# How do you do an F-test for R-square

#### LizR

##### New Member
Hi,

How do you do an F-test to test significance of R-square in EXCEL? I have done a linear regression and get R-square = 0.412, how can I test if this is significantly different from 0?
:yup:
Thanks

Liz

#### Englund

##### TS Contributor
I guess you could look up Wald test or likelihood ratio test.

#### noetsi

##### No cake for spunky
You can test whether the model has signficant predictive power (the F test does this for example in regression and ANOVA). I am not sure if that could be seen as a test of R squared or not. R shows how much variation is explained, whether that is "signficant" or not is a substantive decision not a statistical one.

Hiearchical regression does test (through an F test) if the R squared value increases signficantly as you add variables.

#### JesperHP

##### TS Contributor
R^2 could be defined as $$\frac{SSE}{SST} = \frac{\sum (\hat y_i - \bar y)^2}{\sum (y_i - \bar y)^2}$$. Hence R^2 measures the explained variance relative to the total variance. In computing an F-test you compare two models one restricted one undrestricted. Restricting all parameters to be equal to 0 except the constantterm is what you want to do if you want to test whether R^2 is significantly different from 0. The reason is that when you only have a constant in youre regressionmodel the constant takes on the value of the sample mean and hence the predicted value for any observation equal the mean, which means that $$R^2 = \frac{\sum (\hat y_i - \bar y)^2}{\sum (y_i - \bar y)^2} = 0$$ because $$\hat y_i = \bar y$$ that is predicted value is equal to mean.

To compute the F-statistic in this case us formula:

$$F = \frac{R^2/k}{(1-R^2)/(n-k-1)}$$ where k is number of restricted parameters, n number of observations and R^2 is from the unrestricted model that is youre value of 0.412.

Use F distribution F(k,n-k-1) to compute P-value

Last edited:

#### Dason

##### Ambassador to the humans
Note that that is equivalent to doing a 'typical' Full versus Reduced F test. And I think Jesper meant to square the individual terms in the denominator in the R^2 formula.

#### noetsi

##### No cake for spunky
My argument would be (and this is I feel a growing argument inside statistics) that there is a signficant difference between testing whether R squared is zero in the population, what test of Rsquared normally do, and whether the amount of variation explained is actually meaningful. R squared could be be .001 in the population (and with enough power your statistical test will capture this). Is that a meaningful explained variation? No statistical test tells you that, but it is often treated as if it does.

#### JesperHP

##### TS Contributor
yes I meant to square the inidividual terms thx for pointing out

#### LizR

##### New Member
thanks all - ok, so what gets squared in denominator??? ; (

#### JesperHP

##### TS Contributor
I have corrected the mistyping so everything is as it should be. Anyways the error was "only" in the explanatory formulas hence the F-test statistic which is all you need to calculate is exactly as it should be and always was.