Lecture 3: Inference in SLR

Transcription

1 Lecture 3: Inference in SLR STAT 51 Spring 011 Background Reading KNNL:

2 Topic Overview This topic will cover: Review of hypothesis testing Inference about 1 Inference about 0 Confidence Intervals Prediction Intervals 3-

3 Review: Significance Tests One Sample T-test Take a sample of size n from some (normal) population: H 0 : 0 Y t H : sy a 0 Compare t to a critical value from the students-t distribution (table B.) with (typically)

4 Review: Significance Tests () One Sample T-test: Can turn the test statistic into a confidence interval for 1 /, n1 Y t s Y Generally a confidence interval takes the form Point Est. ± Crit. Value * SE Two Sample T-test: Compares the means of two samples. 3-4

5 Significance Levels The significance level is the probability of making a Type I error and rejecting the null hypothesis when it is in fact true (false positive). The most common significance level that we will use is The corresponding confidence level is 1. So for 0.05 our confidence level will be 95%. 3-5

6 P-Values The p-value for a test is the probability (under the null hypothesis) of observing a test statistic that is at least as extreme as the one that is actually observed. We reject the null if P-value Mathematically, the p-value is Pr H T t, where T ~ tn 0 1 Graphically, the p-value is twice the area in the upper tail of the tn 1 distribution (above the observed t ). 3-6

7 Conclusions Conclude H a means there is sufficient evidence in the data to conclude that H 0 is false, and hence we can assume H a is true. Fail to Reject H 0 means there is insufficient evidence in the data to conclude that either H 0 or H a is true or false, so we default to assuming that H 0 is true. Unless prepared to make further justification (power) it is not appropriate to conclude H

8 Power of a Test The probability of a Type II error (failing to reject H 0 when H a is in fact true or a false negative) is often denoted (not to be confused with regression coefficients). The power of a test is 1. This is the probability that H 0 will be rejected given that H a is true. Power calculations involve the non-central t- distribution (generally use a computer). 3-8

9 β Inference 1 Recall that X X Y Y SS b1 X X SS i i XY i X s are constant, Y s are normally distributed. Using probability theory it can thus be shown that (page 4-43) b ~ Normal, b where b 1 SS X X 3-9

10 Test for H :β As in the case of the one-sample t-test, we can develop the test statistic for testing H 0 : 1 0 vs. H a : 1 0: t b 1 where sb 1 0 s b MSE 1 SSX This statistic has a t-distribution with n degrees of freedom (not n 1 because we are also estimating 0). 3-10

11 Test for H :β Reject H 0 if t tcrit, where tcrit t(1 ; n ). SAS will give us both the value of the t- statistic and the P-value. If the P-value is smaller than, reject in favor of H : 0 a

12 Confidence Interval for β 1 The 1001 % CI for 1 is b 1 tcrits b1 where tcrit t(1 ; n ). In terms of hypothesis testing, if the CI does not contain 0, then we reject H 0 : 1 0 and conclude that Ha : 1 0 is true. 3-1

13 Power In cases where we fail to reject, it is important to know the power of the test for H 0 : 1 0. There are two important questions we must answer before we can determine power: 1. What size difference is important?. Guess for the variance? Note that power calculations should be done prior to collection of data if possible. 3-13

14 Power () The power to detect a difference of size d is calculated using the non-central t distribution. In addition to and the degrees of freedom, we need the noncentrality parameter: 1 1 b SS 1 / X Power for some values of, can be looked up in Table B5. SAS also has a procedure for computing power (for any values). 3-14

15 β Inference 0 Similar to inference for 1 b ~ Normal, b X b0 n SS X where To test 0 k : b0 k t where 0 s b 0 1 X s b MSE n SS X 3-15

16 Test for H :β 0 0 k The statistic has a t-distribution with n degrees of freedom; compare it with the appropriate t-critical value. SAS gives both statistic and p-value for testing 0 0; to test 0 k, obtain and use a confidence interval. The 1001 % CI for 0 is b t s b 0 crit 0 Remember: If X = 0 is not within the scope of the model, inference may be meaningless!! 3-16

