Gluteal muscle activation during rehabilitation exercises in female field hockey players.

BACKGROUND: Field hockey, a team sport played by both men and women at both recreational and professional levels, requires maintaining a forward flexed posture putting stress on the lumbar spine. Hence, it is necessary to assess the muscles supporting the lumbar spine, especially those surrounding the hip, to inform strengthening exercises for this population. OBJECTIVES: To establish the best body weight rehabilitation exercises shown to produce high muscle activation (≥ 61%MVIC - maximal voluntary isometric contraction) for both the gluteus maximus (Gmax) and medius (Gmed) muscles. Four exercises fell into this category. METHOD: Surface electromyography (sEMG) was used to record the muscle activation of Gmax and Gmed of four body weight rehabilitation exercises in 26 high-performance female field hockey players. The %MVIC activation data of both Gmax and Gmed were analysed using a three-way ANOVA. RESULTS: The single-leg squat generated the highest %MVIC activation of both Gmax (125.65%MVIC) and Gmed (126.30%MVIC). The only statistically significant difference for Gmax was between the single-leg squat and plank with hip extension (p = 0.0487). No statistically significant difference was observed for Gmed between the four body weight rehabilitation exercises (p = 0.6285). CONCLUSION: The four exercises generated similar %MVIC activation levels. The single-leg squat produced the highest observed %MVIC of Gmax and Gmed in high-performance female field hockey players and is, therefore, recommended. CLINICAL IMPLICATIONS: Implementation of the findings could result in benefits during prehabilitation, injury prevention programmes and the later stages of rehabilitation for high-performance female field hockey players.

A robust method for the assessment of average bioequivalence in the presence of outliers and skewness.

PURPOSE: In this paper, we propose a robust Bayesian method for the assessment of average bioequivalence based on data from conventional crossover studies. We evaluate and motivate empirically the need for robust methods in bioequivalence studies by comparing the results of robust and conventional statistical methods in a large data pool of bioequivalence studies. METHODS: Robustness of the statistical methodology is achieved by replacing the normal distributions for residuals in the linear mixed model with skew-t distributions. In this way, the statistical model can accommodate skew and heavy-tailed data, particularly outliers, yielding robust statistical inference without the need for excluding outliers from the analysis. We performed a simulation study to investigate and compare the performance of the robust and conventional models. RESULTS: Our study shows that in some trials, the distribution of residuals is skew and heavy-tailed. In the presence of outliers, the 90% confidence intervals for the ratio of geometric means tend to be narrower for the robust methods than for the conventional method. Our simulation study shows that the robust method has suitable frequentist properties and yields more precise confidence intervals and higher statistical power than the conventional maximum likelihood method when outliers are present in the data. CONCLUSIONS: As a sensitivity analysis, we recommend the fit of robust models for handling outliers that are occasionally encountered in crossover design bioequivalence data.

Morphological and skill-related fitness components as potential predictors of injury in elite netball players: A cohort study.

BACKGROUND: A limited number of studies on the epidemiology of injuries and fitness profiles of netball players in South Africa have been conducted, but no research on the potential morphological and skill-related fitness predictors of injuries could be located. OBJECTIVES: We investigated whether morphological or skill-related factors measured in the pre-season could predict injuries sustained in-season. METHOD: In our cohort study, 77 under-18 (U18), U19, U21 and senior elite netball players underwent pre-season testing including anthropometry, balance, flexibility, explosive power, upper and lower body strength, core strength, speed and agility testing. A questionnaire was used to collect demographic data, elite-level experience and injury history. Injuries in pre-season, training and matches were recorded during the subsequent 2017-2018 season using an injury profile sheet. RESULTS: Amongst the 77 players who underwent pre-season fitness tests, 33 players (42.9%) had at least one injury. Regarding player morphology, a significant association of body mass and body fat percentage with injury risk was found in a simple logistic regression. In a multiple logistic regression analysis, only fat percentage (p = 0.0508) remained a significant predictor of injury at the 10% significance level, with higher fat percentage being associated with lower injury risk. CONCLUSION: Heavier players and players with a higher fat percentage had a decreased injury risk. CLINICAL IMPLICATIONS: As a result of the apparent protective effect of heavier weight of players, referees should more strictly enforce the no-contact rule in netball. Further research on functional movement screening as a tool for potential prediction of injury in netball is recommended.

