lecture8 path analysis

Path AnalysisPath Analysis

• Primary goal– to explain the associations among variables

with our a priori model(s)– i.e., we are trying to explain why variables are

correlated using a "temporally-sequenced" model

– draw and test a mathematical model•with underlying equations

• Variables can be based on any type of data• We care about (a) overall model fit and (b)

relations within the model

Path AnalysisPath Analysis• Two traditions to path analysis

– old tradition•the model explains R•solves equations one at a time

–using OLS regression•no determination of overall model fit

– new tradition•the model explains and R•equations of a model are solved simultaneously

–using ML estimation in EQS, LISREL, AMOS, etc.

•determination of overall fit

Path AnalysisPath Analysis• Advantages of path analysis

– ability to test overall models and individual parameters

– ability to test models with multiple DVS– ability to model (multiple) mediator variables

(processes)• Primary disadvantage of path analysis

– cannot reduce the impact of measurement error•only have observed variables•do not have multiple indicators of a latent

variable

Path-Analytic ModelPath-Analytic Model

• Always start with a path diagram

Challenge (1)

Depression (5)

Threat (2)

Problem-Focused (3)

p31

r12

p54

p53

e5

e3

Emotion-Focused (4)

e4

p42

The ProcessThe Process• Model specification based on the path diagram

– you write equations to specify each endogenous variable•three equations comprise the model

–PF = p31(Challenge) + e3

–EF = p42(Threat) + e4

–Depression = p53(PF) + p54(EF) + e5

– this model attempts to explain the variance-covariance matrix () or correlation matrix (R)

Types of effectsTypes of effects

• Direct effects

Challenge (1)

Depression (5)

Threat (2)

Problem-Focused (3)

p51r12

p54

p53

e5

e3

Emotion-Focused (4)

e4

p42

p31

Types of effectsTypes of effects• Indirect effects

– the magnitude of an indirect effect is determined by multiplying compound paths

• ChallengeDepression = p31 * p53

Challenge (1)

Depression (5)

Threat (2)

Problem-Focused (3)

p42

r12

p54

p53

e5

e3

Emotion-Focused (4)

e4

p31

p51

Types of effectsTypes of effects

• Unanalyzed association

Challenge (1)

Depression (5)

Threat (2)

Problem-Focused (3)

p42

r12

p54

p53

e5

e3

Emotion-Focused (4)

e4

p31

p51

Path AnalysisPath Analysis• Calculating and using implied correlations

– we can calculate the correlations among the variables in R based on our model•for each pair of variables there is a correlation

implied by the model– the ultimate goal is to compare observed R to

implied R– use tracing rules to calculate implied R

•we highlight all possible routes between pairs of variables

–multiply compound paths within a route–add up all possible routes

Path AnalysisPath Analysis

– using tracing rules to calculate implied R•3 primary rules to calculate the implied

correlations–No loops: you cannot go through the

same variable twice in a single route–No going forward and then backward

•You can go backward first and then forward

–A maximum of 1 unanalyzed association per route

Tracing Rule 1Tracing Rule 1

D

C

E

A

B

•No loops: cannot go through the same variable twice•Implied rA,B = ACB (YES!!!)•Implied rA,B = ACDECB (NO!!!)

Tracing Rule 2Tracing Rule 2

A D

B

C

•No going forward then backward•Implied rB,C = BAC (YES!!!)•Implied rB,C = BDC (NO!!!)

Tracing Rule 3Tracing Rule 3

A D

B

C

•A maximum of 1 curved error per route•Implied rD,F = DACF (YES!!!)•Implied rD,F = DABCF (NO!!!)

E

F

4-Variable Model4-Variable Model

Challenge (1)

Depression (4)Problem-

Focused (3)

Challenge Threat PFThreat -.70 PF .50 -.45 Depress -.25 .50 -.60

Observed R =

p43 = -.685

p32 = -.430

p31 = .155

Threat (2)

r12 = -.700

4-Variable Model4-Variable Model• implied rchallenge,pf = p31 + (r12*p32) = .456

• implied rthreat,pf = p32 + (r12*p31) = -.539

Challenge (1)

Depression (4)Problem-

Focused (3)

p43 = -.685

p32 = -.430

p31 = .155

Threat (2)

r12 = -.700

4-Variable Model4-Variable Model• implied rchallenge,dep = (p31*p43) + (r12*p32*p43) =

-.312

• implied rthreat,dep = (p32*p43) + (r12*p31*p43) = .369

Challenge (1)

Depression (4)Problem-

Focused (3)

p43 = -.685

p32 = -.430

p31 = .155

Threat (2)

r12 = -.700

4-Variable Model4-Variable Model• Now we compare our implied

correlations to our observed correlations Challenge Threat PF

Challenge 1Threat -.70 1PF .50 -.45 1Depress -.25 .50 -.60

Observed R =

Challenge Threat PFChallenge 1Threat -.70 1PF .46 -.54 1Depress -.31 .37 -.68

Implied R =

4-Variable Model4-Variable Model

• generally you want values in residual R to be around .05• Q, subsequently for us Χ2, gives us a summary index of

residual R– and a p-value as well– we want the p-value to indicate nonsignificance!

•WHY???

Challenge Threat PFChallenge 1Threat 0 1PF .04 .09 1Depress .06 .13 .08

Residual R =

Types of ModelsTypes of Models• recursive: models that only contain

unidirectional relations

Challenge (1)

Depression (3)

Problem-Focused (2)

Types of ModelsTypes of Models

– nonrecursive: models that allow for bidirectional relations

Challenge (1)

Depression (3)

Problem-Focused (2)

Practical IssuesPractical Issues

• Just because your model fits well does NOT mean you have huge effects!– we are simply reproducing R in path

analysis• Beware of the existence of equivalent models

– models beyond your a priori model may fit equally as well

• Report R-squared values for each equation– index of effect size

Practical issuesPractical issues• You can modify your model called trimming a

model in path analysis– same caveats exist

• Standardized vs. Unstandardized Path Coefficients– standardized gives relations in comparable

units•allows for the determination of relative

importance of predictors– use unstandardized when you care about the

original units of measurement

Practical issuesPractical issues• Assumptions of path analysis

– No measurement error– No specification error

•causal ordering and variables are correct?!– Multicollinearity

• Sample size– generally a 10-1 ratio for both

•sample size to number of observed variables•sample size to number of estimated

parameters

Practical issuesPractical issues• Other (descriptive) indices of model fit

– CFI, SRMR, RMSEA, GFI, AIC, ECVI, etc., etc., etc,

– more to come on these when we discuss SEM

Final Tracing ExampleFinal Tracing Example

X1

X2

X3

X4

•implied r31= p31 + (r12*p32)•why not X1 to X4 to X3?

p31

p42

r12 p43

p41

p32

Final Tracing ExampleFinal Tracing Example

X1

X2

X3

X4

•implied r32= p32 + (r12*p31)

p31

p42

r12 p43

p41

p32

Final Tracing ExampleFinal Tracing Example

X1

X2

X3

X4

•implied r41= p41 + (r12*p42) + (p31*p43) + (r12* p32*p43)

p31

p42

r12 p43

p41

p32

Final Tracing ExampleFinal Tracing Example

X1

X2

X3

X4

•implied r42= p42 + (r12 * p41) + (p32*p43) + (r12* p31*p43)

p31

p42

r12 p43

p41

p32

Final Tracing ExampleFinal Tracing Example

X1

X2

X3

X4

•implied r43= p43 + (p31*p41) + (p31*r12*p42) + (p32*r12*p41) +(p32*p42)

p31

p42

r12 p43

p41

p32