III. Inferential Tools

Transcription

1 III. Inferential Tools A. Introduction to Bat Echolocation Data (10.1.1) 1. Q: Do echolocating bats expend more enery than non-echolocating bats and birds, after accounting for mass? 2. Strategy: (i) Explore (resolve need for transformation) (ii) Is 3 parallel lines model ok? Log of energy expenditure non echolocating bats birds echolocating bats Log of mass (iii) If so, answer question by comparing ne-bats line to the two others. St 412/512 page 45

2 3. A note about different parameterizations for a factor a. Example: the factor TYPE has 3 levels (e-bats, birds, ne-bats) b. Consider the model µ(y x,type)= x + TYPE. We have some choices about which of the 3 indicator variables to include c. If we include two indicator variables, the level whose indicator isn t included is the reference level ; i.e. if µ(y x,type) = β 0 + β 1 x + (β 2 I type2 + β 3 I type3 ) where I type2 and I type3 are indicator variables for levels 2 and 3 of type, then β 0 is the intercept for level 1; β 2 is the amount by which the mean of y is greater for level 2 than for level 1 (after accounting for x) and β 3 is the amount by which the mean of y is greater for level 3 than for level 1 (Display 10.5) d. Another parameterization is (β 1 I type2 + β 2 I type2 + β 3 I type3 ) + β 4 x (without a β 0 ). In this, β 1, β 2, β 3 are the 3 intercepts St 412/512 page 46

3 If necessary, change data type to factor To include a factor so that the first level is the reference level ( treatment contrast in S-PLUS), include this term in the formula St 412/512 page 47

4 Note: R 2 and the F-statistic do not make sense when the intercept is dropped. To include a factor so that each level gets its own intercept, drop the overall intercept by using the model formula term - 1 St 412/512 page 48

5 B. Least Squares Estimation 1. µ(y x 1,x 2 ) = β 0 + β 1 x 1 + β 2 x 2 ; var(y x 1,x 2 ) = σ 2 Unknown parameters: Regression coefficients Variance (about regression) 2. Fitted (or predicted) values, yˆi : y = βˆ + βˆ x + βˆ x for i = 1,2,...,n ˆi 0 1 i 1 2 i 2 3. Residuals, res i = y i - yˆi 4. Least squares estimators, ( βˆ 0, βˆ 1, βˆ 2), are chosen to minimize the sum of squared residuals (matrix algebra formula) 2 5. σˆ =(sum of squared residuals)/(n-p) [p=number of β s] St 412/512 page 49

6 C. t-tests and CI s for individual β s 1. Note: a matrix algebra formula for SE( βˆ j ) is also available 2. If distribution of Y given X s is normal, then βˆ j βj t ratio= SE( βˆ ) has a t-distribution on n-p degrees of freedom 3. For testing the hypothesis H 0 : β 2 = 7; compare t stat = βˆ 2 7 SE( βˆ ) to a t-distribution on n-p degrees of freedom. 4. The p-value for the test of H 0 : β j = 0 is standard output j 2 St 412/512 page 50

7 5. It s often useful to think of H 0 : β 2 = 0 (for example) as Full model: µ(y x 1,x 2,x 3 ) = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 Reduced model: β 0 + β 1 x 1 + β 3 x 3 (Is the β 2 x 2 term needed in a model with the other x s?) 6. 95% confidence interval for β j : βˆ ± t (.975) SE( βˆ ) j n p j 7. The meaning of a coefficient (and its significance) depends on what other X s are in the model (Section ) 8. The t-based inference works well even without normality D. t-tests and CIs for Bat Data 1. From Display 10.6 results: The data are consistent with the hypothesis of no energy differences between echolocating and St 412/512 page 51