Selected wetland soil properties correlate to Rift Valley fever livestock mortalities reported in 2009-10 in central South Africa.

Outbreaks of Rift Valley fever have devastating impacts on ruminants, humans, as well as on regional and national economies. Although numerous studies on the impact and outbreak of Rift Valley fever exist, relatively little is known about the role of environmental factors, especially soil, on the aestivation of the virus. This study thus selected 22 sites for study in central South Africa, known to be the recurrent epicenter of widespread Rift Valley fever outbreaks in Southern Africa. Soils were described, sampled and analyzed in detail at each site. Of all the soil variables analyzed for, only eight (cation exchange capacity, exchangeable Ca2+, exchangeable K+, exchangeable Mg2+, soluble Ca2+, medium sand, As, and Br) were statistically identified to be potential indicators of sites with reported Rift Valley fever mortalities, as reported for the 2009-2010 Rift Valley fever outbreak. Four soil characteristics (exchangeable K+, exchangeable Mg2+, medium sand, and Br) were subsequently included in a discriminant function that could potentially be used to predict sites that had reported Rift Valley fever-associated mortalities in livestock. This study therefore constitutes an initial attempt to predict sites prone to Rift Valley fever livestock mortality from soil properties and thus serves as a basis for broader research on the interaction between soil, mosquitoes and Rift Valley fever virus. Future research should include other environmental components such as vegetation, climate, and water properties as well as correlating soil properties with floodwater Aedes spp. abundance and Rift Valley fever virus prevalence.

A robust Bayesian mixed effects approach for zero inflated and highly skewed longitudinal count data emanating from the zero inflated discrete Weibull distribution.

This article proposes a Bayesian mixed effects zero inflated discrete Weibull (ZIDW) regression model for zero inflated and highly skewed longitudinal count data, as an alternative to mixed effects regression models that are based on the negative binomial, zero inflated negative binomial, and conventional discrete Weibull (DW) distributions. The mixed effects ZIDW regression model is an extension of a recently introduced model based on the DW distribution and uses the log-link function to specify the relationship between the linear predictors and the median counts. The ZIDW approach offers a more robust characteristic of central tendency, compared to the mean count, when there is skewness in the data. A matrix generalized half-t (MGH-t) prior distribution is specified for the random effects covariance matrix as an alternative to the widely used Wishart prior distribution. The methodology is applied to a longitudinal dataset from an epilepsy clinical trial. In a data contamination simulation study, we show that the mixed effect ZIDW regression model is more robust than the competing mixed effects regression models when the data contain excess zeros or outliers. The performance of the ZIDW regression model is also assessed in a simulation study under the specification of, respectively, the MGH-t and Wishart prior distributions for the random effects covariance matrix. It turns out that the highest posterior density intervals under the MGH-t prior for the fixed effects maintain nominal coverage when the true variability between random slopes over time is small, whereas those under the Wishart prior are generally conservative.

A generalized Bayesian nonlinear mixed-effects regression model for zero-inflated longitudinal count data in tuberculosis trials.

In this paper, we investigate Bayesian generalized nonlinear mixed-effects (NLME) regression models for zero-inflated longitudinal count data. The methodology is motivated by and applied to colony forming unit (CFU) counts in extended bactericidal activity tuberculosis (TB) trials. Furthermore, for model comparisons, we present a generalized method for calculating the marginal likelihoods required to determine Bayes factors. A simulation study shows that the proposed zero-inflated negative binomial regression model has good accuracy, precision, and credibility interval coverage. In contrast, conventional normal NLME regression models applied to log-transformed count data, which handle zero counts as left censored values, may yield credibility intervals that undercover the true bactericidal activity of anti-TB drugs. We therefore recommend that zero-inflated NLME regression models should be fitted to CFU count on the original scale, as an alternative to conventional normal NLME regression models on the logarithmic scale.

