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CFA

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• Rationale: Adjust, repair measurement model as needed without violating the integrity of the structural model test. Often we have no particular theoretic interest in measurement, except as a means of testing theory at the construct level.

• Not without controversy, however.

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• Chi-square structural model minus chi-square measurement model with df(s)-df(m) degrees of freedom.

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Reliability (Really validity?) (∑λ)2 / (∑λ)2 + ∑θ

AVE(∑λ2) / (∑λ2) + ∑θ

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• Discriminant Validity• AVE > φ2

• φ not equal to 1.0

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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

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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

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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

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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

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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)

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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.

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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

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Holland’s 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 CT YYD

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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.

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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,

NO CAUSATION WITHOUT MANIPULATION

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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.

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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.

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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.

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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.

X Y

e

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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 (X=1) or the control placebo(X=0). 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.

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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.Y =b0 + b1 X + b2 W2 + b3 W3 + b4 W4 + e

X

W2

W3

W4

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e

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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.

X

e

YW 2

X

e

YW 2

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Missing Data Mechanisms

• Terms suggested by Rubin– Rubin (1976), Little & Rubin (1987)

• MISSING COMPLETELY AT RANDOM (MCAR)– Which data point is missing cannot be predicted

by any variable, measured or unmeasured.• Prob(M|Y)=Prob(M)

– The missing data pattern is ignorable. Analyzing available complete data is just fine.

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Missing Data Mechanisms

• MISSING AT RANDOM (MAR)– Which data point is missing is systematically

related to subject characteristics, but these are all measured

• Conditional on observed variables, missingness is random

• Prob(M|Y)=Prob(M|Yobserved)– E.g. Lower educated respondents might not

answer a certain question.– Missingness can be treated as ignorable

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Missing Data Mechanisms

• NOT MISSING AT RANDOM (NMAR)– Data are missing because of process related to

value that is unavailable• Someone was too depressed to come report about

depression• Abused woman is not allowed to meet interviewer

– Missing data pattern is not ignorable.– Whether missing data are MAR or NMAR can not

usually be established empirically.

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Approaches to Missing Data

• Listwise deletion– If a person is missing on any analysis variable, he is

dropped from the analysis.• Pairwise deletion

– Correlations/Covariances are computed using all available pairs of data.

• Imputation of missing data values.• Model-based use of complete data

– E-M (estimation-maximization approach)• SEM-based FIML

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EM and FIML

• Use available data to infer sample moment matrix.

• Uses information from assumed multivariate distribution

• Patterns of associations can be structured or unstructured.

• Now implemented in AMOS, EQS, Mplus

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Example of CFA with Means Modelparameter Complete n=400 Listwise FIML (EQS)

V1 = factor 1 1 1.000 1.000 1.000V2 = factor 1 .9 0.894 0.0060 0.965 0.0270 0.900 0.0060V3 = factor 1 .9 0.901 0.0060 0.996 0.0290 0.915 0.0060V4 = factor 1 .8 0.800 0.0060 0.890 0.0240 0.808 0.0060V5 = factor 1 .8 0.798 0.0050 0.889 0.0230 0.807 0.0050V6 = factor 1 .7 0.690 0.0050 0.751 0.0240 0.693 0.0060V7 = factor 2 1 1.000 1.000 1.000V8 = factor 2 .9 0.910 0.0110 0.941 0.0440 0.903 0.0110V9 = factor 2 .9 0.907 0.0120 0.957 0.0500 0.899 0.0130V10 = factor 2 .8 0.815 0.0110 0.838 0.0430 0.811 0.0110V11 = factor 2 .7 0.707 0.0110 0.702 0.0330 0.702 0.0120V12 = factor 2 .5 0.514 0.0090 0.523 0.0370 0.508 0.0100F1 = mean 100 99.174 0.6910 83.484 2.4660 98.438 0.6380F2 = mean 50 48.765 0.7010 42.629 2.4520 48.999 0.6410D1-F1 variance 100 105.250 8.7600 96.575 29.9810 115.293 9.4120D2-F2 variance 100 118.870 9.9000 117.810 36.1000 119.463 9.8380D2-F2 covariance 60 70.570 7.4200 55.810 25.4400 71.273 7.6540

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Multiple Imputation

• Substitute expected values plus noise for missing values.

• Repeat >5 times.• Combine estimates and standard errors using

formulas described by Rubin (1987). See also Schafer & Grahm (2002) Missing data: Our view of the state of the art. Psychological Methods, 7: 147-177.

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Communicating SEM Results

• Keeping up with the expert recommendations– Psychological Methods– Specialty journals

• Structural Equation Models• Multivariate Behavioral Research• Applied Psychological Measurement• Psychometrika

• Two kinds of audiences– Researchers interested in the substance of the empirical

contribution– Experts in SEM

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Talking Points of Hoyle&Panter, McDonald&Ho

• Model specification– Theoretical justification– Identifiability

• Measurement Model• Structural Model

• Model estimation– Characteristics of data

• Distribution form• Sample size• Missing data

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Talking Points of Hoyle&Panter, McDonald&Ho

• Model estimation– Estimation method: ML, GLS, ULS, ADF– Goodness of estimates and standard errors

• Model Selection and Fit Statistics• Alternative and Equivalent Models• Reporting Results

– Path diagrams– Tabular information– Use software conventions?