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In multiple regression, if you have just an ANOVA table, and nothing else, no specific data, how can you do a partial F test on X1, given X2 is already in the model?

So, you have the ANOVA table:

source        df   SS    MS     F
-----------------------------------
regression    2    1.44  0.72   9.72
error         3    0.22  0.07
total         5    1.66
-----------------------------------

All values are filled in. With only this information, how can you do the partial F test where:

  • F = MSR(X1|X2) / MSE(X1, X2)
  • MSR(X1|X2) = SSR(X1, X2) - SS(X2) = 1.44 - ????
  • MSE (X1, X2) = MSE = 0.07

SSR(X1, X2) can be obtained from the table (SS regression) MSE(X1, X2) can also be obtained from the table (just MSE) but I cannot get SS(X2) from the table, as far as I know......

As far as I know, you need specific X and Y values to do this. Any other way from just the table?

user
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1 Answers1

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As @PatrickCoulombe hints, you can't conduct a partial F-test with (only) the information in that ANOVA table. Let's assume you want to conduct the partial F-test for X1, where your full model includes both X1 and X2. In that case, you would need the ANOVA table for the full model, and the ANOVA table for the reduced model, which would only include X2. The reason you can't find the number to put in place of the "????" is because that number isn't listed in the ANOVA table you have access to--you need the reduced model ANOVA table as well.

The equations you list in the question aren't quite right. Your equation for the F ratio is right, and MSE is right, but your equation for MSR is actually for SS(X1|X2). Having calculated that, you get the MS(X1|X2) by dividing by the appropriate degrees of freedom, which is the degrees of freedom for those regressors that were dropped / you are testing (in your case, I'm guessing the df for X1=1). You calculate the F by dividing the two MSs, as you list; the realized F can be assessed against the theoretical distribution for F with numerator degrees of freedom equal to the df for dropped predictors, and denominator df equal to the df(residual) in the full model.

For a fuller understanding of this topic, it might help you to read my answers here: Testing for moderation with continuous vs categorical moderators, and possibly here: how to interpret type I (sequential) ANOVA and MANOVA.

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gung - Reinstate Monica
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  • Ok! Great.. I actually do have the ANOVA tables for X1 and X2 separately, so I have 2 additional tables. How could I go about completing the partial F-test for X1, given X2 is already in the model if I have the ANOVA tables for X1 and X2 separately? – user Oct 16 '13 at 01:45
  • That's more or less in the other answers I link to, but you might have to read between the lines. But I'll edit so that it's explicit here. – gung - Reinstate Monica Oct 16 '13 at 01:47
  • I have one additional question for you. Given all 3 of these models, the X1X2 model I posted above and two separate ones for X1 and X2, how can you tell which is best, given the R^2 and MSE values for each? – user Oct 16 '13 at 02:19
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    You could use theoretical knowledge that you have of the subject matter, but to a first approximation there isn't a "best", at least that we will be able to discern. Note, eg, that the MSE *must* go down as you add variables, & R2 *must* go up. – gung - Reinstate Monica Oct 16 '13 at 02:26
  • From the values I calculated, I have: for X1: R2 = 8%, MSE=5900; for X2: R2 = 80.1%, MSE = 1880, for X1X2: R2=80.6%, MSE=1940. Which model do you feel is best? It appears to me as if it's a clear race between X2 and X1X2(combined), but which? – user Oct 16 '13 at 03:43