SAMPLE SIZE FOR THE EXACT CONDITIONAL TEST UNDER INVERSE SAMPLING

Inverse sampling is a sampling design in which one continues sampling subjects until one obtains a predetermined number of index subjects. This paper derives a procedure for calculation of the minimum required number of index subjects on the basis of the exact conditional test under inverse sampling. This paper studies quantitatively the effect on power calculations of the number of index subjects. To facilitate use of inverse sampling in study designs, this paper further provides a table that summarizes, in a variety of situations, the minimum required number of index subjects for powers equal to 0⋅90 and 0⋅80 at 0⋅05-level. It also includes a discussion on use of the approximation sample size formula derived on the basis of a variance-stabilizing transformation and large sample theory.     

Notes on conditional confidence limits under inverse sampling

When the number of subjects in a two-by-two table is small or moderate, we may commonly use the exact conditional distribution with all marginals fixed to derive the conditional confidence limits on the underlying parameter. Under inverse sampling, in which we continue to sample subjects until we obtain exactly a pre-determined number of subjects falling into a specific category, this paper notes that derivation of a confidence interval, which has the coverage probability equal to or larger than a nominal 1 – α confidence level, for relative risk and relative difference in cohort studies is straightforward. This paper further finds that, when the underlying disease is rare, we can similarly apply an inverse sampling to produce an approximate 1 – α conditional confidence limits on attributable risk in case-control studies as well. When the number of subjects is small and the test statistic derived on the basis of large sample theory is not strictly adequate for use, this paper also presents an exact hypothesis testing procedure for the above parameters in the corresponding study designs.     

Exact and asymptotic inference in clinical trials with small event rates under inverse sampling

In this paper, we discuss statistical inference for a 2 × 2 table under inverse sampling, where the total number of cases is fixed by design. We demonstrate that the exact unconditional distributions of some relevant statistics differ from the distributions under conventional sampling, where the sample size is fixed by design. This permits us to define a simple unconditional alternative to Fisher's exact test. We provide an asymptotic argument including simulations to demonstrate that there is little power loss associated with the alternative test when the expected event rates are very small. We then apply the method to design a clinical trial in cataract surgery, where a rare side effect occurs in one in 1000 patients. The objective of the trial is to demonstrate that adjuvant treatment with an antibiotic will reduce this risk to one in 2000. We use an inverse sampling design and demonstrate how to set this up in a sequential manner. Particularly simple stopping rules can be defined when using the unconditional alternative to Fisher's exact test. Copyright © 2015 John Wiley & Sons, Ltd.     

DIFFERENTIAL RECALL BIAS AND SPURIOUS ASSOCIATIONS IN CASE/CONTROL STUDIES

Consider a case/control study designed to investigate a possible association between exposure to a putative risk factor and development of a particular disease. Let E denote the information required to specify a subject's exposure to the risk factor. We examine the effect that errors in the recorded values of E (which we denote by E^*) have on inferences of an association between disease and the risk factor. We concentrate on situations where the errors in recorded exposure are such that exposure is underestimated for controls and overestimated for cases. This phenomenon is referred to as differential recall bias and may lead to spurious inferences of an association between exposure and disease. We describe how the standard inferential techniques used in the analysis of data from case/control studies may be adjusted to take account of specified mechanisms whereby E is distorted to produce E^*. Such adjustments may be used to determine the sensitivity of an analysis to the phenomenon of differential recall bias and to quantify the extent of such bias that would be required to overturn the conclusions of the analysis. There remains the matter of judging whether a given distortion mechanism is reasonable in a particular context. This emphasizes the need for investigators to take account of differential recall bias in validation studies of exposure assessment techniques. The methodology developed here is applied to a recent major study investigating the possible association between lung cancer and exposure to environmental tobacco smoke. The log-odds ratio of 0⋅23 based on recorded exposure differs significantly from 0 (p<0⋅02). However, the association is rendered non-significant by a very modest degree of differential recall bias. For example, if 3⋅8 per cent of exposed controls report no exposure, 3⋅8 per cent of unexposed cases report exposure, and all other subjects report exposure accurately, the log-odds ratio drops to 0⋅07 and the corresponding p-value increases to 0⋅49.     

AN INDEX FOR ASSESSING BLINDNESS IN A MULTI-CENTRE CLINICAL TRIAL: DISULFIRAM FOR ALCOHOL CESSATION—A VA COOPERATIVE STUDY

This paper considers an index to assess the success of blinding with application to a clinical trial of disulfiram. The index increases as the success of blinding increases, accounts for uncertain responses, and is scaled to an interval of 0⋅0 to 1⋅0, 0⋅0 being complete lack of blinding and 1⋅0 being complete blinding.     

