Transcript
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
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