mid-P Value Research Articles

Longevity and Clinical Effectiveness of the Humoral and Cellular Responses to SARS-CoV-2 Vaccination in Hemodialysis Patients

Longevity and Clinical Effectiveness of the Humoral and Cellular Responses to SARS-CoV-2 Vaccination in Hemodialysis Patients

LEED-CI V3 AND V4 GOLD PROJECTS FOR OFFICE SPACES: THE DIFFERENCE BETWEEN SHANGHAI AND CALIFORNIA

ABSTRACT The Leadership in Energy and Environmental Design (LEED) rating system is currently progressing from version 3 (V3) to 4 (V4) with the aim of reducing environmental damage in the global construction sector. The LEED Commercial Interiors (LEED-CI) subsystem is widely used for office spaces, particularly in Shanghai and California. Comparing Shanghai and California in terms of LEED-CI projects can shed light on the ability of the LEED-CI subsystem to adapt to local green building standards and local environmental conditions. The aim of this study was to assess the difference between V3 and V4 LEED-CI Gold office projects using a comparison between Shanghai and California. The US Green Building Council and the Green Building Information Gateway databases were used to source LEED-CI V3 and V4 Gold office projects. The natural logarithm of the odds ratio and Fisher’s exact 2 × 2 tests with a mid p-value were used to evaluate the binary data, while the Cliff’s δ and exact Wilcoxon–Mann–Whitney tests were used to evaluate the ordinal data. The results show no difference between Shanghai and California in terms of LEED-CI V3 Gold office projects in the five main LEED-CI categories. The shift from V3 to V4 LEED-CI Gold office projects is increasing the difference between Shanghai and California in terms of the following LEED-CI categories: location and transportation, water efficiency, energy and atmosphere, and materials and resources. Therefore, as compared to V3, V4 for LEED-CI Gold office projects demonstrated greater adaptability to local green building standards and environmental conditions.

An IM-based efficient test for non inferiority of the odds ratio between two independent binomial proportions

In drug development, non inferiority tests are often employed to determine the odds ratio between two independent binomial proportions. Some traditional test statistics for non inferiority based on different statistical schools of thought are available, but they can produce markedly different behaviors. Cornfield's exact conditional test is criticized for its conservativeness, while the Wald and score tests may be applicable for large samples but tend to be liberal with small samples. Although the mid-p value method has recently gained wide acceptance for its better behavior than those of other tests, it has almost no solid theoretical foundation attached to it. In the view of the fiducial school of thought, the fiducial test based on Fisher's fiducial argument also tends to perform unsatisfactorily for small samples. In this paper, we suggest an IM-based efficient test using the randomized plausibility function based on the Inferential Model. We prove that our new method is not only valid but also efficient, i.e., it maintains a nominal type I error rate, and we show, via simulations, that the proposed test also has satisfactory power when compared with those of other competing tests. A numerical example illustrates the proposed method.

Updated confidence intervals for the COVID-19 antibody retention rate in the Korean population.

With the ongoing rise of coronavirus disease 2019 (COVID-19) pandemic across the globe, interests in COVID-19 antibody testing, also known as a serology test has grown, as a way to measure how far the infection has spread in the population and to identify individuals who may be immune. Recently, many countries reported their population based antibody titer study results. South Korea recently reported their third antibody formation rate, where it divided the study between the general population and the young male youths in their early twenties. As previously stated, these simple point estimates may be misinterpreted without proper estimation of standard error and confidence intervals. In this article, we provide an updated 95% confidence intervals for COVID-19 antibody formation rate for the Korean population using asymptotic, exact and Bayesian statistical estimation methods. As before, we found that the Wald method gives the narrowest interval among all asymptotic methods whereas mid p-value gives the narrowest among all exact methods and Jeffrey’s method gives the narrowest from Bayesian method. The most conservative 95% confidence interval estimation shows that as of 00:00 November 23, 2020, at least 69,524 people were infected but not confirmed. It also shows that more positive cases were found among the young male in their twenties (0.22%), three times that of the general public (0.051%). This thereby calls for the quarantine authorities’ need to strengthen quarantine managements for the early twenties in order to find the hidden infected people in the population.

