Testing Inequality Constrained Hypotheses in SEM Models

Example
Dekovic, Wissink, & Meijer (2004)
Testing Inequality Constrained
Hypotheses in SEM Models
Rens van de Schoot
Aspects of the parent adolescent relationship:
Positive quality of the relationship
Negative quality of the relationship
Disclosure (how much adolescents tell the parents)
Antisocial behavior
Example 1
Example
Informative hypothesis:
disclosure is the strongest predictor
of antisocial behavior, followed by
either a negative or positive relation
with the parent
Different types of hypotheses
Informative hypotheses:
H I1 : μ 2 < μ 4 < μ 3 < μ 5 < μ 1
H I2 : μ 1 = μ 2 > μ 4 > {μ 3 , μ 5 }
Traditional null (H 0 ), nothing is going on:
H 0 : μ 1 = μ 2 = μ 3 = μ 4 = μ 5
Different types of tests
We consider two types of hypothesis
tests (Silvapulle & Sen, 2004):
Type A is of the form
H0 : μ 1 = μ 2 = μ 3 = μ 4 = μ 5
H1: μ 2 < μ 4 < μ 3 < μ 5 < μ 1
Unconstrained:
H U : μ 1 , μ 2 , μ 3 , μ 4 , μ 5
Type B is of the form
H0 : μ 2 < μ 4 < μ 3 < μ 5 < μ 1
H1: μ 1 , μ 2 , μ 3 , μ 4 , μ 5

What to do with informative
hypotheses?
What to do with informative
hypotheses?
Difficult to evaluate informative
hypothesis using traditional null
hypotheses testing:
H0 : μ 1 = μ 2 = μ 3 = μ 4 = μ 5
H1: not H0
Why:
Not always interested in null hypothesis
‘accepting’ alternative hypothesis -> no
answer
No direct relation between H0 and
informative hypothesis
What to do with informative hypotheses?
Classical model selection criteria?
Information criteria (ICs):
Akaike's IC, or AIC, (Akaike, 1973,1981)
Bayesian IC, or BIC, (Schwarz, 1978)
Deviance IC, or DIC, (Spiegelhalter et al, 2002)
What should we then do?
Parametric bootstrap method in Mplus
Van de Schoot, R., Hoijtink, H. & Deković, M.
(2010). Testing Inequality Constrained Hypotheses
in SEM Models. Structural Equation Modeling, 17,
443–463.
Van de Schoot , R. & Strohmeier, D. (under
review). Testing inequality constrained hypotheses
in SEM to Increase Power: An illustration
contrasting classical hypothesis testing with a
parametric bootstrap approach
Bootstrap procedure with
plug-in p-values
SEM is often used in practice
Order restricted inference in SEM is new
Testing hypothesis not available
Solution: parametric bootstrap
Likelihood ratio test
However: calibration of plug-in p-value
needed
In Mplus:
Mplus & constraints
MODEL:
anti on pos (b1)
neg (b2)
dis (b3);
MODEL CONSTRAINT:
b1 < b3;
b2 < b3 ;

Mplus & constraints
Parametric bootstrap: Step 1
DATA: FILE = data.dat;
VARIABLE:
NAMES ARE anti pos neg dis dis;
MODEL:
anti on pos (b1)
neg (b2)
dis (b3);
MODEL CONSTRAINT:
b1 = b2;
b2 = b3;
SAVE:
RESULTS ARE D:\vb_1\results_data_H0_vb1.txt;
ESTIMATES ARE D:\vb_1\estimates_vb1.txt;
DATA: FILE = data.dat;
Step 2
VARIABLE: NAMES ARE anti pos neg dis;
MODEL:
anti on pos (b1)
neg (b2)
dis (b3);
MODEL CONSTRAINT:
b1 < b3;
b2 < b3 ;
SAVE: RESULTS ARE D:\vb_1\results_data_H1_vb1.txt;
MONTECARLO:
NAMES ARE anti pos neg dis;
Step 3
NOBSERVATIONS = 603;
NREPS = 1000;
POPULATION = D:\vb_1\estimates_vb1.txt;
REPSAVE= ALL;
SAVE= D:\vb_1\rep_nieuw_vb1_*.txt;
RESULTS = D:\vb_1\results_vb1.txt;
MODEL POPULATION: anti on pos neg dis;
MODEL: anti on pos neg dis;
Step 4
DATA: FILE = D:\vb_1\rep_nieuw_vb1_list.txt;
TYPE = MONTECARLO;
VARIABLE: NAMES ARE anti pos neg dis ;
MODEL:
anti on pos (b1)
neg (b2)
dis (b3);
MODEL CONSTRAINT:
b1 = b3; b2 = b3 ;
SAVE: RESULTS ARE D:\vb_1\results_simulatie_H0_vb1.txt;
Step 5
DATA: FILE = D:\vb_1\rep_nieuw_vb1_list.txt;
TYPE = MONTECARLO;
VARIABLE: NAMES ARE anti pos neg dis ;
MODEL:
anti on pos (b1)
neg (b2)
dis (b3);
MODEL CONSTRAINT:
b1 < b3; b2 < b3 ;
SAVE: RESULTS ARE D:\vb_1\results_simulatie_H1_vb1.txt;

Data sets
Results are 4 data sets:
Likelihood of data under H0
Likelihood of data under H1
1000 times results of simulated
datasets generated under H0 and
evaluated under H0
1000 times results of simulated
datasets generated under H0 but
evaluated under H1
Data sets
Likelihood ratio test:
Between data H0 and H1
For each simulated data set
Count the number of simulated results
that have a larger number then the data
result
Function in R
Bootstrap procedure with
plug-in p-values
Example of informative hypotheses
Different types of hypotheses
What to do with informative hypotheses?
Why not to use classical null hypothesis testing?
Why not to use classical model selection criteria?
What should we then do?
Technical Intermezzo:
What is bootstrapping?
What is a p-value and how to obtain one?
Mplus & constraints
Bootstrap procedure with plug-in p-values
Why the alpha level needs to be calibrated.
Why the alpha level needs to be
calibrated
Frequency Properties of the p-values
Asymptotically uniform [0,1] under H0
Prob(p < α* | H0) = α
Double bootstrap procedure to
estimate α*
Calibration of plug-in p-value
Results
Hypothesis test type A
Type B

Calibration of plug-in p-value
Distribution is not uniform
5th percentile of generated plug-in p-
values has a plug-in p-value of .038
α* = .038 instead of traditional .05
Prob(p < .038 | H0) = .05
Bootstrap procedure with
plug-in p-values
Example of informative hypotheses
Different types of hypotheses
What to do with informative hypotheses?
Why not to use classical null hypothesis testing?
Why not to use classical model selection criteria?
What should we then do?
Technical Intermezzo:
What is bootstrapping?
What is a p-value and how to obtain one?
Mplus & constraints
Bootstrap procedure with plug-in p-values
Why the alpha level needs to be calibrated.
Conclusion
Contact
Inequality constraints can be tested in
SEM
Calibration is needed!
Rens van de Schoot
www.fss.uu.nl/ms/schoot
A.g.j.vandeschoot@uu.nl
Implement this procedure in the
software of Mplus