Robust Bayesian nonlinear mixed-effects modeling of time to positivity in tuberculosis trials.

Early phase 2 tuberculosis (TB) trials are conducted to characterize the early bactericidal activity (EBA) of anti-TB drugs. The EBA of anti-TB drugs has conventionally been calculated as the rate of decline in colony forming unit (CFU) count during the first 14 days of treatment. The measurement of CFU count, however, is expensive and prone to contamination. Alternatively to CFU count, time to positivity (TTP), which is a potential biomarker for long-term efficacy of anti-TB drugs, can be used to characterize EBA. The current Bayesian nonlinear mixed-effects (NLME) regression model for TTP data, however, lacks robustness to gross outliers that often are present in the data. The conventional way of handling such outliers involves their identification by visual inspection and subsequent exclusion from the analysis. However, this process can be questioned because of its subjective nature. For this reason, we fitted robust versions of the Bayesian nonlinear mixed-effects regression model to a wide range of TTP datasets. The performance of the explored models was assessed through model comparison statistics and a simulation study. We conclude that fitting a robust model to TTP data obviates the need for explicit identification and subsequent "deletion" of outliers but ensures that gross outliers exert no undue influence on model fits. We recommend that the current practice of fitting conventional normal theory models be abandoned in favor of fitting robust models to TTP data.

Robust fit of Bayesian mixed effects regression models with application to colony forming unit count in tuberculosis research.

Early bactericidal activity of tuberculosis drugs is conventionally assessed using statistical regression modeling of colony forming unit (CFU) counts over time. Typically, most CFU counts deviate little from the regression curve, but gross outliers due to erroneous sputum sampling are occasionally present and can markedly influence estimates of the rate of change in CFU count, which is the parameter of interest. A recently introduced Bayesian nonlinear mixed effects regression model was adapted to offer a robust approach that accommodates both outliers and potential skewness in the data. At its most general, the proposed regression model fits the skew Student t distribution to residuals and random coefficients. Deviance information criterion statistics and compound Laplace-Metropolis marginal likelihoods were used to discriminate between alternative Bayesian nonlinear mixed effects regression models. We present a relatively easy method to calculate the marginal likelihoods required to determine compound Laplace-Metropolis marginal likelihoods, by adapting methods available in currently available statistical software. The robust methodology proposed in this paper was applied to data from 6 clinical trials. The results provide strong evidence that the distribution of CFU count is often heavy tailed and negatively skewed (suggesting the presence of outliers). Therefore, we recommend that robust regression models, such as those proposed here, should be fitted to CFU count.

Statistical reporting of clinical pharmacology research.

Research in clinical pharmacology covers a wide range of experiments, trials and investigations: clinical trials, systematic reviews and meta-analyses of drug usage after market approval, the investigation of pharmacokinetic-pharmacodynamic relationships, the search for mechanisms of action or for potential signals for efficacy and safety using biomarkers. Often these investigations are exploratory in nature, which has implications for the way the data should be analysed and presented. Here we summarize some of the statistical issues that are of particular importance in clinical pharmacology research.

Effect of uncertainty in total parasite infestation on accuracy and precision of estimates of ectoparasiticide efficacy.

In animal studies of ectoparasiticide efficacy the total number of parasites with which experimental animals are infested is not always equal to the intended number of parasites (usually n=50 per experimental animal in the case of ticks, and n=50 or n=100 in the case of fleas). That is, in the practical implementation of a study protocol, the infestation of experimental animals may be subject to variability so that total infestation is not known precisely. The purpose of the present study is to assess the impact of this variability on the accuracy and precision of efficacy estimates. The results of a thorough simulation study show clearly that uncertainty in total parasite infestation - of the magnitude encountered in well-controlled animal studies - has virtually no effect on the accuracy and precision of estimators of ectoparasiticide efficacy.

