Checking Assumptions

Transkript

1 Merlise Clyde Duke University November 16, 2016

2 Linear Model Linear Model: Y = µ + ɛ Assumptions: µ C(X) µ = Xβ ɛ N(0 n, σ 2 I n ) Focus on Wrong mean for a case or cases Wrong distribution for ɛ Cases that influence the mean If µ i x T i β then expected value of e i = Y i Ŷ i is not zero;

3 Standardized residuals standardized residuals r i = e i / σ 2 (1 h ii ) Under correct model have mean 0 and scale 1 Use estimate of σ 2 The distribution of r i is not a t if h ii (leverage) is close to 1, then Ŷ i is close to Y i so e i is approximately 0 Variance is also almost 0, so standardize value may not flag outliers

4 Predicted Residuals Estimates without Case (i): Predicted residual with variance ˆβ (i) = (X T (i) X (i)) 1 X T (i) Y (i) = ˆβ (XT X) 1 x i e i 1 h ii e (i) = y i x T i e i ˆβ (i) = 1 h ii σ2 var(e (i) ) = 1 h ii Standardized predicted residual is e (i) var(e (i) ) = e i/(1 h ii ) σ/ 1 h ii = e i σ 1 h ii these are the same as standardized residual!

5 External estimate of σ 2 Estimate ˆσ 2 (i) using data with case i deleted SSE (i) = SSE e2 i 1 h ii ˆσ (i) 2 = MSE SSE (i) (i) = n p 1 Externally Standardized residuals t i = e (i) ˆσ 2 (i) /(1 h ii) ˆβ (i) = y i x T i ˆσ (i) 2 /(1 h ii) ( ) n p 1 1/2 = r i n p ri 2 May still miss extreme points with high leverage, but will pick up unusual y i s

6 Distribution of Externally Studentized Residual t i = e (i) ˆσ 2 (i) /(1 h ii) ˆβ (i) = y i x T i ˆσ (i) 2 /(1 h ii) St(n p 1)

7 Outlier Test H 0 : µ i = x T i β versus H a : µ i = x T i β + α i Show that t-test for testing H 0 : α i = 0 is equal to t i if p-value is small declare the ith case to be an outlier: E[Y i ] not given by Xβ but Xβ + δ i α i Can extend to include multiple δ i and δ j to test that case i and j are both outliers Extreme case µ = Xβ + I n α all points have their own mean! Control for multiple testing

8 Multiple Testing look to see if the max t i is larger than expected under the null Need distribution of the max of Student t random variables (simulate) Conservative is Bonferroni Correction: For n tests of size α the probability of falsely labeling at least one case as an outlier is no greater than nα; e.g. with 21 cases and α = 0.05, the probability is no greater than 1.05! Adjust α = α/n so that the probability of falsely labeling at least one point an outlier is α α/n = with n = 21

9 Cook s Distance Cook s Distance measure of how much predictions change with ith case deleted D i = Y (i) Y 2 pˆσ 2 = r i 2 p h ii 1 h ii = (ˆβ (i) ˆβ) T X T X(ˆβ (i) ˆβ) pˆσ 2 Flag cases where D i > 1 or large relative to other cases Influential Cases

10 Stackloss Data Air.Flow Water.Temp Acid.Conc stack.loss

11 Case 21 Leverage (compare to p/n =.19 ) p-value t 21 is Bonferroni adjusted p-value is Cooks Distance.69 Other points? Masking? Refit without Case 21 Other analyses have suggested that cases (1, 2, 3, 4, 21) are outliers

12 Multiple Outliers Hoeting, Madigan and Raftery (in various permutations) consider the problem of simultaneous variable selection and outlier identification. This is implemented in the library(bma) in the function MC3.REG. This has the advantage that more than 2 points may be considered as outliers at the same time. The function uses a Markov chain to identify both important variables and potential outliers, but is coded in Fortran so should run reasonably quickly. Can also use BAS or other variable selection programs

13 Using BAS library(mass) data(stackloss) n = nrow(stackloss) stack.out = cbind(stackloss, diag(n)) library(bas) BAS.stack = bas.lm(stack.loss ~., data=stack.out, prior="hyper-g-n", a=3, modelprior=tr.beta.binomial(1, 1,15), method="mcmc", MCMC.it=20000)

14 Output > summary(bas.stack) Intercept Air.Flow Water.Temp Acid.Conc [1,] [2,] [3,] [4,] [5,] [1,] [2,] [3,] [4,] [5,] BF PostProbs R2 dim logmarg [1,] [2,] [3,] [4,] [5,]

15 BAS Predictions under BMA Residuals Residuals vs Fitted Model Search Order Cumulative Probability Model Probabilities Model Dimension log(marginal) Model Complexity Marginal Inclusion Probability Intercept Air.Flow ater.temp Acid.Conc Inclusion Probabilities

16 BAS Log Posterior Odds Intercept Air.Flow ater.temp Acid.Conc Model Rank

17 Diagnostics Renormalized likelihood Estimates vs MCMC frequencies BAS.stack$probs.MCMC BAS.stack$probne0

18 To Remove or Not? For suspicious cases, check data sources for errors Check that points are not outliers because of wrong mean function or distributional assumptions Investigate need for transformations (use EDA at several stages) Influential cases - report results with and without cases (results may change - are differences meaningful?) Outlier test - suggests alternative population for the case(s); if not influential may in keep analysis, but will inflate ˆσ 2 and interval estimates Document how you handle any case deletions - reproducibility! Robust Regression Methods

19 Stackloss Added Variable Plot Added Variable Plots stack.loss others stack.loss others Air.Flow others Water.Temp others stack.loss others Acid.Conc. STA721 others Linear Models

20 Stackloss Data Again Residuals vs Fitted Normal Q Q Residuals Standardized residuals Fitted values Theoretical Quantiles Standardized residuals Scale Location Standardized residuals Residuals vs Leverage 4 1 Cook's distance Fitted STA721 values Linear Models Leverage