metheval - uni-bielefeld.de...metheval Lehrstuhl für Methodenlehre und Evaluationsforschung •Institut für Psychologie •FSU Jena jena.de 1 / 58 AnIRTmodelwithperson-speciﬁcitemdifﬁculties

methevalLehrstuhl für Methodenlehre und Evaluationsforschung • Institut für Psychologie • FSU Jena

www.metheval.unijena.de 1 / 58

An IRT model with person-specific item difficulties

Rolf Steyer

Friedrich Schiller Universität Jena

Institut für Psychology

Lehrstuhl für Methodenlehre und Evaluationsforschung

SEM Meeting, Berlin

26. Februar 2015

Abstract

Abstract

Logit and Probit

Rasch model

Live Satisfaction

1parameter probit model

The random experiment

Personspecific item

difficultiesPersonspecific item

difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

Components of the Theory

of Causal Effects

Identification of Causal

Effects

Generalized ANCOVA

EffectLite

Theory of Causal Effects


In the Rasch model we assume that the item difficulties are identical for

all persons. In empirical applications, model tests usually show that the

assumptions of this model do not hold. This also applies to the more

liberal Birnbaum model. I present a generalization of the Rasch model

in which it is assumed that the item difficulties are person-specific, i.e.,

in which each item has a latent trait and a latent difficulty variable. The

price for such a model is that there are several occasions of

measurement with time-invariant item-difficulty factors. The model is

applied to the life satisfaction scale of the Freiburger Personality

Inventory (FPI). Using the good-bad scale of the multidimensional

mood state questionnaire (MDBF), it is also investigated if and how the

model can be extended to items with more than two answer categories.

Logit and Probit Transformations

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



0

2

4

6

−2

−4

−6

0.5 1.0

logit

probit

tra

nsf

orm

ed

va

lue

of

p

p

Figure 1. Graphs of the logit (red) and probit (blue) transformations of p

Assumptions of the Rasch model

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



We consider the random experiment of sampling a person and observing thepattern of responses to m dichotomous items Yi with values 0 or 1. We define

logiti := ln( P (Yi = 1 |U )

1−P (Yi = 1 |U )

)

, i = 1, . . . ,m,

where U denotes the person variable, the value of which is the sampled person.

Assumption 1: Rasch homogeneity

∃βi j ∈R : logiti = logit j +βi j , ∀i , j = 1, . . . ,m

Defining ξ := logit1 and βi :=−βi 1, for all i = 1, . . . ,m, yields

logiti = ξ−βi , ∀i = 1, . . . ,m,

and solving for P (Yi = 1 |U ),

P (Yi = 1 |U ) =exp(ξ−βi )

1+exp(ξ−βi ), ∀ i = 1, . . . ,m

Assumption 2: U -conditional independence

P (Yi = 1 |U ,Y1, . . . ,Yi−1,Yi+1, . . . ,Ym ) = P (Yi = 1 |U ), ∀ i = 1, . . . ,m

Live Satisfaction of the FPI

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



The live satisfaction questionnaire consists of the following 12 items, each oneto be answered with stimmt (true) or stimmt nicht (not true).

003 Ich habe (hatte) einen Beruf, der mich voll befriedigt023 Ich lebe mit mir selbst in Frieden und ohne innere Konflikte029 Wenn ich noch mal geboren würde, dann würde ich nicht anders leben

wollen058 In meinem bisherigen Leben habe ich kaum das verwirklichen können,

was in mir steckt088 Ich bin immer guter Laune094 Oft habe ich alles gründlich satt100 Ich bin selten in bedrückter, unglücklicher Stimmung112 Ich grüble viel über mein bisheriges Leben nach119 Ich bin mit meinen gegenwärtigen Lebensbedingungen oft unzufrieden128 Alles in allem bin ich ausgesprochen zufrieden mit meinem bisherigen

Leben131 Meine Partnerbeziehung (Ehe) ist gut138 Meistens blicke ich voller Zuversicht in die Zukunft

Assumptions of the 1-parameter probit model

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



We consider the random experiment of sampling a person and observing thepattern of responses to m dichotomous items Yi with values 0 or 1. We define

probiti := Φ−1(

P (Yi = 1 |U ))

,

where Φ−1 denotes the inverse distribution function of a standard normaldistribution and U the person variable, the value of which is the sampled person.

Assumption 1: Homogeneity

∃γi j ∈R : probiti = probit j +γi j , ∀i , j = 1, . . . ,m

Defining η :=probit1 and γi :=−γi 1, for all i = 1, . . . ,m, yields

probiti = η−γi , ∀i = 1, . . . ,m,

and solving for P (Yi = 1 |U ),

P (Yi = 1 |U ) = Φ(η−γi ), ∀i = 1, . . . ,m

Assumption 2: U -conditional independence

P (Yi = 1 |U ,Y1, . . . ,Yi−1,Yi+1, . . . ,Ym ) = P (Yi = 1 |U ), ∀ i = 1, . . . ,m

The random experiment, the logit and probit transformations

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



We consider the random experiment of sampling a person and

observing the pattern of responses to m dichotomous items Yi t with

values 0 or 1 at n times points t . We define

logiti t := ln( P (Yi t = 1 |Ut ,St )

1−P (Yi t = 1 |Ut ,St )

)

, i = 1, . . . ,m, t = 1, . . . ,n,

where Ut denotes the person variable at time t , the value of which is

the sampled person at time t and St the situation variable at time t , the

value of which is the situation in which assessment takes place at time

t . Note that logiti t is a random variable that is a function of the random

variable (Ut ,St ), i.e., logiti t = f (Ut ,St ).

Analogously, we define

probiti t := Φ−1(

P (Yi t = 1 |Ut ,St ))

, i = 1, . . . ,m, t = 1, . . . ,n,

where Φ−1 denotes the inverse distribution function of a standard

normal distribution.

