Scatter Diagrams and Correlation With 4.3: Coefficient of Determination

4.1

2 Variables ●In many studies, we measure more than one variable for each individual

●Some examples are

Rainfall amounts and plant growth

Exercise and cholesterol levels for a group of people

Height and weight for a group of people

●In these cases, we are interested in whether the two variables have some kind of a relationship

2 Variables ●When we have two variables, they could be related in one of several different ways

They could be unrelated

One variable (the explanatory or predictor variable) could be used to explain the other (the response or dependent variable)

One variable could be thought of as causing the other variable to change

●In this chapter, we examine the second case … explanatory and response variables

Lurking Variable •Sometimes it is not clear which variable is the explanatory variable and which is the response variable

•Sometimes the two variables are related without either one being an explanatory variable

•Sometimes the two variables are both affected by a third variable, a lurking variable, that had not been included in the study

Example of a Lurking Variable ●A researcher studies a group of elementary school children

Y = the student’s height

X = the student’s shoe size

●It is not reasonable to claim that shoe size causes height to change

●The lurking variable of age affects both of these two variables

More Examples ●Rainfall amounts and plant growth

Explanatory variable – rainfall

Response variable – plant growth

Possible lurking variable – amount of sunlight

●Exercise and cholesterol levels

Explanatory variable – amount of exercise

Response variable – cholesterol level

Possible lurking variable – diet

Scatter Diagram •The most useful graph to show the relationship between two quantitative variables is the scatter diagram

•Each individual is represented by a point in the diagram

•The explanatory (X) variable is plotted on the horizontal scale

•The response (Y) variable is plotted on the vertical scale

Scatter Diagram •An example of a scatter diagram

•Note the truncated vertical scale!

Relations ●There are several different types of relations between two variables

A relationship is linear when, plotted on a scatter diagram, the points follow the general pattern of a line

A relationship is nonlinear when, plotted on a scatter diagram, the points follow a general pattern, but it is not a line

A relationship has no correlation when, plotted on a scatter diagram, the points do not show any pattern

Positive vs. Negative •Linear relations have points that cluster around a line

•Linear relations can be either positive (the points slants upwards to the right) or negative (the points slant downwards to the right)

Positive ●For positive (linear) associations

Above average values of one variable are associated with above average values of the other (above/above, the points trend right and upwards)

Below average values of one variable are associated with below average values of the other (below/below, the points trend left and downwards)

●Examples

“Age” and “Height” for children

“Temperature” and “Sales of ice cream”

Negative ●For negative (linear) associations

Above average values of one variable are associated with below average values of the other (above/below, the points trend right and downwards)

Below average values of one variable are associated with above average values of the other (below/above, the points trend left and upwards)

●Examples

“Age” and “Time required to run 50 meters” for children

“Temperature” and “Sales of hot chocolate”

Nonlinear •Nonlinear relations have points that have a trend, but not around a line

•The trend has some bend in it

Not Related •When two variables are not related

•There is no linear trend

•There is no nonlinear trend

•Changes in values for one variable do not seem to have any relation with changes in the other

Examples ●Examples of nonlinear relations

“Age” and “Height” for people (including both children and adults)

“Temperature” and “Comfort level” for people

●Examples of no relations

“Temperature” and “Closing price of the Dow Jones Industrials Index” (probably)

“Age” and “Last digit of telephone number” for adults

Linear Correlation Coefficient •The linear correlation coefficient is a measure of the

strength of linear relation between two quantitative variables

•The sample correlation coefficient “r” is

•This should be computed with software (and not by hand) whenever possible

Linear Correlation Coefficient ●Some properties of the linear correlation coefficient

r is a unitless measure (so that r would be the same for a data set whether x and y are measured in feet, inches, meters, or fathoms)

r is always between –1 and +1

Positive values of r correspond to positive relations

Negative values of r correspond to negative relations

Linear Correlation Coefficient ●Some more properties of the linear correlation coefficient

The closer r is to +1, the stronger the positive relation … when r = +1, there is a perfect positive relation

The closer r is to –1, the stronger the negative relation … when r = –1, there is a perfect negative relation

The closer r is to 0, the less of a linear relation (either positive or negative)

Examples ●Examples of positive correlation

●In general, if the correlation is visible to the eye, then it is likely to be strong

•Examples of positive correlation

Strong Positive r = .8

Moderate Positive r = .5

Very Weak r = .1

Negative ●Examples of negative correlation

●In general, if the correlation is visible to the eye, then it is likely to be strong

Strong Negative r = –.8

Moderate Negative r = –.5

Very Weak r = –.1

●Examples of negative correlation

Nonlinear ●Nonlinear correlation

●Has an r = 0.1, but the difference is that the nonlinear relation shows a clear pattern (or lack of)

Correlation… ●Correlation is not causation!

●Just because two variables are correlated does not mean that one causes the other to change

●There is a strong correlation between shoe sizes and vocabulary sizes for grade school children

Clearly larger shoe sizes do not cause larger vocabularies

Clearly larger vocabularies do not cause larger shoe sizes

●Often lurking variables result in confounding

Coefficient of Determination •R2 – coefficient of determination, measures the proportion of total

variation in the response variable that is explained by the least-squares

regression line.

Example •Weight of Car Vs. Miles Per Gallon

•Y = -.007036x + 44.8793

•R = -964086

•R2 = 929461

93% of the variability in miles per gallon can be explained by its linear relationship with the weight.

7% of miles per gallon would be explained by other factors

Calculators •Draw a scatter diagram

Age HDL Cholesterol 38 57

42 54

46 34

32 56

55 35

52 40

61 42

61 38

26 47

38 44

66 62

30 53

51 36

27 45

52 38

49 55

39 28

AGE VS. HDL CHOLESTEROL A doctor wanted to determine

whether a relation exists between a male’s age and his HDL (so-called

good) cholesterol. He randomly selected 17 of his patients and

determined their HDL cholesterol levels. He obtained the following

data.

• New Document

• Insert Lists & Spreadsheet

• Column A (age) Column B (HDL)

• Type in Data

• Insert Data & Statistics (Ctrl I)

• Put “age” on x-axis (explanatory)

• Put “HDL” on y-axis (response)

• Observe Data (does there appear to be a relationship)

• Menu

• 6:regression

• Linear Regression

Insert Calculator Page (Ctrl I) Run Linear Regression

Menu 6: Statistics 1: Stat Calculations 3: Linear Regression X List “age” Y List “HDL” ENTER Record equation, r-value, and r2 - value