Statistical Theory Commons^™

Some Reflections On Significance Testing, Thomas R. Knapp

Journal of Modern Applied Statistical Methods

This essay presents a variation on a theme from my article “The use of tests of statistical significance”, which appeared in the Spring, 1999, issue of Mid-Western Educational Researcher.

Null Distribution Of The Likelihood Ratio Statistic For Feed-Forward Neural Networks, Douglas Landsittel, Harshinder Singh, Vincent C. Arena, Stewart J. Anderson

Journal of Modern Applied Statistical Methods

Despite recent publications exploring model complexity with modern regression methods, their dimensionality is rarely quantified in practice and the distributions of related test statistics are not well characterized. Through a simulation study, we describe the null distribution of the likelihood ratio statistic for several different feed-forward neural network models.

Combining Quantum Mechanical Calculations And A Χ^2 Fit In A Potential Energy Function For The Co_2 + O^+ Reaction, Ellen F. Sawilowsky

Journal of Modern Applied Statistical Methods

In order to compute a highly accurate statistical rate constant for the CO2 + O⁺ reaction, it is necessary to first calculate the potential energy of the system at many different geometric configurations. Quantum mechanical calculations are very time-consuming, making it difficult to obtain a sufficient number to allow for accurate interpolation. The number of quantum mechanical calculations required can be significantly reduced by using known relations in classical physics to calculate energy for configurations where the oxygen is relatively far from the CO2. A chi-squared fit to quantum mechanical points is obtained for these configurations, and the resulting …

Type I Error Rates For Rank-Based Tests Of Homogeneity Of Slopes, Alan J. Klockars, Tim P. Moses

Journal of Modern Applied Statistical Methods

The purpose of this study was to explicate two issues concerning the standard and rank based test of homogeneity of slopes. Two alternative ranking methods intended to address nonnormality and additive treatment effect patterns were developed and compared in terms of their ability to control Type I error. The results replicated previous findings of inflated Type I error rates with leptokurtic curves and with rank based tests with some patterns of additive treatment effects. The new nonparametric procedures generally control Type I error although they were slightly inflated with skewed distributions.

Exploration Of Distributions Of Ratio Of Partial Sum Of Sample Eigenvalues When All Population Eigenvalues Are The Same, Moonseong Heo

Journal of Modern Applied Statistical Methods

This paper explores empirically the first two moments of ratio of the partial sum of the first two sample eigenvalues to the sum of all eigenvalues when the population eigenvalues of a covariance matrix are all the same. Estimation of the first two moments can be practically crucial in assessing non-randomness of observed patterns on planar graphical displays based on lower rank approximations of data matrices. For derivation of the moments, exact and large sample asymptotic distributions of the sample ratios are reviewed but neither can be applicable to derivation of the moments. Therefore, I rely on simulations, where data …

Chronic Disease Data And Analysis: Current State Of The Field, Ralph D'Agostino Sr., Lisa M. Sullivan

Journal of Modern Applied Statistical Methods

Chronic disease usually spans years of a person’s lifetime and includes a disease free period, a preclinical, or latent period, where there are few overt signs of disease, a clinical period where the disease manifests and is eventually diagnosed, and a follow-up period where the disease might progress steadily or remain stable. It is often of interest to investigate the relationship between risk factors measured at a point in time (usually during the disease free or preclinical period), and the development of disease at some future point (e.g., 10 years later). We outline some popular designs for the identification of …

Twenty Nonparametric Statistics And Their Large Sample Approximations, Gail F. Fahoome

Journal of Modern Applied Statistical Methods

Nonparametric procedures are often more powerful than classical tests for real world data which are rarely normally distributed. However, there are difficulties in using these tests. Computational formulas are scattered throughout the literature, and there is a lack of availability of tables and critical values. The computational formulas for twenty commonly employed nonparametric tests that have large-sample approximations for the critical value are brought together. Because there is no generally agreed upon lower limit for the sample size, Monte Carlo methods were used to determine the smallest sample size that can be used with the respective large-sample approximation. The statistics …

Trimming, Transforming Statistics, And Bootstrapping: Circumventing The Biasing Effects Of Heterescedasticity And Nonnormality, H. J. Keselman, Rand R. Wilcox, Abdul R. Othman, Katherine Fradette

