Stat 529 (Winter 2011) A simple linear regression (SLR) case study. Mammals brain weights and body weights

Transcription

1 Stat 529 (Winter 2011) A simple linear regression (SLR) case study Reading: Sections , 8.6, 8.7 Mammals brain weights and body weights Questions of interest Scatterplots of the data Log transforming both scales Regression output - both weights on log scale There is not just one residual in a regression model! Residual plots to check the model fit Measuring influence A formal test for outliers Outliers in the Forbes and Mammals dataset 1

2 Mammals brain weights and body weights We often wonder about patterns that we perceive in the world around us. Animals have widely varying appearance, and yet there seem to be a number of commonalities between species. Often, we see patterns emerge in the form of groups of animals having a similar appearance or exhibiting similar behavior. Graphical exploration of data and fits of relatively simple models can help us describe the patterns that we see around us. To see how this works, we will examine the relationship between brain weight (the response variable) and body weight (the explanatory variable) in mammals. The data set that we will look at consists of brain weights (measured in grams) and body weights (measured in kilograms) for 62 species of mammals. The data is available on the class website. 2

3 Questions of interest What general patterns emerge when we look at the relationship between brain weight and body weight across a number of species? Is brain weight proportional to body weight? Are there any unusual species? Do humans have unusually large brains, once we adjust for our body size? 3

4 Scatterplots of the data (We take a log to the base 10 transformation of each variable). 4

5 Fitting the SLR model in MINITAB Stat Regression Regression Response: Log(Brain Weight) Predictors: Log(Body Weight) Click on Storage: Tick Residuals, Standardized residuals, Deleted t residuals and Fits. Press OK. Press OK again. On the worksheet: Rename the RESI1 column to residuals. Rename the SRES1 column to std residuals. Rename the TRES1 column to deleted t residuals. Rename the FITS1 column to fitted values. 5

6 Part of the regression output (both variables on the log scale) Regression Analysis: Log(Brain Weight) versus Log(Body Weight) The regression equation is Log(Brain Weight) = Log(Body Weight) Predictor Coef SE Coef T P Constant Log(Body Weight) S = R-Sq = 92.1% R-Sq(adj) = 91.9% Unusual Observations Log(Body Log(Brain Obs Weight) Weight) Fit SE Fit Residual St Resid R X R R R denotes an observation with a large standardized residual. X denotes an observation whose X value gives it large influence. 6

7 The additive model relating log brain weight and log body weight To think of the model in this case let Y be the brain weight, and let X the body weight. Under the (normally distributed) additive model we assume that given the (log) body weight, (log) brain weights for the different species are independent. Our regression model says that the distribution of the log brain weight, log Y, given a log body weight, log X, is normal with a mean µ log Y log X estimated by µ log Y log X = log X, and a standard deviation σ that we estimate by σ = We will check the model assumptions using residuals diagnostics. 7

8 The multiplicative model in this case Under this model, the distribution of log Y given log X is symmetric and the mean is equal to the median: η log Y log X = µ log Y log X = log X. On the original scale the estimated median brain weight given log body weight is η Y log X log X = 10 = ( 10 log X) = X , which is the same as the estimated median brain weight given body weight, η Y X. This says that roughly brain weight is proportional to the body weight raised to the power of 3/4. 8

9 Another way of interpreting on the original scale Let us calculate the ratio of the estimated median brain weights for a body weight of (log X) + 1 to the estimated median brain weights for a body weight of log X: η Y (log X)+1 η Y log X = ((log X)+1) log X = Thus for a one unit increase in the log body weight, we estimate that the median brain weight increases by a factor of = As in the two sample problem we can calculate a confidence interval for the ratio of medians. 9

10 There is not just one residual in a regression model! We have already defined the residual; it is ê i = Y i Ŷi = Y i β 0 β 1 X i. Problem: the variance of the residual is not constant it depends on the X i value. Instead we can define the standardized residual: ê i s.e.(ê i ) (sometimes called internally studentized residuals). If the model fits correctly, these standardized residuals should be 10

