arXiv subjects

Art B. Owen

Publications and source records attributed to Art B. Owen.

At least 73 records · Page 4Linked to original sources

arXiv2018

Propensity Score Methods for Merging Observational and Experimental Datasets

This project considers how one might augment a limited amount of data from randomized controlled trial (RCT) with more plentiful data from an observational database (ODB), in order to estimate a causal effect. In our motivating setting, the ODB has better external validity, while the RCT has genuine randomization. We work with strata defined by the propensity score in the ODB. Subjects from the RCT are placed in strata defined by the propensity they would have had, had they been in the ODB. Our first method simply spikes the RCT data into their corresponding ODB strata. Our second method takes a data-driven convex combination of the ODB and RCT treatment effect estimates within each stratum. Using the delta method and simulations we show that the spike-in method works best when the RCT covariates are drawn from the same distribution as in the ODB. Our convex combination method is more robust than the spike-in to covariate-based inclusion criteria that bias the RCT data. We apply our methods to data from the Women's Health Initiative, a study of thousands of postmenopausal women which has both observational and experimental data on hormone therapy (HT). Using half of the RCT to define a gold standard, we find that a version of the spiked-in estimate yields stable estimates of the causal impact of HT on coronary heart disease.

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arXiv2017

Permutation p-value approximation via generalized Stolarsky invariance

It is common for genomic data analysis to use $p$-values from a large number of permutation tests. The multiplicity of tests may require very tiny $p$-values in order to reject any null hypotheses and the common practice of using randomly sampled permutations then becomes very expensive. We propose an inexpensive approximation to $p$-values for two sample linear test statistics, derived from Stolarsky's invariance principle. The method creates a geometrically derived set of approximate $p$-values for each hypothesis. The average of that set is used as a point estimate $\hat p$ and our generalization of the invariance principle allows us to compute the variance of the $p$-values in that set. We find that in cases where the point estimate is small the variance is a modest multiple of the square of the point estimate, yielding a relative error property similar to that of saddlepoint approximations. On a Parkinson's disease data set, the new approximation is faster and more accurate than the saddlepoint approximation. We also obtain a simple probabilistic explanation of Stolarsky's invariance principle.

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arXiv2017

A randomized Halton algorithm in R

Randomized quasi-Monte Carlo (RQMC) sampling can bring orders of magnitude reduction in variance compared to plain Monte Carlo (MC) sampling. The extent of the efficiency gain varies from problem to problem and can be hard to predict. This article presents an R function rhalton that produces scrambled versions of Halton sequences. On some problems it brings efficiency gains of several thousand fold. On other problems, the efficiency gain is minor. The code is designed to make it easy to determine whether a given integrand will benefit from RQMC sampling. An RQMC sample of n points in $[0,1]^d$ can be extended later to a larger n and/or d.

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arXiv2017

Statistically efficient thinning of a Markov chain sampler

It is common to subsample Markov chain output to reduce the storage burden. Geyer (1992) shows that discarding $k-1$ out of every $k$ observations will not improve statistical efficiency, as quantified through variance in a given computational budget. That observation is often taken to mean that thinning MCMC output cannot improve statistical efficiency. Here we suppose that it costs one unit of time to advance a Markov chain and then $θ>0$ units of time to compute a sampled quantity of interest. For a thinned process, that cost $θ$ is incurred less often, so it can be advanced through more stages. Here we provide examples to show that thinning will improve statistical efficiency if $θ$ is large and the sample autocorrelations decay slowly enough. If the lag $\ell\ge1$ autocorrelations of a scalar measurement satisfy $ρ_\ell\geρ_{\ell+1}\ge0$, then there is always a $θ<\infty$ at which thinning becomes more efficient for averages of that scalar. Many sample autocorrelation functions resemble first order AR(1) processes with $ρ_\ell =ρ^{|\ell|}$ for some $-1<ρ<1$. For an AR(1) process it is possible to compute the most efficient subsampling frequency $k$. The optimal $k$ grows rapidly as $ρ$ increases towards $1$. The resulting efficiency gain depends primarily on $θ$, not $ρ$. Taking $k=1$ (no thinning) is optimal when $ρ\le0$. For $ρ>0$ it is optimal if and only if $θ\le (1-ρ)^2/(2ρ)$. This efficiency gain never exceeds $1+θ$. This paper also gives efficiency bounds for autocorrelations bounded between those of two AR(1) processes.

