# Wald vs. LR $\chi^2$ tests in SPSS generalized linear models

I'm using the generalized linear models function in SPSS with a normal distribution and identity link function. If I choose the likelihood ratio $\chi^2$ statistic, I get the same results as the Univariate GLM, which is not surprising. However if I use the SPSS's default Wald $\chi^2$, I get vastly different $\chi^2$ and p-values in the "test of model effects" table.

What is the difference between what the two stats are telling me, and how can I tell which is appropriate to use?

My dataset is a continuous response variable with three factors, one including a nested term, and a covariate.

It's hard to be definitive without knowing all the details of your model (such as sample size), but I would remark that the likelihood ratio, Wald, and score estimators are only asymptotically equivalent. That is, they agree as N $\rightarrow$ $\infty$.

The Wald estimator is generally considered to be the least reliable in terms of Type I and Type II error. In GLM applications, the likelihood ratio is to be preferred. Additionally, the Wald is not always consistent under transformations. Numerous studies abound that confirm these features, and I have even run simulations of my own using ecological data.

Regarding the limitations of the Wald test, see, for example:

Fears, Thomas R.; Benichou, Jacques; and Gail, Mitchell H. (1996); A reminder of the fallibility of the Wald statistic, The American Statistician 50:226–7

However, there are situations in which the Wald might behave better. I am not an expert on these statistical approaches, but here's an example:

Yanqing Yi and Xikui Wang (2011). Comparison of Wald, Score, and Likelihood Ratio Tests for Response Adaptive Designs, Journal of Statistical Theory and Applications, 10(4): 553-569

Hope that helps,
Brenden