note10 - Chapter 8 Nonparametric Bootstrap Methods 4...

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Unformatted text preview: Chapter 8: Nonparametric Bootstrap Methods 4. Two-Sample Inference • the problem of making a confidence interval for the difference between the means of two populations using the bootstrap method • Notation Y ij = μ i + ǫ ij ∼ F i ( y ) j = 1 , 2 , . . ., n i i = 1 , 2 • t-Pivot Method Assuming Equality of Error Distributions – ǫ ij ∼ F ( ǫ ) for j = 1 , 2 , . . ., n i i = 1 , 2 ⇒ F 1 ( y ) = F 2 ( y −△ ) where △ = μ 1 − μ 2 – if F ( ǫ ) is a cdf of normal distribution, homogeneity of variances – let t = ¯ Y 1 − ¯ Y 2 − ( μ 1 − μ 2 ) S p radicalbig 1 /n 1 + 1 /n 2 where S 2 p = 2 summationdisplay i =1 n i summationdisplay j =1 ( Y ij − ¯ Y i ) 2 n 1 + n 2 − 2 ⇒ t ǫ = ¯ ǫ 1 − ¯ ǫ 2 S ǫp radicalbig 1 /n 1 + 1 /n 2 where S 2 ǫp = 2 summationdisplay i =1 n i summationdisplay j =1 ( ǫ ij − ¯ ǫ i ) 2 n 1 + n 2 − 2 – if ǫ ij ∼ N (0 , σ 2 ), − t . 975 ,df = n 1 + n 2 − 2 < − ( μ 1 − μ 2 ) S p radicalbig 1 /n 1 + 1 /n 2 < t . 975 ,df = n 1 + n 2 − 2 – t as a t-pivot for making inferences for μ 1 − μ 2 – Bootstrap interval: if the errors do not have a normal distribution (a) compute e ij = Y ij − ¯ Y i (b) from the combined observed errors, randomly select n 1 and n 2 errors with replacement for group 1 and group 2 respectively: e ij,b ⇒ compute t ǫ,b using e ij,b ’s: t e,b with b = 1 , . . ., N repetition (c) the 95% bootstrap confidence interval: ¯ Y 1 − ¯ Y 2 − t e, . 975 S p radicalbigg 1 n 1 + 1 n 2 < μ 1 − μ 2 < ¯ Y 1 − ¯ Y 2 − t e, . 025 S p radicalbigg 1 n 1 + 1 n 2 where t e, . 025 and t e, . 975 denote the 2 . 5th and 97 . 5th percentiles of the bootstrap distribution of t e,b 1 Example The mouse data (Efron and Tibshirani, 1993). Sixteen mice were randomly assigned to a treatment group or a control group. Shown are their survival times, in days, following a test surgery. Treatment 94 197 16 38 99 141 23 Control 52 104 146 10 51 30 40 27 46--------------------------------------------- Basic Statistics from the original data Treatment Control Mean 86.86 56.22 Standard Deviation 66.77 42.48 Sample Size...
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note10 - Chapter 8 Nonparametric Bootstrap Methods 4...

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