17 Robustness In cases where the errors are not quite normal, the CIs and significance tests for 1 and 0 are still generally reasonable approximations. We say that these tests are robust with respect to minor violations of the normality assumption. 3-17

18 SAS Coding PROC REG data=diamonds; model price=weight /clb; RUN; clb option in PROC REG requests the confidence limits for b 1 and b 0. You can also specify alpha=0.xxx to change the significance level (default = 0.05) 3-18

19 SAS Output Parameter Std Variable DF Estimate Error t Value Pr > t Intercept <.0001 weight <.0001 Variable DF 95% Confidence Limits Intercept weight

20 Summary of Inference SLR Model Yi 0 1Xi i ~ Normal 0, are independent, random i errors Y ~ Normal X, i 0 1 i 3-0

21 Summary of Inference Parameter Estimates For 1: b X X Y Y SS i i XY 1 X SS i X X For 0: b0 Y b1x For : s SSE e MSE df n E i 3-1

22 Summary of Inference 1001 % Confidence Intervals b t s b 1 crit 1 b t s b 0 crit 0 Where tcrit t(1 ; n ). 3-

23 Summary of Inference Significance tests H 0 : 1 0 vs. H a : 1 0: b1 0 t t( n ) under H 0 s b 1 H 0 : 0 0 vs. H a : 0 0: b0 0 t t( n ) under H 0 s b Reject H 0 if the P-value is small (<) 0 3-3

24 CI for the Mean Response The mean response when ˆh 0 1 X Y b b X h X is Y ˆh is a normal random variable (since the parameter estimates are linear combos of the Y i and these are normal). To develop a confidence interval we can obtain a formula for the standard error from. and b 0 b 1 h 3-4

25 Standard Error The variance associated to Y ˆh is ˆ Var Y Var b X Var b h 0 h 1 1 n Substitute MSE for X h SS X X to get the estimated variance. Take the square root to get the sy ˆh 3-5

26 Confidence Interval for EY h Recall: Point Est. ± Crit. Value * SE Confidence Limits are Yˆ t s Yˆ h crit h Where tcrit t(1 ; n ) 3-6

27 Prediction Intervals Predicting a new observation for X Xh is different from estimating the mean response in that there is additional variation associated to the normal curve EY that is centered at h Hence two components to sy ˆh, new Variance associated to the estimated mean response. Variance associated to the new obs. 3-7

28 Prediction Intervals () The variance associated to Y ˆh, new is ˆ Var Y Var Yˆ h, new h 1 X 1 n As before, substitute MSE for the square root to get sy equivalently, s pred. X SSX and take, or h ˆh, new 3-8

29 Prediction Intervals (3) The 1001 % prediction interval for a new observation at X X is given by Y t s pred ˆh crit Where tcrit t(1 ; n ) h 3-9

30 CI s and PI s in SAS PROC REG data=diamonds; model price=weight /clm cli; clm produces CI s for the mean response cli produces prediction intervals Intervals produced for each data point including those with missing values 3-30

31 SAS Output Predicted Std Error Obs Wt Price Value Mean Predict 95% CL Mean Obs Wt 95% CL Predict Residual

32 Comparing Standard Errors s b MSE 1 SSX 1 X s b0 MSE n SS X 1 ˆ h s Yh MSE n SS X 1 Xh s pred MSE 1 n SS X X X X 3-3

33 Minimizing Standard Errors Can sometimes design experiments to minimize standard errors Increase sample size Increase SS X by spreading out the values of the predictor variable Arrange for the predictor of interest to be X X h 3-33

34 Upcoming in Lecture 4... We will look at one more example illustrating the use of SAS. We ll discuss the Working-Hotelling Confidence Band (.6), details of the ANOVA table (.7.9) and clean up a few details in