Bonferroni Inequality Research Articles

On the exact distribution of maximally selected rank statistics

The construction of simple classification rules is a frequent problem in medical research. Maximally selected rank statistics allow the evaluation of cutpoints, which provide the classification of observations into two groups by a continuous or ordinal predictor variable. The computation of the exact distribution of a maximally selected rank statistic is discussed and a new lower bound of the distribution is derived based on an extension of an algorithm for the exact distribution of a linear rank statistic. Therefore, the test based on the upper bound of the P-value is of level α. For small to moderate sample sizes the lower bound of the exact distribution is a substantial improvement compared to approximations based on an improved Bonferroni inequality or based on the asymptotic Gaussian process. The lower bound of the distribution is compared to the exact distribution by means of a simulation study and the proposal is illustrated by three clinical studies.

Maximally Selected Rank Statistics for Dose-Response Problems

We consider the bivariate situation of some quantitative, ordinal, binary or censored response variable and some quantitative or ordinal exposure variable (dose) with a hypothetical effect on the response. Data can either be the outcome of a planned dose-response experiment with only few dose levels or of an observational study where, for example, both exposure and response variable are observed within each individual. We are interested in testing the null hypothesis of no effect of the dose variable vs. a dose-response function depending on an unknown ‘threshold’ parameter. The variety of dose-response functions considered ranges from no observed effect level (NOEL) models to umbrella alternatives. Here we discuss generalizations of the method of Lausen & Schumacher (Biometrics, 1992, 48, 73–85)which are based on combinations of two-sample rank statistics and rank statistics for trend. Our approach may be seen as a generalization of a proposal for change-point problems. Using the approach of Davies (Biometrika, 1987, 74, 33–43)we derive and approximate the asymptotic null distribution for a large number of thresholds considered. We use an improved Bonferroni inequality as approximation for a small number of thresholds considered. Moreover, we analyse the small sample behaviour by means of a Monte-Carlo study. Our paper is illustrated by examples from clinical research and epidemiology.

THE EFFECTS OF NONNORMALITY ON THE UPPER PERCENTILES OF T2max STATISTIC IN ELLIPTICAL DISTRIBUTIONS

In this paper, we consider the effects of nonnormality on the upper percentiles of T 2 max statistic in elliptical distributions. Some approximations based on the Bonferroni inequalities and asymptotic expansion procedure are given under the elliptical distribution setup. In order to achieve the purpose, asymptotic expansions for the distributions of univariate and bivariate Hotelling’s T 2 type statistics are derived by a perturbation method when each population has the elliptical distribution. Finally, the accuracy of the approximations is investigated by Monte Carlo simulations for selected values of parameters.

Improved Inclusion-Exclusion Identities and Bonferroni Inequalities with Reliability Applications

This paper establishes a connection between the theory of convex geometries, the principle of inclusion-exclusion, and the topological concept of an abstract tube. In particular, it is shown that convex geometries give rise to improved inclusion-exclusion identities and improved Bonferroni inequalities. In this way, several known results from the literature are rediscovered in a concise and unified way. The results are applied in identifying a new class of hypergraphs for which the reliability covering problem can be solved in polynomial time.

On the assessment of tolerance limits under inverse gaussian distribution

Inverse Gaussian distribution has been used in a wide range of applications in modeling duration and failure phenomena. In these applications, one-sided lower tolerance limits are employed, for instance, for designing safety limits of medical devices. Tang and Chang (1994) proposed lowersided tolerance limits via Bonferroni inequality when parameters in the inverse Gaussian distribution are unknown. However, their simulation results showed conservative coverage probabilities, and consequently larger interval width. In their paper, they also proposed an alternative to construct lesser conservative limits. But simulation results yielded unsatisfactory coverage probabilities in many cases. In this article, the exact lower-sided tolerance limit is proposed. The proposed limit has a similar form to that of the confidence interval for mean under inverse Gaussian. The comparison between the proposed limit and Tang and Chang's method is compared via extensive Monte Carlo simulations. Simulation results suggest that the proposed limit is superior to Tang and Chang's method in terms of narrower interval width and approximate to nominal level of coverage probability. Similar argument can be applied to the formulation of two-sided tolerance limits. A summary and conclusion of the proposed limits is included.