Score confidence intervals and sample sizes for stratified comparisons of binomial proportions

In a series of articles, Gart and Nam construct the efficient score tests and confidence intervals with or without skewness correction for stratified comparisons of binomial proportions on the risk difference, relative risk, and odds ratio effect metrics. However, the stratified score methods and their properties are not well understood. We rederive the efficient score tests, which reveals their theoretical relationship with the contrast-based score tests, and provides a basis for adapting the method by using other weighting schemes. The inverse variance weight is optimal for a common treatment effect in large samples. We explore the behavior of the score approach in the presence of extreme outcomes when either no or all subjects in some strata are responders, and provide guidance on the choice of weights in the analysis of rare events. The score method is recommended for studies with a small number of moderate or large sized strata. A general framework is proposed to calculate the asymptotic power and sample size for the score test in superiority, noninferiority and equivalence clinical trials, or case-control studies. We also describe a nearly exact procedure that underestimates the exact power, but the degree of underestimation can be controlled to a negligible level. The proposed methods are illustrated by numerical examples.     

Some design issues of strata-matched non-randomized studies with survival outcomes

Non-randomized studies for the evaluation of a medical intervention are useful for quantitative hypothesis generation before the initiation of a randomized trial and also when randomized clinical trials are difficult to conduct. A strata-matched non-randomized design is often utilized where subjects treated by a test intervention are matched to a fixed number of subjects treated by a standard intervention within covariate based strata. In this paper, we consider the issue of sample size calculation for this design. Based on the asymptotic formula for the power of a stratified log-rank test, we derive a formula to calculate the minimum number of subjects in the test intervention group that is required to detect a given relative risk between the test and standard interventions. When this minimum number of subjects in the test intervention group is available, an equation is also derived to find the multiple that determines the number of subjects in the standard intervention group within each stratum. The methodology developed is applied to two illustrative examples in gastric cancer and sarcoma.     

Deviance estimates of sample size for equivalence tests in vaccine trials

This paper proposes a sample size procedure for both equivalence and conventional tests for the comparison of two binomial proportions, based on the signed square root of the deviance. When the comparison is based on the odds ratio, I describe an alternate ‘close’ conditional exact method that gives results that support those given by the deviance method. I summarize the advantages of the deviance-based method and also show that in general equivalence situations the sample size estimate depends upon the measure of comparison selected, odds ratio, risk ratio or risk difference.     

Confidence limits for the population prevalence rate based on the negative binomial distribution

This paper shows that the extension of the simple procedure of George and Elston in calculation of confidence limits for the underlying prevalence rate to accommodate any finite number of cases in inverse sampling is straightforward. To appreciate the fact that the length of the confidence interval calculated on the basis of the first single case may be too wide for general utility, I include a quantitative discussion on the effect due to an increase in the number of cases requested in the sample on the expected length of confidence intervals. To facilitate further the application of the results presented in this paper, I present a table that summarizes in a variety of situations the minimum required number of cases for the ratio of the expected length of a confidence interval relative to the underlying prevalence rate to be less than or equal to a given value. I also include a discussion on the relation between Cleman'S confidence limits on the expected number of trials before the failure of a given device and those presented here.     

A simple method to estimate sample sizes for safety equivalence studies using inverse sampling

Safety equivalence studies may be required to demonstrate that a new procedure or process is at least as safe as a previous one. They usually involve low or very low outcome rates that are often not precisely determined, making patient-based sample sizing uncertain. Using a reverse sampling approach, a method is derived from standard equations to estimate the number of events that need to be observed to demonstrate equivalence using the confidence interval approach. For instance, for a one-sided (nonsuperiority) hypothesis, 5% alpha risk, and 80% power, almost 100 events need to be observed in each study arm to demonstrate equivalence within 30%, or 250 events for 20% equivalence. The number of patients to be included can be derived directly from expected event rates.     

