Bonferroni Inequality Research Articles

THE POWER OF THE K‐INDEX (OR PMIR) TO DETECT COPYING

ABSTRACTAt Educational Testing Service, the K‐Index is used to assess unusual agreement between the incorrect responses of two test takers on a multiple‐choice test. Since the Fall of 1996, the value of the K‐Index used for this assessment has been adjusted, based on the Bonferroni inequality. The resulting value is referred to as the Probability of Matching Incorrect Responses (PMIR). If a test taker copies from someone (a Source) with no incorrect responses, the PMIR is useless (i.e., has zero power) for detecting unusual agreement. Similarly, if a Source has relatively few incorrect responses, the power of the PMIR will be low. We propose a framework within which to study the power of the PMIR. The power function derived within this framework exhibits quite complex behavior; however, we conclude that the power of the PMIR to detect substantial amounts of copying is quite low, even when the Source has a relatively large number of incorrect responses.

Outliers in Multivariate Regression Models

Likelihood ratio tests for detecting a single outlier in multivariate linear models are considered, where an observation is called an outlier if there has been a shift in the mean. The test statistics are the maximum of n nonindependent statistics, where n is the number of observations. Relevant distributions to use upper and lower Bonferroni's inequalities are given.

Asymptotic comparison of step-down and step-up multiple test procedures based on exchangeable test statistics

In this paper interest is focused on some theoretical investigations concerning the comparison of two popular multiple test procedures, so-called step-down and step-up procedures, in terms of their defining critical values. Such procedures can be applied, for example, to multiple comparisons with a control. In the definition of the critical values for these procedures order statistics play a central role. For $k \epsilon N_0$ fixed we consider the joint cumulative distribution function (cdf) $P(Y_{1:n} \leq c_1,\dots, Y_{n-k:n} \leq c_{n-k})$ of the first $n - k$ order statistics and the cdf$P(Y_{n-k:n} \leq c_{n-k})$ of the ($k + 1$)th largest order statistic $Y_{n-k:n}$ of $n$ random variables $Y_1,\dots, Y_n$ belonging to a sequence of exchangeable real-valued random variables. Our interest is focused on the asymptotic behavior of these cdf's and their interrelation if $n$ tends to $\infty$. It turns out that they sometimes behave completely differently compared with the iid case treated in Finner and Roters so that positive results are only possible under additional assumptions concerning the underlying distribution. We consider different sets of assumptions which then allow analogous results for the exchangeable case. Recently, Dalal and Mallows derived a result concerning the monotonicity of a certain set of critical values in connection with the joint cdf of order statistics in the iid case. We give a counterexample for the exchangeable case underlining the difficulties occurring in this situation. As an application we consider the comparison of certain step-down and step-up procedures in multiple comparisons with a control. The results of this paper yield a more theoretical explanation of the superiority of the step-up procedure which has been observed earlier by comparing tables of critical values. As a by-product we are able to quantify the tightness of the Bonferroni inequality in connection with maximum statistics.

Multiple comparison procedures with the average for exponential location parameters

In this article, multiple comparison procedures with the average of exponential location parameters when sample sizes are equal are under investigation. A subset selection approach and a simultaneous confidence interval approach with minimum expected length are considered for exponential distributions with common known or unknown scale parameter. These procedures will have broad applications in selecting a subset which includes all better-than-the-average treatments in experimental design and/or in identifying all better-than-the-average, worse-than-the-average and not-much-difference-from-the-average products in agriculture, business, manufacturing and other industries. Some numerical approximation approaches using Bonferroni inequality are proposed in this article. Statistical tables to implement these procedures for the case of equal sample size are provided for use in practice. A simulation result indicates that Bonferroni approximation performs better than the confidence interval with equal-tail probability. Computer software programs for calculating the percentage points and for simulation are available from the authors.

A graphical approach to obtaining confidence limits ofCpk

The process capability index Cpk has been widely used as a process performance measure. In practice this index is estimated using sample data. Hence it is of great interest to obtain confidence limits for the actual index given a sample estimate. In this paper we depict graphically the relationship between process potential index (Cp), process shift index (k) and percentage non-conforming (p). Based on the monotone properties of the relationship, we derive two-sided confidence limits for k and Cpk under two different scenarios. These two limits are combined using the Bonferroni inequality to generate a third type of confidence limit. The performance of these limits of Cpk in terms of their coverage probability and average width is evaluated by simulation. The most suitable type of confidence limit for each specific range of k is then determined. The usage of these confidence limits is illustrated via examples. Finally a performance comparison is done between the proposed confidence limits and three non-parametric bootstrap confidence limits. The results show that the proposed method consistently gives the smallest width and yet provides the intended coverage probability. © 1997 John Wiley & Sons, Ltd.

