Which of the following statements relating to Correlation and Regression are true? (a) The Coefficient of Correlation is independent of change of origin and scale. (b) The Coefficient of Correlation between the two variables is the arithmetic average of the two Regression Coefficients. (c) The probable error of the Coefficient Correlation is 0.6745 times its standard error. (d) Coefficient of Correlation multiplied by the ratio between the standard deviations of the two variables denotes the slope of the regression line. Code:
(a), (c) and (d)
Understanding Properties of Correlation and Regression
Let's analyze each statement regarding the properties of the Coefficient of Correlation and Regression Coefficients to determine which ones are true. Understanding these concepts is crucial for quantitative analysis.
Analysis of Statement (a): Independence of Origin and Scale
The Coefficient of Correlation, often denoted by \(r\), measures the strength and direction of the linear relationship between two variables. A key property of the Coefficient of Correlation is its independence from changes in origin and scale.
- Change of Origin: If a constant value is added to or subtracted from all observations of a variable, the Coefficient of Correlation remains unchanged. For example, if we analyze the correlation between temperature in Celsius and sales, changing the origin would be like shifting to Kelvin by adding 273.15. The relationship strength doesn't change.
- Change of Scale: If all observations of a variable are multiplied or divided by a non-zero constant, the Coefficient of Correlation remains unchanged. For example, if we measure height in inches and then convert to centimeters by multiplying by 2.54, the correlation between height and weight remains the same.
This property holds because the Coefficient of Correlation is calculated using standardized values (z-scores), which are unaffected by these linear transformations.
Therefore, statement (a) is true.
Analysis of Statement (b): Correlation Coefficient as Arithmetic Average of Regression Coefficients
Let the regression coefficient of Y on X be \(b_{yx}\) and the regression coefficient of X on Y be \(b_{xy}\). The formulas relating these to the correlation coefficient \(r\), and the standard deviations \(\sigma_x\) and \(\sigma_y\) are:
- \(b_{yx} = r \frac{\sigma_y}{\sigma_x}\)
- \(b_{xy} = r \frac{\sigma_x}{\sigma_y}\)
The arithmetic average of the two regression coefficients is \(\frac{b_{yx} + b_{xy}}{2} = \frac{1}{2} \left( r \frac{\sigma_y}{\sigma_x} + r \frac{\sigma_x}{\sigma_y} \right) = \frac{r}{2} \left( \frac{\sigma_y}{\sigma_x} + \frac{\sigma_x}{\sigma_y} \right)\).
For this to be equal to \(r\), we would need \(\frac{1}{2} \left( \frac{\sigma_y}{\sigma_x} + \frac{\sigma_x}{\sigma_y} \right) = 1\). This implies \(\frac{\sigma_y}{\sigma_x} + \frac{\sigma_x}{\sigma_y} = 2\). Let \(k = \frac{\sigma_y}{\sigma_x}\). Then \(k + \frac{1}{k} = 2\), which means \(k^2 - 2k + 1 = 0\), or \((k-1)^2 = 0\). This only holds if \(k=1\), meaning \(\sigma_y = \sigma_x\).
So, the arithmetic average of the regression coefficients is equal to the Coefficient of Correlation only when the standard deviations of the two variables are equal (\(\sigma_x = \sigma_y\)). It is not true in general.
However, it is worth noting that the geometric mean of the two regression coefficients is related to the correlation coefficient: \(\sqrt{b_{yx} \times b_{xy}} = \sqrt{\left(r \frac{\sigma_y}{\sigma_x}\right) \left(r \frac{\sigma_x}{\sigma_y}\right)} = \sqrt{r^2} = |r|\). The geometric mean is equal to the absolute value of the correlation coefficient.
Therefore, statement (b) is false.
Analysis of Statement (c): Probable Error of Correlation Coefficient
In statistics, the probable error (PE) is a measure of the reliability of a statistic, indicating the range within which the true value is likely to fall. For the Coefficient of Correlation \(r\), the probable error is approximately related to its standard error (SE).
The standard error of the Coefficient of Correlation is given by \(SE_r = \frac{1-r^2}{\sqrt{n-2}}\), where \(n\) is the number of observations.
The probable error of \(r\) is approximately given by \(PE_r = 0.6745 \times SE_r\).
The constant 0.6745 is derived from the properties of the normal distribution; it represents the distance from the mean within which 50% of the values fall. This is a standard formula used in statistical inference for the correlation coefficient.
Therefore, statement (c) is true.
Analysis of Statement (d): Correlation Coefficient, Standard Deviations, and Regression Slope
Let's look again at the formulas for the regression coefficients (slopes of the regression lines):
- Slope of the regression line of Y on X is \(b_{yx} = r \frac{\sigma_y}{\sigma_x}\). This can be interpreted as the Coefficient of Correlation (\(r\)) multiplied by the ratio of the standard deviation of Y (\(\sigma_y\)) to the standard deviation of X (\(\sigma_x\)).
- Slope of the regression line of X on Y is \(b_{xy} = r \frac{\sigma_x}{\sigma_y}\). This is the Coefficient of Correlation (\(r\)) multiplied by the ratio of the standard deviation of X (\(\sigma_x\)) to the standard deviation of Y (\(\sigma_y\)).
