The Ladder of Powers

1. The Ladder of Powers Nonlinear Scatterplots Using the Power Transformation Ladder

2. Straightening Relationships • The fundamental assumption for working with a linear model is that the relationship you are modeling is, in fact, linear. Often it is hard to tell from the scatterplot you are looking at before you fit the regression model. Sometimes you can’t see a bend in the relationship until you plot the residuals.

3. Straightening Relationships • Here again are the weights and fuel efficiency of 38 cars. What do you see?

4. Straightening Relationships • Obviously, heavier cars tend to take more fuel to move. The scatterplot shows a strong negative linear association between weight of a car and its fuel efficiency (r = 0.911). The linear regression equation is • Interpret the coefficients in this equation.

5. Straightening Relationships • Anything surprising you see in the residual plot?

6. Straightening Relationships • The scatterplot of the residuals versus the weights should have no pattern. What we see is a bend, starting high on the left, dropping down in the middle of the graph, and then rising again at the right. • Graphs of residuals can reveal patterns such as this that are hard to see in the original scatterplot.

7. Straightening Relationships • How might we try to correct this deficency in our model? • Instead of looking at miles per gallon we could do like most of the rest of the world and consider the number of gallons per mile, or more typically, the number of gallons per 100 miles.

8. Straightening Relationships • Here again are the weights and fuel efficiency measured in gallons per 100 miles of 38 cars. What do you see now?

9. Straightening Relationships • Do you see any surprises in this residual plot?

10. Letting the River Flow • As white-water rafters know, a river flows more slowing close to its banks due to friction. To study the nature of the relationship between water velocity and the distance from the bank, data were gathered on velocity (in centimeters per second) of a river at different distances (in meters) from the shore.

11. Letting the River Flow • Graph this data and describe the scatterplot. • Do a linear regression and examine the residual plot. Anything distinctive?

13. Letting the River Flow • Let’s re-express the x values by replacing each x by its square root.

14. Letting the River Flow • Graph the square root of the distance from the bank versus the velocity. Describe the scatterplot. • Do a linear regression and write the new regression equation. Examine the residual plot. Anything distinctive?

16. Ladder of Powers • So transforming x values using the square root function worked well. In general, how we can choose a transformation that will result in a linear pattern? • The “ladder of powers” has provided some guidance to statisticians.

17. Ladder of Powers • The table following displays the most commonly used power transformations. The power 1 corresponds to no transformation at all. Using the power 0 would transform every value to 1, which is certainly not very informative, so statisticians use the logarithmic transformation in its place in the ladder of transformations.

18. Ladder of Powers

19. Ladder of Powers • Which of these transformations have we already looked at?

20. Ladder of Powers • The figure following is designed to suggest where on the ladder we should go to find an appropriate transformation. The four curves, labeled 1, 2, 3, and 4, represent shapes of curved scatterplots that are commonly encountered.

21. Ladder of Powers • Sketch the diagram.

22. Ladder of Powers • Suppose a scatterplot has a bend in it like the one labeled 1. Then, to straighten the plot, we would use a power of x that is up the ladder from the no transformation row (x2 or x3) and/or a power on y that is also up the ladder from 1. Thus, we might be led to squaring x, maybe cubing each y, and plotting the transformed ordered pairs.

23. Ladder of Powers • If the curvature in the original scatter looks like curved segment 2 in the diagram, a power up the ladder from no transformation for x (x2 or x3) and or a power down the ladder for y (e.g., or log y) should be used.