Partial Correlation Example

In this post, we will learn about Partial Correlation and will perform on a data as Partial Correlation Example. In multiple correlations, there are more than 2 variables, (3 variables and above) also called multivariable, in partial correlation there are 3 or more variables, partial correlation is defined as the degree of the linear relationship between any two variables, in a set of multivariable data, by keeping the effect of all other variables as a constant.

Introduction to Partial Correlation Coefficient

Like Pearson’s Correlation, Partial correlation measures the strength and direction of the relationship between two variables while controlling for (or removing the influence/effect of) one or more additional variables. It helps isolate the direct association between the two variables of interest, independent of other factors.

Suppose, you are interested in studying the correlation between exercise frequency and heart health while controlling for age, partial correlation removes the effect of age to reveal the pure relationship between exercise and heart health. Partial correlation is denoted as $r_{12.3}$, where 1 and 2 are the variables of interest, and 3 is the controlled variable.

Partial Correlation Formula

For three variables say $X_1, X_2, X_3$ then the partial correlation measures the relation between $X_1$ and $X_2$ by removing the influence of $X_3$ is the partial correlation $X_1$ and $X_2$. And is given as

$$r_{12 \cdot 3}= \frac{ r_{12} – r_{13} r_{23}} {\sqrt{(1-r_{13}^2)(1- r_{23}^2)} }$$

If we want to find the partial correlation between $X_1$ and $X_3$ then

$$r_{13\cdot 2}= \frac{ r_{13} – r_{12} r_{32}}{ \sqrt{(1- r_{12}^2)(1- r_{32}^2)}}$$

If we want to find the partial correlation between $X_2$ and $X_3$ then

$$r_{23\cdot 1}= \frac{r_{23} – r_{21} r_{31}}{\sqrt{(1- r_{21}^2)(1- r_{31}^2)}}$$

Partial Correlation Graphical Representation

Partial correlation is a statistical measure of the relationship between two variables while controlling for (excluding or eliminating) the effects of one or more additional variables. For three variables, say $X, Y,$ and $Z$ is

Partial Correlation Example

Partial Correlation is used when researchers want to determine the strength and direction of the relationship between two variables without the influence of other variables. This is particularly useful in multivariate analysis where multiple variables may be interrelated. The partial correlation coefficient ranges from $-1$ to $+1$, with $-1$ indicating a perfect negative correlation, $+1$ indicating a perfect positive correlation, and 0 indicating no correlation.

Partial Correlation Example

For the Partial Correlation Example, consider the following data with some basic computation.

$X_1$$X_2$$X_3$$X_1X_2$$X_1X_3$$X_2X_3$$X_1^2$$X_2^2$$X_3^2$
741287449161
1272842414144494
148411256321966416
179515385452898125
201282401609640014464
Total7040206173321911078354110

First compute $r_{21}, r_{13}, r_{23}, r_{12}, r_{31}$, and $r_{32}$.

\begin{align}
r_{12} &= \frac{n\Sigma (x_1 x_2 ) – (\Sigma x_1)(\Sigma x_2 )} {\sqrt{\left[n\Sigma x_1 ^2 -(\Sigma x_1)^2\right] \left[n \Sigma x_2^2 – (\Sigma x_2 )^2\right]}}\\
&= \frac{5(617)-(70)(40)} {\sqrt{\left[5 (1078)-(70)^2\right]\left[5(354)-(40)^2\right]} } = 0.987\\
r_{13} &= \frac{n\Sigma(x_1 x_3 ) – (\Sigma x_1)(\Sigma x_3 )}{\sqrt{\left[n\Sigma x_1^2 – (\Sigma x_1 )^2\right]\left[n \Sigma x_3^2 – (\Sigma x_3 )^2\right]}}\\
&= \frac{5(332)-(70)(20)}{\sqrt{\left[5 (1078)-(70)^2\right]\left[5(110)-(20)^2\right]}}= 0.959\\
r_{23} &= \frac{n\Sigma(x_2 x_3 )-(\Sigma x_2 )(\Sigma x_3 )}{\sqrt{\left[n\Sigma x_2^2 -(\Sigma x_2 )^2\right]\left[n\Sigma x_3^2 -(\Sigma x_3 )^2\right]}}\\
& = \frac{5(191)-(40)(20)}{\sqrt{\left[5(354)-40^2\right]\left[5(110)-20^2\right]}}= 0.971\\
r_{12\cdot 3} &= \frac{r_{12} – r_{13} r_{23} } {\sqrt{(1 – r_{13}^2) (1 – r_{23}^2) }}\\
& = \frac{0.987-(0.959)(0.971)} {\sqrt{(1-(0.959)^2)(1-(0.971)^2)}}\\
&=\frac{0.05659}{0.0681} = 0.8305
\end{align}