LEED-EB Gold Projects for Office Spaces in Large Buildings Transitioning from Version 3 (v3) to 4 (v4): Similarities and Differences between Finland and Spain

This study aims to assess the similarities and differences between Finland and Spain in terms of Leadership in Energy and Environmental Design for Existing Buildings (LEED-EB) Gold large office building-type projects transitioning from version 3 (v3) to version 4 (v4). The percentages of the average scores are used here to assess the achievements of the LEED-EB data. The natural logarithm of the odds ratio lnθ and Fisher′s exact 2 × 2 tests with a mid p-value are used to evaluate dichotomous data, while the exact Wilcoxon–Mann–Whitney and Cliff′s δ effect size tests are used to evaluate ordinary data. The results for LEED-EB Gold large offices demonstrate similar certification strategies in Finland and Spain. These results may be useful to LEED-EB practitioners in Finland and Spain for facilitating the selection of appropriate certification strategies in line with identified high-performance credits for large offices.

Confidence intervals for the COVID-19 neutralizing antibody retention rate in the Korean population

The coronavirus disease 2019 (COVID-19), caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has become a global pandemic. No specific therapeutic agents or vaccines for COVID-19 are available, though several antiviral drugs, are under investigation as treatment agents for COVID-19. The use of convalescent plasma transfusion that contain neutralizing antibodies for COVID-19 has become the major focus. This requires mass screening of populations for these antibodies. While several countries started reporting population based antibody rate, its simple point estimate may be misinterpreted without proper estimation of standard error and confidence intervals. In this paper, we review the importance of antibody studies and present the 95% confidence intervals COVID-19 antibody rate for the Korean population using two recently performed antibody tests in Korea. Due to the sparsity of data, the estimation of confidence interval is a big challenge. Thus, we consider several confidence intervals using Asymptotic, Exact and Bayesian estimation methods. In this article, we found that the Wald method gives the narrowest interval among all Asymptotic methods whereas mid p-value gives the narrowest among all Exact methods and Jeffrey’s method gives the narrowest from Bayesian method. The most conservative 95% confidence interval estimation shows that as of 00:00 on September 15, 2020, at least 32,602 people were infected but not confirmed in Korea.

1159. Multidisciplinary Leadership Rounds Are Associated with Decreased Urinary Catheter and Central Venous Catheter Device Utilization at a Tertiary Care, Academic Hospital

BackgroundOptimizing use of urinary catheters (UCs) and central venous catheters (CVCs) is crucial to prevent catheter-associated urinary tract infections (CAUTIs), central line-associated bloodstream infections (CLABSIs), and other complications. Despite education and adoption of catheter removal protocols, indwelling devices not meeting approved indications were still noted.MethodsTwice a week, UC and CVC surveillance rounds were conducted by a team of directors from nursing, vascular access, infection prevention, and hospital epidemiology. Different hospital units were selected each week in random distribution. Rounds emphasized face-to-face discussion with nurses and device observations to identify any removal opportunities and appropriate maintenance. Device utilization was monitored using CDC National Healthcare Safety Network (NHSN) standardized utilization ratio (SUR) and CAUTIs and CLABSIs were monitored using NHSN definitions. Relative ratios of SURs during pre-intervention (pre-INT) and post-intervention (post-INT) time periods for UCs and CVCs were compared using an exact binomial test and mid-P 95% confidence interval (CI). CAUTI and CLABSI rates were compared using Fisher’s exact test using mid-P value.ResultsA baseline time period A of 12 months pre-INT (June 2017-May 2018) was used to compare with the 10-month post-INT time period B (June 2018-March 2019). The UC SURs for periods A and B were 0.813 and 0.696 (Figure 1). The relative ratio shows a post-INT UC SUR that was 85.6% of the pre-INT period (95% CI: 84.1%, 87.2%, P < 0.001). CAUTI rates for periods A and B were not statistically significantly different at 2.276 vs. 2.164/1000 catheter days (P = 0.803). The CVC SURs for periods A and B were 1.244 and 1.081 (Figure 2). The relative ratio shows a post-INT CVC SUR that was 86.9% of the pre-INT period (95% CI: 85.7%, 88.0%, P < 0.001). CLABSI rates for periods A and B were statistically significantly different at 1.27 vs. 0.804/1000 central line days (P = 0.0335).ConclusionLeadership rounds were associated with a significant decrease in utilization of UCs and CVCs. A significant decrease was noted in CLABSI rates but not in CAUTI rates. Multidisciplinary oversight improved adherence to existing policies and should be considered for optimizing device utilization. Disclosures All authors: No reported disclosures.

