More on Two Variable Relationships

 Teaching Tips

• There is a tradeoff between interpretability and a perfect fit. When in doubt, it's often best to prefer an interpretable model over a perfect fit.

• Also, keep in mind the principle of parsimony. Simple and good is preferred over complicated and perfect.

• Students will want to know which variable to transform or transform first. The "ladder of transformations" can help with this. (See YMS p. 201 or POD p. 250). Technology can also help. Fathom lets you easily try on different transforms to see which makes the scatterplot most linear.

• Students may ask why they just can't use the pre-programmed options on the TI calculator (for example "log", "exponent"). The pedagogical answer is that we see value in having students see how the relationship changes as different transforms are tried. The statistical reason is that we don't want to encourage students to simply try to find the transformation with the highest r-squared (or r) value. The correlation coefficient is only part of the story, and statisticians prefer a model with a lower r-squared value if the residual plot looks better.

• The "ladder of transformations" (YMS p. 201 or POD p. 250) might be too much information from some students. This is okay, but make sure they understand how to use a log, square-root, or inverse transform.

• Students might find your discussion easier to follow if you identify the variables by name (e.g. "height") rather than a letter label (e.g. "x"). Thus, when you transform, it will be easier for students to follow "log height" as a transformed variable.

• Interpreting the coefficients in a model that includes transformed variables challenges the students. Be sure that their interpretations are in the proper units. For example, if the model is "degrees Farenheit" = 3+ 4*log(time) then the coefficient 4 relates differences in degrees with differences in log(time) and not time. That is, a 10-fold increase in time corresonds to a 4 degree difference in temperature.

• Be careful inventing exponential examples on the fly because sometimes the transformation also requires a shift. In other words, y = a + exp(bx) needs to be shifted by a-units before you can take a log transform.