*Introduction to Analysis-of-Variance Procedures* |

## Linear Hypotheses

When models are expressed in the framework of
linear models, hypothesis tests are expressed
in terms of a linear function of the parameters.
For example, you may want to test that .In general, the coefficients for
linear hypotheses are some set of *L*s:

Several of these linear functions can
be combined to make one joint test.
These tests can be expressed in one matrix equation:

For each linear hypothesis, a sum of squares
(SS) due to that hypothesis can be constructed.
These sums of squares can be calculated
either as a quadratic form of the estimates

or, equivalently, as the increase in sums of squares for error
(SSE) for the model constrained by the null hypothesis

This SS is then divided by appropriate degrees of freedom and
used as a numerator of an *F* statistic.

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