Bounds for reliability of k-within connected-( r, s)-out-of-( m, n) failure systems

A k-within linear connected-( r, s)-out-of-( m, n) failure system is a two-dimensional grid whose components are ordered like the elements of an ( m, n)-matrix. A k-within circular connected-( r, s)-out-of ( m, n) failure system consists of the intersection points of m circles centered at the same point with n rays starting from that point and crossing the circles. The components of both systems either operate or fail. By definition, a k-within (linear or circular) connected-( r, s)-out-of-( m, n) failure system fails if at least one ( r, s)-submatrix contains k or more failed components. These systems are used as mathematical models for design and operation of many engineering systems. For systems with statistically independent and identically distributed components, a lower and upper bound of system reliability are derived using improved Bonferroni inequalities. These bounds are easy to compute and provide good estimates for system reliability. New bounds for the reliability of other connected-( r, s)-out-of-( m, n) failure systems existing in the current literature are also obtained. Several failure systems with various values of the parameters k, r, s, m, n and p are used as numerical examples for comparison and illustrative purposes.

Calculating mean system-failure frequency with prescribed accuracy

To calculate mean system-failure frequency (MSFF) for good engineering designs, an approximation with prescribed accuracy is possible, starting from mincuts, viz, from a sum-of-products form of the fault-tree Boolean-function. Bonferroni-type inequalities, for which a new proof is included, are used. There is a far-reaching similarity between certain kinds of bounds for MSFF (of coherent systems) and for system unavailability. However, this similarity Is not complete. For most real systems, omitting the less-important mincuts yields lower bounds not only for unavailability but also for MSFF. Because an equivalent of the first Bonferroni inequality also holds with MSFF, it is possible to determine an upper bound of the contribution of the deleted mincuts or, the other way around: given a maximum error, determine a set of less-important mincuts, which can be deleted prior to a standard (exact) analysis of the rest. Since 1977 there is a valuable insight that Bonferroni-type inequalities hold also for MSFF (and since 1995 for mean electronic-system life). However, if the difference between the first upper and lower bounds is not small enough, then the investigation of further bounds might be rather cumbersome. However, there is a straight-forward mincut-based approximation to MSFF. As for approximate values of system mean time to failure (MTTF) and mean time to repair (MTTR), upper and lower bounds can be readily found via MSFF=unavailability/MTTR=availability/MTTF using upper and lower bounds for unavailability and MSFF in an obvious way. Of course, these bounds might not be sufficiently tight initially.

Pairwise Comparisons Using Trimmed Means or M‐Estimators when Working with Dependent Groups

AbstractFor J dependent groups, let θj, j = 1, …, J, be some measure of location associated with the jth group. A common goal is computing confidence intervals for the pairwise differences, θj — θk, j < k, such that the simultaneous probability coverage is 1 — α. If means are used, it is well known that slight departures from normality (as measured by the Kolmogorov distance) toward a heavy‐tailed distribution can substantially inflate the standard error of the sample mean, which in turn can result in relatively low power. Also, when distributions differ in shape, or when sampling from skewed distributions with relatively light tails, practical problems arise when the goal is to obtain confidence intervals with simultaneous probability coverage reasonably close to the nominal level. Extant theoretical and simulation results suggest replacing means with trimmed means. The Tukey‐McLaughlin method is easily adapted to the problem at hand via the Bonferroni inequality, but this paper illustrates that practical concerns remain. Here, the main result is that the percentile t bootstrap method, used in conjunction with trimmed means, gives improved probability coverage and substantially better power. A method based on a one‐step M‐estimator is also considered but found to be less satisfactory.

Bonferroni Inequalities and Negative Cycles in Large Complete Signed Graphs

In this paper the problem of characterizing extremal graphs K n relatively to the number of negative p-cycles, when the number of negative edges is fixed, is solved for large n.This number can be expressed as an alternating sum for which the Bonferroni inequalities hold. Finally, the asymptotic value of the probability that a p-cycle of K n is negative is found as n→∞, if the negative edges induce a subgraph the components of which are paths or cycles.

