Quantitative

Hypothesis Testing

Vincent A. Voelz E-mail: vvoelz@gmail.com

†

Math Bio Boot Camp 2006

University of California at San Francisco, San Francisco, CA 94143

August 30, 2006

Contents

1 Introduction to Statistical tests 1.1 1.2 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p-values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 4

2 Parametric Tests 2.1 2.2 2.3 2.4 2.5 2.6 Parametric versus Non-parametric Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The χ2 -test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The ‘Student’ t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The two-sample t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Likelihood Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Kolmogorov-Smirnov test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 5 6 7 7 8

3 Non-parametric tests 3.1 3.2 Wilcoxon rank-sum test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permutation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

9 10 10

1

INTRODUCTION TO STATISTICAL TESTS

2

3.3 3.4

Bootstrapping: resampling with replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . Jackknifing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 11

1

Introduction to Statistical tests

The underlying concept in statistics is quite easy to understand. A statistic is simply a computed value, θ({x1 , x2 , ...}), that characterizes a particular data sample {x1 , x2 , ...}. If we have some model of the theoretical probability distribution of this statistic, P (θ), we can assess its...