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Topic 5: Power

Topic 5: Power. Outline. Review estimation and inference for simple linear regression Power / Sample Size Estimation Slope Intercept. Simple Linear Normal Error Regression Model. Y i = b 0 + b 1 X i + e i e i is a Normally distributed random variable with mean 0 and variance σ 2

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Topic 5: Power

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  1. Topic 5: Power

  2. Outline • Review estimation and inference for simple linear regression • Power / Sample Size Estimation • Slope • Intercept

  3. Simple Linear Normal Error Regression Model • Yi = b0 + b1Xi + ei • eiis a Normally distributed random variable with mean 0 and variance σ2 • eiandejare uncorrelated → indep

  4. Parameter Estimators • β1: • β0: • σ2:

  5. 95% Confidence Intervals for β0 and β1 b0 ± tcs(b0) and b1 ± tcs(b1) where tc = t(.975, n-2), the upper 97.5 percentile of the t distribution with n-2 degrees of freedom

  6. Significance tests for β0 and β1 • H0: β0 = 0, Ha: β0  0 t* =b0/s(b0) • H0: β1 = 0, Ha: β1≠ 0 t* =b1/s(b1) Reject H0 if the P-value is small (<.05)

  7. Power • The power of a significance test is the probability that the null hypothesis is to be rejected when, in fact, it is false. • This is 1-P(Type II error) • This probability depends on the particular value of the parameter in Ha.

  8. Power for β1 • H0: β1 = 0, Ha: β1  0 t* = b1/s(b1) • When H0 true, t* ~ t(n-2) • We reject H0 when |t*| t(1-/2,n-2)

  9. Power for β1 • To compute power, we need to find P(|t*| t(1-/2,n-2)) for arbitrary values of β1 • Note: When β1 = 0, calculation gives α

  10. Power for β1 • When H0 false, t*~ t(n-2,d). • This refers to the noncentral t distribution • δ= β1/ σ(b1) – noncentrality parameter • Need to assume values to get σ(b1) • Often use prior info or pilot study data

  11. Power for β1 • Need to assume values for s2,n, and • KNNL use tables, see pg 51 • We will use SAS

  12. Example of Power for β1 • From KNNL pg 51 • They assume σ2=2500, n=25, and based on s=48.82 and other results from pg 20 • Results in

  13. Example of Power for β1 • Suppose β1 were 1.5 • We can calculate δ= β1/ σ(b1) and use the distribution t~ t(n-2,δ) to find P(|t*|  t(1-/2,n-2)) • We will use a function to calculate this probability

  14. SAS CODE data a1; n=25; sig2=2500; ssx=19800; alpha=.05; beta1=1.5; sig2b1=sig2/ssx; df=n-2; delta=beta1/sqrt(sig2b1); t_c=tinv(1-alpha/2,df); power=1-probt(t_c,df,delta) +probt(-t_c,df,delta); output; proc print data=a1; run;

  15. SAS OUTPUT Obs n sig2 ssx alpha 1 25 2500 19800 0.05 sig2b1 df beta1 delta 0.12626 23 1.5 4.22137 t_c power 2.06866 0.98121

  16. SAS CODE *Computes power for range of beta1; data a2; n=25; sig2=2500; ssx=19800; alpha=.05; sig2b1=sig2/ssx; df=n-2; t_c=tinv(1-alpha/2,df); do beta1=-2.0 to 2.0 by .05; delta=beta1/sqrt(sig2b1); power=1-probt(t_c,df,delta) +probt(-t_c,df,delta); output; end;

  17. SAS CODE title1 'Power for the slope in Simple linear regression'; symbol1 v=none i=join; proc gplot data=a2; plot power*beta1; proc print data=a2; run;

  18. Background Reading • File knnl051.sas contains the SAS code used in this Topic (addresses example on page 51) • Chapter 2 • 2.4 : Estimation of E(Yh) • 2.5 : Prediction of new observation

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