Power of testing proportions in small two-sample studies when sample sizes are equal

In planning experiments having two groups of equal but small size, investigators face the uncertainty of power calculations that rely on asymptotic methods. This paper presents a method for determining power for two-sided tests. I compare two randomization tests, Fisher's exact test (FET), and the mid-P (MID), with the uncorrected chi-square test (CHI). Results show power as a function of relative risk for these methods, and assesses their relative power and type I error rates. MID is shown to have intermediate power between CHI, which is the most powerful, and FET, the least powerful. Situations are shown in which CHI and MID occasionally exceed the nominal level of α.     

Confidence intervals based on some weighting functions for the difference of two binomial proportions

In this paper, we propose two new methods for computing confidence intervals for the difference of two independent binomial proportions in small sample cases. Several test‐based exact confidence intervals have been developed to guarantee the nominal coverage probability in small sample cases. However, these methods are sometimes unnecessarily too conservative because they use the exact p‐value for constructing confidence intervals by maximizing the tail probability to account for the worst configuration. In order to reduce conservatism, our new methods adopt the p‐value weighted by two types of functions instead of the maximum p‐value. Our proposed methods can be regarded as quasi‐exact methods. The performance evaluation results showed that our methods are much less conservative than the exact method. Compared with other existing quasi‐exact methods, generally, our methods possess coverage probabilities closer to the nominal confidence level and shorter expected confidence widths. In particular, the beta weighing method provides the most reasonable balance between accurate coverage probability and short interval width in small sample cases. Copyright © 2014 John Wiley & Sons, Ltd.     

P value functions: An underused method to present research results and to promote quantitative reasoning

Null hypothesis significance testing has received a great amount of attention in recent years in the light of the reproducibility crisis of science. Recently, there have been calls to retire the dichotomization of study results into “significant” or “not significant” depending on whether the P value crosses some threshold or not. Ways of improving the interpretation of P values and confidence intervals are therefore needed. We illustrate the use of P value functions, which display confidence limits and P values for any confidence level for a parameter. P value functions accessibly display a wealth of information: point estimate for the parameter, one-sided and two-sided confidence limits at any level, and one-sided and two-sided P values for any null and non-null value and the counternull value. Presenting several recent examples from the literature, we show how P value functions can be applied to present evidence and to make informed statistical inferences without resorting to dichotomization. We argue that P value functions are more informative than commonly used summaries of study results such as single P values or confidence intervals. P value functions require minimal retraining, are easily interpreted, and show potential to fix many of the common misinterpretation of P values and confidence intervals. To facilitate the adoption of P value functions, we released an R package for creating P value functions for several commonly used estimates.     

Weighted profile likelihood‐based confidence interval for the difference between two proportions with paired binomial data

Inference on the difference between two binomial proportions in the paired binomial setting is often an important problem in many biomedical investigations. Tang et al. (2010, Statistics in Medicine) discussed six methods to construct confidence intervals (henceforth, we abbreviate it as CI) for the difference between two proportions in paired binomial setting using method of variance estimates recovery. In this article, we propose weighted profile likelihood‐based CIs for the difference between proportions of a paired binomial distribution. However, instead of the usual likelihood, we use weighted likelihood that is essentially making adjustments to the cell frequencies of a 2 × 2 table in the spirit of Agresti and Min (2005, Statistics in Medicine). We then conduct numerical studies to compare the performances of the proposed CIs with that of Tang et al. and Agresti and Min in terms of coverage probabilities and expected lengths. Our numerical study clearly indicates that the weighted profile likelihood‐based intervals and Jeffreys interval (cf. Tang et al.) are superior in terms of achieving the nominal level, and in terms of expected lengths, they are competitive. Finally, we illustrate the use of the proposed CIs with real‐life examples. Copyright © 2014 John Wiley & Sons, Ltd.     

