Linear Hypotheses

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 $\beta_2 - \beta_3 = 0$ .In general, the coefficients for linear hypotheses are some set of Ls:

$H_0\colon L_0 \beta_0 + L_1 \beta_1 + ... + L_k \beta_k = 0$

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

$H_0\colon {L {\beta}} = 0$

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

${SS}({L{\beta}} = 0) = ({Lb})^' (L(X^'X)^{ -} L^')^{ -1}({Lb})$

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

${SS}({L \beta} = 0) = {SSE(constrained)} - {SSE(full)}$

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

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