뒤로Chapter 9: Correlation and Regression – Study Notes
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Correlation and Regression
Introduction
This chapter introduces the concepts of correlation and regression, which are fundamental tools in statistics for analyzing the relationship between two or more variables. The focus is on understanding, calculating, and interpreting correlation coefficients, as well as distinguishing correlation from causation.
Correlation
Definition and Types of Correlation
Correlation refers to a statistical relationship between two variables. The data are typically represented as ordered pairs (x, y), where x is the independent (explanatory) variable and y is the dependent (response) variable. Correlation can be visualized using a scatter plot, which helps determine whether a linear relationship exists between the variables.
Positive Correlation: As x increases, y tends to increase.
Negative Correlation: As x increases, y tends to decrease.
No Correlation: No apparent relationship between x and y.
Nonlinear Correlation: The relationship between x and y is not linear.
Scatter Plots
A scatter plot is a graphical representation of paired data. The independent variable x is plotted on the horizontal axis, and the dependent variable y is plotted on the vertical axis. Scatter plots are used to visually assess the type and strength of the relationship between two variables.
Example: GDP and CO2 Emissions
The table below shows the gross domestic product (GDP) and carbon dioxide (CO2) emissions for 10 countries. The scatter plot of this data suggests a positive linear correlation: as GDP increases, CO2 emissions tend to increase.
GDP (in trillions of dollars), x | CO2 emissions (in millions of metric tons), y |
|---|---|
1.7 | 620.1 |
2.4 | 475.2 |
3.0 | 457.6 |
1.2 | 389.7 |
4.1 | 810.8 |
2.3 | 352.9 |
0.9 | 235.0 |
1.8 | 297.8 |
2.9 | 413.9 |
5.4 | 1216.5 |
Example: Hours of Exercise and GPA
The following table and scatter plot show the relationship between hours of exercise per week and student GPA. The plot suggests no linear correlation between the variables.
Hours of exercise, x | 12 | 3 | 0 | 6 | 10 | 2 | 18 | 14 | 15 | 5 |
|---|---|---|---|---|---|---|---|---|---|---|
GPA, y | 3.6 | 4.0 | 3.9 | 2.5 | 2.4 | 2.2 | 3.7 | 3.0 | 1.8 | 3.1 |
Example: Old Faithful Geyser Data
The table below shows the duration of eruptions and the time until the next eruption for Old Faithful. The scatter plot indicates a positive linear correlation: as the duration increases, the time until the next eruption also increases.
Duration, x | Time, y | Duration, x | Time, y |
|---|---|---|---|
1.80 | 56 | 3.78 | 79 |
1.82 | 58 | 3.83 | 85 |
1.90 | 62 | 3.88 | 80 |
1.93 | 56 | 4.10 | 89 |
1.98 | 57 | 4.27 | 90 |
2.05 | 57 | 4.30 | 89 |
2.13 | 60 | 4.43 | 80 |
2.30 | 67 | 4.47 | 86 |
2.37 | 61 | 4.53 | 89 |
2.82 | 73 | 4.55 | 86 |
3.13 | 76 | 4.60 | 92 |
3.27 | 77 | 4.63 | 91 |
3.65 | 77 |
Correlation Coefficient
Definition and Interpretation
The correlation coefficient (denoted as r for a sample and ρ for a population) measures the strength and direction of a linear relationship between two variables. The value of r ranges from –1 to 1:
r = 1: Perfect positive linear correlation
r = –1: Perfect negative linear correlation
r ≈ 0: No linear correlation
Formula for the Correlation Coefficient
The formula for the sample correlation coefficient r is:
where n is the number of data pairs, x and y are the variables, and the summations are taken over all data pairs.
Calculating the Correlation Coefficient
To calculate r:
Find the sum of the x-values and y-values.
Multiply each x-value by its corresponding y-value and sum the results.
Square each x-value and y-value, then sum the squares.
Substitute these sums into the formula above.
Using Technology to Find r
Statistical software and calculators (such as MINITAB, Excel, TI-84 Plus, and StatCrunch) can compute the correlation coefficient efficiently. For example, the Old Faithful data yields r ≈ 0.979, indicating a strong positive linear correlation.
Testing the Significance of the Correlation Coefficient
Using Critical Values from a Table
To determine if the observed correlation is statistically significant, compare the absolute value of r to a critical value from a table (e.g., Table 11 in Appendix B). If |r| is greater than the critical value, the correlation is significant at the chosen significance level (α).
Steps for Using the Table
Determine the number of data pairs (n).
Specify the significance level (α).
Find the critical value in the table for n and α.
If |r| > critical value, the correlation is significant.
Hypothesis Testing for the Population Correlation Coefficient (ρ)
Formulating Hypotheses
Hypothesis tests can be one-tailed or two-tailed, depending on the research question:
Left-tailed test: vs
Right-tailed test: vs
Two-tailed test: vs
The t-Test for the Correlation Coefficient
The t-test can be used to test the significance of r. The test statistic is:
This statistic follows a t-distribution with (n – 2) degrees of freedom.
Steps for the t-Test
State the null and alternative hypotheses.
Specify the significance level (α).
Calculate the degrees of freedom (d.f. = n – 2).
Find the critical value(s) from the t-distribution table.
Calculate the test statistic t.
Make a decision: if t is in the rejection region, reject H0.
Interpret the result in context.
Correlation vs. Causation
Distinguishing Correlation from Causation
It is important to note that a significant correlation does not imply a cause-and-effect relationship. Possible explanations for a correlation include:
Direct cause-and-effect (x causes y)
Reverse cause-and-effect (y causes x)
A third variable (lurking variable) influencing both x and y
Coincidence
Lurking variables are variables not included in the study that may affect the observed relationship.
