bayesian evaluation of informative hypotheses in sem using mplus rens van de schoot...
TRANSCRIPT
Informative hypotheses
Null hypothesis testing
Difficult to evaluate specific expectations using classical null hypothesis testing:
– Not always interested in null hypothesis
– ‘accepting’ alternative hypothesis no answer
– No direct relation
– Visual inspection
– Contradictory results
Null hypothesis testing Theory
Expectations
Testing:
– H0: nothing is going on
vs.
– H1: something is going on, but we do not know what…
= catch-all hypothesis
Evaluating Informative Hypotheses Theory
Expectations
Evaluating informative hypotheses:
- Ha: theory/expectation 1
vs.
- Hb: theory/expectation 2
vs.
- Hc: theory/expectation 3
etc.
√√
Informative Hypotheses
Hypothesized order constraints between
statistical parameters
Order constraints: < > Statistical parameters: means, regression
coefficients, etc.
Why???
Direct support for your expectation
Gain in power Van de Schoot & Strohmeier, (2011), Testing
informative hypotheses in SEM Increases Power. IJBD vol. 35 no. 2 180-190
7
Default Bayes factors
Default Bayes factors
Default Bayes factors
Bayes factors for informative hypo’s
As was shown by Klugkist et al. (2005, Psych.Met.,10,
477-493), the Bayes factor (BF) of HA versus Hunc can be written as
where fi can be interpreted as a measure for model fit and ci as a measure for model complexity of Ha.
, =,
i
iuncA c
fBF
Bayes factors for informative hypo’s
Model Complexity, ci :– Can be computed before observing
any data. – Determining the number of
restrictions imposed on the means
– The more restriction, the lower ci
Bayes factors for informative hypo’s
Model fit, fi :– After observing some data, – It quantifies the amount of
agreement of the sample means with the restrictions imposed
Bayes factors for informative hypo’s
Bayesian Evaluation of Informative Hypotheses in SEM using Mplus
– Van de Schoot, Hoijtink, Hallquist, & Boelen (in press). Bayesian Evaluation of inequality-constrained Hypotheses in SEM Models using Mplus. Structural Equation Modeling
– Van de Schoot, Verhoeven & Hoijtink (under review). Bayesian Evaluation of Informative Hypotheses in SEM using Mplus: A Black Bear story.
Example: Depression
15
Data
(1) females with a high score on negative coping strategies (n = 1429),
(2) females with a low score on negative coping strategies (n = 1532),
(3) males with a high score on negative coping strategies (n = 1545),
(4) males with a low score on negative coping strategies (n = 1072),
16
Model
17
*40./*41./*47./44. *
Experienced a negative life
event
Depression Time 1
Depression Time 2
001./*04./03./08. *
*19./*18./*22./13. *
*73./*63./*71./61. *
84./83./77./78.
Expectations
“We expected that the relation between life events on Time 1 is a stronger predictor of depression on Time 2 for girls who have a negative coping strategy than for girls with a less negative coping strategy and that the same holds for boys. Moreover, we expected that this relation is stronger for girls with a negative coping style compared to boys with a negative coping style and that the same holds for girls with a less negative coping style compared to boys with a less negative copings style.”
18
Expectations
Hi1 : (β1 > β2) & (β3 > β4)
Hi2 : β1 > (β2, β3) > β4)
19
Model
20
*40./*41./*47./44. *
Experienced a negative life
event
Depression Time 1
Depression Time 2
001./*04./03./08. *
*19./*18./*22./13. *
*73./*63./*71./61. *
84./83./77./78.
Bayes Factor
21
i
i
c
fBF Hu vs.Hi
Hu vs.Hi2
Hu vs.Hi1Hi2 vs.Hi1 BF
BFBF
Step-by-step
22
we need to obtain estimates for fi and ci
Step 1. The first step is to formulate an inequality constrained hypothesis
Step 2. The second step is to compute ci. For simple order restricted hypotheses this can be done by hand.
Step-by-step
23
Count the number of parameters in the inequality constrained hypothesis – in our example: 4 (β1 β2 β3 β4)
Order these parameters in all possible ways: – in our example there are 4! = 4x3x2x1= 24
different ways of ordering four parameters.
Step-by-step
24
Count the number of possible orderings that are in line with each of the informative hypotheses:
– For Hi1 (β1 > β2) & (β3 > β4) that are 6 possibilities;
– For Hi2 β1 > (β2, β3) > β4) that are 2 possibilities;
Step-by-step
25
Divide the value obtained in step 3 by the value obtained in step 2:
– c i1 = 6/24 = 0.25
– c i2 = 2/24 = 0.0833
Note that Hi2 is the most specific hypothesis and receives the smallest value for complexity.
Step-by-step
26
Step 3. Run the model in Mplus:
Mplus syntax
DATA: FILE = data.dat;VARIABLE:NAMES ARE lif1 depr1 depr2 groups;MISSING ARE ALL (-9999);
KNOWNCLASS is g(group = 1 group = 2 group = 3 group = 4);
CLASSES is g(4);27
Mplus syntax
ANALYSIS:TYPE is mixture;
ESTIMATOR = Bayes; PROCESSOR= 32;
28
Mplus syntax
MODEL:%overall%depr2 on lif1;depr2 on depr1;lif1 with depr1;[lif1 depr1 depr2]; lif1 depr1 depr2;
29
Mplus syntax
!save the parameter estimates for each iteration:
SAVEDATA: BPARAMETERS are
c:/Bayesian_results.dat;
30
31
Using MplusAutomation
R syntax
To install MplusAutomation:
R: install.packages(c("MplusAutomation"))R: library(MplusAutomation)
Specify directory:R: setwd("c:/mplus_output")
33
R syntax
Locate output file of Mplus: R: btest <- getSavedata_Bparams("output.out")
Compute f1:
R: testBParamCompoundConstraint (btest, "( STDYX_.G.1...DEPR2.ON.LIF_1 > STDYX_.G.2...DEPR2.ON.LIF_1) & STDYX_.G.3...DEPR2.ON.LIF_1 > TDYX_.G.4...DEPR2.ON.LIF_1)")
34
R syntax
Compute f2:
R: testBParamCompoundConstraint(btest, "( STDYX_.G.1...DEPR2.ON.LIF_1 > STDYX_.G.2...DEPR2.ON.LIF_1) & (STDYX_.G.3...DEPR2.ON.LIF_1 > STDYX_.G.4...DEPR2.ON.LIF_1)& (STDYX_.G.1...DEPR2.ON.LIF_1 > STDYX_.G.3...DEPR2.ON.LIF_1)& STDYX_.G.2...DEPR2.ON.LIF_1 > STDYX_.G.4...DEPR2.ON.LIF_1)")
35
Results
fi1 = .7573
c i1 = 0.25
fi2 = .5146
c i2 = 0.0833
36
Results
BF1 vs unc = .7573 / .25 = 3.03
BF2 vs unc = .5146 / .0833 = 6.18
37
Results
BF1 vs unc = .7573 / .25 = 3.03
BF2 vs unc = .5146 / .0833 = 6.18
BF 2 vs 1 = 6.18 / 3.03 = 2.04
38
Conclusions
Excellent tool to include prior knowledge if available
Direct support for you expectations!
Gain in power