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A Tale of Two Matrix Factorizations
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| Pengarang | : | Paul Fogell |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 67 (No. 4) |
| Halaman | : | 207-218 |
| Abstrak | : | In statistical practice, rectangular tables of numeric data are commonplace, and are often analyzed using dimension-reduction methods like the singular value decomposition and its close cousin, principal component analysis (PCA). This analysis produces score and loading matrices representing the rows and the columns of the original table and these matrices may be used for both prediction purposes and to gain structural understanding of the data. In some tables, the data entries are necessarily nonnegative (apart, perhaps, from some small random noise), and so the matrix factors meant to represent them should arguably also contain only nonnegative elements. This thinking, and the desire for parsimony, underlies such techniques as rotating factors in a search for “simple structure.” These attempts to transform score or loading matrices of mixed sign into nonnegative, parsimonious forms are, however, indirect and at best imperfect. The recent development of nonnegative matrix factorization, or NMF, is an attractive alternative. Rather than attempt to transform a loading or score matrix of mixed signs into one with only nonnegative elements, it directly seeks matrix factors containing only nonnegative elements. The resulting factorization often leads to substantial improvements in interpretability of the factors. We illustrate this potential by synthetic examples and a real dataset. The question of exactly when NMF is effective is not fully resolved, but some indicators of its domain of success are given. It is pointed out that the NMF factors can be used in much the same way as those coming from PCA for such tasks as ordination, clustering, and prediction. Supplementary materials for this article are available online. |
Applications of Multiple Systems Estimation in Human Rights Research
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 67 (No. 4) |
| Halaman | : | 191-200 |
| Abstrak | : | Multiple systems estimation (MSE) is becoming an increasingly common approach for exploratory study of underreported events in the field of quantitative human rights. In this context, it is used to estimate the number of people who died as a result of political unrest when it is believed that many of those who died or disappeared were never reported. MSE relies upon several assumptions, each of which may be slightly or significantly violated in particular applications. This article outlines the evolution of the application of MSE to human rights research through the use of three case studies: Guatemala, Peru, and Colombia. Each of these cases presents distinct challenges to the MSE method. Motivated by these applications, we describe new methodology for assessing the impact of violated assumptions in MSE. Our approach uses simulations to explore the cumulative magnitude of errors introduced by violation of the model assumptions at each stage in the analysis. |
Simultaneous Confidence Intervals for Several Quantiles of an Unknown Distribution
| Pengarang | : | A. J. Hayter |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 68 (No. 1) |
| Halaman | : | 56-62 |
| Abstrak | : | Given a sample of independent observations from an unknown continuous distribution, it is standard practice to construct a confidence interval for a specified quantile of the distribution using the binomial distribution. Furthermore, confidence bands for the unknown cumulative distribution function, such as Kolmogorov’s, provide simultaneous confidence intervals for all quantiles of the distribution, which are necessarily wider than the individual confidence intervals at the same confidence level. The purpose of this article is to show how simultaneous confidence intervals for several specified quantiles of the unknown distribution can be calculated using probabilities from a multinomial distribution. An efficient recursive algorithm is described for these calculations. An experimenter may typically be interested in several quantiles of the distribution, such as the median, quartiles, and upper and lower tail quantiles, and this methodology provides a bridge between the confidence intervals with individual confidence levels and those that can be obtained from confidence bands. Some examples of the implementation of this nonparametric methodology are provided, and some comparisons are made with some parametric approaches to the problem. |
The Gram-Schmidt Construction as a Basis for Linear Models
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 68 (No. 1) |
| Halaman | : | 52-55 |
| Abstrak | : | The Gram-Schmidt construction, with a little extension, can be used to establish results in linear algebra, multiple regression analysis, and the theory of linear models. This article describes and illustrates how it serves to develop the basic results required for statistical inference in the Gauss–Markov model. For upper-level theory courses, the method’s advantage is that it requires less background and fewer results in linear algebra than are usually required. For applications-oriented courses, it makes it possible to describe relations and computations simply and explicitly. |
Prior Elicitation: Interactive Spreadsheet Graphics With Sliders Can Be Fun, and Informative
| Pengarang | : | Geoffrey Jones |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 68 (No. 1) |
