Biostatistics: Parameter Estimation and Hypothesis Testing, Exams of Biostatistics

Download Biostatistics: Parameter Estimation and Hypothesis Testing and more Exams Biostatistics in PDF only on Docsity! BIOM601 Summer 2009 Hypothesis Testing Lecture 6 (June 8, 2009) BIOM601 Summer 2009 Key Elements of Inference • Parameter Estimation – Frequentist approach • Based exclusively on sample information • Ordinary least squares or maximum likelihood – Bayesian approach • Define prior, observe data, obtain posterior • Hypothesis Testing – Statistical tools – Assumptions BIOM601 Summer 2009 Establishing a Hypothesis • Hypotheses are established for parameters – μ, σ2, β1 • Hypotheses are tested with statistics – , s2, b1 • Null hypothesis (H0): status quo, no difference, no effect – H0: σ2=30 – H0: μ1= μ2 y BIOM601 Summer 2009 The Alternative Hypothesis • Alternative hypothesis (Ha): indicator of a difference, presence of an effect – Ha: σ2>30 – Ha: μ1 ≠ μ2 – Ha: At least one pair σ2i=σ2j, i≠j • One-tailed or two-tailed option – Ha: μ1 ≠ μ2 if no anticipated directionality – Ha: μ1 > μ2 if μ1 cannot be smaller than μ2 BIOM601 Summer 2009 Testing the Hypothesis • Choice of the test statistic • Statistics are random variables • Test statistics have sampling variances • Determine the probability of the statement • Decision is always made for H0 – Accept H0 – Reject H0 (consequently accept Ha) – “Accepting H0”: failure to reject? BIOM601 Summer 2009 Decision Errors • Type I Error – Reject a true hypothesis – Probability α (known) • Type II Error – Accept a false hypothesis – Probability β (unknown) • α and β inversely proportional – α chosen arbitrarily – β influenced by sample size, experimental design BIOM601 Summer 2009 Decision Truth ↓ Accept H0 Reject H0 H0 true Correct Type I Error H0 false Type II Error Correct Decision Error Table BIOM601 Summer 2009 Test Statistics • Parameters (function of parameters) have a corresponding test statistic • Test statistic built based on sampling distribution • Test statistic unique! • Assumption on the response variable (Y) Parameter Sampling distribution μ Student’s t μ1 - μ2 Student’s t σ2 Chi-square σ2a / σ2b Snedecor’s F β1 Student’s t BIOM601 Summer 2009