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Comparison of the slope of regression lines

The test for checking the equality of the coefficients of linear regression equation, which come from 2 independent populations

This test is used to verify the hypothesis determining the equality of 2 coefficients of the linear regression equation $\beta_1$ and $\beta_2$ in analysed populations.

Basic assumptions:

$\beta_1$ and $\beta_2$ come from 2 samples which are chosen randomly from independent populations,
$\beta_1$ and $\beta_2$ describe the strength of dependence of the same features: $X$ and $Y$ ,
both sample sizes ( $n_1$ and $n_2$ ) are known,
standard deviations for the values of both features in both samples ( $sd_{x_1}, sd_{y_1}$ and $sd_{x_2}, sd_{y_2}$ ) are known,
the Pearson product-moment correlation coefficients of both samples ( $r_{p_1}$ and $r_{p_2}$ ) are known.

Hypotheses:

$\begin{array}{cl} \mathcal{H}_0: & \beta_1 = \beta_2, \\ \mathcal{H}_1: & \beta_1 \ne \beta_2. \end{array}$

The test statistic is defined by:

$\begin{displaymath} t=\frac{\beta_1 -\beta_2}{\sqrt{\frac{s_{yx_1}^2}{sd_{x_1}^2(n_1-1)}+\frac{s_{yx_2}^2}{sd_{x_1}^2(n_2-1)}}}, \end{displaymath}$

where:

$\displaystyle s_{yx_1}=sd_{y_1}\sqrt{\frac{n_1-1}{n_1-2}(1-r_{p_1}^2)}$ ,

$\displaystyle s_{yx_2}=sd_{y_2}\sqrt{\frac{n_2-1}{n_2-2}(1-r_{p_2}^2)}$ .

The test statistic has the t-Student distribution with $n_1+n_2-4$ degrees of freedom.

The p-value, designated on the basis of the test statistic, is compared with the significance level $\alpha$ :

$\begin{array}{ccl} $ if $ p \le \alpha & \Longrightarrow & $ reject $ \mathcal{H}_0 $ and accept $ \mathcal{H}_1, \\ $ if $ p > \alpha & \Longrightarrow & $ there is no reason to reject $ \mathcal{H}_0. \\ \end{array}$

The settings window with the comparison of correlation coefficients can be opened in Statistics menu → Parametric tests → Comparison of correlation coefficients.

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Comparison of the slope of regression lines

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