Warm Up Go ahead and start wrapping up your Guess My Age projects. Discuss methods with other members in your group. “Fine Tune” anything you need to change.

A medical researcher wishes to determine how the dosage (in mg) of a drug affects the heart rate of the patient. Dosage Heart rate 0.125 95 0.2 90 0.25 93 0.3 92 0.35 88 0.4 80 0.5 82 Find the correlation coefficient & interpret it. Find & interpret the slope. Find & interpret the y-intercept. Give the least squares regression line.              

Correlation & Regression – Non Linear Emphasis

Objective Find prediction equations for linear, quadratic, and exponential models.

Relevance To be able to find an equation to best represent quantitative data with 2 variables and use it to make predictions and to understand slope and y-intercept in context of the problem.

Non Linear

Non Linear Regression Shapes…… Positive Quadratic Regression: Negative Quadratic Regression:

Non Linear Regression Shapes…… Positive Exponential Regression: Negative Exponential Regression:

Quadratic and Exponential on Calculator……

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. Residuals – positive and negative deviations from the least squares line. Each residual is the difference between the observed y value and the corresponding predicted y value. If the residuals have a curved pattern then it is NOT linear.

Residuals Variation in the y values can be effectively explained when the residuals are small – close to the line. Remember Residual = observed – predicted  

Residual Plot  

Scatter Plot vs. Residual Plot

Height vs Shoe size – residual plot Good residual plot – shows relatively no pattern.

Good or Bad (Residual Plot Shown) Linear Not Linear

Notice A residual plot in effect turns the regression line horizontal. It magnifies the deviations of the points from the line, making it easier to see unusual observations and patterns. That is why if the regression line captures the overall pattern of the data, there should be NO pattern in the residuals!

Example……The scatter plot could possibly be linear 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

Resids: 2nd Stat Change y-list to resids after running a linear correlation regression – 2nd stat resid: Notice the curved pattern in the residuals. It is either quadratic or exponential.

This is a quadratic regression….. Equation:

Example 2……Is it linear? x y 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 – we will see that it is exponential. (Positive)

Here is the equation you should use for predictions:

Practice Assignment…… Worksheet