A generalized Rasch model with person-specific item difficulties

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



Trivially,

logiti t = logit1t + (logiti t − logit1t ), ∀ i = 1, . . . ,m, t = 1, . . . ,n . (1)

Assumption 1. Time-invariant logit differencesWe assume

logiti t − logit1t = logiti s − logit1s , ∀ i = 1, . . . ,m, s, t = 1, . . . ,n . (2)

Furthermore, we define ξt := logit1t and ηi :=−(logiti t − logit1t ) for alli = 1, . . . ,m, t = 1, . . . ,n. Using these definitions, rearranging Equation (1) yields

logiti t = ξt −ηi , ∀ i = 1, . . . ,m, t = 1, . . . ,n

Solving for P (Yi t = 1 |Ut ,St ) yields

P (Yi t = 1 |Ut ,St ) =exp(ξt −ηi )

1+exp(ξt −ηi ), ∀ i = 1, . . . ,m, t = 1, . . . ,n (3)

Assumption 2. (Ut ,St )-conditional independence

∀ i = 1, . . . ,m, t = 1, . . . ,n : (4)

P[

Yi t = 1 |U1, . . . ,Un ,S1, . . . ,Sn ,(

Y j s , ( j , s) 6= (i , t ))]

= P (Yi t = 1 |Ut ,St ) (5)

A generalized one-dimensional probit model with person-specific item

difficulties

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



Trivially,

probiti t = probit1t + (probiti t −probit1t ), ∀ i = 1, . . . ,m, t = 1, . . . ,n . (6)

Assumption 1. Time-invariant probit differencesWe assume

probiti t −probit1t = probiti s −probit1s , ∀ i = 1, . . . ,m, s, t = 1, . . . ,n . (7)

Furthermore, we define ξt := probit1t and ηi :=−(probiti t −probit1t ) for alli = 1, . . . ,m, t = 1, . . . ,n. Using these definitions, rearranging Equation (6) yields

probiti t = ξt −ηi , ∀ i = 1, . . . ,m, t = 1, . . . ,n

Solving for P (Yi t = 1 |Ut ,St ) yields

P (Yi t = 1 |Ut ,St ) = Φ(ξt −ηi ), ∀ i = 1, . . . ,m, t = 1, . . . ,n (8)


∀ i = 1, . . . ,m, t = 1, . . . ,n :

P[

Yi t = 1∣

∣

∣U1, . . . ,Un ,S1, . . . ,Sn ,(

Y j s , ( j , s) 6= (i , t ))

]

= P (Yi t = 1 |Ut ,St )(9)

The random experiment, the logit and probit transformations

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



We consider the random experiment of sampling a person and observing thepattern of responses to m items Yi t with c +1 response categories 0,1, . . . ,c at ntimes points t . We define

logiti t y := ln( P (Yi t ≥ y |Ut ,St )

1−P (Yi t ≥ y |Ut ,St )

)

, y = 1, . . . ,c , i = 1, . . . ,m, t = 1, . . . ,n,

where Ut denotes the person variable at time t , the value of which is thesampled person at time t and St the situation variable at time t , the value ofwhich is the situation in which assessment takes place at time t . Note thatlogiti t is a random variable that is a function of the random variable (Ut ,St ), i.e.,logiti t = f (Ut ,St ).

Analogously, we define

probiti t y := Φ−1(

P (Yi t ≥ y |Ut ,St ))

, y = 1, . . . ,c , i = 1, . . . ,m, t = 1, . . . ,n,

where Φ−1 denotes the inverse distribution function of a standard normaldistribution.

A generalized one-dimensional probit model for c +1 response

categories with person-specific item difficulties

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



Trivially, for all y = 1, . . . ,c, i = 1, . . . ,m, t = 1, . . . ,n,

probiti t y = probit1t y + (probiti t y −probit1t y ). (10)

Assumption 1. Time-invariant probit differences

∀ y = 1, . . . ,c, i = 1, . . . ,m, s, t = 1, . . . ,n :

probiti t y −probit1t y = probiti s y −probit1s y .(11)

Assumption 2. Time-invariant constant probit differences between categories

∀x, y = 1, . . . ,c, i = 1, . . . ,m, t = 1, . . . ,n :

∃βi x y ∈R : probiti t y = probiti t x +βi x y .(12)

Furthermore, for all i = 1, . . . ,m, t = 1, . . . ,n, we define the latent variables

ξt :=probit1t1 and ηi :=−(probiti t1 −probit1t1). (13)

Inserting these definitions for y = 1 into Equation (10) yields

probiti t1 = ξt −ηi , ∀ i = 1, . . . ,m, t = 1, . . . ,n . (14)

Choosing x = 1 and inserting βi y :=−βi1y , for all y = 2, . . . ,c and i = 1, . . . ,m into Equation

(12) yields

probiti t y = ξt −ηi −βi y , ∀y = 2, . . . ,c, i = 1, . . . ,m, t = 1, . . . ,n . (15)

Solving for P (Yi t ≥ y |Ut ,St ) results in

∀ y = 1, . . . ,c, i = 1, . . . ,m, t = 1, . . . ,n :

P (Yi t ≥ y |Ut ,St ) = Φ(ξt −ηi −βi y ), with βi y = 0 for y = 1.(16)

Outline

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite




∀ y = 1, . . . ,c, i = 1, . . . ,m, t = 1, . . . ,n :

P[

Yi t ≥ y∣

∣

∣U1, . . . ,Un ,S1, . . . ,Sn ,(

Y j s , ( j , s) 6= (i , t))

]

= P (Yi t ≥ y |Ut ,St )

(17)

Outline

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



Introduction

Some Examples

The Components of the Theory of Causal Effects

Definition of Causal Effects

Identification of Causal Effects

Introduction

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



Introduction

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite



Talking about causality in empirical statistical research we talk about:

Regressions (i.e., conditional expectations) such as E(Y |X ),

E(Y |X , Z ), or their representations by a path diagram, graph, etc.

