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9.5: Re-Expression of Curved Relationships

9.5: Re-Expression of Curved Relationships. Objective : To determine whether or not a curved relationship can be salvaged and re-expressed into a linear relationship. If so, complete the re-expression. Re-Expressing Data.

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9.5: Re-Expression of Curved Relationships

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  1. 9.5: Re-Expression of Curved Relationships Objective: To determine whether or not a curved relationship can be salvaged and re-expressed into a linear relationship. If so, complete the re-expression.

  2. Re-Expressing Data • We cannot use a linear model unless the relationship between the two variables is linear. Often re-expression can save the day, straightening bent relationships so that we can fit and use a simple linear model. • Two simple ways to re-express data are with logarithms and reciprocals.

  3. Re-Expressing Data (cont.) Original Scatterplot Residual Plot A look at the residuals plot shows a problem: • The relationship between fuel efficiency (in miles per gallon) and weight (in pounds) for late model cars looks fairly linear at first:

  4. Re-Expressing Data (cont.) Re-Expressed Scatterplot Re-Expressed Residual Plot A look at the residuals plot for the new model seems more reasonable: • We can re-express fuel efficiency as gallons per hundred miles (a reciprocal-multiplying each by 100) and eliminate the bend in the original scatterplot:

  5. Ladder of Powers • There is a family of simple re-expressions that move data toward our goals in a consistent way. This collection of re-expressions is called the Ladder of Powers. • The Ladder of Powers orders the effects that the re-expressions have on data. • The farther you move from “1” in the ladder of powers, the greater the effect on the original data.

  6. Power Name Comment 2 Square of data values Try with unimodal distributions that are skewed to the left. 1 Raw data Data with positive and negative values and no bounds are less likely to benefit from re-expression. ½ Square root of data values Counts often benefit from a square root re-expression. “0” We’ll use logarithms here Measurements that cannot be negative often benefit from a log re-expression. –1/2 Reciprocal square root An uncommon re-expression, but sometimes useful. –1 The reciprocal of the data Ratios of two quantities (e.g., mph) often benefit from a reciprocal. Ladder of Powers (cont.)

  7. Examples Using the ladder of powers, what transformations might you use as a starting point? • You want to model the relationship between the number of birds counted at a nesting site and the temperature (in degrees Celsius). The scatterplot of counts vs. temperature shows an upwardly curving pattern, with more birds spotted at higher temperatures. • You want to model the relationship between prices for various items in Paris and in Hong Kong. The scatterplot of Hong Kong prices vs. Parisian prices shows a generally straight pattern with a small amount of scatter. • You want to model the population growth of the US over the past 200 years. The scatterplot shows a strongly upwardly curved pattern.

  8. Plan B: Attack of the Logarithms • When none of the data values is zero or negative, logarithms can be a helpful ally in the search for a useful model. • Try taking the logs of both the x- and y-variable. • Then re-express the data using some combination of x or log(x) vs. y or log(y).

  9. Plan B: Attack of the Logarithms • When none of the data values is zero or negative, logarithms can be a helpful ally in the search for a useful model. • Try taking the logs of both the x- and y-variable. • Then re-express the data using some combination of x or log(x) vs. y or log(y).

  10. Plan B: Attack of the Logarithms (cont.)

  11. Why Not Just Use a Curve? • If there’s a curve in the scatterplot, why not just fit a curve to the data?

  12. Why Not Just Use a Curve? (cont.) • The mathematics and calculations for “curves of best fit” are considerably more difficult than “lines of best fit.” • Besides, straight lines are easy to understand. • We know how to think about the slope and the y-intercept.

  13. Example • Let’s try an example to note the techniques to re-express that tuition rates verses the years at Arizona State University:

  14. What Can Go Wrong? • Beware of multiple modes. • Re-expression cannot pull separate modes together. • Watch out for scatterplots that turn around. • Re-expression can straighten many bent relationships, but not those that go up then down, or down then up.

  15. What Can Go Wrong? (cont.) • Watch out for negative data values. • It’s impossible to re-express negative values by any power that is not a whole number on the Ladder of Powers or to re-express values that are zero for negative powers. • Watch for data far from 1. • Data values that are all very far from 1 may not be much affected by re-expression unless the range is very large. If all the data values are large (e.g., years), consider subtracting a constant to bring them back near 1.

  16. Assignments • Day 1: 9.5 Problem Set Online # 1, 2, 5, 6, 7 • Day 2: 9.5 Problem Set Online # 14, 15, 16

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