Simple Alternative Hypothesis Research Articles

Asymptotic relative submajorization of multiple-state boxes

Pairs of states, or “boxes” are the basic objects in the resource theory of asymmetric distinguishability (Wang and Wilde in Phys Rev Res 1(3):033170, 2019. https://doi.org/10.1103/PhysRevResearch.1.033170), where free operations are arbitrary quantum channels that are applied to both states. From this point of view, hypothesis testing is seen as a process by which a standard form of distinguishability is distilled. Motivated by the more general problem of quantum state discrimination, we consider boxes of a fixed finite number of states and study an extension of the relative submajorization preorder to such objects. In this relation, a tuple of positive operators is greater than another if there is a completely positive trace nonincreasing map under which the image of the first tuple satisfies certain semidefinite constraints relative to the other one. This preorder characterizes error probabilities in the case of testing a composite null hypothesis against a simple alternative hypothesis, as well as certain error probabilities in state discrimination. We present a sufficient condition for the existence of catalytic transformations between boxes, and a characterization of an associated asymptotic preorder, both expressed in terms of sandwiched Rényi divergences. This characterization of the asymptotic preorder directly shows that the strong converse exponent for a composite null hypothesis is equal to the maximum of the corresponding exponents for the pairwise simple hypothesis testing tasks.

Likelihood Ratio Testing under Measurement Errors.

We consider the likelihood ratio test of a simple null hypothesis (with density ) against a simple alternative hypothesis (with density ) in the situation that observations are mismeasured due to the presence of measurement errors. Thus instead of for we observe with unobservable parameter and unobservable random variable . When we ignore the presence of measurement errors and perform the original test, the probability of type I error becomes different from the nominal value, but the test is still the most powerful among all tests on the modified level. Further, we derive the minimax test of some families of misspecified hypotheses and alternatives. The test exploits the concept of pseudo-capacities elaborated by Huber and Strassen (1973) and Buja (1986). A numerical experiment illustrates the principles and performance of the novel test.

Using Simple Alternative Hypothesis to Increase Statistical Power in Sparse Categorical Data

There are numerous statistical hypothesis tests for categorical data including Pearson's Chi-Square goodness-of-fit test and other discrete versions of goodness-of-fit tests. For these hypothesis tests, the null hypothesis is simple, and the alternative hypothesis is composite which negates the simple null hypothesis. For power calculation, a researcher specifies a significance level, a sample size, a simple null hypothesis, and a simple alternative hypothesis. In practice, there are cases when an experienced researcher has deep and broad scientific knowledge, but the researcher may suffer from a lack of statistical power due to a small sample size being available. In such a case, we may formulate hypothesis testing based on a simple alternative hypothesis instead of the composite alternative hypothesis. In this article, we investigate how much statistical power can be gained via a correctly specified simple alternative hypothesis and how much statistical power can be lost under a misspecified alternative hypothesis, particularly when an available sample size is small.

Chance Hypotheses Testing

This paper considers the problem of testing hypotheses about hybrid distributions which are models representing situations where impreciseness (explained through fuzzy measure) and randomness (explained through probability measure) coexist. A criterion similar to the Neyman-Pearson criterion is proposed for testing a simple chance null hypothesis against a simple chance alternative hypothesis. The suggested criterion has been applied for testing hypotheses about hybrid triangular Bernoulli distribution and hybrid Poisson distribution. Optimal properties of the resulting tests have also been investigated.

R2VIM: A new variable selection method for random forests in genome-wide association studies.

BackgroundMachine learning methods and in particular random forests (RFs) are a promising alternative to standard single SNP analyses in genome-wide association studies (GWAS). RFs provide variable importance measures (VIMs) to rank SNPs according to their predictive power. However, in contrast to the established genome-wide significance threshold, no clear criteria exist to determine how many SNPs should be selected for downstream analyses.ResultsWe propose a new variable selection approach, recurrent relative variable importance measure (r2VIM). Importance values are calculated relative to an observed minimal importance score for several runs of RF and only SNPs with large relative VIMs in all of the runs are selected as important. Evaluations on simulated GWAS data show that the new method controls the number of false-positives under the null hypothesis. Under a simple alternative hypothesis with several independent main effects it is only slightly less powerful than logistic regression. In an experimental GWAS data set, the same strong signal is identified while the approach selects none of the SNPs in an underpowered GWAS.ConclusionsThe novel variable selection method r2VIM is a promising extension to standard RF for objectively selecting relevant SNPs in GWAS while controlling the number of false-positive results.Electronic supplementary materialThe online version of this article (doi:10.1186/s13040-016-0087-3) contains supplementary material, which is available to authorized users.