Real-Life Examples of Partial Coefficient

The following are some real-life examples of partial correlation to illustrate its application in controlling for confounding variables.

  • Exercise and Health: You may want to analyze the correlation between exercise frequency and heart health while controlling for age. It is because age can affect both exercise habits and heart health, so partial correlation removes its influence to reveal the true relationship between exercise and heart health.
  • Advertising and Sales: Suppose, you want to examine the relationship between advertising spending and sales revenue while controlling for seasonality (e.g., holiday sales). It is because seasonal factors can impact both advertising and sales, so partial correlation helps determine the direct effect of advertising on sales.
  • Education and Income: You may want to study the relationship between education level and income while controlling for work experience. It is because work experience may influence both education and income, so partial correlation helps isolate the direct relationship between education and income, independent of experience.
  • Student Performance: You want to analyze the relationship between hours spent studying and exam scores while controlling for prior academic performance. Because prior academic performance may influence both study habits and exam results, partial correlation reveals the direct effect of studying on exam scores.
  • Smoking and Lung Cancer: You are interested in studying the correlation between smoking and lung cancer risk while controlling for air pollution exposure. It is because air pollution can independently affect lung cancer risk, so partial correlation isolates the impact of smoking alone.
  • Diet and Weight Loss: You want to study the correlation between calorie intake and weight loss while controlling for physical activity levels. Because, physical activity affects both calorie intake and weight loss, so partial correlation helps isolate the direct effect of diet on weight loss.

Partial correlation is commonly used in statistical analysis, especially in fields like psychology, social sciences, and any area where multivariate relationships are analyzed. In short, partial correlation provides a clearer picture of the relationship between two variables by accounting for confounding influences.

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Correlation Regression MCQs 6

The post is about a Quiz on Correlation Regression MCQs with Answers. There are 20 multiple-choice questions covering topics related to correlation and regression analysis, coefficient of determination, testing of correlation and regression coefficient, Interpretation of regression coefficients, and the method of least squares, etc. Let us start with Correlation Regression MCQs with answers.

Online Multiple-Choice Questions about Correlation and Regression Analysis with Answers

1. If the correlation coefficient $r=1.00$ then

 
 
 
 

2. The strength of the linear relationship between two numerical variables may be measured by the

 
 
 
 

3. If the coefficient of determination is 0.49, the correlation coefficient may be

 
 
 
 

4. The correlation coefficient

 
 
 
 

5. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is

 
 
 
 

6. The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by

 
 
 
 

7. Testing for the existence of correlation is equivalent to

 
 
 
 

8. If the correlation coefficient ($r=1.00$) then

 
 
 
 

9. Which one of the following statements is true?

 
 
 
 

10. If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a

 
 
 
 

11. The true correlation coefficient $\rho$ will be zero only if

 
 
 
 

12. Which one of the following situations is inconsistent?

 
 
 
 

13. What do we mean when a simple linear regression model is “statistically” useful?

 
 
 
 

14. Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$

 
 
 
 

15. In a simple linear regression problem, $r$ and $\beta_1$

 
 
 
 

16. Which of the following does the least squares method minimize?

 
 
 
 

17. The $Y$ intercept ($b_0$) represents the

 
 
 
 

18. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0:\rho =04$ against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is

 
 
 
 

19. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is

 
 
 
 

20. The slope ($b_1$) represents

 
 
 
 

Online Correlation & Regression MCQs with Answers

Online Correlation Regression MCQs

  • The $Y$ intercept ($b_0$) represents the
  • The slope ($b_1$) represents
  • Which of the following does the least squares method minimize?
  • What do we mean when a simple linear regression model is “statistically” useful?
  • If the correlation coefficient $r=1.00$ then
  • If the correlation coefficient ($r=1.00$) then
  • Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$
  • Testing for the existence of correlation is equivalent to
  • The strength of the linear relationship between two numerical variables may be measured by the
  • In a simple linear regression problem, $r$ and $\beta_1$
  • The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is
  • The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0:\rho =04$ against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is
  • If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a
  • The correlation coefficient
  • If the coefficient of determination is 0.49, the correlation coefficient may be
  • The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by
  • Which one of the following situations is inconsistent?
  • Which one of the following statements is true?
  • The true correlation coefficient $\rho$ will be zero only if
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Properties of Correlation Coefficient

The coefficient of correlation is a statistic used to measure the strength and direction of the linear relationship between two Quantitative variables.

Properties of Correlation Coefficient

Understanding these properties helps us to interpret the correlation coefficient accurately and avoid misinterpretations. The following are some important Properties of Correlation Coefficient.

  • The correlation coefficient ($r$) between $X$ and $Y$ is the same as the correlation between $Y$ and $X$. that is the correlation is symmetric with respect to $X$ and $Y$, i.e., $r_{XY} = r_{YX}$.
  • The $r$ ranges from $-1$ to $+1$, i.e., $-1\le r \le +1$.
  • There is no unit of $r$. The correlation coefficient $r$ is independent of the unit of measurement.
  • It is not affected by the change of origin and scale, i.e., $r_{XY}=r_{YX}$. If a constant is added to each value of a variable, it is called a change of origin and if each value of a variable is multiplied by a constant, it is called a change of scale.
  • The $r$ is the geometric mean of two regression coefficients, i.e., $\sqrt{b_{YX}\times b_{XY}}$.
    In other words, if the two regression lines of $Y$ on $X$ and $X$ on $Y$ are written as $Y=a+bX$ and $X=c+dy$ respectively then $bd=r^2$.
  • The sign of $r_{XY}, b_{YX}$, and $b_{XY}$ is dependent on covariance which is common in the three as given below:
  • $r=\frac{Cov(X, Y)}{\sqrt{Var(X) Var(Y)}},\,\, b_{YX} = \frac{Cov(Y, X)}{Var(X)}, \,\, b_{XY}=\frac{Cov(Y, X)}{Var(Y)}$

Hence, $r_{YX}, b_{YX}$, and $b_{XY}$ have the same sign.

  • If $r=-1$ the correlation is perfectly negative, meaning as one variable increases the other increases proportionally.
  • If $r=+1$ the correlation is perfectly positive, meaning as one variable increases the other decreases proportionally.
  • If $r=0$ there is no correlation, i.e., there is no linear relationship between the variables. However, a non-linear relationship may exist but it does not necessarily mean that the variables are independent.
Properties of Correlation Coefficient

Examples of Correlation Coefficient

The following are some real-life examples of correlation coefficients (ranging from -1 to +1) to illustrate relationships between variables:

Positive Correlation ($r$ close to +1)

  • The relationship between study time and exam scores: As students spend more time studying, their exam scores tend to increase. A correlation coefficient of $r = 0.85$ indicates a strong positive relationship.
  • The relationship between advertising spending and sales revenue: Companies that invest more in advertising often see higher sales. A correlation coefficient of $r = 0.70$ suggests a strong positive link.

Negative Correlation ($r$ close to -1)

  • The relationship between hours spent on social media and academic performance: As students spend more time on social media, their grades may decline. A correlation coefficient of $r = -0.65$ indicates a moderate negative relationship.
  • The relationship between temperature and heating costs: As outdoor temperatures rise, heating costs tend to decrease. A correlation coefficient of $r = -0.90$ shows a strong negative correlation.

