Correlation with a Non - Linear EmphasisDay 2

Correlation measures the strength of the linear association between 2 quantitative variables.

Before you use correlation, you must check several conditions:

Quantitative Variables Condition: Are both variables quantitative?

Straight Enough Condition: Is the form of the scatterplot straight enough that a linear relationship makes sense? If the relationship is not linear, the correlation will be misleading.

Outlier Condition: Outliers can distort the correlation dramatically. If an outlier is present it is often good to report the correlation with and without that point.

A hidden variable that stands behind a relationship and determines it by simultaneously affecting the other two variables is called a lurking (confounding) variable.

Scatterplots and correlation coefficients NEVER prove causation.

CausationnCorrelatioWarning :

Don’t ever assume the relationship is linear just because the correlation coefficient is high.

In order to determine whether a relationship is linear or not linear, we must always look at the residual plot.

Residuals A residual is the vertical distance

between a data point and the graph of a regression equation.

The Residual is positive if the data point is above the

graph. negative if the data point is below the

graph. Is 0 only when the graph passes through

the data point.

What should you look for to tell if it is not linear?......Sometimes a high “r” value for

linear regression is deceptive. You must look at the scatter plot AND you must look at the residual pattern it makes.

If the residuals have a curved pattern then it is NOT linear.

To prove linearity A scatterplot of the residuals vs. the x-

values should be the most boring scatterplot you’ve ever seen.

It shouldn’t have any interesting features, like a direction or a shape.

It should stretch horizontally, with about the same amount of scatter throughout.

It should show no bends. It should show no outliers.

Some Non Linear Regression Shapes……Positive

Quadratic Regression:

Negative Quadratic Regression:

More Non Linear Regression Shapes……Positive

Exponential Regression:

Negative Exponential Regression:

Quadratic and Exponential on GDC……Quadratic: Exponential:

Example……The scatter plot could possibly be linear. You must check the residual pattern.

x y

5 16.3

10 9.7

15 8.1

20 4.2

45 1.9

25 3.4

60 1.3

Change y-list to resid after running a linear correlation regression – 2nd stat resid:

Notice the curved pattern in the residuals.

.

2

:

REGRESSION

LINEARadoingAFTERStatnd

byfoundareRESIDSNOTE

NOTE!!!!!! Just because the curved pattern on the

residuals looks like a quadratic we cannot determine that until we check the “r” value of other curved functions and see how well the data fits.

You should also consider “real-life” implications when deciding.

When you see that the residuals are curved you must check the correlation coefficient for the exponential and the quadratic to choose the stronger correlation.

A check on the exponential regression yield an r – value of -0.956. (Strong Negative but check out the quadratic….)

This is a quadratic regression…..

Equation: y=.00946x² - 0.839x+18.5

r = 0.966This value is even stronger than the exponential.

Example 2……Is it linear?

x y

0 1

-3 0.125

-4 0.0625

3 8

4 16

5 32

Look at the residuals……

There is a curved pattern in the residuals. It is NOT linear – it is either quadratic or exponential. (Positive)

Use the “r” value to help you decide.

And the Winner is…..Here is the

equation you should use for predictions:

y = 1(2)x

Homework Follow the flowchart.