Review of statistical procedures for determining the equivalency of two treatments

The absence of a significant difference in a classical efficacy trial testing the null hypothesis of equality between N and S does not allow us to conclude that the treatments are equivalent. Testing the null hypothesis of N not equivalent to S requires: specifying the definition of "equivalence" by choosing delta L, the upper allowable value of the actual difference between two equivalent treatments. The appropriate statistic D which evaluates the difference between N and S, has a non central distribution under the null hypothesis of inequivalence (Ko:[E(D)] greater than or equal to delta L, two-sided test). Under the null hypothesis for a two-sided test, parameters of noncentral distribution have to be estimated, and the critical p-value is obtained using some approximation. Confidence interval of the true difference delta can also provide a decision rule. Specific calculation of the minimum number of subjects is required when designing an equivalence trial.     

Tests for equivalence or non-inferiority for paired binary data.

Assessment of therapeutic equivalence or non-inferiority between two medical diagnostic procedures often involves comparisons of the response rates between paired binary endpoints. The commonly used and accepted approach to assessing equivalence is by comparing the asymptotic confidence interval on the difference of two response rates with some clinical meaningful equivalence limits. This paper investigates two asymptotic test statistics, a Wald-type (sample-based) test statistic and a restricted maximum likelihood estimation (RMLE-based) test statistic, to assess equivalence or non-inferiority based on paired binary endpoints. The sample size and power functions of the two tests are derived. The actual type I error and power of the two tests are computed by enumerating the exact probabilities in the rejection region. The results show that the RMLE-based test controls type I error better than the sample-based test. To establish an equivalence between two treatments with a symmetric equivalence limit of 0.15, a minimal sample size of 120 is needed. The RMLE-based test without the continuity correction performs well at the boundary point 0. A numerical example illustrates the proposed procedures.     

The partial testing design: a less costly way to test equivalence for sensitivity and specificity

We propose a new, less costly, design to test the equivalence of digital versus analogue mammography in terms of sensitivity and specificity. Because breast cancer is a rare event among asymptomatic women, the sample size for testing equivalence of sensitivity is larger than that for testing equivalence of specificity. Hence calculations of sample size are based on sensitivity. With the proposed design it is possible to achieve the same power as a completely paired design by increasing the number of less costly analogue mammograms and not giving the more expensive digital mammograms to some randomly selected subjects who are negative on the analogue mammogram. The key idea is that subjects who are negative on the analogue mammogram are unlikely to have cancer and hence contribute less information for estimating sensitivity than subjects who are positive on the analogue mammogram. To ascertain disease state among subjects not biopsied, we propose another analogue mammogram at a later time determined by a natural history model. The design differs from a double sampling design because it compares two imperfect tests instead of combining information from a perfect and imperfect test. © 1998 John Wiley & Sons, Ltd.     

ASSESSMENT OF SMALL HEALTH RISKS BASED ON EXACT SAMPLE SIZES

Exact sample sizes and critical numbers of cases for the rejection of a known event probability (10^-2 to 10^-6) in favour of an increased probability (1⋅5- to 50-fold) at levels {α; β} = {0⋅05; 0⋅10} and {α; β} = {0⋅10; 0⋅05} are presented. The numbers are thoroughly validated using the characteristics of the confidence interval for the unknown true event probability. Equivalence is shown to be obtainable for the tolerated maximal value of the relative risk and the upper limit of the confidence interval for the true event probability. Also demonstrated is the use of the tables for planned actions to reduce given empirical risks. In addition, use of the tables is shown for judging results from given data sets.     

Sample size calculations for risk equivalence testing in pharmacoepidemiology

Equivalence testing has been widely discussed and is commonly used in pharmacokinetics (bioequivalence) and clinical trials (therapeutic equivalence). It can also be applied to pharmacoepidemiology, where the aim may be to test with a known risk (one-group design) or with another drug (two-group design). Whether the approach is two-sided or one-sided, predefined equivalence limits are required. The definition of the equivalence region can be based on either risk difference or risk ratio. Risk equivalence testing is complicated by the binary nature of the outcome, its low frequency, and by the absence of commonly defined equivalence limits for differences or ratios. In this context, we consider usable formulae for sample sizes. In most cases, at least when the risk studied is large enough (above 1/1,000), it appears that these formulae result in sample sizes that may be acceptable for practical purposes. For example, demonstrating equivalence with a known risk of 0.01, a 20% maximal risk difference, and a one-sided test ( = 0.05 and β = 0.2) requires: under the one-group design (known risk), 15,309 patients; and under the two-group design, 30,617 patients per group. This approach is the appropriate way to conclude equivalence, rather than the commonly used approach of difference testing and concluding equivalence when the null hypothesis of equality is not rejected.     

