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Unformatted text preview: Public Health 6450 – Fall 2011 Andrew Mugglin and Lynn Eberly Division of Biostatistics School of Public Health University of Minnesota [email protected] Part 10 Review Revisting Important Concepts Errors in Significance Testing Power in Significance Testing Where are we going? Previously: • Tests and CIs for μ : σ known (use z and σ , Chapter 6.16.2) • Tests and CIs for μ : σ unknown (use t and s , Chapter 7.1) Current topics: • Revisiting some important concepts (Chapter 6.2) • Errors and power in significance testing (Chapter 6.36.4) Mugglin and Eberly PubH 6450 Fall 2011 Part 10 2 / 33 Review Revisting Important Concepts Errors in Significance Testing Power in Significance Testing Test statistics and pvalues The pvalue can be computed in one of two ways. For example, consider H : μ = μ vs. H : μ > μ : • Extreme ¯ X values: p = Pr( ¯ X as or more extreme  H true) • Extreme test statistic z (or t ) values: p = Pr( z as or more extreme  H true) p = Pr( t as or more extreme  H true) These approaches are equivalent because the test statistic z is a function of ¯ X (we standardize ¯ X to a Z in order to do this calculation). We do likewise for a t teststatistic. Mugglin and Eberly PubH 6450 Fall 2011 Part 10 3 / 33 Review Revisting Important Concepts Errors in Significance Testing Power in Significance Testing Test statistics and pvalues Each of these can be represented graphically: μ Distribution of X X pvalue Distribution of test stat. z z pvalue Distributions under H : μ = μ for testing H a : μ μ Mugglin and Eberly PubH 6450 Fall 2011 Part 10 4 / 33 Review Revisting Important Concepts Errors in Significance Testing Power in Significance Testing Significance tests come from rejection rules Similarly, a significance test (comparing a pvalue to α ) is equivalent to a rejection rule (comparing the test statistic to a Z or t n 1 value): Again for H : μ = μ vs. H : μ > μ , these are all equivalent: • Reject H when the pvalue is small ( < α ). • Reject H when ¯ X is extreme. • Reject H when the test statistic is extreme. What is ‘extreme’ for the test statistic z (or t )? Mugglin and Eberly PubH 6450 Fall 2011 Part 10 5 / 33 Review Revisting Important Concepts Errors in Significance Testing Power in Significance Testing Significance tests come from rejection rules (cont’d) First, choose α and find the z * that cuts off the righttail AUC: Pr( Z > z * ) = α . Then, reject H : μ = μ when the test statistic z = ¯ X μ σ/ √ n > z * or equivalently when p < α . Is p < α ? Is z > z * ? D is trib u tio n o f te s t s ta t. z z x α D is trib u tio n o f te s t s ta t. z z p  v a lu e D is trib u tio n s u n d e r H : μ = μ fo r te s tin g H a : μ μ Mugglin and Eberly PubH 6450 Fall 2011 Part 10 6 / 33 Review Revisting Important Concepts Errors in Significance Testing Power in Significance Testing Metabolic changes due to HAART Scientific question of interest: Do HIV+ patients who start a...
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 Fall '10
 AndyMugglin
 Type I and type II errors, significance testing, Eberly

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