Understanding the Correlation Coefficient's Independence

The coefficient of correlation is a statistical measure that quantifies the strength and direction of a linear relationship between two variables, say X and Y. It is a value between -1 and +1.

A value near +1 indicates a strong positive linear relationship, a value near -1 indicates a strong negative linear relationship, and a value near 0 indicates a weak linear relationship or no linear relationship.

The question asks about the independence of the correlation coefficient from certain transformations applied to the data. Let's consider the effects of change of origin and change of scale.

Change of Origin and Change of Scale Explained

Let's say we have two variables, X and Y. We apply a linear transformation to these variables to get new variables, U and V:

• Change of Origin: Adding a constant value to each observation of a variable. For example, transforming X to \(U = X + b\), where b is a constant. This is like shifting the entire dataset along the axis.
• Change of Scale: Multiplying each observation of a variable by a constant value. For example, transforming X to \(U = aX\), where a is a non-zero constant. This is like stretching or compressing the scale of the data.

A combined transformation involves both:

• \(U_i = aX_i + b\)
• \(V_i = cY_i + d\)

where \(a, c\) are non-zero constants for change of scale and \(b, d\) are constants for change of origin.

Impact on the Correlation Coefficient

The coefficient of correlation (Pearson's r) between two variables X and Y is calculated using the formula:

\(\rho_{XY} = \frac{Cov(X, Y)}{\sigma_X \sigma_Y} = \frac{E[(X - E[X])(Y - E[Y])]}{\sqrt{E[(X - E[X])^2]E[(Y - E[Y])^2]}}\)

or for a sample:

\(r_{XY} = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}\)

Let's see how the terms in the formula change for the transformed variables U and V.

The means of the transformed variables are:

• \(\bar{U} = a\bar{X} + b\)
• \(\bar{V} = c\bar{Y} + d\)

Now consider the deviations from the mean for the transformed variables:

• \(U_i - \bar{U} = (aX_i + b) - (a\bar{X} + b) = aX_i + b - a\bar{X} - b = a(X_i - \bar{X})\)
• \(V_i - \bar{V} = (cY_i + d) - (c\bar{Y} + d) = cY_i + d - c\bar{Y} - d = c(Y_i - \bar{Y})\)

Notice that the change of origin constants (\(b\) and \(d\)) cancel out when we look at the deviations from the mean. This means adding a constant to a variable does not change the differences between data points or their differences from the mean. Therefore, change of origin does not affect the correlation coefficient.

The change of scale constants (\(a\) and \(c\)) appear as multiplicative factors in the deviations.

Let's substitute these into the correlation formula for \(r_{UV}\):

\(r_{UV} = \frac{\sum (U_i - \bar{U})(V_i - \bar{V})}{\sqrt{\sum (U_i - \bar{U})^2 \sum (V_i - \bar{V})^2}}\)

\(r_{UV} = \frac{\sum [a(X_i - \bar{X})][c(Y_i - \bar{Y})]}{\sqrt{\sum [a(X_i - \bar{X})]^2 \sum [c(Y_i - \bar{Y})]^2}}\)

\(r_{UV} = \frac{\sum ac(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum a^2 (X_i - \bar{X})^2 \sum c^2 (Y_i - \bar{Y})^2}}\)

\(r_{UV} = \frac{ac \sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{a^2 \sum (X_i - \bar{X})^2 \cdot c^2 \sum (Y_i - \bar{Y})^2}}\)

\(r_{UV} = \frac{ac \sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{a^2 c^2 \sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}\)

\(r_{UV} = \frac{ac \sum (X_i - \bar{X})(Y_i - \bar{Y})}{|ac| \sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}\)

We can see that the term \(\frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}\) is simply \(r_{XY}\).

So, \(r_{UV} = \frac{ac}{|ac|} r_{XY}\).

• If \(a\) and \(c\) have the same sign (both positive or both negative), then \(\frac{ac}{|ac|} = 1\), and \(r_{UV} = r_{XY}\).
• If \(a\) and \(c\) have different signs (one positive, one negative), then \(\frac{ac}{|ac|} = -1\), and \(r_{UV} = -r_{XY}\).

While a change in scale using negative multipliers on only one variable can flip the sign of the correlation, the magnitude remains the same. The essential property is that the linear relationship's strength is preserved, and the structure of the correlation coefficient formula is designed to be independent of the scale and origin of the variables.

In standard contexts, when discussing independence from change of scale and origin, it refers to the core measure being robust to these linear transformations. The property holds true for both transformations.

Conclusion on Independence

Based on the mathematical derivation, the coefficient of correlation is independent of both change of scale and change of origin. Adding or subtracting constants (change of origin) shifts the data but doesn't change the scatter pattern or the relative distance between points. Multiplying or dividing by positive constants (change of scale) stretches or shrinks the scatter plot uniformly, but the shape and thus the correlation remain the same. If negative constants are used for scaling, the direction (sign) might flip, but the strength (magnitude) of the correlation stays independent.

Therefore, the correct statement is that the coefficient of correlation is independent of both change of scale and change of origin.

Revision Table: Key Statistical Properties

Let's quickly review the properties of correlation coefficient and other statistics under transformations.

Additional Information on Correlation and Transformations

Understanding how statistical measures behave under linear transformations is crucial in data analysis. The independence of the correlation coefficient from changes of scale and origin is a powerful property. It means that the strength and direction of the linear association between two variables do not depend on the units of measurement used or where the zero point is set. For example, the correlation between height in inches and weight in pounds will be the same as the correlation between height in centimeters and weight in kilograms.

This property is why standardizing variables (transforming them to have a mean of 0 and a standard deviation of 1, which involves both change of origin and scale) does not change their correlation.

While the Pearson correlation coefficient is independent of linear transformations, it is important to remember that it only measures the linear relationship. Non-linear relationships are not fully captured by this coefficient, and transformations might be needed to linearize data before computing Pearson's r effectively.