G89.2247%20Structural%20Equation%20Models

1
G89.2247Structural Equation Models

• Overview of course
• Setting time for lab session
• Statistical thinking about causality
• SEM as "Causal Models
• Matrix Algebra Tools

2
Goals of Course

• Introduce you to basic concepts and techniques of
  structural equation models
• Help you develop a critical perspective regarding
  what is and is not learned using SEM
• Provide skills for your continued self-education
• Provide a context for you to apply SEM methods to
  a sophisticated problem of your own choosing

3
What are Structural Equation Models?

• Systems of linear equations that describe a
  network of relations among variables.
• Structural, not simply predictive relations
• Implied systems of nonlinear equations that
  describe patterns of variances and covariances
  among variables.
• Output of software systems such as LISREL, EQS,
  AMOS, and MPlus.

4
Why are SEM methods useful?

• Hoyles (1994) review tells us that SEM can
  address
• Questions about causal process
• Basic questions of measurement
• Questions about causal process when variables are
  not well measured
• SEM methods share most of the strengths of OLS
  multiple regression
• SEM models can be used to impress your family,
  friends and colleagues, if not reviewers and
  editors

5
An example of a structural equation model
StressgtDistress

• Extreme stress is known to lead to psychological
  breakdown (Battle fatigue, PTSD)
• Severe stress is believed to cause depression,
  anxiety disorder, psychosis

Stress
Distress1
Distress2
6
Stress causes distress and psychopathology

• To what extent is this common belief true?
• How much stress is needed to cause distress?
• For a unit change in stress, how much do we
  expect distress to increase?
• How do we account for the many persons who
  experience stress who manage to function without
  psychopathology?
• Is the purported causal process universal, or
  does it operate only in a subset of the
  population?

7
Causal Inference Issues

• Causal inference is often illusive in social and
  behavioral sciences
• Prototypes of Causal Effects seem to implicate
  primary (single) causes.
• billiard balls
• bacteria or viruses
• In reality, effects usually have multiple causes
• For distress
• Stressors
• Personal dispositions
• Familial factors
• Social environment
• Biological environment

8
Causal Inference, continued

• Effects of causes are not always constant
• social buffers
• developmental stages
• immune system interventions
• synergistic causal effects
• stochastic variation in causal factor strength
• stochastic measurement factors

9
David Hume's framework for Causality

• If E is said to be the effect of C, then
• 1) C and E must have temporal and spatial
  contiguity ASSOCIATION
• 2) C must precede E temporally DIRECTION
• 3) There must be CONSTANT CONJUNCTION If C,
  then E for all situations

10
Although still influential, Hume's analysis is
known to have limitations.

• Analysis of any cause C must be isolated from
  competing causes (ISOLATION)
• Constant conjunction is too restrictive
  stochastic processes affect causal relations, and
  mechanisms may vary across situations.
• Causal relations may be expressed in terms of
  expectations over stochastic variation

11
Formal causal analyses have led to important
advances

• Robert Koch, the Nobel Prize winning
  bacteriologist, investigated bacteria as causes
  of disease using three principles
• The organism must be found in all cases of the
  disease in question. (association)
• The organism must be isolated and grown in pure
  culture (isolation)
• When inoculated with the isolated organism,
  susceptible subjects must reproduce the disease
  (direction and hedged constant conjunction)

12
Causal Process in Time

• In the behavioral, social, and biological
  sciences, the units of observation cannot be
  trusted to stay the same over time.
• For example, in Koch's inoculation test, how do
  we know that the subject had not been infected by
  chance?
• For studies of distress, we expect both stress
  and distress to change over time.

13
Statisticians developed the randomized experiment
to address causal issues

• Randomly assign subjects to one of two
  conditions, Treatment (T) or Control (C),
• Administer treatment and control procedures
• Measure outcome variable Y (assumed to reflect
  the process of interest) blind to treatment group
• Infer effect of treatment from difference in
  group means

14
Hollands formal analysis of randomized
experiments

• Suppose Y(u) is a measurement on subject u that
  reflects the process that is supposed to be
  affected by treatment, T.
• If subject u is given treatment T, then YT(u) is
  observed.
• If subject u is given a control treatment, C,
  then YC(u) is observed.
• We would like to compare YT(u) with YC(u), but
  only one of these can be available as u is either
  in T or C.
• Let the desired comparison be called D YT(u) -
  YC(u).
• Holland calls this the Effect of cause T
• Although D can not be observed, its average can
  be estimated by computing

15
Between-subject is substituted for within-subject
information.

• Within subject analyses are intuitively
  appealing, but require strong assumptions about
  constancy over time.
• When D?0, then ASSOCIATION is established.
• Randomization prior to treatment deals with the
  causal issue of DIRECTION.
• It also partially supports ISOLATION (double
  blind trials, manipulation checks help address
  other aspects of isolation).
• Randomization does not establish CONSTANT
  CONJUNCTION. The effect is only established for
  the specific experimental conditions used in the
  study.