Diagnostic accuracy of phenotypic assays for determining antimicrobial resistance status in Staphylococcus pseudintermedius isolates from canine clinical cases

This study evaluated the diagnostic test accuracy of disc diffusion relative to broth-microdilution for clinical Staphylococcus pseudintermedius isolated from dogs in Australia (n = 614). Accuracy of disc diffusion and broth-microdilution for oxacillin relative to mecA real-time PCR was also assessed. Each isolate had paired minimum inhibitory concentration and zone diameter values for ten antimicrobial agents. Data was dichotomised using Clinical and Laboratory Standards Institute susceptible and resistant clinical breakpoints. Test accuracy was reported using relative diagnostic sensitivity (RSe), specificity (RSp), likelihood ratio pairs, diagnostic odds ratio, and area-under-the receiver-operating characteristic (ROC AUC) analysis. Disc diffusion was found to have high test accuracy for most antimicrobials (ROC AUC range: 0.96 – 0.99) except rifampicin (ROC AUC = 0.80). The RSp of disc diffusion was high for all antimicrobials (range, 97.1%–100%). However, RSe was considerably variable (range, 35.7%–98.8%), particularly for amoxicillin-clavulanic acid (51.5%, 95% CI, 38.9%, 64.0%), cefoxitin (35.7%, 95% CI, 12.8%, 64.9%), and cephalothin (43.6%, 95% CI, 27.8%, 60.4%). When disc diffusion and broth-microdilution were compared to mecA real-time PCR, the overall accuracy of both assays was similar (ROC AUC, 0.99 respectively). However, the RSe for broth-microdilution (96.1%, 95% CI, 88.9%, 99.2%) was significantly higher than for disc diffusion (86.8%, 95% CI, 77.1%, 93.5%) (McNemars mid-p value 0.01). Overall, these findings demonstrate that for most antimicrobials, disc diffusion performed according to CLSI guidelines can be used to differentiate clinical S. pseudintermedius isolates that might otherwise be assessed by broth-microdilution, provided consideration is given to the performance estimates reported here.

Cotargeting AR signaling and cell cycle: A randomized phase II study of androgen deprivation therapy with or without palbociclib in RB-positive metastatic hormone sensitive prostate cancer (mHSPC).

