1. 1.7 Nonlinear Regression

2. Consider the following:
A data set for 2 variables has a linear correlation coefficient of Does this mean we can never get a strong correlation between the variables?

3. We can get a strong correlation, but not using a linear model!

4. Nonlinear Regression Not all relationships between data are linear
Use nonlinear regression to find curve of best fit
How strong is the correlation?
Correlation coefficient, r, determines linearity, so can’t use that
Use the coefficient of determination, r2
Works for all regression curves

5. Coefficient of determination, r2
Compares deviation from mean of estimated Y to deviation from mean of Y (actual data point)
Variation: differences in deviation from mean
If perfect fit, the variation in X accounts for all the variation in Y, r2 = 1
Perfect (quadratic/cubic/exponential) correlation
If poor fit, the variation in X accounts for only some (or none) of the variation in Y, then r2 is close to 0

6. Warning!
Sometimes more than one type of regression curve can provide a good fit for data (r2 close to 1)
Sometimes a bad model but r2 very close to 1
random fluctuations in the data
Sometimes the model is just silly
To be effective model, curve must be useful for extrapolating beyond the data
How can we tell if it’s a good model?

7. Residuals
Vertical distance between y-value of data point and y-value on curve of best fit
Graph these differences
If the pattern is more or less randomly scattered, then the model (linear, quadratic, exponential, etc.) is a good one
If distinct pattern, then there is probably a better model

8. To Summarize Not all relationships are linear
Coefficient of determination, r2
0 ≤ r2 ≤ 1
If r2 is close to 1, y values change because x values change
Does not mean that it’s a good model
If r2 is close to 0, y values change very little because x values change
Residuals
Can indicate whether it is not a good model

9. To the handouts!