Collapsing regression of nominal data to single significance value I'm working with a dataset of combined continuous and categorical data, and I'm constructing a regression model based on both types of data simultaneously, using dummy variables. The model will look something like this:
$y = \alpha + \beta_1 x_1 + \beta_2 x_2$
where, for example, $x_1 \in [0,1]$ is continuous, and $x_2 \in \textrm{{a, b, c}}$ is nominal. Because of the dummy variables, I will obtain different significance values for each of the three nominal categories. My question is, is there a simple, commonly used method for collapsing the three significances into a single significance value, taking into account both the p-value and coefficient value? I say both p-value and coefficient value because if one term has a very high significance but a very $\beta$, I imagine that should be factored in.
From some discussion with others in my group, we determined that this method (if extant) would have to be some non-linear transformation of the $\beta$ and p-value. The simplest method we came up with was simply select the minimum p-values with an optional weighting term to account for $\beta$. Are there any accepted ways of accomplishing this?
 A: The F test in classical regression is designed for just this issue; testing the significance of a group of variables.  
This situation is very common and you are right, it makes no sense to test the coefficients in front of the dummy variables in x2 one at a time.  Most stats packages will do the dummy coding of x2  automatically for you and the ANOVA table will do an F-test for the full set of x2, if it is the last variable (or group of variables) specified in your model setting.
The F test can be constructed by the sum of squares "explained" by your set of variables (additional to what the remaining variables have already explained) divided by the degrees of freedom they use up; divided by the sum of squares of the residuals divided by the degrees of freedom they have left.  You can do this to test all the variables in the model like you say; or just a subset of them.
Here is some R code with simulated data and the default ANOVA table.  See how the F statistic of 18.8 is generated from the sum of squares "explained" by the set of dummy variables.
> x1 <- rnorm(1000)
> x2<- factor(sample(c("A", "B", "C"), 1000, replace=T))
> y <- 5 + 2*x1 + 3*(x2=="B") + 4*(x2=="C") +rnorm(1000,0,10)
> model <- lm(y ~ x1 + x2)
> coef(model)
(Intercept)          x1         x2B         x2C 
      4.272       1.807       3.154       4.589 
> anova(model)
Analysis of Variance Table

Response: y
           Df Sum Sq Mean Sq F value  Pr(>F)    
x1          1   3708    3708    37.0 1.7e-09 ***
x2          2   3778    1889    18.8 9.3e-09 ***
Residuals 996  99887     100                    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
> (3778/2) /(99887/996)
[1] 18.84

It's worth noting too that this test using the F statistic is only valid if the residuals are normally distributed with constant variance.
A: You can enter x1 and the dummies representing x2 in two separate stages of a regression, in which case you'll get results for the statistical significance of each variable's F test separately.  Also, if by any chance you use SPSS you can use the "test" command to see how much of a contribution a set of dummies makes (along with that set's F and p).  I'm sure other software lets you do the same thing.