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arXiv2017

Quasi-Monte Carlo for an Integrand with a Singularity along a Diagonal in the Square

Quasi-Monte Carlo methods are designed for integrands of bounded variation, and this excludes singular integrands. Several methods are known for integrands that become singular on the boundary of the unit cube $[0,1]^d$ or at isolated possibly unknown points within $[0,1]^d$. Here we consider functions on the square $[0,1]^2$ that may become singular as the point approaches the diagonal line $x_1=x_2$, and we study three quadrature methods. The first method splits the square into two triangles separated by a region around the line of singularity, and applies recently developed triangle QMC rules to the two triangular parts. For functions with a singularity `no worse than $|x_1-x_2|^{-A}$ for $0<A<1$ that method yields an error of $O( (\log(n)/n)^{(1-A)/2})$. We also consider methods extending the integrand into a region containing the singularity and show that method will not improve up on using two triangles. Finally, we consider transforming the integrand to have a more QMC-friendly singularity along the boundary of the square. This then leads to error rates of $O(n^{-1+ε+A})$ when combined with some corner-avoiding Halton points or with randomized QMC, but it requires some stronger assumptions on the original singular integrand.

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arXiv2017

On Shapley value for measuring importance of dependent inputs

This paper makes the case for using Shapley value to quantify the importance of random input variables to a function. Alternatives based on the ANOVA decomposition can run into conceptual and computational problems when the input variables are dependent. Our main goal here is to show that Shapley value removes the conceptual problems. We do this with some simple examples where Shapley value leads to intuitively reasonable nearly closed form values.

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arXiv2016

Admissibility in Partial Conjunction Testing

Meta-analysis combines results from multiple studies aiming to increase power in finding their common effect. It would typically reject the null hypothesis of no effect if any one of the studies shows strong significance. The partial conjunction null hypothesis is rejected only when at least $r$ of $n$ component hypotheses are non-null with $r = 1$ corresponding to a usual meta-analysis. Compared with meta-analysis, it can encourage replicable findings across studies. A by-product of it when applied to different $r$ values is a confidence interval of $r$ quantifying the proportion of non-null studies. Benjamini and Heller (2008) provided a valid test for the partial conjunction null by ignoring the $r - 1$ smallest p-values and applying a valid meta-analysis p-value to the remaining $n - r + 1$ p-values. We provide sufficient and necessary conditions of admissible combined p-value for the partial conjunction hypothesis among monotone tests. Non-monotone tests always dominate monotone tests but are usually too unreasonable to be used in practice. Based on these findings, we propose a generalized form of Benjamini and Heller's test which allows usage of various types of meta-analysis p-values, and apply our method to an example in assessing replicable benefit of new anticoagulants across subgroups of patients for stroke prevention.

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arXiv2016

Multibrand geographic experiments

In a geographic experiment to measure advertising effectiveness, some regions (hereafter GEOs) get increased advertising while others do not. This paper looks at running $B>1$ such experiments simultaneously on $B$ different brands in $G$ GEOs, and then using shrinkage methods to estimate returns to advertising. There are important practical gains from doing this. Data from any one brand helps to estimate the return of all other brands. We see this in both a frequentist and Bayesian formulation. As a result, each individual experiment could be made smaller and less expensive when they are analyzed together. We also provide an experimental design for multibrand experiments where half of the brands have increased spend in each GEO while half of the GEOs have increased spend for each brand. For $G>B$ the design is a two level factorial for each brand and simultaneously a supersaturated design for the GEOs. Multiple simultaneous experiments also allow one to identify GEOs in which advertising is generally more effective. That cannot be done in the single brand experiments we consider.