Omitted variables, other causal factors, etc.

We cannot understand and reasonably discuss causality unless (a) and

(b) have a common mathematical representation.

This means: X ,Y , Z , the regressions E(Y |X ) and E(Y |X , Z ), as well as

the omitted variables are random variables on the same probability

space. This space and its time structure, as well as the relations

between the variables in our regressions on one side and the omitted

variables on the other side need a mathematical representation.

Some Examples

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I

Direct Effects: Data

Direct Effects II


of Causal Effects



Joe and Ann With Self-Selection

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



Table 1: Joe and Ann With Self-Selection

Outcomes ω Observables Regressions

Un

it

Tre

atm

en

t

Su

cc

ess

P(ω

)

Pe

rso

nv

ari

ab

leU

Tre

atm

en

tv

ari

ab

leX

Ou

tco

me

va

ria

ble

Y

E(Y

|X,U

)=

P(Y

=1|X

,U)

E(Y

|X)=

P(Y

=1|X

)

E(X

|U)=

P(X

=1|U

)

EX=

0(Y

|U)=

τ0

EX=

1(Y

|U)=

τ1

EU

(Y|X

)

Ad

just

ed

Re

gre

ssio

n

(Joe, no, −) .144 Joe 0 0 .70 .60 .04 .70 .80 .45(Joe, no, +) .336 Joe 0 1 .70 .60 .04 .70 .80 .45(Joe, yes, −) .004 Joe 1 0 .80 .42 .04 .70 .80 .60(Joe, yes, +) .016 Joe 1 1 .80 .42 .04 .70 .80 .60(Ann, no, −) .096 Ann 0 0 .20 .60 .76 .20 .40 .45(Ann, no, +) .024 Ann 0 1 .20 .60 .76 .20 .40 .45(Ann, yes, −) .228 Ann 1 0 .40 .42 .76 .20 .40 .60(Ann, yes, +) .152 Ann 1 1 .40 .42 .76 .20 .40 .60

Joe and Ann With Self-Selection – Compressed

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



Table 2: Joe and Ann With Self-Selection – Compressed

U P(U

=u

)

EX=

0(Y

|U)=

τ0

EX=

1(Y

|U)=

τ1

P(X

=1|U

)

P(U

=u|X=

0)

P(U

=u|X=

1)

Joe 1/4 .70 .80 .04 .80 .05Ann 1/4 .20 .40 .76 .20 .95

Joe and Ann With Random Assignment

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



Table 3: Joe and Ann With Random Assignment

Outcomes ω Random variables Regressions

Un

it

Tre

atm

en

t

Su

cc

ess

P(ω

)

Pe

rso

nv

ari

ab

leU

Tre

atm

en

tv

ari

ab

leX

Ou

tco

me

va

ria

ble

Y

E(Y

|X,U

)=

P(Y

=1|X

,U)

E(Y

|X)=

P(Y

=1|X

)

E(X

|U)=

P(X

=1|U

)

EX=

0(Y

|U)=

τ0

EX=

1(Y

|U)=

τ1

EU

(Y|X

)

Ad

just

ed

Re

gre

ssio

n

(Joe, no, −) .09 Joe 0 0 .70 .45 .40 .70 .80 .45(Joe, no, +) .21 Joe 0 1 .70 .45 .40 .70 .80 .45(Joe, yes, −) .04 Joe 1 0 .80 .60 .40 .70 .80 .60(Joe, yes, +) .16 Joe 1 1 .80 .60 .40 .70 .80 .60(Ann, no, −) .24 Ann 0 0 .20 .45 .40 .20 .40 .45(Ann, no, +) .06 Ann 0 1 .20 .45 .40 .20 .40 .45(Ann, yes, −) .12 Ann 1 0 .40 .60 .40 .20 .40 .60(Ann, yes, +) .08 Ann 1 1 .40 .60 .40 .20 .40 .60

Joe and Ann Homogeneous

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



Table 4: Joe and Ann Homogeneous

Outcomes ω Observables RegressionsU

nit

Tre

atm

en

t

Su

cc

ess

P(ω

)

Pe

rso

nv

ari

ab

leU

Tre

atm

en

tv

ari

ab

leX

Ou

tco

me

va

ria

ble

Y

E(Y

|X,U

)=

P(Y

=1|X

,U)

E(Y

|X)=

P(Y

=1|X

)

E(X

|U)=

P(X

=1|U

)

( Joe, no, −) .03 Joe 0 0 .70 .70 .80( Joe, no, +) .07 Joe 0 1 .70 .70 .80( Joe, yes, −) .08 Joe 1 0 .80 .80 .80( Joe, yes, +) .32 Joe 1 1 .80 .80 .80(Ann, no, −) .09 Ann 0 0 .70 .70 .40(Ann, no, +) .21 Ann 0 1 .70 .70 .40(Ann, yes, −) .04 Ann 1 0 .80 .80 .40(Ann, yes, +) .16 Ann 1 1 .80 .80 .40

Joe and Ann Homogeneous

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



Table 5: Joe and Ann Homogeneous With Partial Regressions

Outcomes ω Observables Regressions

Un

it

Tre

atm

en

t

Su

cc

ess

P(ω

)

Pe

rso

nv

ari

ab

leU

Tre

atm

en

tv

ari

ab

leX

Ou

tco

me

va

ria

ble

Y

E(Y

|X,U

)=

P(Y

=1|X

,U)

E(Y

|X)=

P(Y

=1|X

)

