Understanding Hypothesis Testing in Biostatistics: Estimation, Testing, and Errors - Prof., Study notes of Biostatistics

Download Understanding Hypothesis Testing in Biostatistics: Estimation, Testing, and Errors - Prof. and more Study notes Biostatistics in PDF only on Docsity! 1 BIOM601 Fall 2008 Hypothesis Testing Lecture 6 (September 18, 2008) BIOM601 Fall 2008 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 Fall 2008 Establishing a Hypothesis • Hypotheses are established for parameters – µ, σ2, β1 • Hypotheses are tested with statistics s2 by– , , 1 • Null hypothesis (H0): status quo, no difference, no effect – H0: σ2=30 – H0: µ1=µ2 – H0: σ21=σ22=σ23=σ24 BIOM601 Fall 2008 The Alternative Hypothesis • Alternative hypothesis (Ha): indicator of a change, a difference exists, 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 either population could have greater µ – Ha: µ1>µ2 when µ1 cannot be smaller than µ2 BIOM601 Fall 2008 2 Testing the Hypothesis • Choice of the test statistic • Statistics are random variables • Test statistics have sampling variances • Determine the probability value • Decision is always made for H0 – Accept H0 – Reject H0 (consequently accept Ha) BIOM601 Fall 2008 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 Fall 2008 Decision Truth ↓ Accept H0 Reject H0 Decision Error Table H0 true Correct Type I Error H0 false Type II Error Correct BIOM601 Fall 2008 Test Statistics • Parameters (function of parameters) have an adequate test statistic • Test statistic built based Parameter Sampling distribution µ Student’s t on sampling distribution • Uniqueness of test statistic • Assumption on random variable (Y) µ1 - µ2 Student’s t σ2 Chi-square σ2a / σ2b Snedecor’s F β1 Student’s t BIOM601 Fall 2008