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index.Rmd
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---
title: "Truncated Adaptive Shrinkage (`truncash`)"
output:
html_document:
toc: false
---
`truncash` (Truncated ASH) is an exploratory project with Matthew, built on [`ashr`].
* [Matthew's initial observation on null, correlated data](voom_null.html)
[Prof. Matthew Stephens](http://stephenslab.uchicago.edu/) did a quick investigation of the p values and z scores obtained for simulated null data (using just voom transform, no correction) from real RNA-seq data of [GTEx](http://www.gtexportal.org/home/). Here is what he found.
"I found something that I hadn’t realized, although is obvious in hindsight: although you sometimes see inflation under null of $p$-values/$z$-scores, the most extreme values are not inflated compared with expectations (and tend to be deflated). That is the histograms of $p$-values that show inflation near $0$ (and deflation near $1$) actually hide something different going on in the very left hand side near $0$. The qq-plots are clearer… showing most extreme values are deflated, or not inflated. This is expected under positive correlation i think. For example, if all $z$-scores were the same (complete correlation), then most extreme of n would just be $N(0,1)$. but if independent the most extreme of n would have longer tails..."
Matthew's initial observation inspired this project. If under positive correlation, the most extreme tend to be not inflated, maybe we can use them to control the false discoveries. Meanwhile, if the moderate are more prone to inflation due to correlation, maybe it's better to make only partial use of their information.
* [Occurrence of extreme observations](ExtremeOccurrence.html)
As [Prof. Michael Stein](https://galton.uchicago.edu/~stein/) pointed during a conversation with Matthew, if the marginal distribution is correct then the expected number exceeding any threshold should be correct. So if the tail is "usually"" deflated, it should be that with some small probability there are many large $z$-scores (even in the tail). Therefore, if "on average" we have the right number of large $z$-scores/small $p$-values, and "usually" we have too few, then "rarely" we should have too many. A simulation is run to check this intuition.
* [Two FWER-controlling procedures on correlated null](StepDown.html)
In order to understand the behavior of $p$-values of top expressed, correlated genes under the global null, simulated from GTEx data, we apply two FWER-controlling multiple comparison procedures, Holm's "step-down" ([Holm 1979]) and Hochberg's "step-up." ([Hochberg 1988])
* [`truncash` Model and first simulations](truncash.html)
* [Pipeline for simulating null data](nullpipeline.html)
Using a toy model to examine and document the pipeline to simulate null summary statistics at each step, including `edgeR::calcNormFactors`, `limma::voom`, `limma::lmFit`, `limma::eBayes`.
* [FDR on Null, Part 1](FDR_Null.html)
* [FDR on Null, Part 2](FDR_null_betahat.html)
Apply two FDR-controlling procedures, BH and BY, as well as two $s$ value models, `ash` and `truncash` to the simulated, correlated null data, and compare the numbers of false discoveries (by definition, all discoveries should be false) obtained. Part 1 uses $z$ scores only, Part 2 uses $\hat \beta$ and moderated $\hat s$.
* [$\hat\pi_0$ estimated in correlated global null](pihat0_null.html)
$\hat\pi_0$ estimated by `ash` and `truncash` with $T = 1.96$ on correlated global null data simulated from GTEx/Liver. Ideally they should be close to $1$.
* [Ordered $p$ values vs critical values](cutoff_null.html)
For various FWER / FDR controlling procedures, and for `truncash`, what matters the most is the behavior of the most extreme observations. Here these extreme $p$ values are plotted along with common critical values used by various procedures, in order to shed light on their behavior.
It's very exploratory. May be related to Extreme Value Thoery and Concentration of Measure. To be continued.
* [Single most extreme observation](SingleExtOb.html)
What will happen if we allow the threshold in `truncash` dependent on data? Let's start from the case when we only know the single most extreme observation.
* [Handling $t$ likelihood](t-likelihood.html)
When moving to $t$ likelihood, or in other words, when taking randomness of $\hat s$ into consideration, things get trickier. Here we review several techiniques currently used in Matthew's lab, regarding $t$ likelihood and uniform mixture priors, based on a discussion with Matthew.
