Transcript Document

Kolmogorov (1933)
Stochastic dependence of events
[ (, A, P) ,
Set (of possible outcomes)
-Algebra (set of possible events)
A is a set of subsets of  with
(a)   A
(b) If A  A, then Ā  A
(c) If A1, A2, ...  A,
then A1  A2  ...  A
P(A  B)]
Probability measure
P: A  [0, 1] with
(a) P() = 1
(b) P(A) ≥ 0  A  A
(c) P(i Ai) = i P(Ai),
if Ai  Aj =   i, j
Conditional Probability
P(A  B)
P( A | B) =
_____________
P(B)
Stochastic dependence of A and B
P(A  B)  P(A)
or
P(A  B)  P(A) P(B)
still Kolmogorov (1933)
Regressive Dependence of Random Variables
[ (, A, P) ,
E(Y  X) ]
same as before
Random variables on (, A, P)
Y:    (real-valued)
X:   X
must be „measurable“
i. e., all events associated with
X and Y are elements in A
Regression or conditional expectation,
i. e. that function of X, the values of
which are the conditional expected
values E(Y  X = x)
Regressive dependence of Y on X
E(Y  X ) ≠ E(Y )
More general framework than before, because
Y and X can be indicator variables for
events A and B, i. e. Y = IA and X = IB.
Kolmogorov´s framework is fine for causal and noncausal dependencies,
but it does not distinguish between the two.
Prototypical examples for noncausal dependencies:
1. Experiment with treatment and control where the more seriously ill
people tend to select themselves into the treatment condition.
2. The size of the older sibbling on the younger one.
Prototypical examples for causal dependencies:
3. Experiment with truely random assignment of units to treatment conditions.
E(Y  X ) can be highly misleading if causally interpreted.
It may indicate a positive dependence when in fact there is a negative
individual causal effect for each and every individual in the population.
(See Steyer et al., 2000) MPR-online, „Causal Regression. Models I“
Structural Prerequisites (Steyer, 1992)
[ (, A, P), E(Y  X ) , (Ct, t T), D ]
same as before (2)
Monotonically nondecreasing family
of -algebras Ct  A
C1
C2
C3
D  A, a sub--algebra of A.
A
used to define preorderedness relation between
events and random variables.
[Random variables generate -algebras  A.]
used to define „potential confounders“ W
(random variables). Their generated algebra is a subset of D.
Pre-orderedness
W  X Y
Causality conditions (Steyer, 1992)
Strict Causality
E(Y  X, W ) = E(Y  X )
for each potential confounder W
Strong Causality
E(Y  X, W ) = E(Y  X ) + f (W )
for each potential confounder W
Weak Causality (= Unconfoundedness)
If W is a potential confounder, then, for each value x of X:
E(Y  X = x ) =
∫ E(Y  X = x,W = w) P (dw)
W
i.e., if W is discrete:
E(Y  X = x) = w E(Y  X = x, W = w) P(W = w)
Sufficient conditions for Weak Causality (Steyer, 1992)
1.
Stochastic independence of X and D implies Weak Causality. [If D is defined to
be generated by U, the random variable, the values of which are the observational
units drawn from the population, then this independence can be deliberately
created via random assignment of units to treatment conditions.]
2.
Both, Strict and Strong Causality Conditions imply Weak Causality.
Necessary conditions for Weak Causality (Steyer, 1992)
1.
See definition of Weak Causality itself (the condition ist directly empirically testable).
2.
If
E(Y  X, W ) = 0 + 1 X + 2 W
then Weak Causality implies
E(Y  X ) = 0 + 1 X
with 1 = 1.
(For statistical tests in the normal distribution case see von Davier, 2001).
More recent work related to the Theory
of Causal Regression Models
Nonorthogonal Analysis of Variance
in cooperation with Wüthrich-Martone
(dissertation just finished May 2001)
Integrating Rubin´s Approach
Steyer et al. (2000a, b, see MPR-online)
Yet to be done
1.
Causal modeling with categorial variables:
- How to test causality?
- How to analyze causal effects?
2.
Extending the theory to the whole distribution of random variables
[instead of only focussing E(Y  X )].
3.
Testing and analyzing causal models with qualitative stochastic regressors
(ANOVA with stochastic regressors).
4.
Developing the theory of nonexperimental design,
i.e., causal analysis in panel studies etc.
5.
Extending the theory to systems of regression models.
How to construct latent variables
Basic Concepts of Classical Test Theory
Primitives
–
–
–
The set of possible events
of the random experiment
Test Score Variables
Projection
 = U  O
Yi :   IR
U:   O
Definition of the Theoretical Variables
–
–
True Score Variable
Measurement Error Variable
i := E(Yi | U )
i := Yi  i
1
Y1
1
2
Y2
2
3
Y3
3
4
Y4
4
5
Y5
5
6
Y6
6
-congenerity
i = ij0 + ij1 j ,
uncorrelated errors
equal error variances
ij0, ij1  IR, ij1 > 0
Cov(i, j) = 0, i  j
Var(i) = Var(j)
-congenerity implies
Yi = i0 + i1  +  i
1
Y1
2
Y2
3
Y3

12  2    2
1

 12  2

 13  2

2      2
2
2
23  2
2




32  2   3 2 
Structural equation models
d1
d2
x1
x2
x11
x1
x 21
f 31
d3
d4
x3
x4
d5
x5
d6
x6
x32
f 21
g 12
x2
x42
x53
z1
g 11
g 22
f 32
g 23
1

z2
21

y11
y1
1
y21
y2
2
y32
y3
3
y42
y4
4
2
x3
x63
LISREL notation, exogeneous and endogenous variables
Structural equation models
–Model equations for structured means
•Measurement model for y: y =  y  Ly   
•Measurement model for x: x =  x  Lx x  d
•Structural model:  =   B   G x  z
Structural equation models
 A(GG  ) A  
=
L x GA

A := L y (I  B)1
A(GLx ) 


L x L x  d 
Category probabilities in IRT models
P(Y=k | x)
1,0
k=0
k=3
0,5
k=1 k=2
0,0
-4
-2
1
2
3
2
4
x