Journal of Modern Applied Statistical Methods

Researchers can adopt different measures of central tendency and test statistics to examine the effect of a treatment variable across groups (e.g., means, trimmed means, M-estimators, & medians. Recently developed statistics are compared with respect to their ability to control Type I errors when data were nonnormal, heterogeneous, and the design was unbalanced: (1) a preliminary test for symmetry which determines whether data should be trimmed symmetrically or asymmetrically, (2) two different transformations to eliminate skewness, (3) the accuracy of assessing statistical significance with a bootstrap methodology was examined, and (4) statistics that use a robust measure of the typical …

A Simulation Study Of The Impact Of Forecast Recovery For Control Charts Applied To Arma Processes, John N. Dyer, B. Michael Adams, Michael D. Conerly

Journal of Modern Applied Statistical Methods

Forecast-based schemes are often used to monitor autocorrelated processes, but the resulting forecast recovery has a significant effect on the performance of control charts. This article describes forecast recovery for autocorrelated processes, and the resulting simulation study is used to explain the performance of control charts applied to forecast errors.

Determining Predictor Importance In Multiple Regression Under Varied Correlational And Distributional Conditions, Tiffany A. Whittaker, Rachel T. Fouladi, Natasha J. Williams

Journal of Modern Applied Statistical Methods

This study examines the performance of eight methods of predictor importance under varied correlational and distributional conditions. The proportion of times a method correctly identified the dominant predictor was recorded. Results indicated that the new methods of importance proposed by Budescu (1993) and Johnson (2000) outperformed commonly used importance methods.

A Program For Generating All Permutations Of {1, 2, ..., N}, Robert Disario

Journal of Modern Applied Statistical Methods

A Visual Basic program that generates all permutations of {1, 2, ..., n} is presented. The procedure for running the program as an Excel macro is described. An application is presented which involves selecting permutations which meet a specific constraint.

A Comparison Of The D’Agostino S_U Test To The Triples Test For Testing Of Symmetry Versus Asymmetry As A Preliminary Test To Testing The Equality Of Means, Kimberly T. Perry, Michael R. Stoline

Journal of Modern Applied Statistical Methods

This paper evaluates the D’Agostino S_U test and the Triples test for testing symmetry versus asymmetry. These procedures are evaluated as preliminary tests in the selection of the most appropriate procedure for testing the equality of means with two independent samples under a variety of symmetric and asymmetric sampling situations. Key words: symmetry; asymmetry; preliminary testing.

Within Groups Multiple Comparisons Based On Robust Measures Of Location, Rand R. Wilcox, H. J. Keselman

Journal of Modern Applied Statistical Methods

Consider the problem of performing all pair-wise comparisons among J dependent groups based on measures of location associated with the marginal distributions. It is well known that the standard error of the sample mean can be large relative to other estimators when outliers are common. Two general strategies for addressing this problem are to trim a fixed proportion of observations or empirically check for outliers and remove (or down-weight) any that are found. However, simply applying conventional methods for means to the data that remain results in using the wrong standard error. Methods that address this problem have been proposed, …

Adaptive Tests For Ordered Categorical Data, Vance W. Berger, Anastasia Ivanova

Journal of Modern Applied Statistical Methods

Consider testing for independence against stochastic order in an ordered 2xJ contingency table, under product multinomial sampling. In applications one may wish to exploit prior information concerning the direction of the treatment effect, yet ultimately end up with a testing procedure with good frequentist properties. As such, a reasonable objective may be to simultaneously maximize power at a specified alternative and ensure reasonable power for all other alternatives of interest. For this objective, none of the available testing approaches are completely satisfactory. A new class of admissible adaptive tests is derived. Each test in this class strictly preserves the Type …

Accounting For Non-Independent Observations In 2×2 Tables, With Application To Correcting For Family Clustering In Exposure-Risk Relationship Studies, Leslie A. Kalsih, Katherine A. Riester, Stuart J. Pocock

Journal of Modern Applied Statistical Methods

Participants in epidemiologic studies may not represent statistically independent observations. We consider modifications to conventional analyses of 2×2 tables, including Fisher’s exact test and confidence intervals, to account for correlated observations in this setting. An example is provided, assessing the robustness of conclusions from a published analysis.

Simulation Study Of Chemical Inhibition Modeling, Pali Sen, Mary Anderson

Journal of Modern Applied Statistical Methods

The combined effects of the activities of different chemicals are of interest of this study. We simulate for the synthetic data, and fit experimental data for three models and estimate the parameters. We assess the fit of the synthetic data and the experimental data by comparing the coefficients of variation for the parameter estimates and identify the best model for the inhibition process.

