# Wald statistic = z-value?

I'm doing an assignment in R for my inference theory class and i'm a bit stuck here.

I have been given a summary of some model and im being asked to confirm that the z-value is equal to the wald statistic.

The wald statistic is according to our book: $\sqrt{I(\theta_{NR)}} \cdot (\theta_{NR}-\theta_0)$

where $\theta_{NR}$ is a theta value of the regression variables which was retrieved from a Newton-Raphson function and $\theta_0$ is a vector of equal length of $\theta_{NR}$ which only contains zeros and $I()$ is the function for the fisher's information matrix.

Well i thought that i could write a function that calculates the fishers matrix and then simply use it in the formula for the wald statistic but it wont produce the right results. Maybe i have interpreted the formula wrong?

any ideas anyone? the code for the fishers matrix is confirmed to be working btw! :)

Maybe you have got it already. Otherwise, check page 99 in the book (i guess we are in the same class haha), where you may find that $se(\theta_{NR})=\frac{1}{\sqrt{I(\theta_{NR})}}$. We have already calculated the standard deviation $se(\theta_{NR})$ in laboration part I. So you may just take $\frac{1}{se(\theta_{NR})}$ and multiply it by $\theta_{NR}$ to get the z-values :)

• hahahah hello to you too classmate!! :) I understand what you're telling me, but im having a hard time understanding why we would get the wrong result by directly calculating the wald statistic with the help of the I() function and then simply multiply it with the newton-raphson theta values @Varren – Janono Dec 8 '16 at 19:53
• This site tries to provide answers that will be generally useful to those who visit the site in the future, not just to the OP. Please expand this answer a bit so that it doesn't depend on a reference to pages in an unnamed and unlinked book or to some specific class exercise that others won't have access to. Writing an answer that is generally useful is an excellent way to confirm your own understanding of the underlying statistical issues. – EdM Dec 8 '16 at 20:21
• @EdM I think your suggestion conflicts with another site policy that applies to homework and self help or homework problems. The answer should provide help (hints) but not complete solutions. The OP designated this post as self-help. So I think the answer should be left alone. – Michael R. Chernick Dec 8 '16 at 23:02
• @MichaelChernick in this case the edit history shows that someone other than the OP added the self-study tag, I think after my comment was written. (Yet another site policy conflict.) But perhaps this answer should stay as is, as you suggest, given the other answer that is available. – EdM Dec 8 '16 at 23:44

A Wald statistic is not endemic to Newton Raphson estimation. It is a particular statistic formed with a parameter estimate and its standard error.

A Z-statitic is something that takes a standard normal distribution.

Wald statistics rarely take standard normal distributions, except in the case of a one sample z-test (variance known) for normally distributed data when the null hypothesis $\mu=0$ is true. The Wald stat, however, has important limit theorems that might relate the two values.

When we do inference, we compare a Wald stat with a distribution it might be expected to take. For most estimates, $W = \hat{\theta}/se(\hat{\theta})$ has an asymptotically normal 0, 1 distribution under the null hypothesis.

As stated your problem, results, and approach do not line up.

• Can you provide an example of a wald estimate where the case you illustrated is not how a wald estimate is used. I have only ever used a Wald statistic in the manner you described (i.e. ratio of parameter to its associated standard error compared to a 0,1 normal). – JWH2006 Jul 20 '18 at 18:43