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An F-test formally tests whether a term contributes to the model.
In most modeling analyses the aim is a model that describes the relationship using as few terms as possible. It is therefore of interest to look at each term in the model to decide if the term is providing any useful information.
An analysis of variance table shows the reduction in error by including each term. An F-test for each term is a formal hypothesis test to determine if the term provides useful information to the model. The null hypothesis states that the term does not contribute to the model, against the alternative hypothesis that it does. When the p-value is small, you can reject the null hypothesis and conclude that the term does contribute to the model.
When a term is not deemed to contribute statistically to the model, you may consider removing it. However, you should be cautious of removing terms that are known to contribute by some underlying mechanism, regardless of the statistical significance of a hypothesis test, and recognize that removing a term can alter the effect of other terms.
A partial F-test is equivalent to testing if all the parameter estimates for the term are equal to zero.
A sequential F-test is often useful when fitting a polynomial regression.