– chap10_false_discoveries.pptx

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Comparison of FWER and FDR

FWER is appropriate when it is important to avoid any error.

But an FWER procedure such as Bonferroni makes many Type II errors and thus, has poor power.

An FWER approach has very a very false discovery rate

FDR is appropriate when it is important to identity positive results, i.e., those belonging to the alternative distribution.

By construction, the false discovery rate is good for an FDR procedure such as the BH approach

An FDR approach also has good power

02/14/2018

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Introduction to Data Mining, 2nd Edition

0.08

0.07

(Probability Density)0.06

0.05

0.04

0.03

0.02

0.01

0.08

0.07

(Probability Density)0.06

0.05

0.04

0.03

0.02

0.01

20 30 40

50 60 70 80

R

20 30 40

50 60 70 80

R

Distribution of test statistic for the alternative hypothesis (rightmost density curve) and null hypothesis (leftmost density curve). Shaded region in right subfigure is .

0.08

0.07

(Probability Density)0.06

0.05

0.04

0.03

0.02

0.01

0.08

0.07

(Probability Density)0.06

0.05

0.04

0.03

0.02

0.01

20 30 40

50 60 70 80

R

20 30 40

50 60 70 80

R

Shaded region in left subfigure is β and shaded region in right subfigure is power.

Confusion table for summarizing multiple hypothesis testing results.

Declared significant

(+ prediction)

Declared not significant

(– prediction)

Total

H1 True

(actual +)

True Positive (TP)

False Negative (FN)

type II error

Positives

(m1 )

H0 True

(actual –)

False Positive (FP)

type I error

True Negative (TN)

Negatives

(m0 )

Positive Predictions

(Ppred)

Negative Predictions

(Npred)

m

1

0.9

(Family−wise Error Rate (FWER))0.8

0.7

Naive Approach

Bonferroni

0.6

0.5

0.4

0.3

0.2

0.1

0.05

0 10 20 30 40 50 60 70 80 90 100

Number of Results, m

Benjamini-Hochberg (BH) FDR algorithm.

1: Compute p-values for the m results.

2: Order the p-values from smallest to largest (p1 to pm ).

(m)3: Compute the significance level for pi as αi = i × α .

4: Let k be the largest index such that pk ≤ αk .

5: Reject H0 for all results corresponding to the first k p-values, pi , 1 ≤ i ≤ k.

0.12

(False Discovery Rate (FDR))0.1

0.08

Naive Approach

Bonferroni

BH Procedure

0.06

0.05

0.04

0.02

0102030405060708090100

Number of stockbrokers, m

False Discovery Rate as a function of m.

1

(Expected Power (True Positive Rate))0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Naive Approach

Bonferroni

BH Procedure

0102030405060708090100

Number of stockbrokers, m

Expected Power as function of m.

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