j_xtmixedlr.hlp

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{smcl}
{* 14mar2005}{...}
{title:Why is my LR test in {cmd:xtmixed} conservative?}

{pstd}
You have performed a likelihood-ratio test comparing two nested models as 
fitted by {helpb xtmixed}, whether it be (1) the test vs. linear regression 
presented at the bottom of {cmd:xtmixed} output, or (2) the comparison of 
two mixed models via {cmd:lrtest}.  What follows below discusses (1), 
but applies equally to (2) as well.


{title:Distribution theory for mixed model comparison tests}

{pstd}
The likelihood-ratio (LR) test presented at the bottom of {cmd:xtmixed} output
is a comparison of the fitted mixed model to plain linear regression.  It
tests whether all random-effects parameters of the mixed model (with the 
exception of the residual variance, a component of linear regression) are
simultaneously zero.

{pstd}
In the case where there is only one random-effects parameter to be tested,
this parameter, a variance component, is restricted to be greater than zero.
Since the null hypothesis is that this parameter is indeed zero, which is on
the boundary of the parameter space, the distribution of the likelihood ratio
test statistic is a 50:50 mixture of a chi2(0) (point mass at zero) and a
chi2(1) distribution.  Therefore, in the one-parameter case significance
levels can be adjusted accordingly and no warning that the test is
conservative is displayed.  See Self and Liang (1987) for the appropriate
theory, or Gutierrez et al. (2001) for a Stata-specific discussion.

{pstd}
When there are more than one random-effects parameters to be tested, however, 
the situation becomes more complicated.  For example, consider a model where
we have two random coefficients with unstructured covariance matrix

{pmore}
{space 16}{c TLC}{space 13}{c TRC}{break} 
{space 16}{c | } {it:v}_11{space 3}{it:v}_12{space 1}{c |}{break}
{space 16}{c | } {it:v}_12{space 3}{it:v}_22{space 1}{c |}{break}
{space 16}{c BLC}{space 13}{c BRC}{break}

{pstd}
Since the "random" component of the mixed model is comprised of three
parameters ({it:v}_11,{it:v}_12,{it:v}_22), on the surface it would seem that
the LR comparison test would be distributed as chi2(3).  However, there are
two complications that need to be considered.  First, the variances {it:v}_11
and {it:v_22} are restricted to be positive, and testing them against zero
presents the same boundary condition described above.  Second, constraints
such as {it:v}_11 = 0 implicitly restrict the covariance {it:v}_12 to be zero
as well, and from a technical standpoint it is unclear how many parameters
need to be restricted in order to reduce the model to linear regression.

{pstd}
It is because of these complications that appropriate and sufficiently 
general computation methods for the more-than-one-parameter case have yet to be
developed.  Theory (e.g. Stram and Lee, 1994) and empirical studies (e.g.
McLachlan and Basford, 1988) have demonstrated that, whatever the distribution
of the LR test statistic, its tail probabilities are bounded above by those of
the chi-squared distribution with degrees of freedom equal to the full number
of restricted parameters (three in the above example).

{pstd}
{cmd:xtmixed} uses this reference distribution, the chi-squared with full
degrees of freedom, in order to produce a conservative test.


{title:How do I interpret the results of this conservative test?}

{pstd}
The reported significance level for the LR test is an upper bound on the
actual significance level.  As such, rejection of the null hypothesis based on
the reported level would imply rejection based on the actual level.


{title:References}

{phang}
Gutierrez, R. G., S. L. Carter, and D. M. Drukker. 2001.
On boundary-value likelihood-ratio tests.  {it:Stata Technical Bulletin}
60:15-18. Reprinted in {it:Stata Technical Bulletin Reprints}, vol. 8, 
pp. 233-236.{p_end}

{phang}McLachlan, G. J. and K. E. Basford. 1988.  {it:Mixture Models}.
New York: Marcel Dekker.{p_end}

{phang}Self, S. G. and K.-Y. Liang. 1987. Asymptotic properties of 
maximum likelihood estimators and likelihood ratio tests under nonstandard
conditions.  {it:Journal of the American Statistical Association} 
82: 605-610.{p_end}

{phang}Stram, D. O. and J. W. Lee. 1994. Variance components testing
in the longitudinal mixed effects model. {it:Biometrics} 50: 1171-1177.{p_end}


{title:Also see}

{psee}
Manual:  {bf:[XT] xtmixed}
{p_end}