Accuracy and precision of alternative estimators of ectoparasiticide efficacy.

While there is consensus that the efficacy of parasiticides is properly assessed using the Abbott formula, there is as yet no general consensus on the use of arithmetic versus geometric mean numbers of surviving parasites in the formula. The purpose of this paper is to investigate the accuracy and precision of various efficacy estimators based on the Abbott formula which alternatively use arithmetic mean, geometric mean and median numbers of surviving parasites; we also consider a maximum likelihood estimator. Our study shows that the best estimators using geometric means are competitive, with respect to root mean squared error, with the conventional Abbott estimator using arithmetic means, as they have lower average and lower median root mean square error over the parameter scenarios which we investigated. However, our study confirms that Abbott estimators using geometric means are potentially biased upwards, and this upward bias is substantial in particular when the test product has substandard efficacy (90% and below). For this reason, we recommend that the Abbott estimator be calculated using arithmetic means.

Efficiency and safety of the combination of moxifloxacin, pretomanid (PA-824), and pyrazinamide during the first 8 weeks of antituberculosis treatment: a phase 2b, open-label, partly randomised trial in patients with drug-susceptible or drug-resistant pulmonary tuberculosis.

BACKGROUND: New antituberculosis regimens are urgently needed to shorten tuberculosis treatment. Following on from favourable assessment in a 2 week study, we investigated a novel regimen for efficacy and safety in drug-susceptible and multidrug-resistant (MDR) tuberculosis during the first 8 weeks of treatment. METHODS: We did this phase 2b study of bactericidal activity--defined as the decrease in colony forming units (CFUs) of Mycobacterium tuberculosis in the sputum of patients with microscopy smear-positive pulmonary tuberculosis-at eight sites in South Africa and Tanzania. We enrolled treatment-naive patients with drug-susceptible, pulmonary tuberculosis, who were randomly assigned by computer-generated sequences to receive either 8 weeks of moxifloxacin, 100 mg pretomanid (formerly known as PA-824), and pyrazinamide (MPa100Z regimen); moxifloxacin, 200 mg pretomanid, and pyrazinamide (MPa200Z regimen); or the current standard care for drug-susceptible pulmonary tuberculosis, isoniazid, rifampicin, PZA, and ethambutol (HRZE regimen). A group of patients with MDR tuberculosis received MPa200Z (DRMPa200Z group). The primary outcome was bactericidal activity measured by the mean daily rate of reduction in M tuberculosis CFUs per mL overnight sputum collected once a week, with joint Bayesian non-linear mixed-effects regression modelling. We also assessed safety and tolerability by monitoring adverse events. This study is registered with ClinicalTrials.gov, number NCT01498419. FINDINGS: Between March 24, 2012, and July 26, 2013 we enrolled 207 patients and randomly assigned them to treatment groups; we assigned 60 patients to the MPa100Z regimen, 62 to the MPa200Z regimen, and 59 to the HRZE regimen. We non-randomly assigned 26 patients with drug-resistant tuberculosis to the DRMPa200Z regimen. In patients with drug-susceptible tuberculosis, the bactericidal activity of MPa200Z (n=54) on days 0-56 (0·155, 95% Bayesian credibility interval 0·133-0·178) was significantly greater than for HRZE (n=54, 0·112, 0·093-0·131). DRMPa200Z (n=9) had bactericidal activity of 0·117 (0·070-0·174). The bactericidal activity on days 7-14 was strongly associated with bactericidal activity on days 7-56. Frequencies of adverse events were similar to standard treatment in all groups. The most common adverse event was hyperuricaemia in 59 (29%) patients (17 [28%] patients in MPa100Z group, 17 [27%] patients in MPa200Z group, 17 [29%] patients. in HRZE group, and 8 [31%] patients in DRMPa200Z group). Other common adverse events were nausea in (14 [23%] patients in MPa100Z group, 8 [13%] patients in MPa200Z group, 7 [12%] patients in HRZE group, and 8 [31%] patients in DRMPa200Z group) and vomiting (7 [12%] patients in MPa100Z group, 7 [11%] patients in MPa200Z group, 7 [12%] patients in HRZE group, and 4 [15%] patients in DRMPa200Z group). No on-treatment electrocardiogram occurrences of corrected QT interval more than 500 ms (an indicator of potential of ventricular tachyarrhythmia) were reported. No phenotypic resistance developed to any of the drugs in the regimen. INTERPRETATION: The combination of moxifloxacin, pretomanid, and pyrazinamide, was safe, well tolerated, and showed superior bactericidal activity in drug-susceptible tuberculosis during 8 weeks of treatment. Results were consistent between drug-susceptible and MDR tuberculosis. This new regimen is ready to enter phase 3 trials in patients with drug-susceptible tuberculosis and MDR-tuberculosis, with the goal of shortening and simplifying treatment. FUNDING: Global Alliance for TB Drug Development.