* [Histogram of correlated $z$ scores, random data sets](correlated_z.html)
* [Histogram of correlated $z$ scores, `ash`-hostile data sets](correlated_z_2.html)
* [Histogram of correlated $z$ scores, `BH`-hostile data sets](correlated_z_3.html)
An implicit key question of this inquiry is: what's the behavior of $z$ scores under dependency? We take a look at their histograms. First for randomly sampled data sets. Second for those most "hostile" to `ash`. Third for those most "hostile" to `BH`. The bottom line is we are reproducing what Efron observed in microarray data, that "the theoretical null may fail" in different ways. Now the key questions are
1. Why the theoretical null may fail? What does it mean by **correlation**?
2. Can `truncash` make `ash` more robust against some of the foes that make the theoretical null fail?
3. Generally, how robust is empirical Bayes? Is empirical Bayes robust or non-robust to certain kinds of correlation?
* [Fitting empirical null with Gaussian derivatives: Theory](gaussian_derivatives.html)
* [Fitting empirical null with Gaussian derivatives: Examples](gaussian_derivatives_2.html)
* [Fitting empirical null with Gaussian derivatives: Numerical issues](gaussian_derivatives_3.html)
* [Fitting empirical null with Gaussian derivatives: Large correlations](gaussian_derivatives_4.html)
* [Fitting empirical null with Gaussian derivatives: Delta function](gd_delta.html)
Inspired by [Schwartzman 2010](http://amstat.tandfonline.com/doi/abs/10.1198/jasa.2010.tm10237), we experiment a new way to tackle "empirical null."
* [Fitting empirical null with Gaussian derivatives: Weight constraints](gaussian_derivatives_5.html)
In Gaussian derivatives decomposition, weights $W_k$ and especially [normalized weights](gaussian_derivatives_5.html#numerical_issues) $W_k^s = W_k\sqrt{k!}$ contain substantial information. In order to produce a proper density, and in order to regularize the fitted density to make it describe a plausible empiricall correlated null, constraints need to be imposed on the weights.
* [True signal vs correlated null: small effects](alternative.html)
* [True signal vs correlated null: larger effects](alternative2.html)
Both true effects and correlation can distort the empirical distribution away from the standard normal $N(0, 1)$, and Gaussian derivatives are presumably able to fit both. Therefore, is there a way to let Gaussian derivatives with [a reasonable number of reasonable weights](gaussian_derivatives_5.html#weight_constraints) automatically identify correlated null from true effects? It works well when effects are *not too small* right now.
* [Fitting $N\left(0, \sigma^2\right)$ with Gaussian derivatives](fitting_normal.html)
Continuing from the empirical studies above, we are looking at why $N\left(0, \sigma^2\right)$ when $\sigma^2$ is small can be satisfactorily fitted by a limited number of Gaussian derivatives.
* [Diagnostic plots for `ASH`](diagnostic_plot.html)
* [Diagnostic plots for `ASH` on correlated null](diagnostic_correlated_z.html)
* [Diagnostic plots for `ASH` on correlated null: `BH` vs $\hat\pi_0$](diagnostic_correlated_z_2.html)
* [Diagnostic plots for `ASH` on correlated null: `BH` vs `lfsr`](diagnostic_correlated_z_3.html)
Under the exchangeability assumption, the goodness of fit of empirical Bayes can be measured by the behavior of $\left\{\hat F_j = \hat F_{\hat\beta_j | \hat s_j}\left(\hat\beta_j\mid\hat s_j\right)\right\}$. Meanwhile, if `ASH` is applied to correlated null $z$ scores, estimated $\hat g$ will not only be different from $\delta_0$; moreover, with this estimated $\hat g$, $\left\{\hat F_j\right\}$ might not behave like $\text{Unif}\left[0, 1\right]$.
* [Diagnostic plots for uniform distribution](diagnostic_plot_2.html)
* [Diagnostic plots for `ASH`](diagnostic_ash.html)
* [`ashr::plot_diagnostic()` in `ashr`](plot_diagnostic.html)
Essentially we hope to tell whether $n$ random samples are uniformly distributed, and the task seems more complicated than what it sounds. There are multiple ways to do it, and some of them have been absorbed into the `ashr` package.
