Understanding Regression Coefficient Independence

The question asks about the properties of the regression coefficient, specifically whether it is independent of changes in 'Scale' or 'Origin'. Let's break down what a regression coefficient represents in the context of simple linear regression.

What is a Regression Coefficient?

In a simple linear regression model relating a dependent variable $Y$ to an independent variable $X$, the model is typically written as:

$\qquad Y = \alpha + \beta X + \epsilon$

Here, $\beta$ (beta) is the regression coefficient (or slope coefficient). It represents the expected change in $Y$ for a one-unit change in $X$, assuming all other factors are held constant. Mathematically, it's related to the covariance and variance of $X$ and $Y$, or the correlation coefficient and standard deviations:

$\qquad \beta = \frac{\text{Cov}(X, Y)}{\text{Var}(X)} = r \frac{\sigma_Y}{\sigma_X}$

where $r$ is the correlation coefficient, $\sigma_Y$ is the standard deviation of $Y$, and $\sigma_X$ is the standard deviation of $X$.

Effect of Change of Origin

A change of origin involves adding or subtracting a constant value to the variables. Let's say we define new variables $X' = X + c$ and $Y' = Y + d$, where $c$ and $d$ are constants. We want to see how the regression coefficient $\beta'$ relating $Y'$ to $X'$ compares to $\beta$.

The covariance of $X'$ and $Y'$ is:

$\qquad \text{Cov}(X', Y') = \text{Cov}(X + c, Y + d)$

Properties of covariance state that adding a constant to variables does not change their covariance:

$\qquad \text{Cov}(X + c, Y + d) = \text{Cov}(X, Y)$

The variance of $X'$ is:

$\qquad \text{Var}(X') = \text{Var}(X + c)$

Properties of variance state that adding a constant to a variable does not change its variance:

$\qquad \text{Var}(X + c) = \text{Var}(X)$

So, the new regression coefficient $\beta'$ is:

$\qquad \beta' = \frac{\text{Cov}(X', Y')}{\text{Var}(X')} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)} = \beta$

This shows that the regression coefficient is independent of the change of origin for both $X$ and $Y$. Changing where the zero point is on the scales of $X$ and $Y$ does not affect the slope of the regression line.

Effect of Change of Scale

A change of scale involves multiplying variables by a constant. Let's say we define new variables $X'' = aX$ and $Y'' = bY$, where $a$ and $b$ are non-zero constants. We want to see how the regression coefficient $\beta''$ relating $Y''$ to $X''$ compares to $\beta$.

The covariance of $X''$ and $Y''$ is:

$\qquad \text{Cov}(X'', Y'') = \text{Cov}(aX, bY)$

Properties of covariance state that multiplying variables by constants results in the covariance being multiplied by the product of the constants:

$\qquad \text{Cov}(aX, bY) = ab \text{Cov}(X, Y)$

The variance of $X''$ is:

$\qquad \text{Var}(X'') = \text{Var}(aX)$

Properties of variance state that multiplying a variable by a constant results in the variance being multiplied by the square of the constant:

$\qquad \text{Var}(aX) = a^2 \text{Var}(X)$

So, the new regression coefficient $\beta''$ is:

$\qquad \beta'' = \frac{\text{Cov}(X'', Y'')}{\text{Var}(X'')} = \frac{ab \text{Cov}(X, Y)}{a^2 \text{Var}(X)} = \frac{b}{a} \frac{\text{Cov}(X, Y)}{\text{Var}(X)} = \frac{b}{a} \beta$

This shows that the new regression coefficient $\beta''$ is equal to the original coefficient $\beta$ multiplied by the ratio of the scale factors ($b/a$). Therefore, the regression coefficient is dependent on the change of scale.

Conclusion

Based on the analysis, the regression coefficient is independent of the change of origin but dependent on the change of scale.

Let's look at the options:

1. (A) Scale only: Incorrect, it depends on scale.
2. (B) Origin only: Correct, it is independent of origin.
3. (C) Both Scale and Origin: Incorrect, it depends on scale.
4. (D) Neither Scale nor Origin: Incorrect, it is independent of origin.

The regression coefficient is independent of the change of origin only.

Regression Coefficient Properties Revision Table

Additional Information on Regression Coefficient and Transformations

Understanding how transformations affect statistical measures is crucial in data analysis. For the regression coefficient, the independence from origin change means that if you shift the entire dataset on the X-axis or Y-axis (like changing units from Celsius to Kelvin, which involves adding 273.15), the slope of the relationship remains the same. The intercept ($\alpha$) will change, but the slope ($\beta$) does not.

However, independence from scale change implies that if you stretch or compress the scales of X or Y (like changing units from meters to centimeters, which involves multiplying by 100), the slope changes proportionally. If you change X from meters to centimeters (multiply by 100, $a=100$) and Y from kilograms to grams (multiply by 1000, $b=1000$), the new slope will be $(1000/100) \times \text{original slope} = 10 \times \text{original slope}$.

This property is different from the correlation coefficient ($r$), which measures the strength and direction of the linear relationship. The correlation coefficient is independent of both change of origin and change of scale (except for the sign if scales are multiplied by negative numbers).