A Nutrient Approach to Prostate Cancer Prevention: The Selenium and Vitamin E Cancer Prevention Trial (SELECT)

The Selenium and Vitamin E Cancer Prevention Trial (SELECT) randomized 35,533 healthy men, >55 yr old (>50 yr if African American), with normal digital rectal exams and prostate specific antigens <4 ng/ml to 1) 200 μg/day l-selenomethionine, 2) 400 IU/day all-rac-alpha-tocopheryl acetate (vitamin E), 3) both supplements, or 4) placebo for 7 to 12 yr. The hypotheses underlying SELECT, that selenium and vitamin E individually and together decrease prostate cancer incidence, derived from epidemiologic and laboratory evidence and significant secondary endpoints in the Nutritional Prevention of Cancer (selenium) and Alpha-Tocopherol Beta-Carotene (vitamin E) trials. In SELECT, prostate cancer incidence did not differ among the 4 arms: hazard ratios [99% confidence intervals (CIs)] for prostate cancer were 1.13 (99% CI = 0.95–1.35, P = 0.06; n = 473) for vitamin E, 1.04 (99% CI = 0.87–1.24, P = 0.62; n = 432) for selenium, and 1.05 (99% CI = 0.88–1.25, P = 0.52; n = 437) for selenium + vitamin E vs. 1.00 (n = 416) for placebo. Statistically nonsignificant increased risks of prostate cancer with vitamin E alone [relative risk (RR) = 1.13, P = 0.06) and newly diagnosed Type 2 diabetes mellitus with selenium alone (RR = 1.07, P = 0.16) were observed. SELECT data show that neither selenium nor vitamin E, alone or together, in the doses and formulations used, prevented prostate cancer in this heterogeneous population of healthy men.     

Improved confidence intervals for a binomial parameter following a group sequential phase II clinical trial

Group sequential procedures are widely employed in phase II clinical trials. It is often desirable to provide an interval estimate of the true response rate upon termination of a group sequential phase II clinical trial. The confidence intervals proposed by Jennison and Turnbull (Technometrics 1983; 25: 49-58) are conservative. Utilization of auxiliary statistics based on the overall disease status is proposed to reduce the discreteness of the underlying distribution. Confidence intervals generated by the proposed methods have confidence coefficients closer to the nominal level and have shorter average lengths than JT.     

Finite sample pointwise confidence intervals for a survival distribution with right‐censored data

We review and develop pointwise confidence intervals for a survival distribution with right‐censored data for small samples, assuming only independence of censoring and survival. When there is no censoring, at each fixed time point, the problem reduces to making inferences about a binomial parameter. In this case, the recently developed beta product confidence procedure (BPCP) gives the standard exact central binomial confidence intervals of Clopper and Pearson. Additionally, the BPCP has been shown to be exact (gives guaranteed coverage at the nominal level) for progressive type II censoring and has been shown by simulation to be exact for general independent right censoring. In this paper, we modify the BPCP to create a ‘mid‐p’ version, which reduces to the mid‐p confidence interval for a binomial parameter when there is no censoring. We perform extensive simulations on both the standard and mid‐p BPCP using a method of moments implementation that enforces monotonicity over time. All simulated scenarios suggest that the standard BPCP is exact. The mid‐p BPCP, like other mid‐p confidence intervals, has simulated coverage closer to the nominal level but may not be exact for all survival times, especially in very low censoring scenarios. In contrast, the two asymptotically‐based approximations have lower than nominal coverage in many scenarios. This poor coverage is due to the extreme inflation of the lower error rates, although the upper limits are very conservative. Both the standard and the mid‐p BPCP methods are available in our bpcp R package. Published 2016. This article is US Government work and is in the public domain in the USA.     

