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What is a partial F-statistic? Is that the same as partial F-test? When would you calculate a partial F-statistic? I'm assuming that this has something to do with comparing regression models, but I'm not following something (?)

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    $\begingroup$ A statistic is not the same thing as a test, no. A z-statistic isn't a z-test, t-statistic isn't a t-test, a chi-squared statistic isn't a chi-squared test... and so a partial F statistic isn't a partial F test. However, a partial F test makes use of a partial F statistic (it's the test statistic in the test). It's partial because it doesn't test if the entire linear model is null, only some components of it. $\endgroup$ – Glen_b -Reinstate Monica Jun 15 '15 at 22:46
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The partial F-test is the most common method of testing for a nested normal linear regression model. "Nested" model is just a fancy way of saying a reduced model in terms of variables included.

For illustration, suppose that you wish to test the hypothesis that $p$ coefficients are zero, and thus these variables can be omitted from the model, and you also have $k$ coefficients in the full model(including the intercept). The test is based on the comparison of the Residual Sum of Squares(RSS) and thus you need to run two separate regressions and save the RSS from each one. For the full model the RSS will be lower since the addition of new vabiables invariably leads to a reduction of the RSS (and an increase in the Explained Sum of Squares, this is closely related to $R^2$). What we are testing therefore is whether the difference is so large that the removal of the variables will be detrimental to the model. Let's be a little more specific. The test takes the following form

$$F=\frac{\frac{RSS_{Reduced}-RSS_{Full}}{p}}{\frac{RSS_{Full}}{n-k}}$$

It can be shown that the variables in the numerator and the denominator when scaled by $\frac{1}{\sigma^2}$ are independent $\chi^2$ variables with degrees of freedom $p$ and $n-k$ respectively, hence the ratio is an F-distributed random variable with parameters $p$ and $n-k$. You reject the null hypothesis that the reduced model is appropriate if the statistic exceeds a critical value from the said distribution which in turn will happen if your model loses too much explanatory power after removing the variables.

The statistic can actually be derived from a likelihood ratio point of view and therefore has some good properties when the standard assumptions of the linear model are met, for instance constant variance, normality and so on. It is also more powerful than a series of individual tests, not to mention that it has the desired level of significance.

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