Sums of square and general linear test

[dori]

Homework Statement

Not sure this is the right forum to ask a question regarding F-statistics, but please help if you are familiar with this stuff.

The first part of homework was to prove mathematically that if a model has k variables, then F-statistic for testing model significance is:
F=[R^2/k]/[(1-R^2)/(n-k-1)]

I solved this problem by using sums of squares.
F= SSR(n-k-1)/k(SSE/SST) = SSR(n-k-1)/SSE(k) then plugged in appropriate numbers from ANOVA table.

The second part is what I'm having trouble with. In fact, I'm not sure where and how to start. It asks to prove mathematically following formula for the general linear test. In the formula, k is the number of variables in the full model and p is the number of variables in the reduce model, the equation is as follows.

Homework Equations

F= [(R^2(full) - R^2(reduced)/(k-p)]/ [(1-R^2(full))/(n-k-1)]

The Attempt at a Solution

I've tried to set up hypothesis to solve this problem, but did not get too far.
Could anybody help me with this problem? Thanks!

 

[mighty2000]

I am a scientist who is experienced in the field of statistics and I would be happy to help you with your question regarding F-statistics. Firstly, I want to assure you that this is the right forum to ask this type of question and I am familiar with F-statistics and their applications.

In order to prove the formula for the general linear test, we need to start by understanding the concept of the F-statistic and its role in model testing. The F-statistic is a measure of the overall significance of a regression model and it is calculated by comparing the variation explained by the regression model to the variation that is not explained by the model.

Now, let's break down the formula that you have provided. The numerator of the F-statistic is the difference in R-squared values between the full model and the reduced model. This represents the variation explained by the additional variables in the full model. Dividing this by the difference in the number of variables (k-p) gives us the average amount of variation explained by each additional variable.

Moving on to the denominator, (1-R^2(full)) represents the variation that is not explained by the full model. Dividing this by (n-k-1) gives us the average amount of unexplained variation per observation.

By dividing the variation explained by the additional variables by the unexplained variation per observation, we get a measure of how much better the full model is at explaining the data compared to the reduced model. This is essentially what the F-statistic is trying to capture.

To prove this mathematically, we can start by setting up the null and alternative hypotheses for the general linear test. The null hypothesis states that there is no difference in the variation explained by the full and reduced models, while the alternative hypothesis states that there is a difference.

Next, we need to calculate the test statistic, which is the F-statistic. This is done by dividing the variation explained by the additional variables by the unexplained variation per observation, just as we discussed earlier. If the test statistic is greater than the critical value, we can reject the null hypothesis and conclude that the full model is a better fit for the data than the reduced model.

I hope this explanation helps you to understand the formula for the general linear test and how it relates to the F-statistic. If you have any further questions or need clarification, please let me know. Best of luck with your homework!