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

Solve problems related to these three methodologies.

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

Next, using SPSS:

• Calculate Pearson's correlation coefficient.
• Construct 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.

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

Next, using SPSS:

• 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 anF-test for an overall comparison of means to see whether any differences are significant.

In addition, in a Microsoft Word document, provide a written interpretation of your results

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.

Table 3: Number of Doctors Verses the Rate of Prematurely Delivered Newborns

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 SPSS:

• 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.

In addition, in a Microsoft Word document, provide a written interpretation of your results.