E(X

|U)=

P(X

=1|U

)

EX=

0(Y

|U)=

τ0

EX=

1(Y

|U)=

τ1

EU

(Y|X

=x

)

Ad

just

ed

Re

gre

ssio

n

( Joe, no, −) .03 Joe 0 0 .70 .70 .80 .70 .80 .70( Joe, no, +) .07 Joe 0 1 .70 .70 .80 .70 .80 .70( Joe, yes, −) .08 Joe 1 0 .80 .80 .80 .70 .80 .80( Joe, yes, +) .32 Joe 1 1 .80 .80 .80 .70 .80 .80(Ann, no, −) .09 Ann 0 0 .70 .70 .40 .70 .80 .70(Ann, no, +) .21 Ann 0 1 .70 .70 .40 .70 .80 .70(Ann, yes, −) .04 Ann 1 0 .80 .80 .40 .70 .80 .80(Ann, yes, +) .16 Ann 1 1 .80 .80 .40 .70 .80 .80

Example

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



Table 6: Four Persons With Self-Selection to Treatment

Outcomes ω Observables RegressionsU

nit

Tre

atm

en

t

Su

cc

ess

P(ω

)

Pe

rso

nv

ari

ab

leU

Se

xZ

Tre

atm

en

tv

ari

ab

leX

Ou

tco

me

va

ria

ble

Y

E(Y

|X,U

)

E(Y

|X,Z

)

E(Y

|X)

P(X

=1|U

)

(Joe, no, −) .0675 Joe m 0 0 .7 .66 .63 .1

(Joe, no, +) .1575 Joe m 0 1 .7 .66 .63 .1

(Joe, yes, −) .0050 Joe m 1 0 .8 .44 .44 .1

(Joe, yes, +) .0200 Joe m 1 1 .8 .44 .44 .1

(Jim, no, −) .0175 Jim m 0 0 .3 .66 .63 .9

(Jim, no, +) .0075 Jim m 0 1 .3 .66 .63 .9

(Jim, yes, −) .1350 Jim m 1 0 .4 .44 .44 .9

(Jim, yes, +) .0900 Jim m 1 1 .4 .44 .44 .9

(Sue, no, −) .0600 Sue f 0 0 .7 .60 .63 .2

(Sue, no, +) .1400 Sue f 0 1 .7 .60 .63 .2

(Sue, yes, −) .0200 Sue f 1 0 .6 .44 .44 .2

(Sue, yes, +) .0300 Sue f 1 1 .6 .44 .44 .2

(Ann, no, −) .0400 Ann f 0 0 .2 .60 .63 .8

(Ann, no, +) .0100 Ann f 0 1 .2 .60 .63 .8

(Ann, yes, −) .1200 Ann f 1 0 .4 .44 .44 .8

(Ann, yes, +) .0800 Ann f 1 1 .4 .44 .44 .8

Simpson’s Paradox

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



Total sample

Treatment

Success no yes(X=0) (X=1)

no (Y=0) 240 232 472yes (Y=1) 360 168 528

600 400 1000

360/600 = .60 168/400 = .42

0.0

0.2

0.4

0.6

0.8

1.0

0.6

0.42

control treatment

Pro

po

rtio

no

fsu

cc

ess

Simpson’s Paradox: Males vs. Females

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



Males (Z=0) Females (Z=1)

Success Control Treatment Control Treatment(X=0) (X=1) (X=0) (X=1)

No (Y=0) 144 4 96 228Yes (Y=1) 336 16 24 152

480 20 120 380

336/480 = .70 16/20 = .80 24/120 = .20 152/380 = .40

0.0

0.2

0.4

0.6

0.8

1.0

0.70.8

control treatment

0.2

0.4

control treatment

Pro

po

rtio

no

fsu

cc

ess

male female

Nonorthogonal ANOVA

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



Table 7: Expectations in three treatment conditions

expectation of Y in treatment

the treatment conditions probabilities

treatment E(Y |X=x) P (X=x)

X=0 (control) 111.25 1/3

X=1 (treatment 1) 100.00 1/3

X=2 (treatment 2) 114.25 1/3

E(Y ) 108.50

Nonorthogonal ANOVA

Abstract

Logit and Probit

Rasch model

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Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



Table 8: Expectations E (Y |X=x, Z=z) in treatment × neediness conditions

neediness

treat-ment low (Z=0) medium (Z=1) high (Z=2) P (X=x)

X=0 120 (20/120) 110 (17/120) 60 (3/120) (40/120)X=1 100 (7/120) 100 (26/120) 100 (7/120) (40/120)X=2 80 (3/120) 90 (17/120) 140 (20/120) (40/120)

P (Z=z) (30/120) (60/120) (30/120)

Note. Probabilities P (X=x, Z=z), P (Z=z), and P (X=x) in parentheses.

Direct Treatment Effect: Path Diagram I

Abstract

Logit and Probit

Rasch model

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Personspecific item


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difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



X

Y

M εM

εY

1.19

−3.75

20

Figure 1: Path diagram with a randomized treatment variable X , an in-

termediate M (post-test motivation), and an outcome variable Y (post-

test achievement).

E(Y |X ) = 130+20 ·X

E(M |X ) = 80+20 ·X

E(Y |X , M ) ≈ 34.9924−3.7528 ·X +1.1876 ·M

Direct Treatment Effect: Expectations, Covariances, and Correlations

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



Table 9: Covariances, Correlations, and Expectations in the Teaching Experi-ment

W Z X M Y

Pre-test achievement W 100.00 .850 .000 .495 .740Pre-test motivation Z 85.00 100.00 .000 .582 .696Treatment (yes/no) X 0.00 0.00 0.25 .727 .597Post-test motivation M 68.00 80.00 5.00 189.00 .893Post-test achievement Y 124.00 116.50 5.00 205.70 280.45

Expectations 100.00 100.00 0.50 90.00 140.00

Note. Correlations (in italics) are rounded.

