AP Statistics. The only statistics you can trust are those you falsified yourself. RE- E X P R E S S I N G D A T A ( P A R T 2 ) C H A P 9

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1 AP Statistics 1 RE- E X P R E S S I N G D A T A ( P A R T 2 ) C H A P 9 The only statistics you can trust are those you falsified yourself. Sir Winston Churchill ( ) (Attribution to Churchill is ironically falsified)

2 Goal of Re-expression 2 Make the distribution of a variable more symmetric: A symmetric distribution can be analyzed much more easily than a skewed distribution.

3 Goal of Re-expression 3 Make the spread of several groups more alike: With similar spreads, distributions are easier to compare.

4 Goal of Re-expression 4 Make the form of a scatterplot more linear: Linear regression is easy non-linear regression is not!

5 Goal of Re-expression 5 Make the scatter in a scatterplot spread out evenly rather than following a fan-shape: An even scatter is a necessary condition for analysis we will learn about later.

6 What Transformation? 6 Ladder of Powers (see p 237) Power 2 Name Square of data values Comment 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. When in doubt, start here: ½ 0-1/2 Square root of data values We ll use logarithms here Reciprocal square root Counts often benefit from a square root reexpression. Measurements that cannot be negative often benefit from a log re-expression. 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.

7 Important Models log yˆ b b x Exponential Model: Original Data Transformed Data y log(y) x This is the zero power on the ladder. It is useful for values that grow (or shrink) by percentages. x

8 Important Models Logarithmic Model: 8 yˆ b b log x 0 1 Original Data Transformed Data y y x log(x) Data with a wide range of x-values or with a scatterplot that is very steep at the left and levels out towards the right.

9 Important Models log Power Model: 9 yˆ b b log x 0 1 Original Data Transformed Data y log(y3) x log(x) The authors of the textbook call this one the Goldilocks Model when steps on the ladder are either too big or too small.

10 Example 10 Below are data from 12 perch caught in a lake in Finland (length in cm and weight in grams). Length (cm) Weight (g) Length (cm) Weight (g)

11 Example 11 In order to create a model to predict weight from length, start by looking at the data: There is a fairly strong, positive, and nonlinear association between weight and length.

12 Example 12

13 Example 13 We need to transform the data (one or both variables) to achieve a more linear relationship. In the biological sciences, power models are fairly common, so we ll start there. Take the logarithm of both variables (either base-10 or base-e log we don t care which). The association between the logs of the variables is quite linear.

14 Example 14 Create a linear model, and then check the residuals to determine if the model may be reasonable. Note you can t use either R or R-squared to determine if your model is reasonable. These statistics are only useful after you assess the model fit. Regression Analysis: log(w) versus log(l) The regression equation is log(w) = log(l) Predictor Coef SE Coef T P Constant log(l) S = R-Sq = 98.9% R-Sq(adj) = 98.7%

15 Example Linear Model remember your calculator doesn t know you are using logtransformed data when it produces the equation. 15 log weight log length The residuals appear to be fairly random, so this linear model is reasonably appropriate.

16 Example 16 Describe what the slope represents: log weight log length For every one-unit increase in the log of length, the log of weight increases by about 3.05.

17 Example 17 Describe what the correlation represents: Predictor Coef SE Coef T P Constant log(l) S = R-Sq = 98.9% R-Sq(adj) = 98.7% The correlation is the square root of R-squared, which is about This indicates there is a very strong, positive, linear relationship between the logs of weight and length.

18 Example 18 Describe what R-squared represents: Predictor Coef SE Coef T P Constant log(l) S = R-Sq = 98.9% R-Sq(adj) = 98.7% About 98.9% of the variability in the log of weight is accounted for by the regression with the log of length.

19 Example 19 Use the model to predict the weight of a perch that is 35 cm long. log weight log length log weight log 35 log weight weight 10 weight The predicted weight for a 35 cm perch is about 446 grams.

20 What Can Go Wrong? 20 Don t expect the re-expressed model to be perfect. Don t use R or R-squared to decide which is the best model. A transformation won t make a multimodal distribution unimodal. You can t transform data into a linear form if the scatterplot rises and falls in a cyclical manner. If your data has values of zero or that are negative, some transformations can t be done (logs, for example). Sometimes, if the negative data are close to zero, you can add a very small constant (1/2 and 1/6 are common) to all data values to make them all positive. If you have data that are dates (years), pick a reference year to be zero, and look at years from the point forward.

21 What Can Go Wrong? 21 Keep the model simple avoid making multiple transformations on the same variable, or mixing quite different transformations on both variables. Stay close to the ladder of powers.

22 Assignment Read Chapter 9 22 Exercises #15, 17-20, 25 xkcd.com