Asymptotic sharpness of product-type inequalities for maxima of random variables with applications in multiple comparisons

Abstract Product-type inequalities for the probability of the intersection of events are valid in a variety of distributional settings and have many applications in probability and statistics. These inequalities can be viewed as an improvement of the classical Bonferroni inequality. In this paper we are concerned with the problem of quantifying the asymptotic sharpness of such inequalities when they are applied to the cdf of maximum statistics. We restrict attention to the maximum of n exchangeable random variables and the range of n iid random variables as well as the studentized range. The asymptotic sharpness for n tending to infinity of the product-type inequality and Bonferroni's inequality will be determined in a variety of distributional settings. Most of them occur in multiple testing problems such as the problem of comparing n treatments with a control or, in case of the range statistic, in connection with the classical problem of pairwise comparisons of n treatments. To obtain the desired results we utilize extreme value theory and derive asymptotic expressions for the quantiles of the distributions of maxima of exchangeable random variables and of certain range statistics. As an application we investigate the true levels of some many–one and range tests based on product-type inequality adjusted critical values and study the expected number of erroneously rejected hypotheses of simultaneous multiple test procedures.

Torsion of the Uterine Adnexa: Pathologic Correlations and Current Management Trends

To evaluate the clinical and pathologic correlates of specimens removed for the diagnosis of adnexal torsion and to evaluate trends in the management of torsion.Cases of ovarian or adnexal torsion (N = 104) were identified retrospectively over a period extending from January 1987 to March 1998 by the coding of ovarian, fallopian tube or adnexal torsion. Statistical evaluation was by chi 2 analysis using the Bonferroni inequality correction when appropriate.Neoplastic and functional tumors of the ovary composed > 90% of the diagnoses at microscopic evaluation, with cancer diagnosed in < 1% of cases. Laparoscopy was attempted in 47 (46%) cases, and adnexasparing procedures were performed in 20 (19%) patients. Patients treated in the latter half of the study were not less likely to undergo laparotomy than those treated in the first half; however, conversion from laparoscopy to laparotomy was significantly less common in the latter half. Patients in this study were more likely to receive an adnexa-sparing operation than historical controls, but there was no improvement in this rate from the first to the second half of this study. A history of previous abdominal surgery was the most common associated condition, but 47% of patients had no known risk factors. Ovarian hyperstimulation, previously omitted in series reports, was an antecedent factor in 9% of patients.Adnexal torsion is most commonly associated with a benign process. A more-conservative approach to the treatment of this process is becoming increasingly common, as seems warranted in light of the low incidence of malignancy. The need for conversion from a laparoscopic to an open approach appears to have been waning over the last decade; that may correlate with an increased comfort level in gynecologists with laparoscopic approaches.

Univariate X¯control charts for individual characteristics in a multinomial model

The early work on multivariate statistical process control was built upon Hotelling's T 2control chart which was developed to simultaneously monitor the means of correlated quality variables. This chart, however, has a drawback, namely, the problem of identifying the responsible variable(s) when an out-of-control signal occurs. One alternative is to use a separate X¯control chart for each individual characteristic with equal risks, based on Bonferroni inequality. In this study, we show that, from an economic perspective, it may be desirable to have unequal type I risks for the individual charts, because of different inspection and restoration costs associated with each variable. We obtain their risk ratios, which are measures of relative importance of the variables monitored. Then, based on these risk ratios, we develop computer algorithms for finding the exact control limits for individual variables from a multinormal distribution, in the sense that the overall type I risk of the charts is equal to the desired value. Numerical studies show that the proposed methods give optimal or near-optimal results from an economic as well as statistical point of view.