Direct Treatment Effect: Path Diagram II

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples

SelfSelection

Compressed Table

Random Assignment

Homogeneous Persons

Homogeneous Persons

Four Persons

Simpson’s Paradox

Males vs. females

Nonorthogonal ANOVA

Nonorthogonal ANOVA

Direct Effects I


Direct Effects II


of Causal Effects



W

Z

X

Y

M εM

εY.90

.50

.80

10

20

85

Figure 2: Path diagram with a randomized treatment variable X , two

pre-tests Z (pre-test motivation) and W (pre-test achievement), an in-

termediate M (post-test motivation), and an outcome variable Y (post-

test achievement).

E(Y |X ) = 130+20 ·X

E(Y |X , M ) ≈ 34.9924−3.7528 ·X +1.1876 ·M

E(Y |X , Z , M ) = 13.50+10 ·X + .50 ·M + .765 ·Z

E(Y |X , Z , M ,W ) = 0+10 ·X +0 ·Z + .50 ·M + .90 ·W

Components of the Theory of Causal Effects

Abstract

Logit and Probit

Rasch model

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Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects

SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples

True Outcome Variables

Another example

Basic Concepts


Effectswww.metheval.unijena.de 30 / 58

Single-Unit Trials

Abstract

Logit and Probit

Rasch model

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Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects

SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples


Another example

Basic Concepts



To which type of empirical phenomenon does the theory refer?

Drawing a person u out of a set of persons. This value u is a value of

the random variable U .

observing the value z of (a possibly multivariate qualitative or

quantitative and possibly fallible) covariate Z of the unit

assigning the unit or observing its assignment to one of several

experimental conditions (represented by the value x of the

treatment variable X ),

recording the numerical value y of the outcome variable Y .

Probability Space and Causality Space

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects

SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples


Another example

Basic Concepts



A probability space (Ω,A,P ) consists of

a set Ω of possible outcomes (of the random experiment)

a σ-algebra A of possible events

a probability measure P : A → [0,1]

On this space we can consider random variables U , X , Y , Z , . . ., all of

which are measurable w. r. t. A .

A filtered space ⟨ (Ω,A,P ), (Ft )t∈T ⟩ consists of:

a probability space (Ω,A,P )

a filtration (Ft )t∈T (w. r. t. which random variables and events can

be ordered)

On this space we can consider random variables U , X , Y , Z , . . ., all of

which are measurable w. r. t. A . Some are measurable w. r. t. some of

the σ-algebras Ft , some are not.

Filtration (Ft )t∈T

Abstract

Logit and Probit

Rasch model

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Personspecific item


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Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects

SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples


Another example

Basic Concepts



Let (Ω,A ) be a measurable space, let T be a set on which there are the

relations =, <, and ≤, and let s, t ∈ T . A family (Ft )t∈T of

sub-σ-algebras Ft of A is called a filtration in A, if Fs ⊂ Ft for all

s ≤ t .


Abstract

Logit and Probit

Rasch model

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Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects

SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples


Another example

Basic Concepts



σ(U ) ⊂F1

U=u ∈F1

X=x ∈F2 =FtX

σ(X ) ⊂F2 =FtX

Y=y ∈F3

σ(Y ) ⊂F3

Figure 3: Venn-diagram of a filtration with T = 1,2,3


Abstract

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Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects

SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples


Another example

Basic Concepts



σ(U ) ⊂F1

U=u ∈F1

Z=z ∈F2

σ(Z ) ⊂F2

X=x ∈F3 =FtX

σ(X ) ⊂F3 =FtX

Y=y ∈F4

σ(Y ) ⊂F4

Figure 4: Venn-diagram of a filtration with T = 1, . . . ,4

Global Covariates

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Rasch model

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Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects

SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples


Another example

Basic Concepts



Let X be a random variable on (Ω,A,P ), let (Ft )t∈T be a filtration in A,

and let CX be a random variable on (Ω,A,P ). Then CX is called a global

covariate if

FtX =σ(X ,CX ), where tX is the smallest element t of T with

σ(X )⊂Ft , t ∈ T .

∀A ∈σ(X )∩CX : P (A) = 0 or P (A) = 1. (That is, σ(X ) and CX are

P-separated.)

Covariates

Abstract

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Rasch model

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Outline

Outline

Introduction

Some Examples


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SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples


Another example

Basic Concepts



A random variable Z on (Ω,A,P ) is called a covariate if σ(Z )⊂CX .

This implies: all events A ∈A that are represented by a covariate, such

as Z=z , are elements of FtX , but they are not an element of σ(X ).

Examples of Global Covariates

Abstract

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Rasch model

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Personspecific item


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Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects

SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples


Another example

Basic Concepts



Examples of global covariates of X

(1) CX =U(2) CX = (U , Z ), where Z is a vector of fallible pre-treatment measures(3) CX = (U , X2), where X2 is a second treatment variable(4) CX = (U , Z , X2).

The most important property in definition of a global covariate of X :σ(CX , X ) = FtX .

True Outcome Variables and True Effect Variables

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Outline

Outline

Introduction

Some Examples


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SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples


Another example

Basic Concepts



E X=x(Y |CX ) denotes the CX -conditional expectation of Y with respect

to the measure P X=x defined by

P X=x (A) = P (A |X=x), ∀A ∈A .

E X=x(Y |CX ) is called P-unique if all versions of E X=x(Y |CX ) are

P-equivalent. [P X=x (A) = 0] ⇒ [P (A) = 0], ∀A ∈CX implies that

E X=x(Y |CX ) is P-unique.