A simulation study of methods for constructing confidence intervals for bioaccumulation factors

Toxicokinetics describe the time course of xenobiotics in an exposed organism. The steady state concentration of the xenobiotic is a characteristic of frequent concern, often summarized in terms of bioaccumulation factors (BAFs). Two experimental protocols are commonly used when estimating BAFs: a continuous exposure experiment and a start-stop exposure experiment. Nonlinear regression is then used for estimating components of the BAF. A variety of methods are available for constructing confidence intervals (CI) for the BAF, including the delta method and Fieller's method. The properties of these CI methods are explored through a simulation study. In particular, the coverage properties of the two CI methods along with their interval widths are examined for a variety of time-course patterns, error distributions, experimental protocols, and sampling designs. Fieller's method is observed to exceed nominal coverage probabilities for all conditions considered (the coverage probability being the probability that a confidence interval contains a parameter it estimates over repeated experiments). The delta method exhibited observed coverage probabilities lower than nominally specified for a number of conditions. For sampling designs in which more of the sampling times are prior to steady state or the stop exposure time, the delta method is observed to provide nominal coverage probabilities with interval widths much narrower than observed with the Fieller's method. Thus, the delta method is preferable when the above sampling time condition is met; however, in all other cases, Fieller's method should be used.     

Bayesian intervals for linkage locations

Intermediate fine mapping has received considerable attention recently, with the goal of providing statistically precise and valid chromosomal regions for fine mapping following initial identification of broad regions that are linked to a disease. The following classes of methods have been proposed and compared in the literature: (1) LOD‐support intervals, (2) generalized estimating equations, (3) bootstrap, and (4) confidence set inference framework. These methods provide confidence intervals either with coverage levels deviating from the nominal confidence levels or that are not fully efficient. Here, we propose a novel Bayesian method for constructing such intervals using affected sibling pair data. The susceptibility gene location is treated as a parameter in this method, with a uniform prior. A Metropolis‐Hastings algorithm is implemented to sample from the posterior distribution and highest posterior density intervals of the disease gene locations are constructed. Correct coverage levels are maintained by our method. Both simulation studies and an application to a rheumatoid arthritis dataset demonstrate the improved efficiency of the Bayesian intervals compared with existing methods. Genet. Epidemiol. 33:604–616, 2009. © 2009 Wiley‐Liss, Inc.     

Correlation Between MTHFR Polymorphisms and Hepatocellular Carcinoma: A Meta-analysis

The correlation between methylenetetrahydrofolate reductase (MTHFR) polymorphisms and hepatocellular carcinoma (HCC) remains controversial. Therefore, we performed this study to better assess the relationship between MTHFR polymorphisms and the likelihood of HCC. A systematic research of PubMed, Medline, and Embase was performed to retrieve relevant articles. Odds ratios (ORs) and 95% confidence intervals (CIs) were calculated. A total of 15 studies with 8,378 participants were analyzed. In overall analyses, a significant association with the likelihood of HCC was detected for the rs1801131 polymorphism with fixed-effect models in recessive comparison (P?=?0.002, OR = 0.62, 95% CI 0.43–0.82). However, no positive results were detected for the rs1801133 polymorphism in any comparison. Further subgroup analyses revealed that the rs1801131 polymorphism was significantly associated with the likelihood of HCC in Asians with both FEMs (recessive model: P?<?0.0001, OR = 0.42, 95% CI 0.29–0.62; allele model: P?=?0.004, OR =1.20, 95% CI 1.06–1.35) and random-effect models (recessive model: P?=?0.002, OR = 0.47, 95% CI 0.29–0.75). Nevertheless, we failed to detect any significant correlation between the rs1801133 polymorphism and HCC. In conclusion, our findings indicated that the rs1801131 polymorphism may serve as a genetic biomarker of HCC in Asians.     