Weak or No Correlation ($r$ close to 0)

  • The relationship between shoe size and IQ: There is no logical connection between shoe size and intelligence. A correlation coefficient of $r = 0.05$ indicates almost no correlation.
  • The relationship between rainfall and stock market performance: Rainfall has no direct impact on stock market trends. A correlation coefficient of $r = -0.10$ suggests a very weak or negligible relationship.

Real-World Applications

  • Healthcare: The correlation between exercise frequency and heart health.
  • Economics: The correlation between unemployment rates and crime rates.
  • Education: The correlation between parental income and children’s academic achievement.
  • Environment: The correlation between carbon emissions and global temperatures.

Independence of Origin and Scale

Theorem: Correlation: Independent of Origin and Scale. Show that the correlation coefficient is independent of origin and scale, i.e., $r_{XY}=r_{uv}$.

Proof: The formula for correlation coefficient is,

$$r_{XY}=\frac{\varSigma(X-\overline{X})((Y-\overline{Y})) }{\sqrt{[\varSigma(X-\overline{X})^2][\varSigma(Y-\overline{Y})^2]}}$$

\begin{align*}
\text{Let}\quad u&=\frac{X-a}{h}\\
\Rightarrow X&=a+hu \Rightarrow \overline{X}=a+h\overline{u} \\
\text{and}\quad v&=\frac{Y-b}{K}\\
\Rightarrow Y&=b+Kv \Rightarrow \overline{Y}=b+K\overline{v}\\
\text{Therefore}\\
r_{uv}&=\frac{\varSigma(u-\overline{u})((v-\overline{v})) }{\sqrt{[\varSigma(u-\overline{u})^2][\varSigma(v-\overline{v})^2]}}\\
&=\frac{\varSigma (a+hu-a-h\overline{u}) (b+Kv-b-K\overline{v})} {\sqrt{\varSigma(a+hu-a-h\overline{u})^2\varSigma(b+Kv-b-K\overline{v})^2}}\\
&=\frac{\varSigma(hu-h\overline{u})(Kv-K\overline{v})}{\sqrt{[\varSigma(hu-h\overline{u})^2][\varSigma(Kv-K\overline{v})^2]}}\\
&=\frac{hK\varSigma(u-\overline{u})(v-\overline{v})}{\sqrt{[h^2 K^2 \varSigma(u-\overline{u})^2] [\varSigma(v-\overline{v})^2]}}\\
&=\frac{hK\varSigma(u-\overline{u})(v-\overline{v})}{hK\,\sqrt{[\varSigma(u-\overline{u})^2] [\varSigma(v-\overline{v})^2]}}\\
&=\frac{\varSigma(u-\overline{u})(v-\overline{v}) }{\sqrt{[\varSigma(u-\overline{u})^2][\varSigma(v-\overline{v})^2]}}=
r_{uv}
\end{align*}

Correlation Coefficient Range

Important Points about Correlation Analysis

  1. Non-causality: Correlation does not imply causation. If two variables are strongly correlated, it does not necessarily mean that changes in one variable cause changes in the other. This is because the correlation only measures the strength and direction of the linear relationship between two quantitative variables, not the underlying cause-and-effect relationship.
  2. Sensitive to Outliers: The correlation coefficient can be sensitive to outliers, as outliers can disproportionately influence the correlation calculation.
  3. Assumption of Linearity: The correlation coefficient measures the linear relationship between variables. It may not accurately capture non-linear relationships between variables.
  4. Scale Invariance: The correlation coefficient is independent of the scale of the data. That is, multiplying or dividing all the values of one or both variables by a constant will not affect the strength and direction of the correlation coefficient. This makes it useful for comparing relationships between variables measured in different units.
  5. Strength vs. Causation: A high correlation does not necessarily imply causation. It is because two variables are strongly correlated does not mean one causes the other. There might be a third unknown factor influencing both variables. Correlation analysis is a good starting point for exploring relationships, but further investigation is needed to establish causality.
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