Regression and Correlation Methods: Correlation, ANOVA, and Least Squares
This is another way of assessing the possible association between a normally distributed variable y and a categorical variable x. These techniques are special cases of linear regression methods. The purpose of the assignment is to demonstrate methods of regression and correlation analysis in which two different variables in the same sample are related.
The following are three important statistics, or methodologies, for using correlation and regression:
• Pearson's correlation coefficient
• ANOVA
• Least squares regression analysis
In this assignment, solve problems related to these three methodologies.
Assignment Part 1: Pearson's Correlation Coefficient
For the problem that demonstrates the Pearson's coefficient, you will use measures that represent characteristics of entire populations to describe disease in relation to some factor of interest, such as age; utilization of health services; or consumption of a particular food, medication, or other products.
To describe a pattern of mortality from coronary heart disease (CHD) in year X, hypothetical death rates from ten states were correlated with per capita cigarette sales in dollar amount per month. Death rates were highest in states with the most cigarette sales, lowest in those with the least sales, and intermediate in the remainder. Observation contributed to the formulation of the hypothesis that cigarette smoking causes fatal CHD. The correlation coefficient, denoted by r, is the descriptive measure of association in correlational studies.
Table 1: Hypothetical Analysis of Cigarette Sales and Death Rates Caused by CHD
|
State
|
Cigarette sales
|
Death rate
|
|
1
|
102
|
5
|
|
2
|
149
|
6
|
|
3
|
165
|
6
|
|
4
|
159
|
5
|
|
5
|
112
|
3
|
|
6
|
78
|
2
|
|
7
|
112
|
5
|
|
8
|
174
|
7
|
|
9
|
101
|
4
|
|
10
|
191
|
6
|
Using the Minitab statistical procedure:
• Calculate Pearson's correlation coefficient.
• Create a two-way scatter plot.
In addition to the above:
• Explain the meaning of the resulting coefficient, paying particular attention to factors that affect the interpretation of this statistic, such as the normality of each variable.
• Provide a written interpretation of your results in APA format.
Visit the media Resources: Dot Plots and Correlation and Resources: Performing Regression Analysis on lecture Correlation and Regression Methods to view an example of Pearson's correlation coefficient.
Assignment Part 2: ANOVA
Let's take hypothetical data presenting blood pressure and high fat intake (less than 3 grams of total fat per serving) or low fat intake (less than 1 gram of saturated fat) of an individual.
Table 2: Blood Pressure and Fat Intake
|
Individual
|
Blood Pressure
|
Fat Intake
|
|
1
|
135
|
1
|
|
2
|
130
|
1
|
|
3
|
135
|
1
|
|
4
|
128
|
0
|
|
5
|
121
|
0
|
|
6
|
133
|
0
|
|
7
|
145
|
1
|
|
8
|
137
|
1
|
|
9
|
148
|
1
|
|
10
|
134
|
0
|
|
11
|
150
|
0
|
|
12
|
121
|
0
|
|
13
|
117
|
1
|
|
14
|
128
|
1
|
|
15
|
121
|
0
|
|
16
|
124
|
1
|
|
17
|
132
|
0
|
|
18
|
121
|
0
|
|
19
|
120
|
0
|
|
20
|
124
|
0
|
Using the Minitab statistical procedure:
• Calculate a one-way ANOVA to test the null hypothesis that the mean of each group is the same.
• Use different variables as grouping variables (fat intake high 1; fat intake low 0) and compare the results.
• Calculate an F-test for an overall comparison of means to see whether any differences are significant.
One-Way ANOVA on lecture Correlation and Regression Methods to view an example of ANOVA.
Assignment Part 3: Least Squares
The following are hypothetical data on the number of doctors per 10,000 inhabitants and the rate of prematurely delivered newborns for different countries of the world.
|
Country
|
Doctors per 100,000
|
Early births per 100,000
|
|
1
|
3
|
92
|
|
2
|
5
|
88
|
|
3
|
5
|
85
|
|
4
|
6
|
86
|
|
5
|
7
|
89
|
|
6
|
7
|
75
|
|
7
|
7
|
70
|
|
8
|
8
|
68
|
|
9
|
8
|
69
|
|
10
|
10
|
50
|
|
11
|
12
|
45
|
|
12
|
12
|
41
|
|
13
|
15
|
38
|
|
14
|
18
|
35
|
|
15
|
19
|
30
|
|
16
|
23
|
6
|
Using the Minitab statistical procedure:
• Apply least squares analysis to fit a regression line to the data.
• Calculate an F-test and a t-test to test for the significance of the regression.
• Test for goodness of fit using R2.