16
Key Feature Treatment is applied to subjects
sampled into group T

• Holland argues that this manipulation is critical
  to guarantee DIRECTION, and ISOLATION.
• Holland and Rubin go on to assert that clear
  causal inference is only possible if manipulation
  is at least conceivable. They propose the motto,

17
NO CAUSATION WITHOUT MANIPULATION

• This motto is not popular with sociologists and
  economists. It explicitly denies causal status
  to personal attributes, such as race, sex, age,
  nationality, and family history.
• Instead, it encourages the investigation of
  processes such as discrimination, physical
  changes corresponding to age, government policy,
  and biochemical consequences of genetic makeup.

18
NO CAUSATION WITHOUT MANIPULATION

• To illustrate, Holland would not say that my
  height causes me to hit my head going into my
  suburban cellar, as my height cannot be
  manipulated.
• My failure to duck, and the dangerous obstruction
  could be shown to be causally related to my
  bumped head.

19
Structural Equation Models

• Researchers of topics such as stress,
  discrimination, poverty, coping and so on cannot
  easily design randomized experiments
• Structural Equation Models (SEM) are often
  presented as a major tool for establishing
  causes.
• The use of SEM is increasing. The number of
  articles has been doubling over 8 year periods.
  Software is more available and accessible.

20
SEM and ISOLATION, ASSOCIATION, and DIRECTION

• Consider a simple SEM model
• Y b1 X e
• For every unit change in X, Y is expected to
  change by b1 units. This equation implies clear
  association of Y and X, and it makes the assumed
  direction underlying the association unambiguous.
  For the equation to be meaningful in terms of
  causation, we must also assume that alternative
  causes of Y are accounted by the independent
  stochastic term, e.
• Bollen calls the requirement that e be
  uncorrelated with X, the pseudo-isolation
  condition.

21
Analysis of Randomized Experiment through SEM

• Y b0 b1 X e
• Let X take one of two values representing whether
  a subject received the treatment (X1) or the
  control placebo(X0). b1 estimates D. Because
  the assignment is randomized, X is expected to be
  uncorrelated with residual causes of Y.
• Randomization justifies the pseudo-isolation
  condition.
• The randomized experiment also reminds us that
  between subject comparisons can be informative
  about average within subject effects. We can
  contemplate what would have happened if a given
  subject had been assigned to a different group.

22
In non-experimental studies, Isolation is
difficult to establish

• We need to specify EVERY causal factor that is
  correlated with X, the causal variable of
  interest.

X
e
W2
Y
W3
W4
23
Stress example continued

• For example, if Y is distress, and X is exposure
  to stress, W2 is some measure of previous
  distress, W3 is social class and W4 is coping
  skills, then all four exogenous (predictor)
  variables might be considered as possible causes
  of distress.
• In this model, variables W2, W3, and W4 are
  introduced as additional causes of Y that are
  distinct from, but not necessarily uncorrelated
  with, X. If the list of covariates is complete,
  then the condition of pseudo-isolation will be
  justified.
• In practice, we never know if the list of
  competing covariates is complete
• SEM analyses become credible as they withstand
  the alternative explanations advanced by their
  critics

24
The effects of model misspecification

• Suppose some W2 is missing in the data set, even
  though we know it is correlated with both Y and
  X. If we know that W is a causal factor for both
  X and Y, then we would portray the model as on
  the right
• If we consider the misspecified model, in which
  W2 is missing, we can see that the estimated
  effect of X will include the indirect effect of
  W2 on Y. The causal impact of X will be
  overestimated in the misspecified model.

25
Estimation of SEM Models

• We will see that the systems of equations imply
  certain patterns of correlations (covariances)
  among the variables in the model
• Estimates are obtained by fitting the sample
  covariance matrix rather than the individual
  observations
• This matrix is computed over persons
• SEM analyses report how well the covariance
  matrix is fit by alternative models.

26
Major strengths of SEM are that

• Proposed causal explanations are made explicit
• Tests of fit allow implausible models to be
  rejected
• Competing models can often be compared, and one
  may emerge as more plausible given the data.

27
Major problems with SEM are that

• Models are often (usually?) misspecified
• Linearity assumption is often made uncritically
• Measurement error distorts analysis
• Important variables may be missing
• Communicating results is challenging
• Novices may overstate claims or make errors in
  complex analyses that are difficult to detect

28
Math Review for SEM

• Matrix notation
• Many of the better books use matrix notation
• New results in journals often use matrices
• Expectation and Variance operators
• Helps distinguish population parameters from
  sample statistics

29
Some Special Matrices

• Data vector
• Data matrix
• Unit vector
• Null vector/matrix
• Identity matrix
• Diagonal matrix
• Symmetric square matrix X'X

30
Review

• Matrix rank/order
• Matrix transpose
• Matrix addition/subtraction
• Scalar multiplication of matrices
• Matrix multiplication
• Matrix Determinant
• Matrix inversion