Plasma Riboflavin Level is Associated with Risk,Relapse, and Survival of Esophageal Squamous Cell Carcinoma

Riboflavin is an essential micronutrient for normal cellular activity, and deficiency may result in disease, such as cancer. We performed a case-control study to explore the association of riboflavin levels with risk and prognosis of esophageal squamous cell carcinoma (ESCC). Plasma riboflavin levels, as measured by enzyme-linked immunosorbent assay (ELISA), in ESCC patients were significantly lower than in those of healthy controls (7.04 ± 6.34 ng/ml vs. 9.32 ± 12.40 ng/ml; P < 0.05). Moreover, there was an inverse relationship between riboflavin level and risk of ESCC (odds ratio (OR) = 0.97, 95% confidence interval (CI) = 0.95–0.99, P = 0.02). The 5-year relapse-free and overall survival rates were significantly lower when riboflavin levels were ≤0.8 ng/ml than >0.8 ng/ml (relapse-free survival rate: 29.4% vs. 54.8%; overall survival rate: 28.6% vs. 55.6%). Plasma riboflavin level was an independent protective factor for both relapse-free (hazard ratio (HR) = 0.325, 95% CI = 0.161–0.657, P = 0.002) and overall survival of ESCC patients (HR = 0.382, 95% CI = 0.190–0.768, P = 0.007). In conclusion, plasma riboflavin levels are significantly related to risk and prognosis of ESCC patients, suggesting that moderate supplementation of riboflavin will decrease risk and prevent recurrence of ESCC and also improve prognosis of ESCC patients.     

Interval estimation of the over‐dispersion parameter in the analysis of one‐way layout of count data

The over‐dispersion parameter is an important and versatile measure in the analysis of one‐way layout of count data in biological studies. For example, it is commonly used as an inverse measure of aggregation in biological count data. Its estimation from finite data sets is a recognized challenge. Many simulation studies have examined the bias and efficiency of different estimators of the over‐dispersion parameter for finite data sets (see, for example, Clark and Perry, Biometrics 1989; 45:309–316 and Piegorsch, Biometrics 1990; 46:863–867), but little attention has been paid to the accuracy of the confidence intervals (CIs) of it. In this paper, we first derive asymptotic procedures for the construction of confidence limits for the over‐dispersion parameter using four estimators that are specified by only the first two moments of the counts. We also obtain closed‐form asymptotic variance formulae for these four estimators. In addition, we consider the asymptotic CI based on the maximum likelihood (ML) estimator using the negative binomial model. It appears from the simulation results that the asymptotic CIs based on these five estimators have coverage below the nominal coverage probability. To remedy this, we also study the properties of the asymptotic CIs based on the restricted estimates of ML, extended quasi‐likelihood, and double extended quasi‐likelihood by eliminating the nuisance parameter effect using their adjusted profile likelihood and quasi‐likelihoods. It is shown that these CIs outperform the competitors by providing coverage levels close to nominal over a wide range of parameter combinations. Two examples to biological count data are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

An investigation of error sources and their impact in estimating the time to the most recent ancestor of spatially and temporally distributed HIV sequences

This is an investigation of significant error sources and their impact in estimating the time to the most recent common ancestor (MRCA) of spatially and temporally distributed human immunodeficiency virus (HIV) sequences. We simulate an HIV epidemic under a range of assumptions with known time to the MRCA (tMRCA). We then apply a range of baseline (known) evolutionary models to generate sequence data. We next estimate or assume one of several misspecified models and use the chosen model to estimate the time to the MRCA. Random effects and the extent of model misspecification determine the magnitude of error sources that could include: neglected heterogeneity in substitution rates across lineages and DNA sites; uncertainty in HIV isolation times; uncertain magnitude and type of population subdivision; uncertain impacts of host/viral transmission dynamics, and unavoidable model estimation errors. Our results suggest that confidence intervals will rarely have the nominal coverage probability for tMRCA. Neglected effects lead to errors that are unaccounted for in most analyses, resulting in optimistically narrow confidence intervals (CI). Using real HIV sequences having approximately known isolation times and locations, we present possible confidence intervals for several sets of assumptions. In general, we cannot be certain how much to broaden a stated confidence interval for tMRCA. However, we describe the impact of candidate error sources on CI width. We also determine which error sources have the most impact on CI width and demonstrate that the standard bootstrap method will underestimate the CI width.     

