arXiv subjects

Jonathan Rosenblatt

Publications and source records attributed to Jonathan Rosenblatt.

5 recordsLinked to original sources

arXiv2016

What's in a pattern? Examining the Type of Signal Multivariate Analysis Uncovers At the Group Level

Multivoxel pattern analysis (MVPA) has gained enormous popularity in the neuroimaging community over the past few years. At the group level, most MVPA studies adopt an "information based" approach in which the sign of the effect of individual subjects is discarded and a non-directional summary statistic is carried over to the second level. This is in contrast to a directional "activation based" approach typical in univariate group level analysis, in which both signal magnitude and sign are taken into account. The transition from examining effects in one voxel at a time vs. several voxels (univariate vs. multivariate) has thus tacitly entailed a transition from directional to non-directional signal definition at the group level. While a directional group-level MVPA approach implies that individuals have similar multivariate spatial patterns of activity, in a non-directional approach each individual may have a distinct spatial pattern. Using an experimental dataset, we show that directional and non-directional group-level MVPA approaches uncover distinct brain regions with only partial overlap. We propose a method to quantify the degree of spatial similarity in activation patterns over subjects. Applied to an auditory task, we find higher values in auditory regions compared to control regions.

q-bio.QM↗

arXiv2015

Modeling and Analysing Respondent Driven Sampling as a Counting Process

Respondent-driven sampling (RDS) is an approach to sampling design and analysis which utilizes the networks of social relationships that connect members of the target population, using chain-referral methods to facilitate sampling. RDS typically leads to biased sampling, favoring participants with many acquaintances. Naive estimates, such as the sample average, which are uncorrected for the sampling bias, will themselves be biased. To compensate for this bias, current methodology suggests inverse-degree weighting, where the "degree" is the number of acquaintances. This stems from the fundamental RDS assumption that the probability of sampling an individual is proportional to their degree. Since this assumption is tenuous at best, we propose to harness the additional information encapsulated in the time of recruitment, into a model-based inference framework for RDS. This information is typically collected by researchers, but ignored. We adapt methods developed for inference in epidemic processes to estimate the population size, degree counts and frequencies. While providing valuable information in themselves, these quantities ultimately serve to debias other estimators, such a disease's prevalence. A fundamental advantage of our approach is that, being model-based, it makes all assumptions of the data-generating process explicit. This enables verification of the assumptions, maximum likelihood estimation, extension with covariates, and model selection. We develop asymptotic theory, proving consistency and asymptotic normality properties. We further compare these estimators to the standard inverse-degree weighting through simulations, and using real-world data. In both cases we find our estimators to outperform current methods. The likelihood problem in the model we present is convex, and thus efficiently solvable. We implement these estimators in an R package, chords, available on CRAN.

stat.ME↗

arXiv2015

On the Optimality of Averaging in Distributed Statistical Learning

A common approach to statistical learning with big-data is to randomly split it among $m$ machines and learn the parameter of interest by averaging the $m$ individual estimates. In this paper, focusing on empirical risk minimization, or equivalently M-estimation, we study the statistical error incurred by this strategy. We consider two large-sample settings: First, a classical setting where the number of parameters $p$ is fixed, and the number of samples per machine $n\to\infty$. Second, a high-dimensional regime where both $p,n\to\infty$ with $p/n \to κ\in (0,1)$. For both regimes and under suitable assumptions, we present asymptotically exact expressions for this estimation error. In the fixed-$p$ setting, under suitable assumptions, we prove that to leading order averaging is as accurate as the centralized solution. We also derive the second order error terms, and show that these can be non-negligible, notably for non-linear models. The high-dimensional setting, in contrast, exhibits a qualitatively different behavior: data splitting incurs a first-order accuracy loss, which to leading order increases linearly with the number of machines. The dependence of our error approximations on the number of machines traces an interesting accuracy-complexity tradeoff, allowing the practitioner an informed choice on the number of machines to deploy. Finally, we confirm our theoretical analysis with several simulations.

stat.ML↗

arXiv2013

Another Argument in Favour of Wilcoxon's Signed Rank Test

The Wilcoxon Signed Rank test is typically called upon when testing whether a symmetric distribution has a specified centre and the Gaussianity is in question. As with all insurance policies it comes with a cost, even if small, in terms of power versus a t-test, when the distribution is indeed Gaussian. In this note we further show that even when the distribution tested is Gaussian there need not be power loss at all, if the alternative is of a mixture type rather than a shift. The signed rank test may turn out to be more powerful than the t-test, and the supposedly conservative strategy, might actually be the more powerful one. Drug testing and functional magnetic imaging are two such scenarios. Wilcoxon' signed rank test will typically be called upon by a researcher when testing for the location of a single population, using a small sample and Gaussianity is dubious. As all insurance policies, it will come with a cost-- power. It is well known, that under a Gaussian setup, the signed rank test is less powerful than, say, a t-test. The works of Pitman and others have reassured us that this power loss is surprisingly small. In this note we argue that the power loss might actually be smaller than typically assumed. In particular, if the deviation from the null Gaussian distribution is of a mixture type and not a shift type, the signed rank test is no longer dominated by the t-test and can actually be more powerful.

math.ST↗

arXiv2013

A Practitioner's Guide to Multiple Testing Error Rates

It is quite common in modern research, for a researcher to test many hypotheses. The statistical (frequentist) hypothesis testing framework, does not scale with the number of hypotheses in the sense that naively performing many hypothesis tests will probably yield many false findings. Indeed, statistical "significance" is evidence for the presence of a signal within the noise expected in a single test, not in a multitude. In order to protect himself from an uncontrolled number of erroneous findings, a researcher has to consider of the type or errors he wishes to avoid and select the adequate procedure for that particular error type and data structure. A quick search of the tag [multiple-comparisons] in the statistics Questions & Answers web site Cross Validates (http://stats.stackexchange.com) demonstrates the amount of confusion this task can actually cause. This was also a point made at the 2009 Multiple Comparisons conference in Tokyo. In an attempt to offer guidance, we review possible error types for multiple testing, and demonstrate them with some practical examples, which clarify the formalism. Finally, we include some notes on the software implementations of the methods discussed. The emphasis of this manuscript is on the error-rates, and not on the procedures themselves. We do try to name several procedures in this manuscript where appropriate. P-value adjustment will not be discussed as it is procedure specific. I.e., it is the choice of a procedure that defines the p-value adjustment, and not the error rate itself. Simultaneous confidence intervals will, also, not be discussed.

stat.ME↗