1. 1 Re-expressing Data  Chapter 6 – Normal Model –What if data do not follow a Normal model?  Chapters 8 & 9 – Linear Model –What if a relationship between two variables is not linear?

2. 2 Re-expressing Data  Re-expression is another name for changing the scale of (transforming) the data.  Usually we re-express the response variable, Y.

3. 3 Goals of Re-expression  Goal 1 – Make the distribution of the re-expressed data more symmetric.  Goal 2 – Make the spread of the re-expressed data more similar across groups.

4. 4 Goals of Re-expression  Goal 3 – Make the form of a scatter plot more linear.  Goal 4 – Make the scatter in the scatter plot more even across all values of the explanatory variable.

5. 5 Ladder of Powers  Power: 2  Re-expression:  Comment: Use on left skewed data.

6. 6 Ladder of Powers  Power: 1  Re-expression:  Comment: No re-expression. Do not re-express the data if they are already well behaved.

7. 7 Ladder of Powers  Power: ½  Re-expression:  Comment: Use on count data or when scatter in a scatter plot tends to increase as the explanatory variable increases.

8. 8 Ladder of Powers  Power: “0”  Re-expression:  Comments: Not really the “0” power. Use on right skewed data. Measurements cannot be negative or zero.

9. 9 Ladder of Powers  Power: –½, –1  Re-expression:  Comments: Use on right skewed data. Measurements cannot be negative or zero. Use on ratios.

10. 10 Goal 1 - Symmetry  Data are obtained on the time between nerve pulses along a nerve fiber.  Time is rounded to the nearest half unit where a unit is of a second. –30.5 represents

11. 11 Time ( sec)

12. 12 Time – Nerve Pulses  Distribution is skewed right.  Sample mean (12.305) is much larger than the sample median (7.5).  Many potential outliers.  Data not from a Normal model.

13. 13 Sqrt(Time)

14. 14 Log(Time)

15. 15 Summary  Time – Highly skewed to the right.  Sqrt(Time) – Still skewed right.  Log(Time) –Fairly symmetric and mounded in the middle. –Could have come from a Normal model.

16. 16 Goal 3 – Straighten Up  What is the relationship between the temperature of coffee and the time since it was poured? –Y, temperature ( o F) –X, time (minutes)

17. 17

18. 18 Cooling Coffee  There is a general negative association – as time since the coffee was poured increases the temperature of the coffee decreases.

19. 19 Linear Model

20. 20 Linear Model Fit  Summary –Predicted Temp = 176.7 – 1.56*Time –On average, temperature decreases 1.56 o F per minute. –R 2 = 0.99, 99% of the variation in temperature is explained by the linear relationship with time.

21. 21 Plot of Residuals

22. 22 Curved Pattern  There is a clear pattern in the plot of residuals versus time. –Under predict, over predict, under predict.  The linear fit is very good, but we can do better.

23. 23

24. 24 Log(Temp) by Time  Summary –Predicted Log(Temp) = 5.1946 – 0.0114*Time –On average, log temperature decreases 0.0114 log( o F) per minute.

25. 25 Plot of Residuals

26. 26 Interpretation  There is a random scatter of points around the zero line.  The linear model relating Log(Temp) to Time is the best we can do.

27. 27 Original Scale?  Predicted Log(Temp) = 5.1946 – 0.0114*Time  Predicted Temp = 180.3*e –0.0114*Time –Predicted temp at time=0, 180.3 o F –The predicted temp in one more minute is the predicted temp now multiplied by e –0.0114 = 0.98866

28. 28 JMP  Method 1 –Create a new column in JMP, Log(Temp): Cols – Formula – Transcendental – Log.

29. 29 JMP  Method 1 (continued) –Fit Y by X  Y – Log(Temp)  X – Time –Fit Linear

30. 30 JMP  Method 2 –Fit Y by X  Y – Temp  X – Time –Fit Special  Transform Y – Log