《多元统计分析》课程教学资源（阅读材料）Multiple hypothesis testing

ONTNU Multiple hypothesis testing-recent developments and future challenges Ingelin Steinsland ingelins@math.ntnu.no Norwegian University of Science and Technology Multiple hypothesis testing-recent developments and future challenges-p.1/28

NTNUMultiple hypothesis testing - recent developments and future challenges Ingelin Steinsland ingelins@math.ntnu.no Norwegian University of Science and Technology Multiple hypothesis testing - recent developments and future challenges – p.1/28

ONTNU Outline Single hypothesis testing Multiple hypothesis testing Quantities and issues 0 False discovery rates ● Future challenges .Within false discovery rates. Multiple hypothesis tests,the right tool? Multiple hypothesis testing-recent developments and future challenges-p.2/28

NTNU Outline Single hypothesis testing Multiple hypothesis testing Quantities and issues False discovery rates Future challenges Within false discovery rates. Multiple hypothesis tests, the right tool? Multiple hypothesis testing - recent developments and future challenges – p.2/28

ONTNU Single hypothesis testing,example Typical question:Does treatment A give the wished effect? Hypothesis: H=0:Non or negative effect. H=1:Positive effect Multiple hypothesis testing-recent developments and future challenges-p.3/28

NTNU Single hypothesis testing, example Typical question: Does treatment A give the wished effect? Hypothesis: H = 0: Non or negative effect. H = 1: Positive effect Multiple hypothesis testing - recent developments and future challenges – p.3/28

ONTNU Single hypothesis testing,example Typical question:Does treatment A give the wished effect? Hypothesis: I=0:Non or negative effect. H-1:Positive effect Collect data. IF the collected data is very unlikely given H=0; H=0 rejected and H=1 accepted. Treatment A has positive effect. ELSE H=0 accepted. ● Treatment A does not have significant positive effect. Multiple hypothesis testing-recent developments and future challenges-p.3/28

NTNU Single hypothesis testing, example Typical question: Does treatment A give the wished effect? Hypothesis: H = 0: Non or negative effect. H = 1: Positive effect Collect data. IF the collected data is very unlikely given H = 0; H = 0 rejected and H = 1 accepted. Treatment A has positive effect. ELSE H = 0 accepted. Treatment A does not have significant positive effect. Multiple hypothesis testing - recent developments and future challenges – p.3/28

ONTNU Single hypothesis testing ●Hypothesis test: pH=0:0∈曰o versus 。H=1:0∈Θ1（Θ0∩Θ1=0） Test statistics:T(X),observed t. Multiple hypothesis testing-recent developments and future challenges-p.4/28

NTNU Single hypothesis testing Hypothesis test: H = 0 : θ ∈ Θ0 versus H = 1 : θ ∈ Θ1 (Θ0 ∩ Θ1 = ∅). Test statistics: T(X), observed t. Multiple hypothesis testing - recent developments and future challenges – p.4/28

ONTNU Single hypothesis testing ●Hypothesis test: pH=0:0∈曰o versus 。H=1:0∈Θ1（Θ0∩Θ1=0） Test statistics:T(X),observed t. ●Rejection region:T oIft∈T reject H=O. .IftT accept H=0. Multiple hypothesis testing-recent developments and future challenges-p.4/28

NTNU Single hypothesis testing Hypothesis test: H = 0 : θ ∈ Θ0 versus H = 1 : θ ∈ Θ1 (Θ0 ∩ Θ1 = ∅). Test statistics: T(X), observed t. Rejection region: Γ If t ∈ Γ reject H = 0. If t ∈/ Γ accept H = 0. Multiple hypothesis testing - recent developments and future challenges – p.4/28

ONTNU Single hypothesis testing 0 Rejection region:D pIft∈T reject H=O. .Ift T accept H=0. Multiple hypothesis testing-recent developments and future challenges-p.4/28

NTNU Single hypothesis testing Γ α f T|H=0 t Rejection region: Γ If t ∈ Γ reject H = 0. If t ∈/ Γ accept H = 0. Multiple hypothesis testing - recent developments and future challenges – p.4/28

ONTNU Single hypothesis testing Two types of errors: accept Ho reject Ho Ho type-I error Hu type-II error .Type I error(false positive),θ∈yet t∈T. TypeⅡerror(false negative),O∈Θ1 yett Multiple hypothesis testing-recent developments and future challenges-p.4/28

NTNU Single hypothesis testing Γ α f T|H=0 t Two types of errors: accept H0 reject H0 H0 type-I error H1 type-II error Type I error (false positive), θ ∈ Θ0 yet t ∈ Γ. Type II error (false negative), θ ∈ Θ1 yet t ∈/ Γ Multiple hypothesis testing - recent developments and future challenges – p.4/28

ONTNU Single hypothesis testing Want to control type I error rate; Pr(t∈TH=0), and minimise type II error rate; Pr(tH=1). ·Power=1-Pr(tTH=1) Multiple hypothesis testing-recent developments and future challenges-p.4/28

NTNU Single hypothesis testing Γ α f T|H=0 t Want to control type I error rate; Pr(t ∈ Γ|H = 0), and minimise type II error rate; Pr(t ∈/ Γ|H = 1). Power = 1 − Pr(t ∈/ Γ|H = 1). Multiple hypothesis testing - recent developments and future challenges – p.4/28

ONTNU Single hypothesis testing 。Significant level a=Pr(t∈TlH=O) Multiple hypothesis testing-recent developments and future challenges-p.4/28

NTNU Single hypothesis testing Γ α f T|H=0 t Significant level α = Pr(t ∈ Γ|H = 0). Multiple hypothesis testing - recent developments and future challenges – p.4/28