251 Background: Palbociclib, a CDK4/6 inhibitor, blocked proliferation and promoted G1 arrest in an Rb and Cyclin D dependent manner in preclinical models of HSPC. Alterations in this pathway contribute to the development of CRPC. We hypothesized that co-targeting AR (ADT) and cell cycle (palbociclib) would improve outcomes including PSA RR at 28 weeks in mHSPC pts. Methods: mHSPC pts with Rb intact tumors based on IHC of metastatic biopsy were stratified and randomized (1:2) to Arm A: ADT or Arm B: ADT+ palbociclib (125mg 3 weeks on, 1 week off). Primary endpoint is confirmed PSA RR (≤ 4 ng/mL) after 28 weeks of therapy. With 20 patients randomized to ADT and 40 to ADT + palbociclib there is a 64.2% power to detect a 20% difference in proportions with a one-sided type I error of 0.10 using the mid p-value method of the Fisher’s exact test. Secondary endpoints include safety/tolerability, biochemical and clinical PFS, PSA and radiographic RR and exploratory biomarkers (circulating DNA, tumor cells and transcriptome). Results: 72 eligible pts (median age 67, PSA 73ng/mL) with newly diagnosed mHSPC were registered and underwent biopsy (41 soft tissue, 31 bone). 64/72 (90%) had adequate tissue for RB assessment and 62/64 (97%) retained RB expression (IHC). 62 pts were stratified by disease extent and early initiation of ADT, and randomized. 60 pts initiated therapy (Arm A: 20; Arm B: 40). Neutropenia was the most common G3/4 AE (Arm A: 0% Arm B: 33%). 80% of pts (Arm A: 16/20, Arm B: 32/40; p = 0.87) on both arms met primary PSA endpoint (≤4ng/mL at 28 weeks). PSA undetectable rate at 28 weeks was Arm A: 50% (10/20) and Arm B: 43% (17/40; p = 0.5). Measurable disease RR: Arm A: 78% and Arm B: 74%. 12-month biochemical PFS was Arm A 69% (95%CI: 43-85%), Arm B 74% (95%CI: 56-85%). Clinical PFS is not mature and correlative biomarker studies are ongoing. Conclusions: Co-targeting of AR signaling and cell cycle in a tissue biomarker preselected trial is feasible in mHSPC. RB loss was rare in this population. PSA RR at 28 weeks was not impacted by addition of palbociclib to ADT. Important clinical PFS data will be reported when mature. Clinical trial information: NCT02059213.

Feasibility of cotargeting cell cycle and androgen signaling in metastatic hormone-sensitive prostate cancer (mHSPC).

151 Background: Most patients (pts) with mHSPC treated with androgen deprivation therapy (ADT) will progress to castration resistance. Androgens drive PC cell proliferation via upregulation of cyclin D which complexes with the cyclin-dependent protein kinases 4/6 (CDK4/6) resulting in phosphorylation of retinoblastoma (Rb) tumor suppressor protein and G1/S progression. Alterations in this pathway contribute to progression to castration-resistance (CR). In preclinical PC models, palbociclib, a novel specific CDK4/ 6 inhibitor, blocked proliferation and promoted G1 arrest in an Rb and Cyclin D dependent manner. Rb is lost in &lt;10% of localized PC, but &gt;30% of CR disease. Rb loss in mHSPC is estimated to be 10-20%. Based on this preclinical data, we designed a randomized phase II clinical trial (NCT02059213) to test the hypothesis that palbociclib + ADT in newly metastatic Rb-positive PC pts will significantly increase ADT efficacy. Methods: mHSPC pts with Rb intact tumors based on metastatic biopsy are stratified by disease extent and randomized (1:2) to ADT or ADT+ palbociclib. Primary endpoint is confirmed prostate specific antigen (PSA) response (≤ 4 ng/mL) after 28 weeks of therapy. With 20 patients randomized to ADT and 40 to ADT + palbociclib there will be a 64.2% power to detect a 20% difference in proportions with a one-sided type I error of 0.10 using the mid p-value method of the Fisher’s exact test. Secondary endpoints: safety and tolerability, rate of undetectable PSA (≤ 0.2ng/mL), biochemical and clinical PFS, overall PSA and radiographic RR, therapeutic biomarkers (circulating DNA and tumor cells, tumor protein and transcriptome) and to establish a repository of mHSPC samples. Results: Since July 2014, 68 patients have been registered and underwent tumor metastatic disease biopsy (36 soft tissue, 32 bone). 60/68 (88%) had adequate tissue for Rb assessment. 58/60 (97%) tumors were Rb positive (IHC). To date, 58/60 patients have been randomized to ADT (19 patients) or ADT + palbociclib (39 patients). Conclusions: Biopsy driven studies are feasible in the mHSPC population. Rb expression was present in 58/60 (97%) patients. This study is ongoing. Support: Movember-PCF Challenge Award, Pfizer. Clinical trial information: NCT02059213.