Linking Rates of Diffusion and Consumption in Relation to Resources

The functional response is a fundamental model of the relationship between consumer intake rate and resource abundance. The random walk is a fundamental model of animal movement and is well approximated by simple diffusion. Both models are central to our understanding of numerous ecological processes but are rarely linked in ecological theory. To derive a synthetic model, we draw on the common logical premise underlying these models and show how the diffusion and consumption rates of consumers depend on elementary attributes of naturally occurring consumer-resource interactions: the abundance, spatial aggregation, and traveling speed of resources as well as consumer handling time and directional persistence. We show that resource aggregation may lead to increased consumer diffusion and, in the case of mobile resources, reduced consumption rate. Resource-dependent movement patterns have traditionally been attributed to area-restricted search, reflecting adaptive decision making by the consumer. Our synthesis provides a simple alternative hypothesis that such patterns could also arise as a by-product of statistical movement mechanics.

Using Parametric Multipoint Lods and Mods for Linkage Analysis Requires a Shift in Statistical Thinking

Multipoint (MP) linkage analysis represents a valuable tool for whole-genome studies but suffers from the disadvantage that its probability distribution is unknown and varies as a function of marker information and density, genetic model, number and structure of pedigrees, and the affection status distribution [Xing and Elston: Genet Epidemiol 2006;30:447–458; Hodge et al.: Genet Epidemiol 2008;32:800–815]. This implies that the MP significance criterion can differ for each marker and each dataset, and this fact makes planning and evaluation of MP linkage studies difficult. One way to circumvent this difficulty is to use simulations or permutation testing. Another approach is to use an alternative statistical paradigm to assess the statistical evidence for linkage, one that does not require computation of a p value. Here we show how to use the evidential statistical paradigm for planning, conducting, and interpreting MP linkage studies when the disease model is known (lod analysis) or unknown (mod analysis). As a key feature, the evidential paradigm decouples uncertainty (i.e. error probabilities) from statistical evidence. In the planning stage, the user calculates error probabilities, as functions of one’s design choices (sample size, choice of alternative hypothesis, choice of likelihood ratio (LR) criterion k) in order to ensure a reliable study design. In the data analysis stage one no longer pays attention to those error probabilities. In this stage, one calculates the LR for two simple hypotheses (i.e. trait locus is unlinked vs. trait locus is located at a particular position) as a function of the parameter of interest (position). The LR directly measures the strength of evidence for linkage in a given data set and remains completely divorced from the error probabilities calculated in the planning stage. An important consequence of this procedure is that one can use the same criterion k for all analyses. This contrasts with the situation described above, in which the value one uses to conclude significance may differ for each marker and each dataset in order to accommodate a fixed test size, α. In this study we accomplish two goals that lead to a general algorithm for conducting evidential MP linkage studies. (1) We provide two theoretical results that translate into guidelines for investigators conducting evidential MP linkage: (a) Comparing mods to lods, error rates (including probabilities of weak evidence) are generally higher for mods when the null hypothesis is true, but lower for mods in the presence of true linkage. Royall [J Am Stat Assoc 2000;95:760–780] has shown that errors based on lods are bounded and generally small. Therefore when the true disease model is unknown and one chooses to use mods, one needs to control misleading evidence rates only under the null hypothesis; (b) for any given pair of contiguous marker loci, error rates under the null are greatest at the midpoint between the markers spaced furthest apart, which provides an obvious simple alternative hypothesis to specify for planning MP linkage studies. (2) We demonstrate through extensive simulation that this evidential approach can yield low error rates under the null and alternative hypotheses for both lods and mods, despite the fact that mod scores are not true LRs. Using these results we provide a coherent approach to implement a MP linkage study using the evidential paradigm.

A power comparison of generalized additive models and the spatial scan statistic in a case-control setting

