1. Transforming to Linearity

2. Transforming to Linearity
There are a number of relationships that follow something along the following form
We can now estimate a regression line through normal means

3. Transforming to Linearity

4. Lung Cancer Problem

5. Lung Cancer Problem

6. Lung Cancer Problem

8. Lung Cancer Problem Ln(1-P(x)) = 0.0154 +0.0099x
But, I want a and b; how do I get them?

9. Lung Cancer Problem Ln(1-P(x)) = 0.0154 +0.0099x
But, I want a and b; how do I get them?

10. Residual
These look gaussian but remember, this is in transform space. What do they look like in normal space?

11. Untransformed Residuals

12. Untransformed Residuals

13. Residual Comparison

14. Yarn Strength
Let’s consider the strength of a certain yarn fiber. 100 samples of yarn are collected and placed in a tensile test machine until the yarn breaks. We plot a histogram of the data.

15. Yarn Histogram
Yarn appears that it might follow a lognormal, weibull, or possibly an exponential distribution. We will assume exponential for now.

16. Transformation
If yarn strength is exponential, then We do not have the actual cdf, but we can estimate it empirically. Specifically, sort the data smallest to largest. For each value of xi, an estimate of the cdf may be given by

17. Empirical cdf .

18. Empirical cdf

19. Empirical cdf

20. Empirical cdf

21. Empirical cdf

22. Residuals
We are probably missing a term. Maybe try Weibull.

23. One Final Thing
If we regress the empirical cdf against sorted xi, we estimate a l=0.514.
If we go to the Prob/Stat web page we note that using MLE estimates gives