Lecture10 - 30 May 2003 Biostatistics 6650-L10 1 Todays...

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30 May 2003 Biostatistics 6650--L10 1
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30 May 2003 Biostatistics 6650--L10 2 Today’s Schedule Hypothesis Testing: Two Sample Tests General approach Two-sample t-test Large sample Binomial
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30 May 2003 Biostatistics 6650--L10 3 General Approach Comparing Two Populations Two scenarios: Dependent populations Independent populations Dependent Paired test Takes advantage of the dependencies (lack of independence) in the data paired t-test for normal X Independent Two-sample test Assumes both populations are independent
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30 May 2003 Biostatistics 6650--L10 4 General Approach Recall…. Two samples are independent if the data for one sample are collected in a manner unrelated to that for the other sample. Can arrive at two independent samples in several ways Draw separate random samples from each of two disjoint populations; e.g. male and female diabetics. Draw a random sample from one population, and then subdivide into two disjoint groups; e.g. sample patients with renal failure and compare those with to those without a history of diabetes Draw a random sample from a single population and randomly assign each unit to one of two treatments.
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30 May 2003 Biostatistics 6650--L10 5 General Approach Testing a hypothesis comparing two independent populations Null Hypothesis: • H o : μ 1 = μ 2 or, H o : μ 1 - μ 2 =0 – Where μ 1 and μ 2 are the population means for population 1 and 2, respectively Alternative Hypothesis: • H a : μ 1 = μ 2 or, H a : μ 1 - μ 2 =0 – Where μ 1 and μ 2 are the population means for population 1 and 2, respectively
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30 May 2003 Biostatistics 6650--L10 6 General Approach • Linear combinations(aX 1 +bX 2 + . .., a and b are constants) of independent normal random variables are normal with The difference in sample means is such a combination with a=1, b=-1. Hence… and 1 2 Random Variable is X X - 2 2 1 2 1 2 1 2 1 2 N( - , ) X X n n σ μ μ - + : 2 2 2 2 1 2 1 2 mean=aμ +bμ . .. and var=a σ +b σ +. .. . 1 2 1 2 2 2 1 1 2 2 Var(X -X )=Var(X )+Var(X )+0 =σ /n +σ /n
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30 May 2003 Biostatistics 6650--L10 7 General Approach The CLT implies the above is true no matter what the distribution of X, as long as “n” is large. This implies that. .. 2 2 1 2 1 2 1 2 1 2 N( - , ) X X n n σ μ μ - + : o obs 1 / 2 1 2 1 2 1 2 1 2 1 2 obs o 2 2 2 2 x -x 1 2 x x 1 2 Reject H at the two-sided significance level if |z | (x -x )-0 x -x x -x z = = = N(0, 1) under H SE σ σ SE +SE + n n Z α - : &
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30 May 2003 Biostatistics 6650--L10 8 General Approach • Approximate 100(1-0)% CI for 0 1 -0 2 is given by: 1 2 2 2 1 2 1 2 1- /2 1 2 1 2 1- /2 ( ) z or ( ) z x x x x n n x x SE α σ - - ± + - ±
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30 May 2003 Biostatistics 6650--L10 9 General Approach Calculate n 1 and n 2 to yield a 100(1-0)% CI for 0
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Lecture10 - 30 May 2003 Biostatistics 6650-L10 1 Todays...

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