Bactericidal activity of pyrazinamide and clofazimine alone and in combinations with pretomanid and bedaquiline.

RATIONALE: New regimens to shorten tuberculosis treatment and manage patients with drug-resistant tuberculosis who are infected with HIV are urgently needed. Experimental and clinical evidence suggests that the new drugs bedaquiline (B) and pretomanid (Pa), combined with an existing drug, pyrazinamide (Z), and a repurposed drug, clofazimine (C), may assist treatment shortening of drug-susceptible and drug-resistant tuberculosis. OBJECTIVES: To evaluate the 14-day bactericidal activity of C and Z in monotherapy and in combinations with Pa and B. METHODS: Groups of 15 treatment-naive, sputum smear-positive patients with pulmonary tuberculosis were randomized to receive combinations of B with Z-C, Pa-Z, Pa-Z-C, and Pa-C, or C or Z alone, or standard combination treatment for 14 days. The primary endpoint was the mean daily fall in log10 Mycobacterium tuberculosis CFU per milliliter sputum estimated by joint nonlinear mixed-effects Bayesian regression modeling. MEASUREMENTS AND MAIN RESULTS: Estimated activities were 0.167 (95% confidence interval [CI], 0.075-0.257) for B-Pa-Z, 0.151 (95% CI, 0.071-0.232) for standard treatment, 0.124 (95% CI, 0.035-0.214) for B-Z-C, 0.115 (95% CI, 0.039-0.189) for B-Pa-Z-C, and 0.076 (95% CI, 0.005-0.145) for B-Pa-C. Z alone had modest activity (0.036; 95% CI, -0.026 to 0.099). C had no activity alone (-0.017; 95% CI, -0.085 to 0.053) or in combinations. Treatments were well tolerated and safe. CONCLUSIONS: B-Pa-Z, including two novel agents without resistance in prevalent M. tuberculosis strains, is a potential new tuberculosis treatment regimen. C had no measurable activity in the first 14 days of treatment. Clinical trial registered with www.clinicaltrials.gov (NCT 01691534).

A Bayesian Nonlinear Mixed-Effects Regression Model for the Characterization of Early Bactericidal Activity of Tuberculosis Drugs.

Trials of the early bactericidal activity (EBA) of tuberculosis (TB) treatments assess the decline, during the first few days to weeks of treatment, in colony forming unit (CFU) count of Mycobacterium tuberculosis in the sputum of patients with smear-microscopy-positive pulmonary TB. Profiles over time of CFU data have conventionally been modeled using linear, bilinear, or bi-exponential regression. We propose a new biphasic nonlinear regression model for CFU data that comprises linear and bilinear regression models as special cases and is more flexible than bi-exponential regression models. A Bayesian nonlinear mixed-effects (NLME) regression model is fitted jointly to the data of all patients from a trial, and statistical inference about the mean EBA of TB treatments is based on the Bayesian NLME regression model. The posterior predictive distribution of relevant slope parameters of the Bayesian NLME regression model provides insight into the nature of the EBA of TB treatments; specifically, the posterior predictive distribution allows one to judge whether treatments are associated with monolinear or bilinear decline of log(CFU) count, and whether CFU count initially decreases fast, followed by a slower rate of decrease, or vice versa.