11 Other residuals Now imagine removing observation i, and refitting the regression model. Let Ŷ ( i) i be the fitted value corresponding to Y i, with s.e.(ê ( i) i ). The deleted-t residual is defined by Y i Ŷ ( i) i s.e.(ê ( i) i ) (sometimes called jackknife or externally studentized residuals). We will show later that can use this in a test for outlying values. 11

12 Residual plots to check the model fit 1. Residuals versus fitted values, or residuals versus X values. Look for any trends in the residuals (indicates mean of residuals are not zero). look for any change in the variability (indicates spread of residuals is not constant). 2. Residuals versus normal scores. Use the standardized residuals. Look for symmetric residuals about a mean of zero (the standardized residuals cannot be normally distributed). 3. Use the deleted-t residuals individually to look for cases that are extreme. 4. Residuals vs. time order, or other variables not in model. Patterns indicate evidence that the residuals are not independent! 12

13 Residuals plots for the mammals dataset Referring to the handout and the model summary, diagnose the fit of the regression model fit to the log transformed mammals dataset. 13

14 Residuals plots for the mammals dataset, continued 14

15 Measuring influence An influential value is an observation that has unusual X values compared to the X values of the rest of the data. They can greatly affect the fit of the regression model. Log(Body Log(Brain Obs Weight) Weight) Fit SE Fit Residual St Resid X (african elephant) X denotes an observation whose X value gives it large influence. 15

16 The regression output (with the influential case) Regression Analysis: Log(Brain Weight) versus Log(Body Weight) The regression equation is Log(Brain Weight) = Log(Body Weight) Predictor Coef SE Coef T P Constant Log(Body Weight) S = R-Sq = 92.1% R-Sq(adj) = 91.9% Unusual Observations Log(Body Log(Brain Obs Weight) Weight) Fit SE Fit Residual St Resid R X R R R denotes an observation with a large standardized residual. X denotes an observation whose X value gives it large influence. 16

17 The regression output (without the influential case) Regression Analysis: Log(Brain Weight) versus Log(Body Weight) The regression equation is Log(Brain Weight) = Log(Body Weight) Predictor Coef SE Coef T P Constant Log(Body Weight) S = R-Sq = 91.4% R-Sq(adj) = 91.2% Unusual Observations Log(Body Log(Brain Obs Weight) Weight) Fit SE Fit Residual St Resid R R R R denotes an observation with a large standardized residual. 17

18 Comparing the models for the means 18

19 A formal test for outliers We can use the deleted t residuals for a formal test. We test H 0 : mean for case i is β 0 + β 1 X i, versus H a : mean for case i is not β 0 + β 1 X i. Under H 0 the deleted t statistic, t = Y i Ŷ ( i) i s.e.(ê ( i) i ), has a t distribution on (n 1) 2 = n 3 degrees of freedom. If we test more than one case we need to adjust for multiple testing. 19

20 Outliers in the Forbes dataset Case 12 of the Forbes dataset. We test H 0 : mean for case 12 is β 0 + β 1 X 12, versus H a : mean for case 12 is not β 0 + β 1 X 12. The t value is on 14 df. The p-value is very small (< 0.001). We reject H 0 and conclude that the mean of case i = 12 is not β 0 + β 1 X 12. We conclude case 12 is an outlier. 20

21 Outliers in the Mammals dataset With n = 62, the d.f. for the deleted-t statistic is Adjusting for 62 comparisons using the Bonferroni method and a familywise error rate of 5%, the individual error rate is The t multiplier is Now scan through the mammals to see if any deleted-t statistic has absolute value large than the Conclusion? 21

22 Outliers in the Mammals dataset, continued name deleted t residual water opossum tenrec musk shrew pig brazilian tapir giant armadillo desert hedgehog kangaroo cow european hedgehog nine-banded armadillo north american opossum galago red fox arctic fox vervet patas monkey chimpanzee ground squirrel owl monkey baboon rhesus monkey human