Abstract IA06: Co-Targeting cell cycle and androgen signaling to personalize therapy for hormone dependent prostate cancer

Abstract Prostate cancer (PC) is an androgen driven and dependent disease and androgen deprivation therapy (ADT) is the standard for patients with metastatic hormone sensitive disease. Despite a high response rate, most patients will progress to castration resistance. The expanding molecular knowledge highlights the biologic heterogeneity and adaptive capacity of PC, and provides the rationale for a preemptive multi-targeted therapeutic strategy. Compelling support for the latter strategy is provided by recent data from two randomized trials demonstrating an unprecedented impact on survival with the addition of docetaxel to ADT. Androgens drive proliferation of PC cells via upregulation of cyclin D which complexes with the cyclin-dependent protein kinases 4/6 (CDK4/6) resulting in phosphorylation of retinoblastoma (Rb) tumor suppressor protein and G1/S progression. Alterations in this pathway contribute to progression to castration-resistance. Palbociclib is a novel specific CDK4/ 6 inhibitor which was recently approved in combination with letrozole for postmenopausal women with ER +, HER2 negative advanced breast cancer. In preclinical prostate cancer models, palbociclib inhibited proliferation and promoted G1 arrest in an Rb and Cyclin D dependent manner. Current data indicate that wild-type Rb expression is present in in 95% vs 75% of patients with newly diagnosed vs metastatic castration resistant PC respectively. We estimate that 80-90% of early metastatic hormone-sensitive PC will harbor intact Rb. Based on the biologic and preclinical data, we developed a multicenter randomized phase II clinical trial (NCT02059213) to test the hypothesis that adding palbociclib to ADT in patients with newly metastatic Rb-positive PC will significantly increase the efficacy of ADT. Patients (n = 60) with new metastatic hormone sensitive PC and Rb intact tumors based on metastatic biopsy are stratified by disease extent and randomized (1:2) to ADT or ADT+ palbociclib. Primary endpoint is confirmed prostate specific antigen (PSA) response (&amp;lt; 4 ng/mL) after 28 weeks of therapy. With 20 patients randomized to ADT and 40 randomized to ADT plus palbociclib there will be a 64.2% power to detect a 20% difference in proportions with a one-sided type I error of 0.10 using the mid p-value method of the Fisher's exact test. Secondary endpoints: safety and tolerability of ADT + palbociclib, rate of undetectable PSA (&amp;lt; 0.2ng/mL), biochemical and clinical progression-free survival, overall PSA and radiographic response rates, assessment of biomarkers which predict therapy response (circulating DNA and tumor cells, tumor protein and transcriptome analysis) and to establish a repository of metastatic hormone sensitive prostate tumor samples. To date 39 patients have been registered and underwent tumor metastatic disease biopsy (soft tissue 26, bone 13). 2 patients were Rb negative and 5 had inadequate tissue. 32 patients have been randomized to ADT (11 patients) or ADT + palbociclib (21 patients). The study is in progress. Support: Movember-PCF Challenge Award, Pfizer. Citation Format: Maha Hussain, Phillip L. Palmbos1, Scott A. Tomlins1, Stephanie Daignault-Newton, Neeraj Agarwal, Przemyslaw Twardowski, Alicia K. Morgans, Emmanuel S. Antonarakis, Karen E. Knudsen, Felix Y. Feng. Co-Targeting cell cycle and androgen signaling to personalize therapy for hormone dependent prostate cancer. [abstract]. In: Proceedings of the AACR Precision Medicine Series: Cancer Cell Cycle - Tumor Progression and Therapeutic Response; Feb 28-Mar 2, 2016; Orlando, FL. Philadelphia (PA): AACR; Mol Cancer Res 2016;14(11_Suppl):Abstract nr IA06.

