- There's a Wald test for nested models where you can look at the effect of dropping parameters.
- Yes, you can use a Wald test in the way Mitchell mentioned. Usually, a likelihood-ratio test is the better choice.
- The Wald test has application in many areas of statistical modelling. Any time a likelihood based approach is used for estimation (e.g., logistic regression, Poisson regression, the partial likelihood of Cox's model) there are three categories of tests that can be used, 1, Wald tests, 2, likelihood ratio tests and the less common, 3, score tests. The latter test category, goes by different names sometimes, For instance the logrank test is the score test corresponding to Cox's model. The score test for logisitic or Poisson regression turns out to be quite similar to a z-test for comparing means (or t-test for large samples size).

A Wald test is distinguished by the property that it is based on the parameter estimates and their covariances (e.g., standard errors) derived using maximum likelihood or least squares (or other), whereas a likelihood ratio tests is based on the likelihood function evaluated at two points.

Wald tests and likelihood ratio tests typically yield very similar results, especially as the sample size increases. - Julie! In Generalized Linear Models context you can verify the significance of the predictor (in regression analysis) using the Wald's test and more commonly the Likelihood Ratio Test (LRT). Both test follow asymptotically a Chi-square distribution to calculate its significance. If you prefer I can send you a spreadsheet with some example of how to calculate the LRT. Best regards.
- "Wald test" is a very general method - in fact, one of the criticisms is that you can invent lots of Wald tests for a particular testing problem and decide which one you like best. I'm sure I can invent an infinite number (well, an unending sequence) of Wald tests for any hypothesis testing problem.

I do agree with a previous responder that the (for a given frequentist model) unique Likelihood Ratio Test (LRT) may be better (the LRT is a.k.a. Wilks' Test; though Wilks and Wald share a common first letter for their family name, they seem to have had rather different personalities; Wald contributed to many fields whereas Wilks was mainly a statistician; perhaps that explains the difference in their methodologies; but in defense of Wald, his tests are more generally applicable). I hope that answers your question. - Just to back this up with another vote, when some version of a likelihood ratio test is available, it seems the way to go. I wasn't aware that one could generate so many Wald tests, but it seems like they survive in certain statistical procedures where little else is available.
- Wald tests are more broadly applicable than LRTs or score tests. For the LRT to be applicable, you need to specify a distributional model for the data generating process, which then gives you the likelihood for the parameters, given the observed data. You cannot do an LRT without a likelihood! Suppose that you do not want to write down a full probability model for the data, but want to estimate parameters related to some moments of the distribution, most commonly, the mean of the distribution. Now, how do you test whether a particular term in the model is warranted or not? This is where Wald-type tests come in handy. The quadratic product, b' V b has a chi-square distribution with K-1 degrees of freedom (asymptotically), where b is the vector of parameter estimates and V is the estimated variance-covariance matrix for parameters, and K is the dimension of b (number of parameters).
- Under the null, Wald, LRT and Rao are asymptotically equivalent . However, if you specify the alternative, things will become quite complicated - but mostly, we do NHST...
- Wald test follows chi square distribution and may be used with normal distribution.It may be used when looking at one model for an estimate with less harm if the test fails.

http://www.ats.ucla.edu/stat/mult_pkg/faq/general/nested_tests.htm - Wald test can be used to test the association between the independent variables (predictors) and the criterion variable (dependent) variable. A Wald test can be used in a great variety of different models including models for dichotomous variables and models for continuous variables.
- Wald's test is better described as a distribution free method for testing the null hypothesis that two samples come from identical populations. In other words it can indeed be used to test the influence of parameters on distribution of patients between 2 groups.
- ..From an elaborate view point Wald's statistic is a test statistic with

a known probability distribution (a chi-square distribution) that is used to test whether the b coefficient for a predictor in a logistic regression model is significantly different from zero. It is analogous to the t-statistic in a regression model in that it is simply

the b coefficient divided by its standard error. The Wald statistic is inaccurate when the regression coefficient (b) is large, because the standard error tends to become

inflated, resulting in the Wald statistic being underestimated. - for complete information about Wald statistic, LM stat (also called Score test), and LR stat read the following chapter written by RF Engle in Handbook of Econometrics

http://www.stern.nyu.edu/rengle/LagrangeMultipliersHandbook_of_Econ__II___Engle.pdf

It will give complete information and is quite easy to understand.

Best Regards,

Rajesh Sharma

## All Answers (13)