A Comparison of Confidence Interval Methods for Habitat Use-Availability Studies

Wildlife managers routinely compute sets of simultaneous confidence intervals to estimate the actual proportion of use of a set of k habitat types. Confidence intervals are determined by assuming that the counts of observed use are from k binomial populations. A set of k intervals is constructed from a large sample approximation for a confidence interval for a single binomial proportion. The simultaneous confidence level is controlled by use of the Bonferroni inequality. The coverage probability of these intervals can be less than the nominal (1 - a) 100% level. This paper presents results of a simulation study comparing the performance of these intervals with 3 alternatives; the usual method with a continuity correction factor, and 2 methods of computing confidence intervals for multinomial proportions. The 2 latter methods are superior and should be used in place of the binomial intervals. J. WILDL. MANAGE. 60(3):653-658

Negative cycles in complete signed graphs

Using an inclusion-exclusion formula for the symmetric difference and its associated Bonferroni inequalities, some extremal and divisibility properties for the number of negative p-cycles of K n are deduced. It is also shown that the asymptotic probability that a p-cycle of K n is negative does not depend on the structure of the subgraph spanned by negative edges provided this subgraph has bounded degrees.

Bonferroni inequalities and deviations of discrete distributions

In the paper we first show how to convert a generalized Bonferroni-type inequality into an estimation for the generating function of the number of occurring events, then we give estimates for the deviation of two discrete probability distributions in terms of the maximum distance between their generating functions over the interval [0, 1].

Bonferroni inequalities and deviations of discrete distributions

In the paper we first show how to convert a generalized Bonferroni-type inequality into an estimation for the generating function of the number of occurring events, then we give estimates for the deviation of two discrete probability distributions in terms of the maximum distance between their generating functions over the interval [0, 1].

Bounds for reliability of k-within two-dimensional consecutive-r-out-of-n failure systems

A k-within two-dimensional consecutive- r-out-of- n failure system is a square grid of side n (containing n 2 components) and fails if, and only if, there is at least one square sub-grid of side r (containing r 2 components) that has at least k (1 ⩽ k ⩽ r 2, 1 < r ⩽ n) failed components. For systems with statistically independent and identically distributed components, a lower and an upper bound of system reliability are derived using improved Bonferroni inequalities. New bounds for the reliability of the two-dimensional consecutive- r-out- n failure system (1 < r ⩽ n) are also obtained.

Dietary Nutrient Profiles of Canadian Baffin Island Inuit differ by Food Source, Season, and Age

Objective To compare the effect of food source (traditional or market), season (six seasons), and age (five age groups) on dietary nutrient patterns of Inuit living in Baffin Island, Canada. Design Twenty-four-hour recall interviews of all residents who had lived ≥3 years in this one community in each of six seasons. Foods that were recalled were divided by source. Setting/subjects The study took place in the Inuit community of Qikiqtarjuaq, which harvests the highest quantity of wildlife per capita of all Baffin communities. Three hundred sixty-six residents contributed a total of 1,410 recalls: 401 from nonpregnant, nonlactating adult women, 74 from pregnant women, 301 from adult men, 451 from children aged 3 to 12 years, and 183 from teenagers aged 13 to 19 years. Participation was voluntary and averaged 65% to 75% of residents. Main outcome measures Energy, total dry weight of food, and dietary nutrients (ie, carbohydrate, protein, total fat, saturated fat, polyunsaturated fat, vitamin A, iron, copper, zinc, calcium, phosphorus, magnesium, and sodium) were measured by food source, season, and age. Nutrient density (nutrient per 1,000 kcal) was calculated in traditional and market food sources. Selected nutrients were computed in total diets, and compared with Recommended Dietary Allowances (RDAs). Statistical analyses performed Tests for normality of the distribution of nutrient intakes (ie, Shapiro-Wilk statistic) were performed followed by nonparametric analyses (ie, Wilcoxon paired-sample t test, Kruskal-Wallis analysis of variance, and adjustment for Bonferroni inequalities resulting from multiple comparisons). Results Most nutrient intakes were significantly different by food source ( P<.05). Traditional food contributed more protein, phosphorus, iron, zinc, copper, magnesium, and vitamin A for several age groups. Market food contributed greater amounts of dry weight, energy, fat, carbohydrate, calcium, and sodium for most age groups. Seasonal variation ( P<.05) existed for nutrients coming from traditional and market food. Of the 10 nutrients assessed for nutrient density, all except calcium and sodium were present in greater amounts in traditional food than in market food ( P<.05). Calcium and vitamin A intakes fell below 66.6% of the RDAs for more than 60% of the population. Conclusions The comprehensive view of nutrient profiles, food source, and seasonality of Inuit diets will assist health professionals in developing nutrition promotion and education programs for all age groups of this population. Traditional food is an essential source of the total annual dietary nutrient intake of Inuit. Results indicated, however, that calcium and vitamin A intake must be improved. J Am Diet Assoc. 1996; 96:155–162.