STATISTICAL ANALYSIS OF ZIDOVUDINE (AZT) EFFECT ON CD4 CELL COUNTS IN HIV DISEASE

We fit a class of random effects linear growth curve models for the square root of CD4 count to serial marker data from 164 HIV-positive individuals with known (or accurately estimated) dates of seroconversion and at least 10 CD4 measurements each (median 16). We do so by adopting a Bayesian viewpoint and using the Markov chain Monte Carlo technique Gibbs sampling. In particular, we examine the effect of the antiretroviral treatment zidovudine on the √CD4 series for the 136 patients who took the drug. Treatment effects are modelled by positing recoveries in √CD4 level proportional to current immuno-competence and changes in slope proportional to current rate of √CD4 loss. Both fixed and random treatment effects are considered and models are criticized and compared using Bayesian predictive methodology and checking data which comprise 424 new observations. Results indicate re-elevation of √CD4 level is associated with treatment but the effect, though significant, is mostly of small magnitude and is possibly transient; models neglecting consideration of treatment fit the checking data almost as well. Best overall model estimates mean rate of √CD4 loss per annum to be 2⋅1 (standard error 0⋅12); mean seroconversion value of √CD4 is 28⋅4 (SE 0⋅65). The estimated variance of individual slopes is 1⋅9 (SE 0⋅28), there being considerable individual variation in rate of CD4 loss, and a recovery in level of 0⋅047 (SE 0⋅014) times current √CD4 level is estimated at treatment uptake.     

Individual bioequivalence: what matters to the patient

Current procedures for assessing the bioequivalence of two formulations are based on the concept of average bioequivalence. That is, they assess whether the average responses between individuals on the two formulations are similar. We show first that average bioequivalence is not sufficient to assure that an individual patient could be expected to respond similarly to the two formulations. To have such reasonable assurance requires a different notion of bioequivalence; individual (or within-subject) bioequivalence. Second, we propose a simple statistical procedure for assessing individual bioequivalence. This decision rule, TIER (test of individual equivalence ratios) requires the specification of the minimum proportion of subjects in the applicable population for which the two formulations being tested must be bioequivalent (a regulatory decision). The TIER rule is summarized in terms of the minimum number of subjects with bioavailability ratios falling within the specified equivalence interval necessary to be able to claim bioequivalence for given sample size and type I error. We recommend that the corresponding lower bounds (one-sided confidence intervals) for the proportion of bioequivalent subjects be calculated.     

Test non-inferiority (and equivalence) based on the odds ratio under a simple crossover trial

For testing the non-inferiority (or equivalence) of a generic drug to a standard drug, the odds ratio (OR) of patient response rates has been recommended to measure the relative treatment efficacy. On the basis of a random effects logistic regression model, we develop asymptotic test procedures for testing non-inferiority and equivalence with respect to the OR of patient response rates under a simple crossover design. We further derive exact test procedures, which are especially useful for the situations in which the number of patients in a crossover trial is small. We address sample size calculation for testing non-inferiority and equivalence based on the asymptotic test procedures proposed here. We also discuss estimation of the OR of patient response rates for both the treatment and period effects. Finally, we include two examples, one comparing two solution aerosols in treating asthma, and the other one studying two inhalation devices for asthmatics, to illustrate the use of the proposed test procedures and estimators.     

How to establish equivalence when data are censored: A randomized trial of treatments for B non-Hodgkin lymphoma

Interest in equivalence trials has been increasing for many years, though the methodology which has been developed for such trials is mainly for uncensored data. In cancer research we are more often concerned with survival. In an efficacy trial, the null hypothesis specifies equality of the two survival distributions, but in an equivalence trial, a null hypothesis of inequivalence H₀ has to be tested. The usual logrank test has to be modified to test whether the true value r of the ratio of hazard rates in two treatment groups is at least equal to a limit value r₀. If prognostic factors have to be taken into account, the Cox model provides tests of H₀, and a useful confidence interval for the adjusted relative risk derived from the regression parameter for the treatment indicator. An equivalence trial of maintenance therapy was carried out in children with B non-Hodgkin lymphoma, and serves as an illustration.     

Sample size requirements for reliability studies

This paper provides exact power contours to guide the planning of reliability studies, where the parameter of interest is the coefficient of intraclass correlation rho derived from a one-way analysis of variance model. The contours display the required numbers of subjects k and number of repeated measurements n that provide 80 per cent power for testing Ho: rho less than or equal to rho 0 versus H1: rho greater than rho 0 at the 5 per cent level of significance for selected values of rho o. We discuss the design considerations of these results.