Hypothesis testing and Estimability

Now assuming Normal distribution of errors one has a Normal distribution for the parameter vector $\beta$ which in this section is supposed to be the BLUE; then to test a hypothesis of the form $c\beta=0$ where $c$ is a $1 \times q$ vector, the following statistic is used:

$$t_o=\frac{c\hat{\beta}}{\sqrt{c \; var(\; \hat{\beta} \;)\; ^tc }} \sim t_{dist}(trace(P_{X\bot}))=t_{dist}(T-rank(X))$$ (19)

with the unbiased estimate¹⁷

$$\hat{\sigma}^2=\frac{ ^t\epsilon V^{-1}\epsilon}{T-rank(X)}$$ (20)

The reader can check that taking $X$ with 2 columns one of 1 everywhere and the other of 1 and -1 whether under condition B or A; $\beta=^t(\mu \quad b)$, choosing $c=(0 \quad 1)$ gives the same $t_0$ shown in the single-subject analysis section. Only $c\beta$ which are estimable can be used in the testing procedure. A linear function of the parameter $c\beta$ is said to be estimable if a linear unbiased estimate exists:

$$\mbox{if } \ell \mbox{ exists, so that } E(^t\ell y)=c\beta$$

and this is if and only if $^t\ell X=c$. This makes $c\widehat{\beta}$ a unique BLUE of $c\beta$. Contrasts are particular estimable linear functions with the additional property that $c1_q=0$, where $1_q=^t(1 1 \cdots 1)$, i.e. the sum of the entries (weights) in the contrast is zero. This definition holds for more general hypothesis testing when $C$ is a $(q-k)\times q$ matrix of $(q-k)$ independent linear estimable functions. The following statistic (with distribution under $H_0$) called the Lawley-Hotelling trace is used to test the general hypothesis $H_0$; $C\beta=0$:

$$LHt=\frac{(rank(P_{X\bot}))}{\nu}trace(HE^{-1}) \approx F_{((q-k)p,Df)}$$ (21)

$$\mbox{with } H = (^t(C\hat{\beta})[C(^t(XV^{-1}X)^-{\;}^tC]^-C\hat{\beta}$$

$$E = ^t(P_{X\bot }Y)V^{-1}P_{X\bot }Y$$

This statistic ¹⁸ is in fact for Multivariate GLM, i.e. when $Y$ is a matrix $n \times p$ of $p$ variables $y_1 \; y_2 \cdots \;y_p$. In our case $p=1$ and $LHt$ reduces to the traditional $F$ ratio statistic:

$$F=\frac{(^t(C\hat{\beta})[C(^t(XV^{-1}X)^-{\;}^tC]^-C\hat{\beta}}{(q-k)\hat{\sigma}^2}\sim F_{(q-k,rank(P_{X\bot}))}$$ (22)

Writing $C\hat{\beta}=0$ as:
$C(^tXV^{-1}X)^-(^tXV^{-1}X)\hat{\beta}=C(^tXV^{-1}X)^-{\;}^tXV^{-1} \hat{\mu}$
$=C(^tXV^{-1}X)^-{\;}^tXV^{-1}P_Xy=C(^tXV^{-1}X)^-{\;}^tXV^{-1}y={\;}^tC_HV^{-1}y=0$; denoting $C_H=X(^tXV^{-1}X)^-{\;}^tC$; $C_H$ generates what is called in the appendix the conditioned model, then the numerator of the F statistic given in the appendix $\left\Vert P_{C_H}y\right\Vert ^2$ is the numerator given here. As an example the $F$ statistic can be used in a single subject analysis with a paradigm having more than two conditions (ON and OFF) but different levels for the ON conditions, like different audio stimuli. Remarks: Notice that the statistic (22) can be written:

$$\frac{(^t(C\hat{\beta})[var(C\hat{\beta)}]^-C\hat{\beta}}{(q-k)}$$

called Wald's type statistic. This form is simple and then advantageous to be used directly in any context. In the Lawley-Hotelling trace statistic, one must also note that $H={\;}^t(P_{C_H}y)V^{-1}P_{C_H}y$, sometimes called the hypothesis statistic ($E$ being the error statistic) and can be derived also using the form $(^t(C\hat{\beta})[var(C\hat{\beta)}]^-C\hat{\beta} \times \sigma^2$: with OLS estimate, $C_H=VX(^tXX)^-{\;}^tC$ generates the conditioned model. When not considering the BLUE of $\beta$ their distributions are then approximated (see next section for estimates of the degrees of freedom for the denominator).