The mid p-value in exact tests for Hardy-Weinberg equilibrium

Exact tests for Hardy-Weinberg equilibrium are widely used in genetic association studies. We evaluate the mid p-value, unknown in the genetics literature, as an alternative for the standard p-value in the exact test. The type 1 error rate and the power of the exact test are calculated for different sample sizes, significance levels, minor allele counts and degrees of deviation from equilibrium. Three different p-value are considered: the standard two-sided p-value, the doubled one-sided p-value and the mid p-value. Practical implications of using the mid p-value are discussed with HapMap datasets and a data set on colon cancer. The mid p-value is shown to have a type 1 error rate that is always closer to the nominal level, and to have better power. Differences between the standard p-value and the mid p-value can be large for insignificant results, and are smaller for significant results. The analysis of empirical databases shows that the mid p-value uncovers more significant markers, and that the equilibrium null distribution is not tenable for both databases. The standard exact p-value is overly conservative, in particular for small minor allele frequencies. The mid p-value ameliorates this problem by bringing the rejection rate closer to the nominal level, at the price of occasionally exceeding the nominal level.

Score and Pseudo-Score Confidence Intervals for Categorical Data Analysis

This article reviews methods for constructing confidence intervals for analyzing categorical data. A considerable literature indicates that the method of inverting score tests performs well for a variety of cases. When the sample size is small or the parameter is near the parameter space boundary, this method usually performs much better than inverting Wald tests and sometimes better than inverting likelihood-ratio tests. For small samples, exact methods are also available. Although these methods can be quite conservative, inverting a score test using the mid P-value provides a sensible compromise that uses the small-sample distribution while reducing the conservatism and only slightly sacrificing the lower bound for the desired confidence level. For some models, score-test-based inferences are impractical, such as when the likelihood function is not an explicit function of the model parameters. For such cases, pseudo-score inference can be based on a Pearson-type chi-squared statistic that compares fitted values for a working model with fitted values when the parameter of interest takes a fixed value. For some simple cases involving proportions and their differences, a different pseudo-score approach that adds artificial observations before forming Wald confidence intervals provides a simple way of approximating score confidence intervals.

The Minimaxity of the Mid P-value under Linear and Squared Loss Functions

The mid-p is defined as the sum of the probabilities of all outcomes more extreme than an observed value, plus half of the probabilities of all outcomes exactly as extreme. On the one hand, it offers greater power than the standard p-value, but on the other, tests based on the mid-p statistic may have greater Type I error than their nominal level. This article investigates the mid p-value's properties under the estimated truth paradigm, which views p-values as estimators of the truth. The mid-p is shown to minimize the maximum risk for one-sided and two-sided tests.

Stage line diagram: An age‐conditional reference diagram for tracking development

This paper presents a method for calculating stage line diagrams, a novel type of reference diagram useful for tracking developmental processes over time. Potential fields of applications include: dentistry (tooth eruption), oncology (tumor grading, cancer staging), virology (HIV infection and disease staging), psychology (stages of cognitive development), human development (pubertal stages) and chronic diseases (stages of dementia). Transition probabilities between successive stages are modeled as smoothly varying functions of age. Age-conditional references are calculated from the modeled probabilities by the mid-P value. It is possible to eliminate the influence of age by calculating standard deviation scores (SDS). The method is applied to the empirical data to produce reference charts on secondary sexual maturation. The mean of the empirical SDS in the reference population is close to zero, whereas the variance depends on age. The stage line diagram provides quick insight into both status (in SDS) and tempo (in SDS/year) of development of an individual child. Other measures (e.g. height SDS, body mass index SDS) from the same child can be added to the chart. Diagrams for sexual maturation are available as a web application at http://vps.stefvanbuuren.nl/puberty. The stage line diagram expresses status and tempo of discrete changes on a continuous scale. Wider application of these measures scores opens up new analytic possibilities.