Testing Equality of Means and Simultaneous Confidence Intervals in Repeated Measures with Missing Data

In this paper, repeated measures with intraclass correlation model is considered when the observations are missing at random. An exact test for the equality of the mean components and simultaneous confidence intervals (Scheffe and Bonferroni inequality types) are given for linear contrasts of the mean components when the missing observations are of a monotone type. When the missing observations are not of the monotone type, the maximum likelihood estimates are obtained numerically by iterative methods given in Srivastava and Carter (1986). These estimators are then used to obtain asymptotic tests and confidence intervals for the equality of mean components and linear contrasts, respectively. An example is given to illustrate the method.

Detecting excessive similarity in answers on multiple choice exams

This paper provides a simple and robust method for detecting cheating. Unlike some methods, non-cheating behaviour and not cheating behaviour is modelled because this requires the fewest assumptions. The main concern is the prevention of false accusations. The model is suitable for screening large classes and the results are simple to interpret. Simulation and the Bonferroni inequality are used to prevent false accusation due to 'data dredging'. The model has received considerable application in practice and has been verified through the adjacent seating method.

Changepoint Tests Designed for the Analysis of Hiring Data Arising in Employment Discrimination Cases

When a complaint of discrimination is made, an employer may respond by hiring more minorities. From a legal viewpoint, the practices in effect during the time period prior to the complaint are more relevant for determining liability than those of the postcharge period. In Gay v. Waiters, the trial judge observed that the data suggested that a change occurred after the charge was filed. Because the data had not been subject to a formal statistical analysis, the court was reluctant to base its decision on this observation. Gastwirth and Freidlin and Gastwirth proposed cumulative-sum-based procedures for the analysis of hiring data following the binomial model. In this article, the procedures are extended to data following the hypergeometric model and to analysis of stratified data. Several datasets that were submitted to the courts in the United States are analyzed by the proposed methods. Because the data are usually reported by year, the ordinary large-sample theory is not sufficiently accurate. Therefore, we obtain the p values of the statistics by simulation. For binomial data, recent improvements in the Bonferroni inequality are used to derive a new upper bound.

Human leukocyte antigen DRB1 1302 protects against bile duct damage and portal lymphocyte infiltration in patients with chronic hepatitis C

Background/Aims: To confirm the immune reaction of hosts in chronic hepatitis C, we examined the association of human leukocyte antigen (HLA) DR with the histopathological outcome including bile duct damage and steatosis, which are characteristic of hepatitis C virus (HCV) infection. Methods: One hundred and fifty-five patients with chronic HCV infection were examined. The pathological appearance of liver biopsy specimens was evaluated by both Knodell's histological activity index and examination of bile duct damage and steatosis. HLA DRB1 was determined by the polymerase chain reaction sequence-specific oligonucleotide probe method. Results: HLA DRB1 1302 was found with significantly higher frequency in patients without than with bile duct damage (34.8% vs. 4.7%, p=0.0001, p corrected by Bonferroni's inequality method=0.002). It was also found more frequently in patients without marked portal lymphocyte infiltration (28.6% vs. 7.7%, p=0.0015, p corrected by Bonferroni's method=0.03). HLA DRB1 1101 was found more frequently in patients without than with piecemeal necrosis ( p=0.004). In contrast, the frequency of HLA DRB1 1502 tended to be higher in patients with than without piecemeal necrosis and marked portal lymphocyte infiltration ( p=0.015 and p=0.03, respectively). HLA DRB1 1201 and 0802 were seen more frequently in bile duct damage-negative ( p=0.02) and piecemeal necrosis-negative patients ( p=0.03), respectively. Interestingly, serum HCV levels of HLA DRB1 1302-positive patients were significantly higher than those of 1302-negative patients (mean: 7.7Meq/ml vs. 3.1Meq/ml, p=0.0007). Conclusion: These findings suggest that some histopathological changes in chronically HCV-infected livers could be caused by the host's immune reaction regulated by HLA DR.