On Distribution Function Estimation Using Double Ranked Set Samples With Application, Walid A. Abu-Dayyeh, Hani M. Samawi, Lara A. Bani-Hani

Journal of Modern Applied Statistical Methods

As a variation of ranked set sampling (RSS); double ranked set sampling (DRSS) was introduced by Al-Saleh and Al-Kadiri (2000), and it has been used only for estimating the mean of the population. In this paper DRSS will be used for estimating the distribution function (cdf). The efficiency of the proposed estimators will be obtained when ranking is perfect. Some inference on the distribution function will be drawn based on Kolomgrov-Smirnov statistic. It will be shown that using DRSS will increase the efficiency in this case.

On The Estimation Of Binomial Success Probability With Zero Occurrence In Sample, Mehdi Razzaghi

Journal of Modern Applied Statistical Methods

The problem of estimating the probability of a rare event when the sample shows no incidence of the event is considered. Several methodologies based on various statistical techniques are described and their relative performances are investigated. A decision theoretic approach for estimation of response probability when the sample contains zero responses is examined in depth. The properties of each method are discussed and an example from teratology is used to provide illustration and to demonstrate the results.

A Longitudinal Follow-Up Of Discrete Mass At Zero With Gap, Joseph L. Musial, Patrick D. Bridge, Nicol R. Shamey

Journal of Modern Applied Statistical Methods

The first part of this paper discusses a five-year systematic review of the Journal of Consulting and Clinical Psychology following the landmark power study conducted by Sawilowsky and Hillman (1992). The second part discusses a five-year longitudinal follow-up of a radically nonnormal population distribution: discrete mass at zero with gap. This distribution was based upon a real dataset.

Jmasm4: Critical Values For Four Nonparametric And/Or Distribution-Free Tests Of Location For Two Independent Samples, Bruce R. Fay

Journal of Modern Applied Statistical Methods

Researchers engaged in computer-intensive studies may need exact critical values, especially for sample sizes and alpha levels not normally found in published tables, as well as the ability to control ‘best-fit’ criteria. They may also benefit from the ability to directly generate these values rather than having to create lookup tables. Fortran 90 programs generate ‘best-conservative’ (bc) and ‘best-fit’ (bf) critical values with associated probabilities for the Kolmogorov-Smirnov test of general differences (bc), Rosenbaum’s test of location (bc), Tukey’s quick test (bc and bf)) and the Wilcoxon rank-sum test (bc).

Locally Efficient Estimation With Bivariate Right Censored Data , Christopher M. Quale, Mark J. Van Der Laan, James M. Robins

U.C. Berkeley Division of Biostatistics Working Paper Series

Estimation for bivariate right censored data is a problem that has had much study over the past 15 years. In this paper we propose a new class of estimators for the bivariate survivor function based on locally efficient estimation. The locally efficient estimator takes bivariate estimators Fn and Gn of the distributions of the time variables T1,T2 and the censoring variables C1,C2, respectively, and maps them to the resulting estimator. If Fn and Gn are consistent estimators of F and G, respectively, then the resulting estimator will be nonparametrically efficient (thus the term ``locally efficient''). However, if either Fn or …

The Analysis Of Placement Values For Evaluating Discriminatory Measures, Margaret S. Pepe, Tianxi Cai

UW Biostatistics Working Paper Series

The idea of using measurements such as biomarkers, clinical data, or molecular biology assays for classification and prediction is popular in modern medicine. The scientific evaluation of such measures includes assessing the accuracy with which they predict the outcome of interest. Receiver operating characteristic curves are commonly used for evaluating the accuracy of diagnostic tests. They can be applied more broadly, indeed to any problem involving classification to two states or populations (D = 0 or D = 1). We show that the ROC curve can be interpreted as a cumulative distribution function for the discriminatory measure Y in the …

Accelerated Hazards Model: Method, Theory And Applications, Ying Qing Chen, Nicholas P. Jewell, Jingrong Yang

U.C. Berkeley Division of Biostatistics Working Paper Series

In an accelerated hazards model, the hazard functions of a failure time are related through the time scale-change, which is often a function of covariates and associated parameters. When the hazard functions have special properties, such as monotonicity in time, the parameters may be clinically meaningful in measuring a treatment effect. This paper reviews methodological and theoretical development of this model. Applications of the accelerated hazards model including sample size calculation in clinical trials, are also explored.

