The analysis of individual and average causal effects: Basic principles and some applications Presid

1
The analysis of individual and average causal
effects Basic principles and some
applicationsPresidential address at the Joint
Conference of the SMABS and the EAM in Budapest
2006
Rolf Steyer Institute of PsychologyDepartment
of Methodology and Evaluation ResearchEmail
rolf.steyer_at_uni-jena.de
Rolf Steyer Institute of PsychologyDepartment
of Methodology and Evaluation ResearchEmail
rolf.steyer_at_uni-jena.de
Rolf Steyer Institute of PsychologyDepartment
of Methodology and Evaluation ResearchEmail
rolf.steyer_at_uni-jena.de
Rolf Steyer Institute of PsychologyDepartment
of Methodology and Evaluation ResearchEmail
rolf.steyer_at_uni-jena.de
Rolf Steyer University of Jena (Germany) Institute
of PsychologyDepartment of Methodology and
Evaluation ResearchEmail rolf.steyer_at_uni-jena.de
www.uni-jena.de/svw/metheval
2
Overview

• Individual and average causal effects (Neyman,
  Rubin)
• Pre-Post Design with Control Group for the
  Analysis of Intervention Effects
• Design for the Analysis of Method Effects
• Examples
• Conclusions

3
Individual and average causal effects (Neyman,
Rubin)

• We define ?0 as follows
• ?0(u) E(Y 0 ? U u)
• and
• ?1(u) E (Y 1 ? U u)
• ICE1-0(u) ?1(u) ? ?0(u)
• individual causal effect of unit u
• ACE1-0 E(?1) E(?0)
• average causal effect

4
Three Design Types

• Between-Group Designs
• Pre-post Designs (not used at all in the
  Neyman-Rubin tradition)
• Between-Group Designs with Pre-Post Measures
  (only the between group comparisons are used in
  the Neyman-Rubin tradition)

5
Utilizing Pre-Post Designs for the Analysis of
Individual Effects

• Pre-post Designs and Between-Group Designs with
  Pre-Post Measures can be used to analyze not only
  average but also individual causal effects.
• The crucial asssumption is that the individual
  pretest distribution is the same as the
  individual posttest (outcome variable)
  distribution under control (no treatment).

6
Pre-Post Design with Control Group for the
Analysis of Intervention Effects
7
Not Yet Identified Individual-Effects Model

8
Not Yet Identified Individual Effects Model
Y0 t0 ?0 Y1 t0 t1 ?
t0 ?1

9
Identified Individual Effects Model with pretests
Y11 and Y21

10
Identified Individual Effects Model with pretests
Y11 and Y21

11
Design for the Analysis of Method Effects
12
Design for the Analysis of Method Effects
t22 ? t12 t21 ? t11 IME2-1

13
An identifíed Individual-Method-Effects Model
t22 ? t12 t21 ? t11 IME2-1

t22 ? E(t22 t12) M.EID
14
Model in treatment group
Treatment
15
Model in control group
16
Model in treatment group (t-values)
17
Model in control group (t-values)
18

• Correlation Matrix of ETA in Control group
  
• IQ ICE IME
• -------- -------- --------
• IQ 1.00
• ICE -0.65 1.00
• IME -0.37 0.43 1.00

19

• Correlation Matrix of ETA in Experimental Group
• IQ ICE IME
• -------- -------- --------
• IQ 1.00
• ICE -0.52 1.00
• IME -0.26 0.15 1.00

20
The effects of negativ item formulation
21
The common factor model

• y ?0 ? h ?
• Cov(h, ?) 0
• E(?) 0
• ?yy ? ? ? ??
• Even if we can identify the loadings, variances
  and covariances occuring in ? ? ? ?? , the
  factors in h are not defined by these equations.
• Factor score indeterminacy.

22

• Even if we know the joint distributions of all
  variables involved we would not be able to define
  the scores of the latent variables in ? .
• Having undefined theoretical variables (factors)
  is from my point of view too latent!
• How can we do better?

23
True score in Classical Test Theory

• The true-score variables are defined
• ? i E(Yi  ? U )

24
Basic decompositions of the test-score variables
in CTT
? 1
Y1
? 1
? 2
Y2
? 2
? 3
Y3
? 3
25
Basic concepts of Classical Test Theory (CTT)

• Primitives
• The set of possible events the random experiment
  ? ?U ? ?O
• Test-score variables Yi ? ? IR
• Observational-unit variable U ? ? ?U
• Definition of the theoretical variables
• True-score variables ?i E(Yi U )
• Measurement error variables ?i Yi ? ?i

26
Properties of True Score and Error Variables
Implied by Their Definition

• Decomposition of the variables Yi ?i
  ?i (1)
• Decomposition of the variances
  Var(Yi) Var(?i) Var(?i) (2)
• Other properties of true-score and error
  variables implied by their definition
  Cov(?i , ?j) 0 (3)
• (3) E(?i) 0 (4)
• 
  E(?i ? U ) 0 (5)
• for each (measurable) mapping of U
  E?i ? f (U ) 0 (6)

27
Assumptions and models in CTT

• (a1) t-equivalent tests ?i ?j
• (a2) essential t-equivalent tests ?i ?j
  ?ij , ?ij ? IR
• (a3) t-congeneric tests ?i ?ij0 ?ij1 ?j
  , ?ij0, ?ij1 ? IR, ?ij1 gt 0
• (b) uncorrelated errors Cov(?i, ?j) 0,
  i ? j
• (c) equal error variances Var(?i) Var(?j)
• Models defined by these assumptions
• Parallel tests (a1), (b) and (c)
• Essentially t-equivalent tests (a2) and (b)
• Congeneric tests (a3) and (b)

28
The congeneric model for 3 manifest variables

29
Model of ?-congeneric tests
30
Model of ?-congeneric tests

• Defining the latent variable in this may implies
  
• the latent variable
• is a linear function of each of the true-score
  variables
• and
• the latent variable is uniquely defined up to
  linear transformations.
• This means the origin (or mean) and the scale
  (or variance) of the latent variable can be fixed
  arbitrarily, but after fixing the scale, the
  latent variable including its scores is
  uniquely defined, even though, in applications,
  we do not know these scores!

31
Primitives in Latent State Trait Theory (LST
theory)
32
Theoretical Variables in LST Theory
33

34

35

36
t22 ? t12 t21 ? t11 IME2-1

37
An identifíed Individual-Method-Effects Model
t22 ? t12 t21 ? t11 IME2-1

t22 ? E(t22 t12) M.EID
38

39

40
Summary and Conclusion

• We can analyze individual (and average) causal
  effects in Pre-post Designs
• The causal interpretation rests on assumptions
• These assumptions can be tested
• Latent variables can be constructed from
  true-scores
• Not a single path in the SEM models represented a
  causal effect

41
Thanks toSven HartensteinUlf KröhneBenjamin
NagengastIvailo PartchevSteffi Pohl
42
Want More?

• Tomorrow morning 900 this conference
• Symposium on causality in Jena July 7 to 9, 2006
  with Tom Cook, Steve West, Don Rubin
  (online participation
  possible see www.uni-jena.de/svw/metheval
• Online video of workshop on the analysis of
  causal effects (same home page)
• Software EffectLite (see www.statlite.com)