4.1 Scatter Diagrams and Correlation

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)

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

4.3 Coefficient of Determination R 2 – 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 = x R = R 2 = % 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 AgeHDL Cholesterol 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 r 2 - value