Using the global covariate CX of X and the conditional expectation

E X=x(Y |CX ) of Y given CX in treatment x, we can define the

(total-effect) true outcome variables τx := E X=x(Y |CX ) and the true

total-effect variables δxx ′ := τx −τx ′ , where x, x ′ denote two values of

the treatment variable X .

These variables τx and δxx ′ are, by definition, unconfounded,

because we condition on all covariates (potential confounfers).

τx and δxx ′ are random variables on the same probability space as

the original random variables X and Y . Under the assumption of

P-uniqueness they have uniquely defined expectations, conditional

expectations, variances, covariances, etc.

The true outcome variables τx play the same role as Rubin’s

potential outcome variables Yx .

Example Illustrating the Basic Concepts

Abstract

Logit and Probit

Rasch model

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Personspecific item


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Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects

SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples


Another example

Basic Concepts



Fundamental parameters Derived parametersU

nit

u

Sa

mp

lin

gp

rob

ab

ilit

y

P(U

=u

)

Co

va

ria

teg

en

de

rZ

τ0

(u)=

E(Y

|X=

0,U

=u

)

τ1

(u)=

E(Y

|X=

1,U

=u

)

P(X

=1|U

=u

)

δ1

2=τ

1(u

)−τ

0(u

)

P(U

=u|X

=0

)

P(U

=u|X

=1

)

EX=

0(Y

|Z)

EX=

1(Y

|Z)

u1 1/6 m 68 81 3/4 13 1/10 3/14 83 92.5

u2 1/6 m 78 86 3/4 8 1/10 3/14 83 92.5

u3 1/6 m 88 100 3/4 12 1/10 3/14 83 92.5

u4 1/6 m 98 103 3/4 5 1/10 3/14 83 92.5

u5 1/6 f 106 114 1/4 8 3/10 1/14 111 122

u6 1/6 f 116 130 1/4 14 3/10 1/14 111 122

Expectations 92.33 102.33 7/12 10 92.33 102.33

E (Y |X=x) 99.80 96.71

Average total effect (ACE) 10.000

Prima facie effect (PFE) −3.086

male female

Conditional total effects 9.50 11.00

Conditional PFEs 9.50 11.00

Basic Concepts of the Theory of Causal Effects

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Rasch model

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Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects

SingleUnit Trials

Causality Space

Filtration

Filtration

Filtration

Global Covariates

Covariates

Global Covariates:

Examples


Another example

Basic Concepts



Primitives

Ω = ΩU ×ΩZ ×ΩX ×ΩY The set of possible outcomes

U : Ω→ΩU Person variable

Z : Ω→Ω′Z Covariate

X : Ω→ 0,1, . . . , J Treatment variable

Y : Ω→R Outcome variable

Theoretical Concepts the Theory of Causal Effects

τx := E X=x(Y |CX ) True outcome variables in treatment

conditions x with respect to total effects

δxx ′ := τx −τx ′ True total effect variables

E(δxx ′ ) Average total effect

E(δxx ′ |Z=z) Conditional total effect given Z=z

E(δxx ′ |X=x∗) Conditional total effect given X=x∗

Identification of Causal Effects

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Rasch model

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Outline

Outline

Introduction

Some Examples


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Effects

Core of the Theory

Sufficient Conditions



Implication Structure

Generalized ANCOVA

EffectLite



Core of the Theory of Total Effects

Abstract

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Some Examples


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Effects

Core of the Theory





Generalized ANCOVA

EffectLite



Notation

X — treatment variable Z = f (CX ) — covariate of X

Y — outcome variable U — person variable

τx := E X=x(Y |CX ) — true outcome variable

δxx ′ := τx −τx ′ — true total-effect variable

E(δxx ′ ) — average total effect

E(δxx ′ |Z=z) — conditional total effect given Z=z

Unbiasedness

. . . of the expectation E(Y |X=x) :⇔ E(Y |X=x) = E(τx )

. . . of the regression E X=x(Y |Z ) :⇔ E X=x(Y |Z ) =P

E(τx |Z )

Unbiasedness of the regression E X=x(Y |Z ) implies:

(1) E[

E X=x(Y |Z )]

= E[

E(τx |Z )]

= E(τx )

Remember : E(δxx ′ ) = E(τx )−E(τx ′ )

and, if V = f (Z ),

(2) E[

E X=x(Y |Z ) |V]

=P

E[

E(τx |Z ) |V]

=P

E(τx |V )

Remember : E(δxx ′ |V ) =P

E(τx |V )−E(τx ′ |V )

Each of These Conditions is Sufficient for Unbiasedness

Abstract

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Effects

Core of the Theory





Generalized ANCOVA

EffectLite



Let CX denote a global covariate of X and Z = f (X ) a covariate.

Z -conditional independence of CX and treatments (X ⊥⊥CX |Z )

P (X=x |CX ) =P

P (X=x |Z ), ∀x

Completeness of the regression (Y ⊢CX |X , Z )

E (Y |X ,CX ) =P

E (Y |X , Z )

Z -Conditional Strong Causality

if W is CX -measurable,then there is a real-valued function h such that

E (Y |X , Z ,W ) =P

E (Y |X , Z ) + h(Z ,W ) and P (X=x |CX ) >P

0, ∀x

Z -conditional independence of true outcomes and treatments( τ⊥⊥X |Z “strong ignorability”)

P (X=x |Z ,τ0,τ1, . . . ,τJ ) =P

P (X=x |Z ) and P (X=x |CX ) >P

0, ∀x

Z -conditional regressive independence of true outcomes from treatments(τ⊢ X |Z )

E (τx |X , Z ) =P

E (τx |Z ) and P (X=x |CX ) >P

0, ∀x

Z -conditional unconfoundedness of the regression E (Y |X , Z )

P Z=z (X=x |CX ) = P Z=z (X=x) or E X=x, Z=z (Y |CX ) = E X=x, Z=z (Y )

for all pairs of values (x, z) of X and Z and P (X=x |CX ) >P

0, ∀x

Proof: Z -Conditional Independence Implies Unbiasedness

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Effects

Core of the Theory





Generalized ANCOVA

EffectLite



If CX =U is discrete and Z = f (U ), then

P (X=x |U , Z ) =P

P (X=x |Z ), ∀x

is equivalent to P (X=x |U ) =P

P (X=x |Z ), ∀x, and to

P (X=x |U=u) = P (X=x |Z=z), ∀x,u, z.