Two-stage multiple comparisons with the average for normal distributions under heteroscedasticity

In this article we propose a two-stage procedure for multiple comparisons with the average for normal distributions under heteroscedasity. One-sided and two-sided confidence intervals are proposed. These intervals can be used to identify a subset which includes all no-worse-than-the-average treatments in an experimental design and to identify better-than-the-average, worse-than-the-average and not-much-different-from-the-average products in agriculture, stock market, medical research, and auto models. An upper limit of critical values are obatined using Bonferroni inequality. But the approximate values are shown to be too conservative compared with the simulation critical values in Table 7. Therefore, simulation critical values should be used in our multiple comparison procedures. Statistical tables and software programs are provided for use in practice.

Univariate X¯ control charts for individual characteristics in a multinormal model

Abstract The early work on multivariate statistical process control was built upon Hotelling's T 2control chart which was developed to simultaneously monitor the means of correlated quality variables. This chart, however, has a drawback, namely, the problem of identifying the responsible variable(s) when an out-of-control signal occurs. One alternative is to use a separate X¯control chart for each individual characteristic with equal risks, based on Bonferroni inequality. In this study, we show that, from an economic perspective, it may be desirable to have unequal type I risks for the individual charts, because of different inspection and restoration costs associated with each variable. We obtain their risk ratios, which are measures of relative importance of the variables monitored. Then, based on these risk ratios, we develop computer algorithms for finding the exact control limits for individual variables from a multinormal distribution, in the sense that the overall type I risk of the charts is equal...

Large-sample confidence intervals for validity and reliability coefficients.

Large-sample confidence intervals (CI) for reliability, validity, and unattenuated validity are presented. The CI for unattenuated validity is based on the Bonferroni inequality, which relies on one CI for test-retest reliability and one for validity. Covered are four reliability-validity situations: (a) both estimates were from random samples; (b) reliability was from a random sample but validity was from a selected sample; (c) validity was from a random sample but reliability was from a selected sample; and (d) both estimates were from selected samples. All CIs were evaluated by using a simulation. CIs on reliability, validity, or unattenuated validity are accurate as long as selection ratio is at least 20% and selected sample size is 100 or larger. When selection ratio is less than 20%, estimators tend to underestimate their parameters.

Multiple comparison procedures for discrete test statistics

Tarone (1990, Biometrics 46, 515–522) generalized the Bonferroni adjustment for multiple comparisons to obtain increased power by taking advantage of discreteness in the test statistics, but this paper shows that his procedure (denoted by T) does not have α-consistency (AC), i.e., a hypothesis that is accepted at a given α-level may be rejected at a lower α-level. Though lack of AC is theoretically very undesirable, practitioners might reasonably disagree about whether a procedure which lacks AC is preferable in a real application to one which has AC but is substantially less powerful. Several new procedures which are uniformly more powerful than T are proposed, with particular emphasis on two of them, one (denoted by T∗) which achieves AC and one (denoted by R) which fails to achieve AC but is usually much more powerful than T∗. The power difference is not surprising since T∗ is based on the Bonferroni inequality, while R is a discrete generalization of Hochberg's (1988, Biometrika 75, 800–802) improvement to Bonferroni. A variation of R (denoted by RMOD) generalizes Rom's (1990, Biometrika 77, 663–665) procedure instead of Hochberg's, and the arguments for preferring either of the two variations over the other are analogous to the arguments for preferring either Rom's or Hochberg's procedure in the continuous case. In the discrete case, unlike the continuous case, it matters whether strong control (SC) of the familywise error rate (FWE) is defined by requiring FWE<α or merely FWE⩽α. Versions of all of the above procedures (T,T∗, and both variations of R) are constructed to correspond to each of these two definitions of SC. In both cases, T∗ (like T) always achieves SC. A set of conditions closely related to SC of the FWE under Hochberg's [Rom's] original procedure, which is known to include the case of independence, is shown to be sufficient to guarantee SC under R [RMOD] as well. All of the proposed procedures are compared, and recommendations are made.

