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Explanatory and Response Variables

- Studying the relationship between two variables.
- Measuring both variables on the same individuals.
- a response variable measures an outcome of a study
- an explanatory variable explains or influences changes in a response variable
- sometimes there is no distinction

Chapter 4

Case study

♦ In a study to determine whether surgery

or chemotherapy results in higher survival

rates for some types of cancer.

♦ Whether or not the patient survived is one

variable, and whether they received

surgery or chemotherapy is the other variable.

♦ Which is the explanatory variable and

which is the response variable?

Chapter 4

Scatterplot

- Graphs the relationship between two quantitative (numerical) variables measured on the same individuals.
- Usually plot the explanatory variable on the horizontal (x) axis and plot the response variable on the vertical (y) axis.

Chapter 4

Relationship between mean SAT verbal score and percent of high school grads taking SAT

Chapter 4

Scatterplot

- Look for overall pattern and deviations from this pattern
- Describe pattern by form, direction, and strength of the relationship
- Look for outliers

Chapter 4

Linear Relationship

Some relationships are such that the points of a scatterplot tend to fall along a straight line -- linear relationship

Chapter 4

Direction

- Positive association

◙ A positive value for the correlation implies a positive association.

◙ large values of X tend to be associated with large values of Y and small values of X tend to be associated with small values of Y.

- Negative association

◙ A negative value for the correlation implies a negative or inverse association

◙ large values of X tend to be associated with small values of Y and vice versa.

Chapter 4

Examples

From a scatterplot of college students, there is a positive associationbetween verbal SAT score and GPA.

For used cars, there is a negative association between the age of the car and the selling price.

Chapter 4

Examples of Relationships

Chapter 4

Measuring Strength & Directionof a Linear Relationship

- How closely does a straight line (non-horizontal) fit the points of a scatterplot?
- The correlation coefficient (often referred to as just correlation): r
- measure of the strength of the relationship: the stronger the relationship, the larger the magnitude of r.
- measure of the direction of the relationship: positive r indicates a positive relationship, negative r indicates a negative relationship.

Chapter 4

Correlation Coefficient

- special values for r :
- a perfect positive linear relationship would have r = +1
- a perfect negative linear relationship would have r = -1
- if there is no linear relationship, or if the scatterplot points are best fit by a horizontal line, then r = 0
- Note: r must be between -1 and +1, inclusive
- both variables must be quantitative; no distinction between response and explanatory variables
- r has no units; does not change when measurement units are changed (ex: ft. or in.)

Chapter 4

Examples of Correlations

Chapter 4

Examples of Correlations

- Husband’s versus Wife’s ages
- r = .94
- Husband’s versus Wife’s heights
- r = .36
- Professional Golfer’s Putting Success: Distance of putt in feet versus percent success
- r = -.94

Chapter 4

Not all Relationships are Linear Miles per Gallon versus Speed

- Linear relationship?
- Correlation is close to zero.

Chapter 4

Not all Relationships are Linear Miles per Gallon versus Speed

- Curved relationship.
- Correlation is misleading.

Chapter 4

Problems with Correlations

- Outliers can inflate or deflate correlations (see next slide)
- Groups combined inappropriately may mask relationships (a third variable)
- groups may have different relationships when separated

Chapter 4

Outliers and Correlation

A

B

For each scatterplot above, how does the outlier affect the correlation?

A: outlier decreases the correlation

B: outlier increases the correlation

Chapter 4

Correlation Calculation

- Suppose we have data on variables X and Y for n individuals:

x1, x2, … , xnand y1, y2, … , yn

- Each variable has a mean and std dev:

Chapter 4

Case Study

Per Capita Gross Domestic Product

and Average Life Expectancy for

Countries in Western Europe

Chapter 4

Case Study

Chapter 4

Case Study

Chapter 4

Case Study

Chapter 4

Summary

- Explanatory variable Vs responsible variable
- Scatterplot: display the relationship between two quantitative variables measured on the same individuals
- Overall pattern of scatterplot showing:

☻ Direction ☻Form ☻ Strength

♦ Correlation coefficientr:Measures only straight-line

linear relationship

♦ R indicate the direction of a linear relationship by sign

R > 0, positive association R< 0, negative association

♦ The range of R:-1 ≤ R ≤ 1,

Chapter 4

Scatterplots

Which of the following scatterplots displays the stronger linear relationship?

- Plot A
- Plot B
- Same for both

Correlation

If two quantitative variables, X and Y, have a correlation coefficient r = 0.80, which graph could be a

scatterplot of the two variables?

- Plot A
- Plot B
- Plot C

Plot A ? Plot B? Plot C?

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