Statement (d) says "Coefficient of Correlation multiplied by the ratio between the standard deviations of the two variables denotes the slope of the regression line". While the phrasing "ratio between the standard deviations of the two variables" is slightly general, it refers to either \(\sigma_y/\sigma_x\) or \(\sigma_x/\sigma_y\).
Multiplying the Coefficient of Correlation (\(r\)) by the ratio \(\frac{\sigma_y}{\sigma_x}\) gives the slope of the regression line of Y on X. Multiplying \(r\) by the ratio \(\frac{\sigma_x}{\sigma_y}\) gives the slope of the regression line of X on Y.
Since multiplying the Coefficient of Correlation by one of these ratios does indeed give the slope of a regression line, the statement is describing a correct method for calculating a regression slope.
Therefore, statement (d) is true.
Summary of Analysis
- Statement (a): True
- Statement (b): False
- Statement (c): True
- Statement (d): True
Based on this analysis, the true statements are (a), (c), and (d).
| Statement | Truth Value | Explanation |
|---|---|---|
| (a) Correlation Coefficient is independent of change of origin and scale. | True | Coefficient of Correlation is based on relative positions of data points, unaffected by linear shifts or scaling. |
| (b) Correlation Coefficient is arithmetic average of Regression Coefficients. | False | This is only true if standard deviations are equal (\(\sigma_x = \sigma_y\)). The geometric mean equals \(|r|\). |
| (c) Probable error of Correlation Coefficient is 0.6745 times its standard error. | True | This is a standard approximation in statistics relating Probable Error and Standard Error based on the normal distribution. |
| (d) Correlation Coefficient multiplied by standard deviation ratio gives regression slope. | True | \(b_{yx} = r \frac{\sigma_y}{\sigma_x}\) and \(b_{xy} = r \frac{\sigma_x}{\sigma_y}\). This correctly describes how to find regression slopes. |
Revision Table: Key Correlation and Regression Concepts
| Concept | Description | Formula (where applicable) |
|---|---|---|
| Coefficient of Correlation (\(r\)) | Measures strength and direction of linear relationship between two variables (-1 to +1). | \(r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}\) or \(r = b_{yx} \frac{\sigma_x}{\sigma_y} = b_{xy} \frac{\sigma_y}{\sigma_x}\) |
| Regression Coefficient \(b_{yx}\) | Slope of the regression line Y on X. Represents change in Y for a unit change in X. | \(b_{yx} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} = r \frac{\sigma_y}{\sigma_x}\) |
| Regression Coefficient \(b_{xy}\) | Slope of the regression line X on Y. Represents change in X for a unit change in Y. | \(b_{xy} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (y_i - \bar{y})^2} = r \frac{\sigma_x}{\sigma_y}\) |
| Probable Error (PE) | A measure of sampling variability, indicating the interval around a sample statistic likely containing the population parameter with 50% probability. | \(PE_r \approx 0.6745 \times SE_r\) |
| Standard Error (SE) | The standard deviation of the sampling distribution of a statistic. | \(SE_r = \frac{1-r^2}{\sqrt{n-2}}\) (for correlation coefficient) |
Additional Information on Correlation and Regression Properties
Understanding the properties of correlation and regression is fundamental in statistics and data analysis. Here are a few more key points:
- The Coefficient of Correlation \(r\) is always between -1 and +1. \(|r|=1\) indicates perfect linear correlation, and \(r=0\) indicates no linear correlation.
- The sign of the correlation coefficient (\(r\)) is the same as the sign of both regression coefficients (\(b_{yx}\) and \(b_{xy}\)). This means if variables are positively correlated, both regression lines will have positive slopes, and vice versa.
- The two regression lines (\(Y\) on \(X\) and \(X\) on \(Y\)) intersect at the mean of the variables \((\bar{x}, \bar{y})\).
- Regression analysis assumes a linear relationship, independence of errors, homoscedasticity (constant variance of errors), and normally distributed errors (especially for inference).
- Correlation measures the degree of association, while regression describes the nature of the relationship (how one variable changes with the other) and can be used for prediction. Correlation does not imply causation.
Important Questions from Correlation and Regression
Variable Y regresses with variable X with the conditions that \(\overline X = 5.50\) , \(\overline Y = 3.50\) and b = 1.50 in the linear regression model (Y = a + bX), where \(\overline Y\) and \(\overline X\) are means of the respective variables and b refers to gradient of line of Y w. r. t X. Which one of the following values of parameter 'a' of the model is correct?
Given below are two statements:
Statement l: One of the assumptions under OLS method states that the regression model is linear in the parameters, though it may or may not be linear in the variables.
Statement ll: The variance of the error, or disturbance, term is not the same regardless of the value of the explanatory variable under OLS.
In light of the above statements, choose the most appropriate answer from the options given below
If two regression coefficients are -0.8 and -0.2, then the value of coefficient of correlation is
If two regression coefficients are 0.8 and 1.2, which one of the following is the value of coefficient of correlation?
For 4 data points of two correlated variables x and y, it is given that
∑ x = 24, ∑ y = 11, ∑ x 2= 202, ∑ xy = 84, ∑ y 2= 39
Fit a least squares line to this data using x as independent variable.
