Last updated: 2019-06-16

Checks: 2 0

Knit directory: stats topics/

This reproducible R Markdown analysis was created with workflowr (version 1.4.0). The Checks tab describes the reproducibility checks that were applied when the results were created. The Past versions tab lists the development history.

R Markdown file: up-to-date

Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.

Repository version: d531670

Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated.

Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use wflow_publish or wflow_git_commit). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated: 


working directory clean

Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.

These are the previous versions of the R Markdown and HTML files. If you’ve configured a remote Git repository (see ?wflow_git_remote), click on the hyperlinks in the table below to view them.

File Version Author Date Message
html d531670 Zhengyang Fang 2019-06-16 Build site.
html a2ad691 Zhengyang Fang 2019-06-16 Host with GitLab.
Rmd ffc3849 Zhengyang Fang 2019-06-16 start

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 \(\alpha\))\(=\alpha\).