# Meta-analyzing Likehood ratio or wald test outputs

Hope your day is going well! I've been reading a lot on the Wald test and likelihood ratio test recently because it is the output of my RNA sequencing data. I was interested in meta-analysis of different datasets and was wondering if I needed to transform the LRT or Wald test statistic to a Z-score first? I prefer using LRT due to lower sample sizes.

I was reading online that it is similar to a Chi-square test statistic, so would I square it? Or can I just use scale() in R? If there are better methods, I'm also open to hearing them.

Thanks so much!

• Please say more about what types of results you are trying to put together with your meta-analysis. Results of RNAseq experiments might be expressed in many ways, for example with respect to overall differences among conditions, differences in specific genes, etc. Studies also might have differences with respect to databases against which the RNAseq data were mapped, normalization methods, etc. Both likelihood-ratio and Wald statistics are distributed as chi-square in the limit of large sample sizes; Wald tests of single parameters sometimes evaluate equivalently a square root in a Z-test. – EdM Feb 15 at 22:52

The simple answer to your question is that a sum of independent $$\chi^2$$-distributed variables is also $$\chi^2$$-distributed, with the sum having degrees of freedom equal to the sum of the degrees of freedom of the individual variables. So if you have $$\chi^2$$ statistics for a set of independent likelihood-ratio tests, then you apply that rule to get the $$\chi^2$$ statistic and the degrees of freedom for their sum.
Just what that would represent, however, is in some question. I suspect that the distribution of a sum of $$\chi^2$$ values will be the least of your problems in attempting meta-analysis of multiple RNAseq studies, as noted in my comment on the question. I found two R-based packages already designed to perform such meta-analyses, metaSeq in Bioconductor and metaRNASeq on CRAN. It's often best to take advantage of prior work like that rather than try to devise your own analysis method. Even if you choose to develop your own method you will benefit from examining those packages to see the issues that need to be addressed to deal with potential differences among studies.