Hello,
I have continuous data about CSAD (cross-sectional absolute deviation) and returns for a long period and I am using linear regression to analyze it. I have two sub-periods. I want to use Wald test to see if beta of the regression model for first sub-period is significantly different from that of the second sub-period, b1=b2, (the model is the same for both sub-periods). What are the procedures and data required to run Wald test in SPSS to test the difference of two betas for two sub-periods (all data are continuous)?
Please see this link - I hope it helps.
https://www.statisticssolutions.com/the-logistic-regression-analysis-in-spss/
Is it so that for each subject in your data you have CSADs from first (let's say p measurements) and second (q measurements) sub-periods? What is the meaning of beta in your model? Is it an intervention grouping variable so that the beta related to control-intervention difference or is it just the regression intercept (constant term)? Also note that using linear reqression does not account for correlation within subjects, which is generally the case if your design for the sub-period data is as described above. To account for this it would be better to use a mixed model (MIXED) or generalized estimating equations (GENLIN) approach.
Assuming you are interested in the interention effect of predictor x (x=0 is control and x=1 is intervention) on y (CSAD) whether there is a difference in the effects in the two periods. Below I outline one way of performing the test with the mixed model, where predictor variables can be declared as fixed factors. MIXED also permits testing specific contrasts through syntax, which is needed for the test.
1) Using Data>Restructure arrange your data in long form. If you are not familiar with this format, see e.g. this video as an example: https://www.youtube.com/watch?v=_4pnUa5nmWs
In the long form each data row contains data from one measurement time for a single subject, so that timepoint data is stacked on top of each other and not next to each other as in the wide form. For the piecewise model the long form subject-wise measurements of y for a single individual can be decomposed in the following manner:
sub-period 1:
y_1 = b01 + b1*x
y_2 = b01 + b1*x
:
y_p = b01 + b1*x
sub-period 2:
y_p+1 = b02 + b2*x
y_p+2 = b02 + b2*x
:
y_p+q = b02 + b2*x,
where b01 is the constant in sub-period 1 and b02 in sub-period 2 and b1 is the intervention-control difference in sub-period 1 and b2 is the intervention-control difference in sub-period 2.
2) To model these effects this an indicator variables is needed that identifies the sub-periods. E.g. d1=0 for sub-period 1 and d1=1 for sub-period 2.
3) Using the indicator you can specify a linear model with the additional modification that in the analysis options you should choose a model that does not include the intercept. Then the only model terms to add in are d1 (main effect) and d1*x (interaction). Note that this model assumes that the intercept and regression coefficient remain constant within each sub-period and that there may be a discontinuity in the intercept, where the sub-period changes from first to second. As SPSS uses the last category as the reference for the predictors, the model estimates four coefficients are estimated reverse order for the intervention effect:
b01 is the constant for sub-period 1 (mean of the *intervention group* in sub-period 1)
b02 is the constant for sub-period 2 (mean of the *intervention group* in sub-period 2)
b1 is the difference control - intervention in sub-period 1
b2 is the difference control - intervention in sub-period 2
Using "EM Means" in MIXED you have the procedure estimate the maringal means for the d1*x interaction, the estimated means for the groups can be computed as:
Control, sub-period 1: b01 + b1
Control, sub-period 2: b02 + b2
Intervention, sub-period 1: b01
Intervention, sub-period 2: b02
4) Finally, we can consider the test of b1 vs. b2. Since b1 = (b01 + b1) - b01 and b2 = (b02 + b2) - b02, the test can be framed as a test of period-group interaction using the TEST-subcommand in MIXED. This subcommand is available through MIXED syntax and can be specified as:
/TEST 'Contrast b2-b1' d1*x -1 1 1 -1
where I assumed use of the default settings in MIXED procedure, so that the interaction terms of the solution are arranged in the form:
[d1=0]*[x=0] (estimated)
[d1=0]*[x=1] (fixed to zero)
[d1=1]*[x=0] (estimated)
[d1=1]*[x=1] (fixed to zero)
(these can be found as row labels in the table "Estimates of fixed effects" for reference).
The test statement can be added as the last line of syntax. Example data and a model specification is included at the end of this response. The example data is already in long form.
You can find more information about piecewise constant (and other) models involving time in the highly readable and accessible text from Singer and Willett titled: "Applied Longitudinal Data Analysis: Modelling Change and Event Occurrence".
Regards,
Timo Törmäkangas
Gerontology Research Center
University of Jyväskylä
Finland
* Generate example data
INPUT PROGRAM.
LOOP #i=1 to 20.
LOOP #j=1 to 5.
COMPUTE id = #i.
COMPUTE t = #j.
if(mod(#i,2)=1) x = 1.
if(mod(#i,2)=0) x = 0.
END CASE.
END LOOP.
END LOOP.
END FILE.
END INPUT PROGRAM.
EXECUTE.
SET SEED=15022013.
COMPUTE y=rv.norm(10,2)+(t-1)*rv.norm(1,0.01).
EXECUTE.
IF (t < 4) d1 = 0.
IF (t > 3) d1 = 1.
EXECUTE.
* Run the mixed model procedure
MIXED y BY d1 x
/FIXED=d1 d1*x | NOINT SSTYPE(3) /METHOD=ML /PRINT=SOLUTION
/RANDOM=INTERCEPT | SUBJECT(id) COVTYPE(ID)
/EMMEANS=TABLES(d1*x)
/TEST 'Contrast b2-b1' d1*x -1 1 1 -1 .
Why do you want a Wald test?. Why not the Usual overall F test for two regression coefficients? If the OLS assumptions are satisfied (from your description you seem to be saying that OLS is appropriate) then the F test should be ok. I am attaching a paper that gives an example of the use of a Wald test. and a link to the F test:
https://www.google.com/search?q=F+test+in+ordinary+linear+models&oq=F+test+in+ordinary+linear+models&aqs=chrome..69i57.28115j0j8&sourceid=chrome&ie=UTF-8
Hope this helps, David Booth
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