Understanding the Regression Coefficient and Transformations

The question asks how subtracting a constant value from both the independent variable (X) and the dependent variable (Y) affects the regression coefficient. Let's analyze this based on the properties of regression coefficients.

In a simple linear regression model, the estimated relationship between Y and X is typically represented as:

\(\hat{Y} = a + bX\)

Here, '\(b\)' is the regression coefficient (or slope), which measures the average change in Y associated with a one-unit increase in X. The formula for the regression coefficient \(b\) is:

\(b = \dfrac{\text{Cov}(X, Y)}{\text{Var}(X)}\)

where \(\text{Cov}(X, Y)\) is the covariance between X and Y, and \(\text{Var}(X)\) is the variance of X.

Effect of Subtracting a Constant

Let's consider the effect of subtracting a constant, say \(c = 60\), from both X and Y. Let the new variables be \(X' = X - c\) and \(Y' = Y - c\).

Effect on Variance

The variance of a variable measures the spread of the data around its mean. If we subtract a constant from every value of a variable, the mean also changes by that constant, but the spread around the new mean remains the same. Therefore, subtracting a constant does not change the variance.

\(\text{Var}(X') = \text{Var}(X - c) = \text{Var}(X)\)

Effect on Covariance

The covariance measures the extent to which two variables change together. The formula for covariance is:

\(\text{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]\)

For the new variables \(X'\) and \(Y'\):

\(\text{Cov}(X', Y') = \text{Cov}(X - c, Y - c)\)

Let \(\mu_X = E[X]\) and \(\mu_Y = E[Y]\). Then \(E[X'] = E[X - c] = E[X] - c = \mu_X - c\) and \(E[Y'] = E[Y - c] = E[Y] - c = \mu_Y - c\).

So, \(\text{Cov}(X', Y') = E[((X - c) - (\mu_X - c))((Y - c) - (\mu_Y - c))]\)

\(\text{Cov}(X', Y') = E[(X - c - \mu_X + c)(Y - c - \mu_Y + c)]\)

\(\text{Cov}(X', Y') = E[(X - \mu_X)(Y - \mu_Y)]\)

This is the same as the original covariance.

\(\text{Cov}(X', Y') = \text{Cov}(X, Y)\)

Effect on Regression Coefficient

Now let's look at the regression coefficient \(b'\) for the transformed variables \(X'\) and \(Y'\):

\(b' = \dfrac{\text{Cov}(X', Y')}{\text{Var}(X')}\)

Substituting the results from the variance and covariance analysis:

\(b' = \dfrac{\text{Cov}(X, Y)}{\text{Var}(X)}\)

This is the same as the original regression coefficient \(b\).

\(b' = b\)

Conclusion

Subtracting a constant value (like 60) from each observation of both the independent variable (X) and the dependent variable (Y) does not change the value of the regression coefficient (slope). This type of transformation is called a shift in origin. Regression coefficients (slopes) are invariant under shifts in origin for both variables.

Therefore, if a constant 60 is subtracted from each of the values of X and Y, the regression coefficient is not changed.

Revision Table: Effect of Transformations on Statistics

Additional Information on Regression Coefficients

The regression coefficient is a key measure in regression analysis. It quantifies the linear relationship between the independent and dependent variables. Understanding how transformations affect it is crucial for data analysis.

• Scale Changes: If X is scaled (e.g., \(X' = cX\)) or Y is scaled (e.g., \(Y' = dY\)), the regression coefficient will change. Specifically, if \(X' = cX\) and \(Y' = Y\), \(b_{YX'} = b_{YX}/c\). If \(X' = X\) and \(Y' = dY\), \(b_{Y'X} = d \cdot b_{YX}\). If \(X' = cX\) and \(Y' = dY\), \(b_{Y'X'} = (d/c) \cdot b_{YX}\).
• Origin Shifts: As shown, adding or subtracting a constant (shift in origin) from X and Y does not change the slope \(b\). It only affects the intercept \(a\). The new intercept \(a'\) for \(Y' = a' + b'X'\) when \(X' = X-c\) and \(Y'=Y-d\) is related to the original intercept \(a\) for \(Y = a + bX\) by \(a' = a + d - bc\).
• Units: The regression coefficient \(b\) has units of (units of Y) / (units of X).
• Relationship with Correlation: The sign of the regression coefficient is the same as the sign of the correlation coefficient. Both indicate the direction of the linear relationship.