[GenABEL-dev] ProbABEL, chi^2, Wald and log-likelihood

[Yurii Aulchenko]

In principle score, Wald, and LRT have to give similar answers in
non-extreme cases. LRT is theoretically the most superior method (if
underlying model assumptions, e.g. normality, hold).  Score / Wald are the
approximations to LRT derived at the point of null/alternative,
respectively. They actually ARE derived from quadratic approximations of
the likleihood function derived at these points :)

As for practical advantages/disadvantages of these, may be someone else
could comment. I remember there are good/bad sides in both...

Re: Wald on 2df - you can not add Walds from individual beta/se, you need
to take the covariance into account. For full treatment of the problem, see

http://www.math.chalmers.se/~wermuth/pdfs/86-95/CoxWer90_An_approximation_to_ML.pdf

For a simple variant, I think our ProbABEL paper does give some details on
score/Wald.

Would that be good idea to put this discussion topic to our "Journal club"?
- these are kind of topics of general interest irrespective of GenABEL.

best,
Yurii

On Thu, Jul 11, 2013 at 11:56 PM, L.C. Karssen <lennart at karssen.org> wrote:

> Dear all,
>
> For the upcoming release of ProbABEL I've run into the following. In the
> past (~ v 0.1-3) the output of ProbABEL had chi^2 values when doing Cox
> regression. These were based on the likelihood ratio test:
>  2 * (loglik -loglik_null) ~ chi_1^2
> However, at some point, when having hamissing data was allowed in
> ProbABEL, we ran into the problem that the null model had to be
> recalculated for cases with missing genotype data. To do that 'simply'
> for each SNP would be time consuming, so the chi^2 values were removed
> from the output and replaced by the loglik values for the full model.
> (At least, that's how I guess it went).
>
> Now, I would like to get them back. This can be done in two ways:
> 1) calculate chi^2 as described above, with some smart way of only
> recalculating the null model when a missing value occurs (this shouldn't
> be often with today's imputed data).
> 2) simply calculate the chi^2 value through the Wald test. We have betas
> and se_betas, so that is easy.
>
> Many of you have more knowledge about statistics than I do, so,
> statistically, are these methods equivalent? Or is one better (more
> precise/unbiased) than the other?
>
>
> Another question:
> While testing the Wald-type implementation I ran into the following:
> I would assume that for the 2df models (where we get beta_SNP_A1A2 and
> beta_SNP_A1A1) the final chi^2 value would be the sum of the individual
> Wald statistics, which would be distributed as chi_2^2 (so 2 df). Is
> that correct? I ask this because if I compare them with the chi^2 values
> from the LRT I get different values. In the example data set I get:
> name      chi^2_Wald        chi^2_LRT
> rs7247199 0.880949           0.452465
> rs8102643 0.0116651          0.512709   <- here we have a missing value!
> rs8102615 1.51434            0.754701
> rs8105536 2.56337            1.33223
> rs2312724 0.492364           0.256649
>
> When running the additive model I do get (almost) the same results:
> name       chi^2_Wald        chi^2_LRT
> rs7247199  0.0101558          0.01012
> rs8102643  0.353168           0.492147  <- here we have a missing value!
> rs8102615  0.0181841          0.0180033
> rs8105536  0.00222781         0.00222216
> rs2312724  0.0412005          0.0401556
>
> Shouldn't the chi_2 values be equal in both cases? FYI: the LRT chi^2
> values are the same as those obtained with ProbABEL v0.1-3.
>
>
> Any suggestions?
> Thanks,
>
> Lennart.
>
> --
> -----------------------------------------------------------------
> L.C. Karssen
> Utrecht
> The Netherlands
>
> lennart at karssen.org
> http://blog.karssen.org
>
> Stuur mij aub geen Word of Powerpoint bestanden!
> Zie http://www.gnu.org/philosophy/no-word-attachments.nl.html
> ------------------------------------------------------------------
>
>
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