Meta-analysis of quantile intervals from different studies with an application to a pulmonary tuberculosis data

After the completion of many studies, experimental results are reported in terms of distribution-free confidence intervals that may involve pairs of order statistics. This article considers a meta-analysis procedure to combine these confidence intervals from independent studies to estimate or construct a confidence interval for the true quantile of the population distribution. Data synthesis is made under both fixed-effect and random-effect meta-analysis models. We show that mean square error (MSE) of the combined quantile estimator is considerably smaller than that of the best individual quantile estimator. We also show that the coverage probability of the meta-analysis confidence interval is quite close to the nominal confidence level. The random-effect meta-analysis model yields a better coverage probability and smaller MSE than the fixed-effect meta-analysis model. The meta-analysis method is then used to synthesize medians of patient delays in pulmonary tuberculosis diagnosis in China to provide an illustration of the proposed methodology.     

Hartung–Knapp method is not always conservative compared with fixed‐effect meta‐analysis

A widely used method in classic random‐effects meta‐analysis is the DerSimonian–Laird method. An alternative meta‐analytical approach is the Hartung–Knapp method. This article reports results of an empirical comparison and a simulation study of these two methods and presents corresponding analytical results. For the empirical evaluation, we took 157 meta‐analyses with binary outcomes, analysed each one using both methods and performed a comparison of the results based on treatment estimates, standard errors and associated P‐values. In several simulation scenarios, we systematically evaluated coverage probabilities and confidence interval lengths. Generally, results are more conservative with the Hartung–Knapp method, giving wider confidence intervals and larger P‐values for the overall treatment effect. However, in some meta‐analyses with very homogeneous individual treatment results, the Hartung–Knapp method yields narrower confidence intervals and smaller P‐values than the classic random‐effects method, which in this situation, actually reduces to a fixed‐effect meta‐analysis. Therefore, it is recommended to conduct a sensitivity analysis based on the fixed‐effect model instead of solely relying on the result of the Hartung–Knapp random‐effects meta‐analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Plasma Carotenoids and Retinol and Overall and Breast Cancer Risk: A Nested Case-Control Study

Experimental studies suggest that carotenoids and retinol may play a role in carcinogenesis, but epidemiological evidence is lacking. We investigated the prospective associations between plasma concentrations of major carotenoids and retinol, and overall and breast cancer risk. A nested case-control study included all first incident cancer cases diagnosed in the SU.VI.MAX cohort between 1994 and 2002 (n = 159 cases, 1 matched control/case). Baseline plasma concentrations of carotenoids and retinol were measured by high-performance liquid chromatography. Conditional logistic regression was used to assess odds ratios for an increase of 0.1 μmol/L [odds ratio (OR)] and 95% confidence intervals (CI). Plasma β-carotene (OR = 0.95, 95% CI = 0.90–0.99, P_trend = 0.04) and β-cryptoxanthin concentrations (OR = 0.89, 95% CI = 0.81–0.99, P_trend = 0.03) were inversely associated with overall cancer risk. Plasma β-cryptoxanthin concentration was inversely associated with breast cancer risk (OR = 0.83, 95% CI = 0.71–0.96, P_trend = 0.02). The OR between plasma lycopene concentration and overall cancer risk was 1.07 (0.99–1.15), P_trend = 0.06. This association turned significant (P_trend = 0.01) when excluding cancer cases diagnosed during the first year of follow-up. This prospective study suggests an inverse association between plasma concentrations of β-cryptoxanthin and both overall and breast cancer risk, and an inverse association between β-carotene and overall cancer risk. The direct association between lycopene concentration and cancer risk deserves further investigation.     