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arXiv2016

Confounder Adjustment in Multiple Hypothesis Testing

We consider large-scale studies in which thousands of significance tests are performed simultaneously. In some of these studies, the multiple testing procedure can be severely biased by latent confounding factors such as batch effects and unmeasured covariates that correlate with both primary variable(s) of interest (e.g. treatment variable, phenotype) and the outcome. Over the past decade, many statistical methods have been proposed to adjust for the confounders in hypothesis testing. We unify these methods in the same framework, generalize them to include multiple primary variables and multiple nuisance variables, and analyze their statistical properties. In particular, we provide theoretical guarantees for RUV-4 and LEAPP, which correspond to two different identification conditions in the framework: the first requires a set of "negative controls" that are known a priori to follow the null distribution; the second requires the true non-nulls to be sparse. Two different estimators which are based on RUV-4 and LEAPP are then applied to these two scenarios. We show that if the confounding factors are strong, the resulting estimators can be asymptotically as powerful as the oracle estimator which observes the latent confounding factors. For hypothesis testing, we show the asymptotic z-tests based on the estimators can control the type I error. Numerical experiments show that the false discovery rate is also controlled by the Benjamini-Hochberg procedure when the sample size is reasonably large.

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arXiv2016

Efficient moment calculations for variance components in large unbalanced crossed random effects models

Large crossed data sets, described by generalized linear mixed models, have become increasingly common and provide challenges for statistical analysis. At very large sizes it becomes desirable to have the computational costs of estimation, inference and prediction (both space and time) grow at most linearly with sample size. Both traditional maximum likelihood estimation and numerous Markov chain Monte Carlo Bayesian algorithms take superlinear time in order to obtain good parameter estimates. We propose moment based algorithms that, with at most linear cost, estimate variance components, measure the uncertainties of those estimates, and generate shrinkage based predictions for missing observations. When run on simulated normally distributed data, our algorithm performs competitively with maximum likelihood methods.

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arXiv2015

Transformations and Hardy-Krause variation

Using a multivariable Faa di Bruno formula we give conditions on transformations $τ:[0,1]^m\to\mathcal{X}$ where $\mathcal{X}$ is a closed and bounded subset of $\mathbb{R}^d$ such that $f\circτ$ is of bounded variation in the sense of Hardy and Krause for all $f\in C^d(\mathcal{x})$. We give similar conditions for $f\circτ$ to be smooth enough for scrambled net sampling to attain $O(n^{-3/2+ε})$ accuracy. Some popular symmetric transformations to the simplex and sphere are shown to satisfy neither condition. Some other transformations due to Fang and Wang (1993) satisfy the first but not the second condition. We provide transformations for the simplex that makes $f\circτ$ smooth enough to fully benefit from scrambled net sampling for all $f$ in a class of generalized polynomials. We also find sufficient conditions for the Rosenblatt-Hlawka-Mück transformation in $\mathbb{R}^2$ and for importance sampling to be of bounded variation in the sense of Hardy and Krause.

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arXiv2015

Single Nugget Kriging

We propose a method with better predictions at extreme values than the standard method of Kriging. We construct our predictor in two ways: by penalizing the mean squared error through conditional bias and by penalizing the conditional likelihood at the target function value. Our prediction exhibits robustness to the model mismatch in the covariance parameters, a desirable feature for computer simulations with a restricted number of data points. Applications on several functions show that our predictor is robust to the non-Gaussianity of the function.

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arXiv2015

Optimal Multiple Testing Under a Gaussian Prior on the Effect Sizes

We develop a new method for frequentist multiple testing with Bayesian prior information. Our procedure finds a new set of optimal p-value weights called the Bayes weights. Prior information is relevant to many multiple testing problems. Existing methods assume fixed, known effect sizes available from previous studies. However, the case of uncertain information is usually the norm. For a Gaussian prior on effect sizes, we show that finding the optimal weights is a non-convex problem. Despite the non-convexity, we give an efficient algorithm that solves this problem nearly exactly. We show that our method can discover new loci in genome-wide association studies. On several data sets it compares favorably to other methods. Open source code is available.

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arXiv2015

A constraint on extensible quadrature rules

When the worst case integration error in a family of functions decays as $n^{-α}$ for some $α>1$ and simple averages along an extensible sequence match that rate at a set of sample sizes $n_1<n_2<\dots<\infty$, then these sample sizes must grow at least geometrically. More precisely, $n_{k+1}/n_k\ge ρ$ must hold for a value $1<ρ<2$ that increases with $α$. This result always rules out arithmetic sequences but never rules out sample size doubling. The same constraint holds in a root mean square setting.