| Halaman | : | 42-51 |
| Abstrak | : | There are several approaches to setting priors in Bayesian data analysis. Some attempt to minimize the impact of the prior on the posterior, allowing the data to “speak for themselves,” or to provide Bayesian inferences that have good frequentist properties. In contrast, this note focuses on priors where scientific knowledge is used, possibly partially informative. There are many articles on the use of such subjective information. We focus on using standard software for eliciting priors from subject-matter specialists, in the form of models such as the binomial, Poisson, and normal.  Our approach uses a common spreadsheet package with the facility to display dynamic pictures of prior distributions as the user toggles scroll bars or “sliders” that manipulate parameters of particular distributions. This allows interactive exploration of the shape of a probability distribution. We have found this a useful tool when eliciting priors for Bayesian data analysis. We present examples to illustrate the scope and flexibility of the method. Supplementary materials for this article are available online. |
Two Useful Reformulations of the Hazard Ratio
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 68 (No. 1) |
| Halaman | : | 36-41 |
| Abstrak | : | The hazard ratio is a standard summary for comparing survival curves yet hazard ratios are often difficult for scientists and clinicians to interpret. Insight into the interpretation of hazard ratios is obtained by relating hazard ratios to the maximum difference and an average difference between survival probabilities. These reformulations of the hazard ratio are useful in classroom discussions of survival analysis and when discussing analyses with scientists and clinicians. Large-sample distribution theory is provided for these reformulations of the hazard ratio. Two examples are used to illustrate the ideas. |
Resurrecting the Third Variable: A Critique of Pearl's Causal Analysis of Simpson's Paradox
| Pengarang | : | Timothy W. Armistead |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 68 (No. 1) |
| Halaman | : | 1-7 |
| Abstrak | : | Pearl argued that Simpson's Paradox would not be considered paradoxical but for statisticians’ unwillingness to acknowledge the role of causality in resolving an instance of it. He proposed using a causal calculus to determine which set of contradictory findings in an instance of the paradox should be accepted—the aggregated data or the data disaggregated by conditioning on the third variable. Pearl used the example of a hypothetical quasi-experiment to argue that when third variables are not causal, one should not condition on them, and—assuming no other sources of confounding—the aggregated data should be accepted. Pearl was precipitate in his argument that it would be inappropriate to condition on the noncausal third variables in the example. Whether causal or not, third variables can convey critical information about a first-order relationship, study design, and previously unobserved variables. Any conditioning on a nontrivial third variable that produces Simpson's Paradox should be carefully examined before either the aggregated or the disaggregated findings are accepted, regardless of whether the third variable is thought to be causal. In some cases, neither set of data is trustworthy; in others, both convey information of value. Pearl's hypothetical example is used to illustrate this argument. |
A Unique Collaboration: Prominent Statisticians’ Survey Work in Greece in 1946
| Pengarang | : | Catherine Michalopoulou |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 68 (No. 3) |
| Halaman | : | 196-203 |
| Abstrak | : | In 1946, Neyman, Jessen, Deming, Kempthorne, Daly, and Blythe conducted a series of sample surveys as sampling experts of the two Allied Missions that were set up to observe the preparation and conduct of the Greek parliamentary elections (March 31) and the revision of electoral rolls for the plebiscite (September 1). This article revisits these surveys, using both published and unpublished sources, and discusses the lessons learned from their history as they relate to current sampling practices. |
Kurtosis as Peakedness, 1905–2014. R.I.P.
| Pengarang | : | Peter H. Westfall |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 68 (No. 3) |
| Halaman | : | 191-195 |
| Abstrak | : | The incorrect notion that kurtosis somehow measures “peakedness” (flatness, pointiness, or modality) of a distribution is remarkably persistent, despite attempts by statisticians to set the record straight. This article puts the notion to rest once and for all. Kurtosis tells you virtually nothing about the shape of the peak—its only unambiguous interpretation is in terms of tail extremity, that is, either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution). To clarify this point, relevant literature is reviewed, counterexample distributions are given, and it is shown that the proportion of the kurtosis that is determined by the central μ ± σ range is usually quite small. |
Two New Elementary Derivations of Geometric Expectation
| Pengarang | : | Liang Hong |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 68 (No. 3) |
| Halaman | : | 188-190 |
| Abstrak | : | This article presents two new elementary derivations of the expectation of the geometric distribution. I also review six existing approaches. I hope that this article will benefit instructors and students in an introductory probability course. |