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I'm having trouble understanding what exactly Type III test statistic does. Here is what I got from my book:

"Type III" tests test for the significance of each explanatory variable, under the assumption that all other variables entered in the model equation are present.

My questions are :

  1. What exactly does "other variable entered in the model equation are present" mean? Let's say I have a Type III test statistics for variable $x_i$, does Type III test tells us whether the coefficient in front of $x_i$ equal to zero or not?

  2. If so, then what's the difference between Type III test statistics and a Wald test? (I believe they are essentially two different things since SAS gives me two different numerical outputs) Currently I have both output for my independent variables (which are all dummy variables). I don't know which p-value to look at to decided which $x$ to drop.

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It may help you to read my answer here: How to interpret type I (sequential) ANOVA and MANOVA to get a fuller sense of the types of sums of squares.

The distinction between the different types of SS only matters if you have more than one variable. It isn't clear that you do from your description, but imagine that you have two independent / predictor variables, $X_1$ and $X_2$. In that case the type III test of $X_1$ tests whether the coefficient for $X_1$ is equal to $0$, after having accounted for the influence of $X_2$ on $Y$. The type III test for $X_2$ is the same.

Type III SS tests are equivalent to the Wald tests that come with standard output in the sense that the $p$-values will always be the same. However, type III SS tests are $F$-tests (i.e., the test statistic is distributed as $F$); they are not Wald tests. A Wald test is a statistical test where the test statistic is calculated by subtracting the null value of the parameter from the estimated parameter and dividing that difference by the standard error of the parameter estimate. Importantly, the distribution of this test statistic must be asymptotically normal (e.g., not $F$). This is, frankly, very technical, and may well be legitimately more detail that you need to be concerned with.

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  • $\begingroup$ So suppose I have a model $y=\beta_0 + \beta_1 x_1 + \beta_2 x_2$ and the Type III test on $\beta_2$ is insignificant, does that mean I should consider dropping $x_2$? $\endgroup$
    – 3x89g2
    Commented Aug 2, 2014 at 1:01
  • $\begingroup$ That's a different question, largely unrelated to this. In general, I say no you shouldn't. If you want, you could read my answer here: Algorithms for automatic model selection. $\endgroup$
    – gung - Reinstate Monica
    Commented Aug 2, 2014 at 1:46
  • $\begingroup$ I've read your answer. Actually I ran the same model on different random samples from the same dataset and many of the p-value of coefficients change drastically. Is that also due to my "stepwise" selection of variables? $\endgroup$
    – 3x89g2
    Commented Aug 2, 2014 at 3:09
  • $\begingroup$ @Misakov, probably. $\endgroup$
    – gung - Reinstate Monica
    Commented Aug 2, 2014 at 14:06

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