Q Lecture Introduction to Regression

Transcription

1 Q

2 Before/After Transformation 2

3 Construction Role of T-ratios Formally, even under Null Hyp: H : 0, ˆ, being computed from k t k SE ˆ ˆ y values themselves containing random error, will sometimes be large ( ) for reasons of random chance only. Also t-ratios k k 0 k In (conceptual) replications, differing from current data by chance alone, the probability of obtaining, by chance, a t-ratio larger ( ) than quoted value is quoted P value 3

4 Role of T-ratios ˆ ˆ Informally, if t is not large (>2 in mag; p ) then coeff of x can be given a value of zero - equivalently x can be dropped from model - k k with little appreciable impact. t k k SE k k Cautio n: applies one variable at a time 4

5 Residuals Regression Analysis: Taste versus Lactic Acid, LAcetic, LH2S Taste = Lactic Acid LAcetic LH2S Predictor Coef SE Coef T P Constant Lactic Acid LAcetic LH2S S = R-Sq = 65.2% R-Sq(adj) = 61.2% Unusual Observations Lactic Obs Acid Taste Fit SE Fit Residual St Resid R Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS Lactic Acid LAcetic LH2S

6 Residuals Unusual Observations Lactic Obs Acid Taste Fit SE Fit Residual St Resid R In fact barely outlying despite 2.63 Recall, one has to be largest!! 6

7 Options with Large Residuals Examine carefully: Why outlying? Anything special about this case/obs? Refit without does its removal change anything important? If delete, then formally Conclusions are based on something like this never happening in future Is this a meaningful statement? 7

8 Residuals, Standardized Residuals, Deleted T-residuals Normal Scores 8

9 Classic Linear Model Y x x... x Transformations i 1 1i 2 2i p pi i 2 where ~ N 0, Var or N 0, SD i Statistical Theory assumes Normally Dist residuals/'errors' makes NO assumptions re dist of Y, X,.. 1 (technical) makes assumption of additivity (crucial) But non-additivity - esp multiplicative models - and non normality often occur together 9

10 Normality of data or errors/resids? 10

11 Extreme example 11

12 Random Variation Additive? 6.00 Exp Decay. Random variation decreases with time Exp Decay on log scale. Random variation constant in time time t time t 12

13 Artificially created data Rescaled Additive model to create data Resids by Exponentiation of line and data t line erors Y = line + e subtraction line Exp y Data created multiplicatively exhibit neither linearity nor Normality in either data or residuals Log transform solves both issues 13

14 Artificially created data Rescaled Additive model to create data Resids by constant Exponentiation by construction of line and data t line erors Y = line + e subtraction 2.5 line Exp y Linear Decay Rand variation 14

15 Artificially created data Rescaled Additive model to create data Resids by Exponentiation of line and data t line erors Y = line + e subtraction line Exp y Exp Decay. Random variation decreases with time time t Exp Decay on log scale. Random variation now seems constant in time time t 15

16 Artificially created data Distributions of obs Y under both models Additive model to create dat Exponentiation of line and data t line Y = line + e line Exp y

17 Resids for artificially created data Rescaled Residuals Exponentiation of line and data Resids by Resids by line Exp y subtraction division Erroneous Not reflecting creation 17

18 Artificially created data Exponentiation of line and data Resids by Resids by line Exp y subtraction division Now reflecting creation, but not Normal 18

19 Artificially created data Exponentiation of line and data Resids by Resids by line Exp y subtraction division Now reflecting creation, but not Normal. However Normal after log transformation 19

20 Artificially created data Exponentiation of line and data Resids by Resids by line Exp y subtraction division Now reflecting creation, but not Normal. However Normal after log transformation 20

21 Plotting Plotting Multiple Regression Fits gpm = wt hp/wt a b1 b2 Given hp/wt hp/wt wt line Fitted line vs wt when hp/wt= wt Excel plotting 21

22 Very active area of advanced research Exam Q will not ask for undiscovered network Direct/Indirect Links Causal Networks Networks Network Models No primary response variable Now can consider Regress Directed Arrows Regression has no useful role Undirected Arrows Regression has some role? Y= wt on X s hp, hp/wt, gpm hp hp/ wt wt gpm 22

23 Undirected Network Models Indicative research methodology hp hp/ wt gpm Regression Analysis: wt versus hp, hp/wt, gpm wt = hp hp/wt gpm T-ratios Deleted stuff... wt Tentative network model Source DF Seq SS hp hp/wt gpm Note order gpm adds relatively little to pred wt when hp and hp/wt already in the model Relatively weak evidence in favour of Can drop link But propose new link hp NB changing the response variable hp/ wt wt gpm Proposed new network model 23

24 Undirected Network Models Direct/Indirect Links Key: No primary response variable Network Order hp hp/ wt wt gpm Recall regression fit completely insensitive to order Both following make the same predictions Regression not the natural tool but is relevant wt = hp hp/wt gpm wt = hp/wt gpm hp 24

25 Not on the exam in 2010 ANOVA and F -tables t-tables PIs and CIs 25