BackgroundA common, important problem in spatial epidemiology is measuring and identifying variation in disease risk across a study region. In application of statistical methods, the problem has two parts. First, spatial variation in risk must be detected across the study region and, second, areas of increased or decreased risk must be correctly identified. The location of such areas may give clues to environmental sources of exposure and disease etiology. One statistical method applicable in spatial epidemiologic settings is a generalized additive model (GAM) which can be applied with a bivariate LOESS smoother to account for geographic location as a possible predictor of disease status. A natural hypothesis when applying this method is whether residential location of subjects is associated with the outcome, i.e. is the smoothing term necessary? Permutation tests are a reasonable hypothesis testing method and provide adequate power under a simple alternative hypothesis. These tests have yet to be compared to other spatial statistics.ResultsThis research uses simulated point data generated under three alternative hypotheses to evaluate the properties of the permutation methods and compare them to the popular spatial scan statistic in a case-control setting. Case 1 was a single circular cluster centered in a circular study region. The spatial scan statistic had the highest power though the GAM method estimates did not fall far behind. Case 2 was a single point source located at the center of a circular cluster and Case 3 was a line source at the center of the horizontal axis of a square study region. Each had linearly decreasing logodds with distance from the point. The GAM methods outperformed the scan statistic in Cases 2 and 3. Comparing sensitivity, measured as the proportion of the exposure source correctly identified as high or low risk, the GAM methods outperformed the scan statistic in all three Cases.ConclusionsThe GAM permutation testing methods provide a regression-based alternative to the spatial scan statistic. Across all hypotheses examined in this research, the GAM methods had competing or greater power estimates and sensitivities exceeding that of the spatial scan statistic.

Choosing an optimal method to combine P‐values

Fisher (1925) was the first to suggest a method of combining the p-values obtained from several statistics and many other methods have been proposed since then. However, there is no agreement about what is the best method. Motivated by a situation that now often arises in genetic epidemiology, we consider the problem when it is possible to define a simple alternative hypothesis of interest for which the expected effect size of each test statistic is known and we determine the most powerful test for this simple alternative hypothesis. Based on the proposed method, we show that information about the effect sizes can be used to obtain the best weights for Liptak's method of combining p-values. We present extensive simulation results comparing methods of combining p-values and illustrate for a real example in genetic epidemiology how information about effect sizes can be deduced.

Impact of controlling the sum of error probability in the sequential probability ratio test

A generalized modified method is proposed to control the sum of error probabilities in sequential probability ratio test to minimize the weighted average of the two average sample numbers under a simple null hypothesis and a simple alternative hypothesis with the restriction that the sum of error probabilities is a pre-assigned constant to find the optimal sample size and finally a comparison is done with the optimal sample size found from fixed sample size procedure. The results are applied to the cases when the random variate follows a normal law as well as Bernoullian law.

Theory and applications of a new methodology for the random sequential probability ratio test

Wald’s [A. Wald, Sequential Analysis, Wiley, New York, 1947] sequential probability ratio test (SPRT) and group sequential probability ratio test (GSPRT) remain relevant in addressing a wide range of practical problems. The area of clinical trials owes a great debt to the theory and methodologies of SPRT and GSPRT. In recent years, there has been a surge of practical applications of these methodologies in many areas including low frequency sonar detection, tracking of signals, early detection of abrupt changes in signals, computer simulations, agricultural sciences, pest management, educational testing, economics, and finance. But, obviously, there are circumstances where sampling one observation at a time may not be practical. In contexts of monitoring “inventory” or “queues”, observations may appear sequentially but in groups where the group sizes may be ideally treated as random variables themselves. For example, one may sequentially record the number of stopped cars ( M i ) and the number of cars ( ∑ j = 1 M i X i j ) without “working brake lights” when a traffic signal changes from green to red, i = 1 , 2 , … . This can be easily accomplished since every “working brake light” must glow bright red when a driver applies brakes. In this example, one notes that (i) it may be reasonable to model M i ’s as random, but (ii) it would appear impossible to record data sequentially on the status of brake lights (that is, X i j = 0 or 1) individually for car #1, and then for car #2, and so on when a group of cars rush to stop at an intersection! In order to address these kinds of situations, we start with an analog of the concept of a best “fixed-sample-size” test based on data { M i , X i 1 , … , X i M i , i = 1 , … , k } . Then, a random sequential probability ratio test (RSPRT) is developed for deciding between a simple null hypothesis and a simple alternative hypothesis with preassigned Type I and II errors α , β . The RSPRT and the best “fixed-sample-size” test with k ≡ k min associated with the same errors α , β are compared. Illustrations of RSPRT include a one-parameter exponential family followed by substantive computer simulations that lead to a broad set of guidelines. Two real applications and data analysis are highlighted.

No-decision classification: an alternative to testing for statistical significance

This paper proposes a new statistical technique for deciding which of two theories is better supported by a given set of data while allowing for the possibility of drawing no conclusion at all. Procedurally similar to the classical hypothesis test, the proposed technique features three, as opposed to two, mutually exclusive data classifications: reject the null, reject the alternative, and no decision. Referred to as No-decision classification (NDC), this technique requires users to supply a simple null and a simple alternative hypothesis based on judgments concerning the smallest difference that can be regarded as an economically substantive departure from the null. In contrast to the classical hypothesis test, NDC allows users to control both Type I and Type II errors by specifying desired probabilities for each. Thus, NDC integrates judgments about the economic significance of estimated magnitudes and the shape of the loss function into a familiar procedural form.

