement ❍ Account for the Differences ❍ Account for the Differences ❍ Item Difficulties ❍ Reparameterization Matrix ❍ Design Matrix ❍ Account for Item Difficulties Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

consider different item difficulties consider subject covariates sex and faculty

Account for Item Difficulties

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement ❍ Account for the Differences ❍ Account for the Differences ❍ Item Difficulties ❍ Reparameterization Matrix ❍ Design Matrix ❍ Account for Item Difficulties Results  Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

consider different item difficulties consider subject covariates sex and faculty ln m(jk)j|l = µ(jk)jl + λO

j − λO k + λOS jl − λOS kl

λO

j = xj · R + xj · I + xj · A + xj · S + xj · E + xj · C

Results 

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  ❍ Model Selection ❍ Worthplot Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Model Selection

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  ❍ Model Selection ❍ Worthplot Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

anova(G+F,G,F)

• Resid. Df Resid. Dev

Df Deviance G+F  . G  .

• 
• .

F  . 

• .

Worthplot

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  ❍ Model Selection ❍ Worthplot Comparison of Results

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

estimated worth

Preferred Interests

0.10 0.15 0.20 female male R C E S I A A C E I R S

Comparison of Results

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results ❍ Sub-Scale Means vs. Categorical Object Covariate ❍ Sub-Scale Means vs. Continous Object Covariate ❍ Categorical Object Covariates vs. Continous Object Covariates ❍ Conclusio ❍ Thank you

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Sub-Scale Means vs. Categorical Object Covariate

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results ❍ Sub-Scale Means vs. Categorical Object Covariate ❍ Sub-Scale Means vs. Continous Object Covariate ❍ Categorical Object Covariates vs. Continous Object Covariates ❍ Conclusio ❍ Thank you

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

estimated worth

Scale Means

0.12 0.14 0.16 0.18 0.20 0.22 female male R A C E I S A C R E S I estimated worth

Preferred Interests

0.10 0.15 0.20 female male R C I S E A A E C I S R

Sub-Scale Means vs. Continous Object Covariate

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results ❍ Sub-Scale Means vs. Categorical Object Covariate ❍ Sub-Scale Means vs. Continous Object Covariate ❍ Categorical Object Covariates vs. Continous Object Covariates ❍ Conclusio ❍ Thank you

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

estimated worth

Scale Means

0.12 0.14 0.16 0.18 0.20 0.22 female male R A C E I S A C R E S I estimated worth

Preferred Interests

0.10 0.15 0.20 female male R C E S I A A C E I R S

Categorical Object Covariates vs. Continous Object Covariates

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results ❍ Sub-Scale Means vs. Categorical Object Covariate ❍ Sub-Scale Means vs. Continous Object Covariate ❍ Categorical Object Covariates vs. Continous Object Covariates ❍ Conclusio ❍ Thank you

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

estimated worth

Preferred Interests

0.10 0.15 0.20 female male R C I S E A A E C I S R estimated worth

Preferred Interests

0.10 0.15 0.20 female male R C E S I A A C E I R S

AIC: . AIC: .

Conclusio

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results ❍ Sub-Scale Means vs. Categorical Object Covariate ❍ Sub-Scale Means vs. Continous Object Covariate ❍ Categorical Object Covariates vs. Continous Object Covariates ❍ Conclusio ❍ Thank you

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

the method of analysis makes a difference further refinement is still needed

Thank you

Introduction Analysing FIT with Paired Comparisons Methods Results  Possible Refinement Results  Comparison of Results ❍ Sub-Scale Means vs. Categorical Object Covariate ❍ Sub-Scale Means vs. Continous Object Covariate ❍ Categorical Object Covariates vs. Continous Object Covariates ❍ Conclusio ❍ Thank you

Thomas Gareiß Dittrich & Hatzinger The BTL for latent variables – 

Thank you!

Almut.Thomas@uni-klu.ac.at