8 non-echolocating bats, after accounting for body size (2- sided p-value =.7) 2. But that doesn t prove that there is no difference. A large p-value means either: (i) there is no difference (H 0 is true) or (ii) there is a difference and this study is not powerful enough to detect it. 3. So: report a confidence interval in addition to the p-value. 95% CI for β 3 :.0787 ± = (.35,.51). Interpretation (back-transform, e.0787 =1.08, e -.35 =.70 and e.51 =1.67): It is estimated that the median energy expenditure for echolocating bats is 1.08 times the median for non-echolocating bats of the same body weight (95% confidence interval:.70 to 1.67 times). St 412/512 page 52

9 E. Review of variance, SD, SE 1. var(y) = Mean{(y-µ) 2 } (population variance) 2. SD(y) = {var(y)} 1/2 3. var(y x) = Mean{[y-µ( µ(y x)] 2 } in subpopulation of y s at x ˆβ β 4. var( βˆ ) = Mean{( - ) 2 } (sampling variance) 5. SD( βˆ ) = {var( β ˆ )} 1/2 6. SE( βˆ ) = Estimate of SD( β ˆ ), usually obtained by using in place of the unknown σ (with associated d.f.) ˆσ St 412/512 page 53

10 F. Reminder about interpolation and extrapolation? 2 Extrapolation involves Y additional speculation? 1 X We can safely make statements about the distribution of y given x-values within the sampled range St 412/512 page 54

11 G. Extra SS F-tests 1. Full model: β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 Reduced model: β 0 + β 1 x 1 i.e. in full model, H 0 : β 2 = β 3 = 0 2. Extra sum of squares = (Sum of squared residuals from reduced model) - (Sum of squared residuals from full model) 3. F-statistic = [Extra SS/Extra # of β s]/ 2 σˆ full 4. Display example: Full: log.energy ~ lmass + TYPE Reduced: log.energy ~ lmass One can fit both models and get the SS Residual from the ANOVA table (as in Display 10.10), or... St 412/512 page 55

12 5. Valuable short-cut in S-PLUS: Save model objects, then use Compare Models (continued, next page) St 412/512 page 56

13 1 Compare with Display Write names of model objects, separated by, St 412/512 page 57

14 H. Special Cases of Extra SS F-test 1. F-test for overall significance of regression Full model: β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 Reduced model: β 0 Asks: are any of the x s useful predictors of y? The calculations are laid out in the ANOVA table 2. F-test for a single coefficient If full model has one more term than the reduced model, the F- test is equivalent to the (2-sided) t-test 3. For µ(y CAT) = CAT (i.e. a single categorical factor), the F-test for overall significance is the one-way ANOVA F-test St 412/512 page 58

15 I. Prediction and prediction intervals 1. Suppose µ ( y x, x ) = β + β x + β x The estimated mean of y at x 1 = 15 and x 2 = 7 (say) is µ ˆ( y x = 15, x = 7) = βˆ + βˆ (15) + βˆ (7) The SE of the estimated mean at x 1 = 15 and x 2 = 7 is SE[ µ ˆ( y x1 = 15, x2 = 7)] = (matrix algebra formula; S-plus knows it and does the calculations in the Predict tab) 4. The predicted value of y at x 1 = 15 and x 2 = 7 is Pred( y x = 15, x = 7) = µ ˆ( y x = 15, x = 7) i.e. predict y to be its mean value St 412/512 page 59

16 5. SE[Pred( y x = 15, x = 7)] = { SE[ µ ˆ( y x = 15, x = 7)] + σˆ } /2 sampling variance in estimating mean of y at those x s variance of an individual y about its (subpopulation) mean 6. 95% Prediction interval: Pred( y x = 15, x = 7) ± t (.975) SE[Pred( y x = 15, x = 7)] 1 2 n p S-PLUS: Create a new data set with columns named x 1 and x 2, with values x 1 *, x 2 * µ ˆ( y x = x, x = x ) * * SE[ µ ˆ( y x = 15, x = 7)] 1 2 SE[Pred] must be calculated from these, σˆ 2 and the formula in 5 above St 412/512 page 60