12.2 - Correlation

12.2 - Correlation

In this course we have been using Pearson's \(r\) as a measure of the correlation between two quantitative variables. In a sample, we use the symbol \(r\). In a population, we use the symbol \(\rho\) ("rho").

Pearson's \(r\) can easily be computed using Minitab Express. However, understanding the conceptual formula may help you to better understand the meaning of a correlation coefficient.

Pearson's \(r\): Conceptual Formula

\(r=\dfrac{\sum{z_x z_y}}{n-1}\)
where \(z_x=\dfrac{x - \overline{x}}{s_x}\) and \(z_y=\dfrac{y - \overline{y}}{s_y}\)

When we replace \(z_x\) and \(z_y\) with the \(z\) score formulas and move the \(n-1\) to a separate fraction we get the formula in your textbook: \(r=\dfrac{1}{n-1}\sum{\left(\dfrac{x-\overline x}{s_x}\right) \left( \dfrac{y-\overline y}{s_y}\right)}\)

If conducting a test by hand, a \(t\) test statistic with \(df=n-2\) is computed: \(t=\dfrac{r- \rho_{0}}{\sqrt{\dfrac{1-r^2}{n-2}}} \)

In this course you will never need to compute \(r\) or the test statistic by hand, we will always be using Minitab Express to perform these calculations. 

MinitabExpress  – Computing Pearson's r

We previously created a scatterplot of quiz averages and final exam scores and observed a linear relationship. Here, we will compute the correlation between these two variables.

  1. Open the data set:
  2. On a PC: Select STATISTICS > Correlation > Correlation 
    On a MAC: Select Statistics > Regression > Correlation
  3. Double click the Quiz_Average and Final in the box on the left to insert them into the Variables box
  4. Click OK

This should result in the following output:

Correlation
Pearson correlation of Quiz_Average and Final = 0.608630
P-Value = <0.0001
Video Walkthrough

Select your operating system below to see a step-by-step guide for this example.

Properties of Pearson's r

  1. \(-1\leq r \leq +1\)
  2. For a positive association, \(r>0\), for a negative association \(r<0\), if there is no relationship \(r=0\)
  3. The closer \(r\) is to 0 the weaker the relationship and the closer to +1 or -1 the stronger the relationship (e.g., \(r=-.88\) is a stronger relationship than \(r=+.60\)); the sign of the correlation provides direction only
  4. Correlation is unit free; the \(x\) and \(y\) variables do NOT need to be on the same scale (e.g., it is possible to compute the correlation between height in centimeters and weight in pounds)
  5. It does not matter which variable you label as \(x\) and which you label as \(y\). The correlation between \(x\) and \(y\) is equal to the correlation between \(y\) and \(x\). 

The following table may serve as a guideline when evaluating correlation coefficients

Absolute Value of \(r\) Strength of the Relationship
0 - 0.2 Very weak
0.2 - 0.4 Weak
0.4 - 0.6 Moderate
0.6 - 0.8 Strong
0.8 - 1.0 Very strong

12.2.1 - Hypothesis Testing

12.2.1 - Hypothesis Testing

In testing the statistical significance of the relationship between two quantitative variables we will use the five step hypothesis testing procedure:

1. Check assumptions and write hypotheses

In order to use Pearson's \(r\) both variables must be quantitative and the relationship between \(x\) and \(y\) must be linear

Research Question Is the correlation in the population different from 0? Is the correlation in the population positive? Is the correlation in the population negative?
Null Hypothesis, \(H_{0}\) \(\rho=0\) \(\rho= 0\) \(\rho = 0\)
Alternative Hypothesis, \(H_{a}\) \(\rho \neq 0\) \(\rho > 0\) \(\rho< 0\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
2. Calculate the test statistic

Use Minitab Express to compute \(r\)

3. Determine the p-value

Minitab Express will give you the p-value for a two-tailed test (i.e., \(H_a: \rho \neq 0\)). If you are conducting a one-tailed test you will need to divide the p-value in the output by 2.

4. Make a decision

If \(p \leq \alpha\) reject the null hypothesis, there is evidence of a relationship in the population.

If \(p>\alpha\) fail to reject the null hypothesis, there is not evidence of a relationship in the population.

5. State a "real world" conclusion.

Based on your decision in Step 4, write a conclusion in terms of the original research question.


12.2.1.1 - Video Example: Quiz & Exam Scores

12.2.1.1 - Video Example: Quiz & Exam Scores

This example uses the EXAM.MTW dataset.


