Statistical test for comparing simple linear regression models with and without slope parameter

Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Univariate normal data ▷ Simple linear regression ▷ F-test for model comparison

Theorem: Consider a simple linear regression model with independent observations

$\label{eq:slr} y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2), \; i = 1,\ldots,n \; ,$

and the parameter estimates

$\label{eq:slr-est} \begin{split} \hat{\beta}_0 &= \bar{y} - \hat{\beta}_1 \bar{x} \\ \hat{\beta}_1 &= \frac{s_{xy}}{s_x^2} \\ \hat{\sigma}^2 &= \frac{1}{n-2} \sum_{i=1}^{n} (y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i)^2 \; . \end{split}$

where $\bar{x}$ and $\bar{y}$ are the sample means of the $x_i$ and $y_i$ , $s_{xy}$ is the sample covariance of the $x_i$ and $y_i$ and $s_x^2$ is the sample variance of the $x_i$ .

Then, the test statistic

$\label{eq:slr-f-comp} F = \frac{s_{xy}^2/s_x^2}{\hat{\sigma}^2/(n-1)}$

follows an F-distribution

$\label{eq:slr-f-comp-dist} F \sim \mathrm{F}(1, n-2)$

under the scenario that the data were generated using a model in which the slope parameter is zero:

$\label{eq:slr-f-comp-h0} H_0: \; \beta_1 = 0 \; .$

Proof: In multiple linear regression, the contrast-based F-test is based on the F-statistic

$\label{eq:mlr-f} F = \hat{\beta}^\mathrm{T} C \left( \hat{\sigma}^2 C^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} C \right)^{-1} C^\mathrm{T} \hat{\beta} / q$

which follows an F-distribution under the null hypothesis that the product of the contrast matrix $C \in \mathbb{R}^{p \times q}$ and the regression coefficients is a zero vector:

$\label{eq:mlr-f-dist-h0} F \sim \mathrm{F}(q, n-p), \quad \text{if} \quad C^\mathrm{T} \beta = 0_q = \left[ 0, \ldots, 0 \right]^\mathrm{T} \; .$

Since simple linear regression is a special case of multiple linear regression, we have the following quantities, if we want to compare the regression model against a model without the slope parameter:

$\label{eq:slr-mlr} \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right], \; \hat{\beta} = \left[ \begin{matrix} \hat{\beta}_0 \\ \hat{\beta}_1 \end{matrix} \right], \; C = \left[ \begin{matrix} 0 \\ 1 \end{matrix} \right], \; X = \left[ \begin{matrix} 1_n & x \end{matrix} \right], \; V = I_n \; .$

Thus, we have the null hypothesis

$\label{eq:slr-f-comp-h0-qed} H_0: \; C^\mathrm{T} \beta = \left[ \begin{matrix} 0 \\ 1 \end{matrix} \right]^\mathrm{T} \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right] = \beta_1 = 0$

and the contrast estimate

$\label{eq:slr-f-comp-CTb} C^\mathrm{T} \hat{\beta} = \left[ \begin{matrix} 0 \\ 1 \end{matrix} \right]^\mathrm{T} \left[ \begin{matrix} \hat{\beta}_0 \\ \hat{\beta}_1 \end{matrix} \right] = \hat{\beta}_1 = \frac{s_{xy}}{s_x^2} \; .$

Moreover, when deriving the distribution of ordinary least squares parameter estimates for simple linear regression with independent observations, we have identified the parameter covariance matrix as

$\label{eq:slr-XTX-inv} (X^\mathrm{T} X)^{-1} = \frac{1}{(n-1) \, s_x^2} \cdot \left[ \begin{matrix} x^\mathrm{T}x/n & -\bar{x} \\ -\bar{x} & 1 \end{matrix} \right] \; .$

Plugging $\eqref{eq:slr-mlr}$ , $\eqref{eq:slr-f-comp-CTb}$ , $\eqref{eq:slr-XTX-inv}$ and $\eqref{eq:slr-est}$ into $\eqref{eq:mlr-f}$ , the test statistic becomes

$\label{eq:slr-f-comp-qed} \begin{split} F &= \hat{\beta}^\mathrm{T} C \left( \hat{\sigma}^2 C^\mathrm{T} (X^\mathrm{T} V^{-1} X)^{-1} C \right)^{-1} C^\mathrm{T} \hat{\beta} / q \\ &= \left( \frac{s_{xy}}{s_x^2} \right) \left( \hat{\sigma}^2 \left[ \begin{matrix} 0 & 1 \end{matrix} \right] \left( \frac{1}{(n-1) \, s_x^2} \cdot \left[ \begin{matrix} x^\mathrm{T}x/n & -\bar{x} \\ -\bar{x} & 1 \end{matrix} \right] \right) \left[ \begin{matrix} 0 & 1 \end{matrix} \right]^\mathrm{T} \right)^{-1} \left( \frac{s_{xy}}{s_x^2} \right) / 1 \\ &= \frac{s_{xy}^2/(s_x^2)^2}{\hat{\sigma}^2/((n-1) \, s_x^2)} \\ &= \frac{s_{xy}^2/s_x^2}{\hat{\sigma}^2/(n-1)} \; . \end{split}$

Finally, because $C = \left[ \begin{matrix} 0 & 1 \end{matrix} \right]^\mathrm{T} \in \mathbb{R}^{2 \times 1}$ and $X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \in \mathbb{R}^{n \times 2}$ , we have $p = 2$ and $q = 1$ , such that from $\eqref{eq:mlr-f-dist-h0}$ it follows that

$\label{eq:slr-f-comp-dist-qed} \begin{split} F \sim \mathrm{F}(1, n-2), \quad \text{if} \quad \beta_1 = 0 \; . \end{split}$

∎

Sources:

original work

Metadata: ID: P453 | shortcut: slr-fcomp | author: JoramSoch | date: 2024-05-24, 13:19.