Efficient power computation for exact and mid-P tests for the common odds ratio in several 2 × 2 tables

When designing a study that may generate a set of sparse 2 × 2 tables, or when confronted with ‘negative’ results upon exact analysis of such tables, we need to compute the power of exact tests. In this paper we provide an efficient approach for computing exact unconditional power for four exact tests on the common odds ratio in a series of 2 × 2 tables. These tests are the traditional exact test; a test based on a probability ordering of the sample space; and two tests based on ordering the sample space according to distance from the mean, or median. For each test, we consider both a conservative version and a mid-P adjusted version. We explore three computational options for power determination: exact power computation, calculation of exact upper and lower bounds for power, and Monte Carlo confidence bounds for power. We present an interactive program implementing these options. For study design, the program may be run several times to arrive at a sample configuration with adequate power.     

Confidence limits for the ratio of two rates based on likelihood scores: non-iterative method

This paper describes a method for creating a confidence interval for the ratio of rates using the score statistic. This non-iterative and easy to apply procedure produces confidence intervals that are suitable for use with Poisson data and simulation results indicate that it is close to the nominal level for a wide range of scenarios.     

Fruits and Vegetables Consumption and Esophageal Squamous Cell Carcinoma: A Case-Control Study

The authors examined the association of food group intakes and the risk of esophageal squamous cell carcinoma (SCC) in a hospital-based case-control study in Iran. In total, 47 patients with esophageal SCC and 96 controls underwent face-to-face private interviews. Usual dietary intake was assessed using a semiquantitative food frequency questionnaire. Multivariate logistic regression was used to estimate odds ratios and 95% confidence intervals. Cases had higher tobacco consumption and symptomatic gastresophageal reflux, whereas controls had higher mean body mass index (25.3 vs. 20.4) and years of education. A protective independent effect was observed for the highest tertile of total fruit consumption (OR: 0.13, CI: 0.04–0.45, P value = 0.001). Within the group of fruits, a significant inverse association was observed for bananas and kiwis (P for trends: 0.03 and 0.02, respectively). The risk of SCC decreased monotonically with increasing intake frequency of oranges (P value for trend = 0.01). The effect of total vegetable consumption on esophageal SCC was not significant, although a reduction in risk was observed in the highest tertile of intake (OR: 0.66, CI: 0.23–1.87, P value = 0.43). The results of the present study suggest a reasonable association between fruit consumption and esophageal SCC in a Middle Eastern high-risk population.     

The Effects of Probiotics in Early Enteral Nutrition on the Outcomes of Trauma

Background: The role of probiotics in trauma patients remains unclear. We undertook a meta‐analysis of published randomized controlled trials (RCTs) to assess the effects of probiotics on the clinical outcomes of trauma patients. Methods: A systematic electronic literature search was conducted to identify RCTs comparing the use of probiotics with a control in trauma patients. Results were expressed as risk ratios (RRs) or standardized mean differences (SMDs) with accompanying 95% confidence intervals (CIs). The primary outcome measurement was the incidence of nosocomial infections. Secondary outcome measurements included the incidence of ventilator‐associated pneumonia (VAP), length of intensive care unit (ICU) stay, and mortality. The meta‐analysis was performed with the fixed‐effect or random‐effect model according to the heterogeneity. Results: Five studies involving 281 patients met our inclusion criteria. The use of probiotics was associated with a reduction in the incidence of nosocomial infections (5 trials; RR, 0.65; 95% CI, 0.45–0.94, P = .02), VAP (3 trials; RR, 0.59; 95% CI, 0.42–0.81, P = .001), and length of ICU stay (2 trials; SMD, ?0.71; 95% CI, ?1.09 to ?0.34, P < .001) but no reduction in mortality (4 trials; RR, 0.63; 95% CI, 0.32–1.26, P = .19). Conclusions: The use of probiotics is associated with a reduction in the incidence of nosocomial infections, VAP, and length of ICU stay but is not associated with an overall mortality advantage. However, the results should be interpreted cautiously due to the heterogeneity among study designs. Further large‐scale, well‐designed RCTs are needed.