12.2.1.2 - Example: Age & Height

12.2.1.2 - Example: Age & Height

Data concerning body measurements from 507 adults retrieved from body.dat.txt for more information see body.txt. In this example we will use the variables of age (in years) and height (in centimeters).

Research question: Is there a relationship between age and height in adults?

1. Check assumptions and write hypotheses

Age (in years) and height (in centimeters) are both quantitative variables. From the scatterplot below we can see that the relationship is linear (or at least not non-linear).

Scatterplot of Height (cm) vs Age

\(H_0: \rho = 0\)
\(H_a: \rho \neq 0\)

2. Calculate the test statistic

From Minitab Express:

Correlation
Pearson correlation of Height (cm) and Age = 0.067883
P-Value = 0.1269

\(r=0.067883\)

3. Determine the p-value

\(p=.1269\)

4. Make a decision

\(p > \alpha\) therefore we fail to reject the null hypothesis.

5. State a "real world" conclusion.

There is not evidence of a relationship between age and height in the population from which this sample was drawn.


12.2.1.3 - Example: Temperature & Coffee Sales

12.2.1.3 - Example: Temperature & Coffee Sales

Data concerning sales at student-run cafe were retrieved from cafedata.xls more information about this data set available at cafedata.txt. Let's determine if there is a statistically significant relationship  between the maximum daily temperature and coffee sales.

1. Check assumptions and write hypotheses

Maximum daily temperature and coffee sales are both quantitative variables. From the scatterplot below we can see that the relationship is linear.

Scatterplot of Coffees vs Max Daily Temperature (F)

\(H_0: \rho = 0\)
\(H_a: \rho \neq 0\)

2. Calculate the test statistic
Correlation
Pearson correlation of Max Daily Temperature (F) and Coffees = -0.741302
P-Value = <0.0001

\(r=-0.741302\)

3. Determine the p-value

\(p<.0001\)

4. Make a decision

\(p \leq \alpha\) therefore we reject the null hypothesis.

5. State a "real world" conclusion.

There is evidence of a relationship between the maximum daily temperature and coffee sales in the population.


12.2.2 - Correlation Matrix

12.2.2 - Correlation Matrix

When examining correlations for more than two variables (i.e., more than one pair), correlation matrices are commonly used. In Minitab Express, if you request the correlations between three or more variables at once, your output will contain a correlation matrix with all of the possible pairwise correlations. For each pair of variables, Pearson's r will be given along with the p value. The following pages include examples of interpreting correlation matrices. 


12.2.2.1 - Video Example: Student Survey

12.2.2.1 - Video Example: Student Survey

This example uses the StudentSurvey.MTW dataset from the Lock5 textbook.


12.2.2.2 - Example: Body Correlation Matrix

12.2.2.2 - Example: Body Correlation Matrix

This correlation matrix was constructed using the body dataset. These data are from the Journal of Statistics Education data archive.

 

Six variables were used: age, weight (kg), height (cm), hip girth, abdominal girth, and wrist girth.

Cell contents grouped by Age, Weight, Height, Hip Girth, and Abdominal Girth; First row: Pearson correlation, Following row: P-Value

Cell contents grouped by Age, Weight, Height, Hip Girth, and Abdominal Girth; First row: Pearson correlation, Following row: P-Value

Correlations
  Age Weight (kg) Height (cm) Hip Girth Abdominal Girth
Weight (kg) 0.207265        
  <0.0001        
Height (cm) 0.067883 0.717301      
  0.1269 <0.0001      
Hip Girth 0.227080 0.762969 0.338584    
  <0.0001 <0.0001 <0.0001    
Abdominal Girth 0.422188 0.711816 0.313197 0.825892  
  <0.0001 <0.0001 <0.0001 <0.0001  
Wrist Girth 0.192024 0.816488 0.690834 0.458857 0.435420
  <0.0001 <0.0001 <0.0001 <0.0001 <0.0001

This correlation matrix presents 15 different correlations. For each of the 15 pairs of variables, the top box contains the Pearson's r correlation coefficient and the bottom box contains the p value. 

The correlation between age and weight is \(r=0.207265\). This correlation is statistically significant (\(p<0.0001\)). That is, there is evidence of a relationship between age and weight in the population.

The correlation between age and height is \(r=0.0678863\). This correlation is not statistically significant (\(p=0.1269\)). There is not evidence of a relationship between age and height in the population.

The correlation between weight and height is \(r=0.717301\). This correlation is statistically significant (\(p<0.0001\)). That is, there is evidence of a relationship between weight and height in the population.

And so on.


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