The definition of conditional probability implies

P (X=x,U=u)

P (U=u)=

P (X=x, Z=z)

P (Z=z), ∀x,u, z.

Dividing both sides by P (X=x, Z=z) and multiplying by P (U=u) yields

P (X=x,U=u)

P (X=x, Z=z)=

P (U=u, Z=z)

P (Z=z), ∀x,u, z.

Because P (X=x,U=u)= P (X=x,U=u, Z=z) this implies

P (X=x,U=u, Z=z)

P (X=x, Z=z)=

P (U=u, Z=z)

P (Z=z), ∀x,u, z.

The definition of conditional probability then yields

P (U=u |X=x, Z=z) = P (U=u |Z=z), ∀x,u, z. (18)

If Z = f (U )

E (Y |X=x, Z=z) =∑

uE (Y |X=x,U=u) ·P (U=u |X=x, Z=z), ∀x,u, z

is always true. Therefore (18) implies

E (Y |X=x, Z=z) =∑

uE (Y |X=x,U=u) ·P (U=u |Z=z), ∀x,u, z,

which is equivalent to E X=x (Y |Z )= E [E X=x (Y |U ) |Z ], ∀x, i.e., unbiasedness of

E X=x (Y |Z ).

Proof: Completeness Implies Unbiasedness

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Effects

Core of the Theory





Generalized ANCOVA

EffectLite



If CX =U and Z = f (U ), then completeness of the regression can be written

E (Y |X ,U ) =P

E (Y |X , Z ).

If CX =U is discrete, then this is equivalent to

E (Y |X=x,U=u) = E (Y |X=x, Z=z), ∀x,u, z. (19)

Now,

E (Y |X=x, Z=z) = E (Y |X=x, Z=z) ·1

= E (Y |X=x, Z=z) ·∑

uP (U=u |Z=z)

=∑

uE (Y |X=x, Z=z) ·P (U=u |Z=z), ∀x,u, z.

Inserting (19) yields

E (Y |X=x, Z=z) =∑

uE (Y |X=x,U=u) ·P (U=u |Z=z), ∀x,u, z,

which is equivalent to E X=x (Y |Z )= E [E X=x (Y |U ) |Z ], ∀x, i.e., unbiasedness of

E X=x (Y |Z ).

Implication Structure Among Conditions of Unbiasedness

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Effects

Core of the Theory





Generalized ANCOVA

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Table 10: Implication Structure Between Causality Conditions

(iii) (iv) (vi) (vii)

(i) Z -conditional independence of X and ⇒ ⇒ ⇒ ⇒

potential confounders (X ⊥⊥CX |Z )

(ii) Completeness of E (Y |X , Z ) ⇒ ⇒ ⇒ ⇒

(Y ⊢CX |X , Z )

(iii) Z -conditional independence of X ⇒ ⇒

and true outcomes (X ⊥⊥τ |Z )

(iv) Z -conditional regressive independence ⇒

of true outcomes from X (τ⊢ X |Z )

(v) Z -conditional strong causality ⇒

of E (Y |X , Z )

(vi) Z -conditional unconfoundedness ⇒

of E (Y |X , Z )

(vii) Z -conditional unbiasedness ⇒

of E (Y |X , Z )

Note: ⇒ indicates that condition in row implies condition in column.

Generalized ANCOVA

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite



Traditional ANCOVA Model

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite



If Z is a univariate covariate, the treatment variable X takes values

0,1, . . . , J , and the random variables 1X=x indicate with their values 1

and 0 whether or not X=x, traditional analysis of covariance assumes

E(Y |X , Z ) = γ00 +γ01 ·Z +

J∑

x=1

γx0 ·1X=x . (20)


Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite






E(Y |X , Z ) = γ00 +γ01 ·Z +

J∑

x=1

γx0 ·1X=x . (20)

For X=0, this equation yields:

E X=0(Y |Z ) = γ00 +γ01 ·Z , (21)


Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite






E(Y |X , Z ) = γ00 +γ01 ·Z +

J∑

x=1

γx0 ·1X=x . (20)

For X=0, this equation yields:

E X=0(Y |Z ) = γ00 +γ01 ·Z , (21)

and for X=x, Equation (20) yields:

E X=x(Y |Z ) = γ00 +γ01 ·Z +γx0 . (22)

Generalized ANCOVA Model

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite



The fundamental equation for generalized analysis of covariance is:

E(Y |X , Z ) = g0(Z )+J

∑

x=1

gx (Z ) ·1X=x , (23)

where the intercept function g0(Z ) and the effect functions gx (Z ) are

unknown functions of the (possibly multivariate, numerical or

non-numerical) covariate Z .


Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite




E(Y |X , Z ) = g0(Z )+J

∑

x=1

gx (Z ) ·1X=x , (23)




Remember, traditional analysis of covariance assumes

E(Y |X , Z ) = γ00 +γ01 ·Z +

J∑

x=1

γx0 ·1X=x . (24)


Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite




E(Y |X , Z ) = g0(Z )+J

∑

x=1

gx (Z ) ·1X=x , (25)





Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite




E(Y |X , Z ) = g0(Z )+J

∑

x=1

gx (Z ) ·1X=x , (25)




If X is discrete this equation is always true as long as no restrictive

assumptions about the intercept and/or effect functions are

introduced.


Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite




E(Y |X , Z ) = g0(Z )+J

∑

x=1

gx (Z ) ·1X=x , (25)




If X is discrete this equation is always true as long as no restrictive

assumptions about the intercept and/or effect functions are

introduced.

Conditioning on the covariate, Equation (25) yields

E Z=z (Y |X ) = g0(z)+J

∑

x=1

gx (z) ·1X=x . (26)

This equation shows that the effects of the treatments may be different

for different values of the covariate.


Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite



Conditioning on the treatment, Equation (25) yields, for X=0

E X=0(Y |Z ) = g0(Z ), (27)


Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite




E X=0(Y |Z ) = g0(Z ), (27)

and for X=x:

E X=x(Y |Z ) = g0(Z )+ gx (Z ). (28)


Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite




E X=0(Y |Z ) = g0(Z ), (27)

and for X=x:

E X=x(Y |Z ) = g0(Z )+ gx (Z ). (28)

Hence,

gx (Z ) (29)

is the (prima facie) effect function, comparing treatment x to treatment

0


Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite




E X=0(Y |Z ) = g0(Z ), (27)

and for X=x:

E X=x(Y |Z ) = g0(Z )+ gx (Z ). (28)

Hence,

gx (Z ) (29)


0 and

E [gx (Z )] (30)

is the average effect of treatment x compared to treatment 0.


Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

Traditional ANCOVA

Generalized ANCOVA

Generalized ANCOVA

Generalized ANCOVA

EffectLite




E X=0(Y |Z ) = g0(Z ), (27)

and for X=x:

E X=x(Y |Z ) = g0(Z )+ gx (Z ). (28)

Hence,

gx (Z ) (29)


0 and

E [gx (Z )] (30)

is the average effect of treatment x compared to treatment 0.

If x = 0, . . . , J denote the values of X , in generalized ANCOVA, we

estimate both the conditional-effect functions gx (Z ) and the average

effects E [gx (Z )], for x = 1, . . . , J .

EffectLite

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite

EffectLite

EffectLite



Scope of EffectLite

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite

EffectLite

EffectLite



EffectLite ...

does not assume homogeneity of variances (in the univariate case with asingle outcome variable) or of covariance matrices (in the multivariate casewith two or more outcome variables) of the outcome variables betweentreatment groups.

Scope of EffectLite

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite

EffectLite

EffectLite



EffectLite ...


allows analyzing mean differences between groups w. r. t. several manifestoutcome variables, one or more latent outcome variables, and a mixture ofthe two kinds of outcome variables.

Scope of EffectLite

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite

EffectLite

EffectLite



EffectLite ...



allows analyzing conditional and average effects w. r. t. several manifestcovariates or w. r. t. one or more latent covariates, and a mixture of the twokinds of covariates.

Scope of EffectLite

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite

EffectLite

EffectLite



EffectLite ...




estimates and tests average effects for non-orthogonal analysis of variancedesigns, provided that the covariates are specified as qualitative indicatorvariables. Other programs typically do not test the average effect at all,orthey do not treat the covariates as stochastic regressors, which usually leadsto invalid tests of the average effect.

Scope of EffectLite

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite

EffectLite

EffectLite



EffectLite ...





produces results which are easily interpretable in the analysis of conditionaland of average effects (mean differences) between groups.

Scope of EffectLite

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite

EffectLite

EffectLite



EffectLite ...





produces results which are easily interpretable in the analysis of conditionaland of average effects (mean differences) between groups.

estimates and tests conditional and average total effects, provided that thecovariate-treatment regression is unbiased.

estimates and tests conditional and average direct effects, provided that thecovariate-intermediate-treatment regression is unbiased.

Scope of EffectLite

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite

EffectLite

EffectLite



EffectLite ...

needs Mplus or LISREL in the background

works with demo version or student version for small models


Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite


Book

Conclusion


Probability and Causality

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite


Book

Conclusion


Probability and Causality

Rolf Steyer

Ivailo Partchev

Ulf Kröhne

Benjamin Nagengast

Christiane Fiege

draft available at www.causal-effects.de

also contains the Causal Effects Explorer

and EffectLite (with manual).

Conclusion

Abstract

Logit and Probit

Rasch model

Live Satisfaction



Personspecific item


difficulties


Personspecific item

difficulties

Outline

Outline

Introduction

Some Examples


of Causal Effects


Effects

Generalized ANCOVA

EffectLite


Book

Conclusion


The random variables consider in a system of regression equations (pathdiagram) and the omitted variables are random variables in the samerandom experiment represented by the probability space (Ω,A,P )

What allows us to give the path diagram a causal interpretation are therelationships between the variables in the regressions (path diagram) andthe omitted variables

The basic idea in defining the true total-effect variable is to condition on allrandom variables that are prior or simultaneous to X except for X itself

The basic idea in defining the true direct-effect variable is to condition on allrandom variables that are prior or simultaneous to M except for X

These (most fine grained) true effect variables are then aggregated(coarsened) by taking expectations (average effects) or conditionalexpectations (conditional effects)

Under the assumption of unbiasedness (which is implied by severalcausality conditions that are empirically falsifiable), these aggregated effects(or effect functions) can be identified by empirically estimable parameters(or functions).

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metheval - uni-bielefeld.de...metheval Lehrstuhl für Methodenlehre und Evaluationsforschung •Institut für Psychologie •FSU Jena jena.de 1 / 58 AnIRTmodelwithperson-speciﬁcitemdifﬁculties