Last updated: 2019-06-19

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General theory

Common concerns about validity of data analysis

  1. question asked \(\neq\) data measured / analysis performed

  2. confounding variables

  3. Analysis assumes a model that is false

  4. The data is used to guide choices made in the analysis

  5. Multiple comparisons

Different types of questions

  1. Binary - hypothesis testing

    • Test if X is associated with Y.
    • Multiple testing issues: are \(X,Y\) fixed in advance? Transforming X or Y?

  2. Parameter estimation

    • Choice of model (similar issues in 1)
    • Assuming a false model

  3. Prediction

    • If model / other choices are data-dependent, may encounter over-fitting issues.

  4. Situations where we aren’t targeting a single number

    • e.g. Clustering (can often overfit)

Topic I. Hypothesis testings

  • Assuming:

    • Null hypothesis is fixed (not data dependent)
    • Can compute valid p-values for testing each null, and the function mapping to p-value is fixed in advance.

  • Formulation of Multiple testing problem:

    • There are a list of null hypothesis \(H_{o1},H_{o2},\dots,H_{on}\), and the p-values for each hypothesis are \(p_1,p_2,\dots,p_n\).

    • A Testing procedure is a function mapping \(\{p_1,\dots,p_n\}\) to a subset of \(\{1,2,\dots,n\}\) (discoveries/rejections).

  • Possible goals to control false positives:

    1. FamilyWise Error Rate (FWER): Bound \(\mathbb P\)(any false discoveries)

    2. Bound \(\mathbb P\)(more than … % false discoveries)

    3. Bound \(\mathbb P\)(# false discoveries)

    4. Bound \(\mathbb P\)(proportion of false discoveries(FDP) )

    5. Bound \(\mathbb P\)(false discoveries rate (FDR) )

1, 4, 5 are more common (they have abbreviation!)

Sub-question 1. Global null hypothesis

  • Goal: test \(H_{o,global}=\) all n nulls are true \((=\bigcap_{i=1,2,\dots,n}H_{oi})\).

    • Notice: if \(H_{o,global}\) is true, then every \(P_i\sim unif(0,1)\) (but may not be independent)

Method 1. Bonferroni correction

Statistic \(= \min\{p_1,p_2,\dots,p_n\}.\)

Reject if \(\min_i \{p_i\}\leq c\), where \(c = \frac \alpha n\), \(\alpha\) is the level of test.

Proof (upperbound of type_I error): \(\mathbb P(\min_i p_i\leq c)\leq \sum_i\mathbb P(p_i\leq c)=nc\), using the property \(\mathbb P(A\cup B)\leq \mathbb P(A)+\mathbb P(B)\).

Very conservative: the bound in the proof is hard to reach.

Method 2. Fisher’s test

Statistic \[F = -2\sum_i \log(p_i).\]

Under \(H_o,global\), \(p_i\sim unif (0,1)\), if additionally \(p_i\)’s are independent, then \(F\sim \chi_{2n}^2\).

Reject if \(F\geq (1-\alpha)\) -quantile of \(\chi^2_{2n}\).

3. Simes test:

  • Reject \(H_{o,global}\) if:

    • At least 1 \(p_i\leq\frac\alpha n\)
    • and/or at least 2 \(p_i\leq\frac{2\alpha} n\)
    • \(\vdots\)
    • and/or at least n \(p_i\leq\frac{n\alpha} n\)

  • Theorem: Under \(H_{o,global}\), if \(p_i\)’s are independent. Then \(\mathbb P\)(Simes rejects at level \(\alpha\))\(=\alpha\).