A Nonrandomized, Nonconservative Version of the Fisher Exact Test

The Fisher exact test has been unjustly dismissed by some as ‘only conditional,’ whereas it is unconditionally the uniform most powerful test among all unbiased tests, tests of size α and with power greater than its nominal level of significance α. The problem with this truly optimal test is that it requires randomization at the critical value(s) to be of size α. Obviously, in practice, one does not want to conclude that ‘with probability x the we have a statistical significant result.’ Usually, the hypothesis is rejected only if the test statistic's outcome is more extreme than the critical value, reducing the actual size considerably. The randomized unconditional Fisher exact is constructed (using Neyman–structure arguments) by deriving a conditional randomized test randomizing at critical values c(t) by probabilities γ(t), that both depend on the total number of successes T (the complete-sufficient statistic for the nuisance parameter—the common success probability) conditioned upon. In this paper, the Fisher exact is approximated by deriving nonrandomized conditional tests with critical region including the critical value only if γ (t) > γ0, for a fixed threshold value γ0, such that the size of the unconditional modified test is for all value of the nuisance parameter—the common success probability—smaller, but as close as possible to α. It will be seen that this greatly improves the size of the test as compared with the conservative nonrandomized Fisher exact test. Size, power, and p value comparison with the (virtual) randomized Fisher exact test, and the conservative nonrandomized Fisher exact, Pearson's chi-square test, with the more competitive mid-p value, the McDonald's modification, and Boschloo's modifications are performed under the assumption of two binomial samples.

Nonconservative exact small-sample inference for discrete data

Exact small-sample methods for discrete data use probability distributions that do not depend on unknown parameters. However, they are conservative inferentially: the actual error probabilities for tests and confidence intervals are bounded above by the nominal level. This article surveys ways of reducing or even eliminating the conservatism. Fuzzy inference is a recent innovation that enables one to achieve the error probability exactly. We present a simple way of conducting fuzzy inference for discrete one-parameter exponential family distributions. In practice, most scientists would find this approach unsuitable yet might be disappointed by the conservatism of ordinary exact methods. Thus, we recommend using exact small-sample distributions but with inferences based on the mid-P value. This approach can be motivated by fuzzy inference, it is less conservative than standard exact methods, yet usually it does well in terms of achieving desired error probabilities. We illustrate for inferences about the binomial parameter.

Comment: Randomized Confidence Intervals and the Mid-P Approach

We enjoyed reading the interesting, thought-provok ing article by Geyer and Meeden. In our comments we will try to place their work in perspective rela tive to the original proposals for exact and random ized confidence intervals for the binomial parameter. We propose a fuzzy version of the original binomial randomized confidence interval, due to Stevens (1950). Our approach motivates an existing nonrandomized confidence interval based on inverting a test using the mid-P value. The mid-P confidence interval provides a sensible compromise that mitigates the effects of con servatism of exact methods, yet provides results that are more easily understandable to the scientist.

Small Sample Confidence Intervals for the Odds Ratio

Numerous methods—based on exact and asymptotic distributions—can be used to obtain confidence intervals for the odds ratio in 2 × 2 tables. We examine ten methods for generating these intervals based on coverage probability, closeness of coverage probability to target, and length of confidence intervals. Based on these criteria, Cornfield’s method, without the continuity correction, performed the best of the methods examined here. A drawback to use of this method is the significant possibility that the attained coverage probability will not meet the nominal confidence level. Use of a mid-P value greatly improves methods based on the “exact” distribution. When combined with the Wilson rule for selection of a rejection set, the resulting method is a procedure that performed very well. Crow’s method, with use of a mid-P, performed well, although it was only a slight improvement over the Wilson mid-P method. Its cumbersome calculations preclude its general acceptance. Woolf's (logit) method—with the Haldane–Anscombe correction— performed well, especially with regard to length of confidence intervals, and is recommended based on ease of computation.

The equivalence of the mid p-value and the expected p-value for testing equality of two balanced Binomial proportions

This paper considers the problem of testing equality between two independent binomial proportions. Hwang and Yang (Statist. Sinica 11 (2001) 807) apply the Neyman–Pearson fundamental lemma and the estimated truth approach to derive optimal procedures, named expected p-values. This p-value has been shown to be identical to the mid p-value in Lancaster (J. Amer. Statist. Assoc. (1961) 223) for the one-sided test. For the two-sided test, the paper proves the usual two-sided mid p-value is identical to the expected p-value in the balanced sample case.