Fwd: st: New package, -swain- : Correct small sample chi-square overidentification test

[John Antonakis <John.Antonakis@unil.ch>]

Austin Nichols kindly alerted me to a typo in the description, with 
respect to the ratio of parameters to sample size problem (i.e,. when 
that ratio is large, and not small as I said in the original e-mail). 
Thus, the description, which is now more explicit should read:

swain corrects the chi-square test of fit for structural equation models 
(with or without latent
variables). The chi-square statistic is asymptotically correct; however, 
it does not behave as expected
in small samples (Kenny & McCoach, 2003) and/or when the model is 
complex (Curren, Bollen, Paxton &
Kirby, 2002). Thus, particularly in situations where the ratio of the 
number of parameters (p)
estimated to sample size (n) is relatively large (i.e., the n to p ratio 
is small), the chi-square test
will tend to overreject correctly specified models.  To obtain a closer 
approximation to the
distribution of the chi-square statistic, Swain (1975) developed a 
correction; this scaling factor,
which converges to 1 asymptotically, is multiplied with the chi-square 
statistic. The resulting
correction better approximates the chi-square distribution resulting in 
more appropriate Type 1 reject
error rates.

I have requested an update to the description and the help file.

Best,
J.

-------- Original Message --------
Subject:     st: New package, -swain- : Correct small sample chi-square 
overidentification test
Date:     Wed, 20 Mar 2013 13:11:53 +0100
From:     John Antonakis <John.Antonakis@unil.ch>
Reply-To:     statalist@hsphsun2.harvard.edu
To:     statalist@hsphsun2.harvard.edu

Hi:

With the usual thanks to Kit Baum, a new package -swain- is available on
SSC.  This package should be interesting to those who estimate
structural equation models via -sem- (using maximum likelihood). It
might also be useful to those estimating models via two or three-stage
least squares, which can be also estimated with -sem-.*

Here is a description of -swain-: Correct small sample chi-square
overidentification test after -sem-

      swain corrects the chi-square test of fit for structural
      equation models (with or without latent variables). The
      chi-square statistic is asymptotically correct; however, it does
      not behave as expected in small samples (Kenny & McCoach, 2003)
      and/or when the model is complex (Curren, Bollen, Paxton & Kirby,
      2002). Thus, particularly in situations where the ratio of the
      number of parameters estimated to sample size is relatively
      small, the chi-square test will tend to overreject correctly
      specified models. To obtain a closer approximation to the
      distribution of the chi-square statistic, Swain (1975) developed
      a correction; this scaling factor, which converges to 1
      asymptotically, is multiplied with the chi-square statistic. The
      resulting correction better approximates the chi-square
      distribution resulting in more appropriate Type 1 reject error.

To install swain, simply type -ssc install swain- from the Stata command
line.

*How to estimate instrumental variable models via -sem-.

E.g. 1

ivregress 2sls y (x = z1 z2) z3

can be estimated in -sem- as:

sem (x <- z1 z2 z3) (y <- x z3) , cov(e.x*e.y)
cov(e.x*e.y) allows cross equations disturbances of x and y to correlate
(and it the Hausman test).

E.g. 2

reg3 (y = x1 z3) (x = z1 z2 z3 x2) (x2 = z4 z3)

can be estimated in -sem- as:

sem (y <- x1 z3) (x <- z1 z2 z3 x2) (x2 <- z4 z3), cov(e._OEn, unstructured)

The cov option above allows all cross-equation disturbances of
endogenous variables correlate. Thus, the Hausman test is the Wald test:

test (_b[cov(e.y,e.x1):_cons] = 0) (_b[cov(e.y,e.x2):_cons]=0)
(_b[cov(e.x1,e.x2):_cons]=0)

Note, the Hansen-Sargan overidentification statistic in 2sls is the
chi-square test of fit in sem (reported as the "LR test of model vs.
saturated model" on the bottom of the output). In small sample size
situations where there are many parameters to be estimated, -swain- will
correct this overidentification statistic.

Another advantage of using -sem- is that there are score tests
(modification indices or Langrange Multiplier tests) available after
estimation (-estat mindices-) that will identify model constraints that
are inconsistent with the data.

Best,
J.

--
__________________________________________

John Antonakis
Professor of Organizational Behavior
Director, Ph.D. Program in Management

Faculty of Business and Economics
University of Lausanne
Internef #618
CH-1015 Lausanne-Dorigny
Switzerland
Tel ++41 (0)21 692-3438
Fax ++41 (0)21 692-3305
http://www.hec.unil.ch/people/jantonakis

Associate Editor
The Leadership Quarterly
__________________________________________


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