[R-meta] Question regarding Generalized Linear Mixed-effects Model for Meta-analysis

[Viechtbauer Wolfgang (SP)]

To be precise, ln(p/(1-p)) doesn't limit the range of the response variable, it actually maps p (which is restricted to 0 to 1) to -Inf to +Inf. It is then via the back-transformation that the final estimate or predicted values become restricted to the 0 to 1 range.

As for articles/books: Just search for 'logit transformation'.

Best,
Wolfgang 

-----Original Message-----
From: Akifumi Yanagisawa [mailto:ayanagis at uwo.ca] 
Sent: Friday, 05 January, 2018 15:54
To: Viechtbauer Wolfgang (SP)
Cc: James Pustejovsky; Michael Dewey; r-sig-meta-analysis at r-project.org
Subject: Re: [R-meta] Question regarding Generalized Linear Mixed-effects Model for Meta-analysis

Thank you for you comments, Wolfgang, Michael, and James. 

Thank you very much for suggesting using ln(p/(1-p)) for response variables, Wolfgang. That’s really nice to hear that I can limit the range of response variables from 0 to 1 by using this function. I will try this approach with my data!

I would like to learn more about this approach. So, If you know any, could you let me know some of the research articles or statistics textbooks that explain how to use this approach? 

Thank you very much. 
Best regards,
Aki

On Jan 3, 2018, at 9:59 AM, Viechtbauer Wolfgang (SP) <wolfgang.viechtbauer at maastrichtuniversity.nl> wrote:
Hi James,

I tried to be clever and derived it myself. But now that I had a bit more time to think about this, I don't think it is applicable for these purposes. The equation gives an estimate of the sampling variance of p if we would repeatedly observe the performance of the same n individuals; that is, under repeated observations, their p_i values would differ, but it assumes that the underlying true probabilities stay the same across repeated observations. But the more appropriate sampling variance would be for repeated observations of n new individuals and their true probabilities would change across repeated observations. The latter type of sampling variance is indeed just estimated by s^2 / n.

So, Aki, please ignore my previous mail. Well, except that you can still analyze ln(p/(1-p)). And the sampling variance of ln(p/(1-p)) would then be estimated with v = 1/(p*(1-p))^2 * s^2 / n.

Best,
Wolfgang