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arXiv2014

Data enriched linear regression

We present a linear regression method for predictions on a small data set making use of a second possibly biased data set that may be much larger. Our method fits linear regressions to the two data sets while penalizing the difference between predictions made by those two models. The resulting algorithm is a shrinkage method similar to those used in small area estimation. We find a Stein-type finding for Gaussian responses: when the model has 5 or more coefficients and 10 or more error degrees of freedom, it becomes inadmissible to use only the small data set, no matter how large the bias is. We also present both plug-in and AICc-based methods to tune our penalty parameter. Most of our results use an $L_2$ penalty, but we obtain formulas for $L_1$ penalized estimates when the model is specialized to the location setting. Ordinary Stein shrinkage provides an inadmissibility result for only 3 or more coefficients, but we find that our shrinkage method typically produces much lower squared errors in as few as 5 or 10 dimensions when the bias is small and essentially equivalent squared errors when the bias is large.

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arXiv2014

Optimal mixture weights in multiple importance sampling

In multiple importance sampling we combine samples from a finite list of proposal distributions. When those proposal distributions are used to create control variates, it is possible (Owen and Zhou, 2000) to bound the ratio of the resulting variance to that of the unknown best proposal distribution in our list. The minimax regret arises by taking a uniform mixture of proposals, but that is conservative when there are many components. In this paper we optimize the mixture component sampling rates to gain further efficiency. We show that the sampling variance of mixture importance sampling with control variates is jointly convex in the mixture probabilities and control variate regression coefficients. We also give a sequential importance sampling algorithm to estimate the optimal mixture from the sample data.

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arXiv2014

Extensible grids: uniform sampling on a space-filling curve

We study the properties of points in $[0,1]^d$ generated by applying Hilbert's space-filling curve to uniformly distributed points in $[0,1]$. For deterministic sampling we obtain a discrepancy of $O(n^{-1/d})$ for $d\ge2$. For random stratified sampling, and scrambled van der Corput points, we get a mean squared error of $O(n^{-1-2/d})$ for integration of Lipshitz continuous integrands, when $d\ge3$. These rates are the same as one gets by sampling on $d$ dimensional grids and they show a deterioration with increasing $d$. The rate for Lipshitz functions is however best possible at that level of smoothness and is better than plain IID sampling. Unlike grids, space-filling curve sampling provides points at any desired sample size, and the van der Corput version is extensible in $n$. Additionally we show that certain discontinuous functions with infinite variation in the sense of Hardy and Krause can be integrated with a mean squared error of $O(n^{-1-1/d})$. It was previously known only that the rate was $o(n^{-1})$. Other space-filling curves, such as those due to Sierpinski and Peano, also attain these rates, while upper bounds for the Lebesgue curve are somewhat worse, as if the dimension were $\log_2(3)$ times as high.

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arXiv2014

Moment based gene set tests

{\bf Motivation:} Permutation-based gene set tests are standard approaches for testing relationshi ps between collections of related genes and an outcome of interest in high throughput expression analyses. Using $M$ random permutations, one can attain $p$-values as small as $1/(M+1)$. When many gene sets are tested, we need smaller $p$-values, hence larger $M$, to achieve significance while accounting for the n umber of simultaneous tests being made. As a result, the number of permutations to be done rises along with the cost per permutation. To reduce this cost, we seek parametric approximations to the permutation distributions for gene set tes ts. {\bf Results:} We focus on two gene set methods related to sums and sums of squared $t$ statistics. Our approach calculates exact relevant moments of a weighted sum of (squared) test statistics under permutation. We find moment-based gene set enrichment $p$-values that closely approximate the permutation method $p$-values. The computational cost of our algorithm for linear statistics is on the order of doing $|G|$ permutations, where $|G|$ is the number of genes in set $G$. For the quadratic statistics, the cost is on the order of $|G|^2$ permutations which is orders of magnitude faster than naive permutation. We applied the permutation approximation method to three public Parkinson's Disease expression datasets and discovered enriched gene sets not previously discussed. In the analysis of these experiments with our method, we are able to remove the granularity effects of permutation analyses and have a substantial computational speedup with little cost to accura cy. {\bf Availability:} Methods available as a Bioconductor package, npGSEA (www.bioconductor.org). {\bf Contact:} {larson.jessica@gene.com} \end{abstract}

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