Using Common Random Numbers for Indifference-Zone Selection and Multiple Comparisons in Simulation

We present a general recipe for constructing experiment design and analysis procedures that simultaneously provide indifference-zone selection and multiple-comparison inference for choosing the best among k simulated systems. We then exhibit two such procedures that exploit the variance-reduction technique of common random numbers to reduce the sample size required to attain a fixed precision. One procedure is based on the Bonferroni inequality and is guaranteed to be statistically conservative. The other procedure is exact under a specific dependence structure, but may be slightly liberal otherwise. Both are easy to apply, requiring only simple calculations and tabled constants. We illustrate the procedures with a numerical example.

Simulation-Based Estimation of Proportions

Simulated estimates for several proportions are needed in many simulation studies. We propose a method for controlling the length of a simulation run so that the proportions estimated satisfy a prespecified precision requirement. The method applies the arcsin transform in order to generate confidence intervals with the desired width. The Bonferroni inequality is applied in a new way to construct a rectangular confidence area for the proportions being estimated. Since we are using the Bonferroni inequality, we can use the existing methods developed for estimating the variance of a single mean. The properties of the method proposed depend on three factors: 1) the arcsin transform, 2) the proposed way of applying the Bonferroni inequality, and 3) the method for estimating the variance. We examine the effects that the transform and the way of applying the Bonferroni inequality have. Empirical results indicate that the method provides estimates that satisfy the prespecified precision requirement, when the spectral method is used to estimate the variances of the proportions. In addition, simulation runs, when estimating either several proportions or the corresponding mean, are about the same lengths.

Controlling the precision of estimated means in interactive simulation

The computing power of modern workstations has made it possible to simulate many queueing systems interactively. Recent development in simulation software has mainly concentrated on interactive facilities. Unfortunately the precision of estimates has widely been overlooked in interactive simulation. In this paper we propose a method for controlling the precision of estimated means during an interactive simulation run. Since in a typical situation of interactive simulation the user is simultaneously interested in several means, we consider both the simultaneous precision and individual precisions of the estimated means. The method is based on the existing methods for estimating standard errors and on the Bonferroni inequality. The Bonferroni inequality is used to obtain a lower bound for the simultaneous precision.

Monitoring Pairwise Comparisons in Multi-Armed Clinical Trials

This paper proposes a method for monitoring multi-armed clinical trials on the basis of pairwise comparisons between arms. The set of pairwise test statistics is examined during the course of the trial in order to make decisions about hypotheses, continuation of treatment arms, and continuation of the trial. Strong control of the Type I error rate is achieved by modifying two-armed group sequential procedures of Pocock (1977, Biometrika 64, 191-199), O'Brien and Fleming (1979, Biometrics 35, 549-556), and Lan and DeMets (1983, Biometrika 70, 659-663) to multi-armed trials. In the fixed-sample situation, these methods reduce to either Dunnett's or Tukey's procedure for multiple comparisons. A simpler, more flexible approximation based on the Bonferroni inequality is suggested, as well as an analogue to a sequentially rejective procedure.

Generalized Bonferroni inequalities

Consider a number of events in a probability space. Let X be a random variable that is the number of events that occur. Given some of the moments of the distribution of X, it is possible to obtain bounds on the probability that at least one event occurs. The best possible bounds are given here. If there are many equiprobable events that are d- wise independent, and d is even, then the probability that at least one event happens is at least 1 — O(µ–d /2), where μ = E(X).