* [Marginal distribution of $z$ scores: Correlated null](marginal_z.html)
* [Marginal distribution of $z$ scores: Correlated non-null](marginal_z_alternative.html)
Take a look at the null $z$ scores we've been experimenting with so far and check whether they truly are marginally $N\left(0, 1\right)$. It's an enhanced version of the simulation on [occurrence of extreme observations](ExtremeOccurrence.html). The non-null $z$ scores are produced for comparison.
* [Can we create correlated null from the raw data?](create_null.html)
This whole project comes from a ubiquitous issue that in the simultaneous inference problem, the deviation from what would be expected under the null can come from both distortion due to correlation and enrichment due to true effects. Ideally, it would be perfect if we could create controls which keep the distortion by correlation unchanged, but remove the true effects. Efron in a series of papers (for example, [Efron et al 2001](http://www.tandfonline.com/doi/abs/10.1198/016214501753382129)) suggests to create the null from the raw data by resampling or permutation. We apply the idea to our data sets, and the results are not promising.
* [Normal means inference with heteroskedastic and correlated noise: Model](ash_gd.html)
The previous investigation is boiled down to a prototypical problem: the classic normal means inference, but with heteroskedastic and especially correlated noise. Gaussian derivatives and `ASH` provide a way to tackle it.
* [Implementation by `Rmosek`: Correlated null](Rmosek.html)
* [Implementation by `Rmosek`: Simulated non-null](gd-ash.html)
`cvxr` is still under active development. We are moving to the more established `Rmosek`, exploring and experimenting this convex optimization package.
* [Improvement on implementation with `Rmosek`: Primal and dual](mosek_reg.html)
* [Improvement on implementation with `Rmosek`: Normalization](mosek_reg_2.html)
* [Improvement on implementation with `Rmosek`: Regularization](mosek_reg_3.html)
Several ways are explored to make fitting Gaussian derivatives more efficient and accurate.
* [Regularized Gaussian Derivatives: Correlated null](mosek_reg_4.html)
* [Regularized Gaussian Derivatives & `ASH`: Simulated true effects with homoscedastic correlated noises](mosek_reg_5.html)
* [Regularized Gaussian Derivatives & `ASH`: Simulated true effects with heteroskedastic correlated noises](mosek_reg_6.html)
Fitting the corrrelated null with Gaussian derivatives has two constraints: non-negativity and orderly decay, and both are intractable. We are using $L_1$ regularization to try to solve both.
* [Simulated signals added to real data: Part 1](real_data_simulation.html)
* [Simulated signals added to real data: Part 2](real_data_simulation_2.html)
* [Simulated signals added to real data: Part 3](real_data_simulation_3.html)
* [Simulated signals added to real data: Part 4](real_data_simulation_4.html)
Using the `seqgendiff` pipeline developed by [David](http://dcgerard.github.io/), [Mengyin](https://github.com/mengyin), and Matthew, we are adding simulated $\pi_0\delta_0 + \left(1 - \pi0\right)N\left(0, \sigma^2\right)$ signals to GTEx/Liver RNA-seq expression matrix, and compare our method with `ASH` and [`SVA`](http://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.0030161).
* [Completely synthetic correlated null](simulated_correlated_null.html)
* [Completely synthetic correlated null: More complicated cases](simulated_correlated_null_2.html)
Following [Gao](http://home.uchicago.edu/gaow/)'s suggestion, we are taking a look at whether Gaussian derivatives can satisfactorily fit synthetic correlated $N\left(0, 1\right)$ $z$ scores. Suggested by Matthew, **it might have something to do with the distribution of eigenvalues in a random matrix**, and the [Wigner semicircle distribution](https://en.wikipedia.org/wiki/Wigner_semicircle_distribution). It's also an interesting comparison with Gaussian mixtures.
* [Simulated true effects with realistic correlated noises: Large and moderate SNR](simulation_real_se.html)
* [Simulated true effects with realistic correlated noises: Small SNR](simulation_real_se_2.html)
The large-scale simulation testing for GD-ASH with simulated effects under different sparsity, strength, and shape, and realistic correlated noises.
[`ashr`]: https://github.com/stephens999/ashr
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