Torsion of the uterine adnexa—Pathologic correlations and current management trends

OBJECTIVE: To evaluate the clinical and pathologic correlates of specimens removed for the diagnosis of adnexal torsion and to evaluate trends in the management of torsion. STUDY DESIGN: Cases of ovarian or adnexal torsion (N = 104) were identified retrospectively over a period extending from January 1987 to March 1998 by the coding of ovarian, fallopian tube or adnexal torsion. Statistical evaluation was by X 2 analysis using the Bonferroni inequality correction when appropriate. RESULTS: Neoplastic and functional tumors of the ovary composed >90% of the diagnoses at microscopic evaluation, with cancer diagnosed in < 1% of cases. Laparoscopy was attempted in 47 (46%) cases, and adnexa-sparing procedures were performed in 20 (19%) patients. Patients treated in the latter half of the study were not less likely to undergo laparotomy than those treated in the first half; however, conversion from laparoscopy to laparotomy was significantly less common in the latter half. Patients in this study were more likely to receive an adnexa-sparing operation than historical controls, but there was no improvement in this rate from the first to the second half of this study. A history of previous abdominal surgery was the most common associated condition, but 47% of patients had no known risk factors. Ovarian hyperstimulation, previously omitted in series reports, was an antecedent factor in 9% of patients. CONCLUSION: Adnexal torsion is most commonly associated with a benign process. A more-conservative approach to the treatment of this process is becoming increasingly common, as seems warranted in light of the low incidence of malignancy. The need for conversion from a laparoscopic to an open approach appears to have been waning over the last decade; that may correlate with an increased comfort level in gynecologists with laparoscopic approaches.

Multiple comparison procedures with the average for k dependent normal means

In Wu and Chen [8], multiple comparison procedures with the average for k independent normal means when variances are known or unknown are proposed. The exact critical values are obtained when common variances are known or unknown and sample sizes are all equal. However, when k normal populations are dependent, the problem becomes more complicated. In this article, we consider the multiple comparison procedures with the average for k normal means when populations are not independent by simultaneous confidence interval and subset selection approaches. These procedures have broad applicability in identifying better-than-the-average, worse-than-the-average and not-much-difference-from the average stocks in Dow-Jone industrial stock market and it can also select a subset which includes all better-than-the-average treatments within its own group in experimental design if treatments are not independent. These procedures can be a very useful screening procedure especially when k is large. The comparison of the efficiency between approximation results by Bonferroni inequality and simulation results by Monte Carlo method has been studied and it shows Bonferroni approximates are efficient for equal correlation cases. An example of eight biggest mutual funds in the United States is also given to illustrate the implementation of these multiple comparison procedures with the average.

Multiple comparison procedures with the average for scale families: normal and exponential populations

In this article, multiple comparison procedures with the average of scale parameters for normal and exponential populations are proposed. A subset selection procedure and a confidence interval procedure are investigated. These procedures will have broad applicability in identifying groups of treatments with smaller than the average, larger than the average and not different from the average variability in various fields. Bonferroni inequality is used to bound the probability of a correct selection and the coverage probability of confidence intervals. The Pearson’s approximation to the sampling distribution of the log chi-square ratio utilized in both subset selection and confidence intervals is shown to be satisfactory compared with the simulation results. Statistical tables to implement these procedures for the case of equal sample sizes are provided for use in practice.

Exact joint forecast regions for vector autoregressive models

Assume that a k-element vector time series follows a vector autoregressive (VAR) model. Obtaining simultaneous forecasts of the k elements of the vector time series is an important problem. Based on the Bonferroni inequality, Lutkepohl (1991) derived the procedures which construct the conservative joint forecast regions for the VAR model. In this paper, we propose to use an exact method which provides shorter prediction intervals than does the Bonferroni method. Three illustrative examples are given for comparison of the various VAR forecasting procedures.