Locally Efficient Estimation Of Regression Parameters Using Current Status Data, Chris Andrews, Mark J. Van Der Laan, James M. Robins

U.C. Berkeley Division of Biostatistics Working Paper Series

In biostatistics applications interest often focuses on the estimation of the distribution of a time-variable T. If one only observes whether or not T exceeds an observed monitoring time C, then the data structure is called current status data, also known as interval censored data, case I. We consider this data structure extended to allow the presence of both time-independent covariates and time-dependent covariate processes that are observed until the monitoring time. We assume that the monitoring process satisfies coarsening at random.

Our goal is to estimate the regression parameter beta of the regression model T = Z*beta+epsilon where the …

Case-Control Current Status Data, Nicholas P. Jewell, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Current status observation on survival times has recently been widely studied. An extreme form of interval censoring, this data structure refers to situations where the only available information on a survival random variable, T, is whether or not T exceeds a random independent monitoring time C, a binary random variable, Y. To date, nonparametric analyses of current status data have assumed the availability of i.i.d. random samples of the random variable (Y, C), or a similar random sample at each of a set of fixed monitoring times. In many situations, it is useful to consider a case-control sampling scheme. Here, …

Why Prefer Double Robust Estimates? Illustration With Causal Point Treatment Studies, Romain Neugebauer, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In point treatment marginal structural models with treatment A, outcome Y and covariates W, causal parameters can be estimated under the assumption of no unobserved confounders. Three estimates can be used: the G-computation, Inverse Probability of Treatment Weighted (IPTW) or Double Robust (DR) estimates. The properties of the IPTW and DR estimates are known under an assumption on the treatment mechanism that we name "Experimental Treatment Assignment" (ETA) assumption. We show that the DR estimating function is unbiased when the ETA assumption is violated if the model used to regress Y on A and W is correctly specified. The practical …

Bivariate Current Status Data, Mark J. Van Der Laan, Nicholas P. Jewell

U.C. Berkeley Division of Biostatistics Working Paper Series

In many applications, it is often of interest to estimate a bivariate distribution of two survival random variables. Complete observation of such random variables is often incomplete. If one only observes whether or not each of the individual survival times exceeds a common observed monitoring time C, then the data structure is referred to as bivariate current status data (Wang and Ding, 2000). For such data, we show that the identifiable part of the joint distribution is represented by three univariate cumulative distribution functions, namely the two marginal cumulative distribution functions, and the bivariate cumulative distribution function evaluated on the …

Current Status Data: Review, Recent Developments And Open Problems, Nicholas P. Jewell, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Researchers working with survival data are by now adept at handling issues associated with incomplete data, particular those associated with various forms of censoring. An extreme form of interval censoring, known as current status observation, refers to situations where the only available information on a survival random variable T is whether or not T exceeds a random independent monitoring time C. This article contains a brief review of the extensive literature on the analysis of current status data, discussing the implications of response-based sampling on these methods. The majority of the paper introduces some recent extensions of these ideas to …

Semiparametric Regression Analysis On Longitudinal Pattern Of Recurrent Gap Times, Ying Qing Chen, Mei-Cheng Wang, Yijian Huang

U.C. Berkeley Division of Biostatistics Working Paper Series

In longitudinal studies, individual subjects may experience recurrent events of the same type over a relatively long period of time. The longitudinal pattern of the gaps between the successive recurrent events is often of great research interest. In this article, the probability structure of the recurrent gap times is first explored in the presence of censoring. According to the discovered structure, we introduce the proportional reverse-time hazards models with unspecified baseline functions to accommodate heterogeneous individual underlying distributions, when the ongitudinal pattern parameter is of main interest. Inference procedures are proposed and studied by way of proper riskset construction. The …

Multiple Hypothesis Testing In Microarray Experiments, Sandrine Dudoit, Juliet Popper Shaffer, Jennifer C. Boldrick

U.C. Berkeley Division of Biostatistics Working Paper Series

DNA microarrays are a new and promising biotechnology which allows the monitoring of expression levels in cells for thousands of genes simultaneously. An important and common question in microarray experiments is the identification of differentially expressed genes, i.e., genes whose expression levels are associated with a response or covariate of interest. The biological question of differential expression can be restated as a problem in multiple hypothesis testing: the simultaneous test for each gene of the null hypothesis of no association between the expression levels and the responses or covariates. As a typical microarray experiment measures expression levels for thousands of …