Stepwise Likelihood Ratio Statistics in Sequential Studies

SummaryIt is well known that in a sequential study the probability that the likelihood ratio for a simple alternative hypothesis H1versus a simple null hypothesis H0 will ever be greater than a positive constant c will not exceed 1/c under H0. However, for a composite alternative hypothesis, this bound of 1/c will no longer hold when a generalized likelihood ratio statistic is used. We consider a stepwise likelihood ratio statistic which, for each new observation, is updated by cumulatively multiplying the ratio of the conditional likelihoods for the composite alternative hypothesis evaluated at an estimate of the parameter obtained from the preceding observations versus the simple null hypothesis. We show that, under the null hypothesis, the probability that this stepwise likelihood ratio will ever be greater than c will not exceed 1/c. In contrast, under the composite alternative hypothesis, this ratio will generally converge in probability to ∞. These results suggest that a stepwise likelihood ratio statistic can be useful in a sequential study for testing a composite alternative versus a simple null hypothesis. For illustration, we conduct two simulation studies, one for a normal response and one for an exponential response, to compare the performance of a sequential test based on a stepwise likelihood ratio statistic with a constant boundary versus some existing approaches.

Improved estimation of accuracy in simple hypothesis versus simple alternative testing

In the hypothesis testing problem, a most common used evidence against the null hypothesis is the p-value. Although there have been many Bayesian criticisms leveled at p-value, Hwang et al. (Ann. Statist. 20 (1992), 490) show the adequacy of using p-value as evidence against the null hypothesis by considering testing as an estimation problem. However, when the parameter space is not the natural space, Woodroofe and Wang (Ann. Statist. 28 (2000) 1561) show that the usual p-value derived by the N–P test is not appropriate to be the evidence against the null hypothesis for the Poisson distribution from an estimation point of view and provide a modified p-value. Although this modified p-value is admissible, it is not the admissible estimator which can dominate the usual p-value. In this paper, we concentrate on the simple hypothesis versus simple alternative hypothesis testing problem. Admissible estimators which dominate the usual p-value are provided.

A note on hypothesis testing for discrete cases

Over the years, students have been taught to rely solely on the Neyman-Pearson Lemma in the hypothesis testing problem of a simple hypothesis against a simple alternative hypothesis to obtain a most powerful test, or equivalently, a best critical region of size β. If a particular β level is not attainable by the lemma, randomization can be employed. However, randomized tests are impractical in reality. In this note, illustrations are given to show that most powerful tests may still be found in discrete situations where the Neyman-Pearson Lemma does not apply and randomized tests are not adopted.

On estimation of linkage test power.

Two methods of estimating linkage test power are proposed. The first method is based on the maximum likelihood (ML) estimate of the recombination fraction and is intended for use with a likelihood ratio test (LRT) in the form of a chi 2 or lod score. The power is estimated through a noncentral chi 2 distribution with a specially chosen noncentrality parameter. The second method uses the LRT constructed for a simple alternative hypothesis regarding the recombination fraction value. The approximate distribution of this test and a method of estimating its power is proposed. Using simulated pedigree data, the power estimates for these two methods were shown to be satisfactory. Comparisons among these two methods and the computer simulation approach of Boehnke [1986] are performed.

Most powerful tests of a simple null hypothesis against a simple alternative hypothesis with posterior significance level

Most powerful tests of a simple null hypothesis against a simple alternative hypothesis with posterior significance level

Hitch-hiking: An alternative to coadaptation for the barley and slender wild oat examples

The associations of alleles at different electrophoretic loci observed in barley and wild oats can be explained by hitch-hiking due to the mating system. In other words, selection at other loci in the genome may have resulted in gametic disequilibrium or a change in gametic disequilibrium at neutral loci in these highly self-fertilised plants. This explanation appears to be a simple alternative hypothesis to account for the change in gametic disequilibrium in barley and the presence of gametic disequilibrium in wild oats. However, it appears difficult to differentiate experimentally between the hitch hiking and the previously proposed coadaptation hypotheses.

Sample Size Precision Function

The paper develops the concept of precision function for a given sample size when testing a simple hypothesis against a simple alternative hypothesis. The construction of sample size precision functions is discussed with reference to the location and sc..

A Note on the Non-equivalence of the Neyman-Pearson and Generalized Likelihood Ratio Tests for Testing a Simple Null versus a Simple Alternative Hypothesis

A Note on the Non-equivalence of the Neyman-Pearson and Generalized Likelihood Ratio Tests for Testing a Simple Null versus a Simple Alternative Hypothesis