The empirical coverage of confidence intervals: point estimates and confidence intervals for confidence levels.

Many confidence intervals calculated in practice are potentially not exact, either because the requirements for the interval estimator to be exact are known to be violated, or because the (exact) distribution of the data is unknown. If a confidence interval is approximate, the crucial question is how well its true coverage probability approximates its intended coverage probability. In this paper we propose to use the bootstrap to calculate an empirical estimate for the (true) coverage probability of a confidence interval. In the first instance, the empirical coverage can be used to assess whether a given type of confidence interval is adequate for the data at hand. More generally, when planning the statistical analysis of future trials based on existing data pools, the empirical coverage can be used to study the coverage properties of confidence intervals as a function of type of data, sample size, and analysis scale, and thus inform the statistical analysis plan for the future trial. In this sense, the paper proposes an alternative to the problematic pretest of the data for normality, followed by selection of the analysis method based on the results of the pretest. We apply the methodology to a data pool of bioequivalence studies, and in the selection of covariance patterns for repeated measures data.

Mixed models for data from thorough QT studies: part 1. assessment of marginal QT prolongation.

We investigate mixed models for repeated measures data from cross-over studies in general, but in particular for data from thorough QT studies. We extend both the conventional random effects model and the saturated covariance model for univariate cross-over data to repeated measures cross-over (RMC) data; the resulting models we call the RMC model and Saturated model, respectively. Furthermore, we consider a random effects model for repeated measures cross-over data previously proposed in the literature. We assess the standard errors of point estimates and the coverage properties of confidence intervals for treatment contrasts under the various models. Our findings suggest: (i) Point estimates of treatment contrasts from all models considered are similar; (ii) Confidence intervals for treatment contrasts under the random effects model previously proposed in the literature do not have adequate coverage properties; the model therefore cannot be recommended for analysis of marginal QT prolongation; (iii) The RMC model and the Saturated model have similar precision and coverage properties; both models are suitable for assessment of marginal QT prolongation; and (iv) The Akaike Information Criterion (AIC) is not a reliable criterion for selecting a covariance model for RMC data in the following sense: the model with the smallest AIC is not necessarily associated with the highest precision for the treatment contrasts, even if the model with the smallest AIC value is also the most parsimonious model.

Comparative evaluation of ectoparasiticide efficacy: non-inferiority and superiority assessment.

We present a statistical framework for the comparative evaluation of ectoparasiticide efficacy in controlled animal studies. Such a comparative evaluation can have one of two objectives: an assessment of non-inferiority of a test parasiticide to a reference parasiticide, or an assessment of superiority of one treatment over another. We show that the observed efficacy of an ectoparasiticide can be viewed as a point estimate of its conditional "kill" probability. Thus concepts used in the comparative evaluation of human antibiotics, which involve the comparison of cure probabilities, can be applied to the situation of comparing parasiticide efficacy. In particular, we define non-inferiority of a test parasiticide to a reference parasiticide in terms of the "test - reference" difference of their efficacies. We outline the hypothesis testing framework and associated statistical decision rules for declaring either non-inferiority or superiority. Both non-inferiority and superiority can be evaluated statistically using confidence intervals. SAS and R code for data analysis is presented, and the methodology is applied to a data set from a controlled animal study.

Mixed models for data from thorough QT studies: part 2. One-step assessment of conditional QT prolongation.

We investigate mixed analysis of covariance models for the 'one-step' assessment of conditional QT prolongation. Initially, we consider three different covariance structures for the data, where between-treatment covariance of repeated measures is modelled respectively through random effects, random coefficients, and through a combination of random effects and random coefficients. In all three of those models, an unstructured covariance pattern is used to model within-treatment covariance. In a fourth model, proposed earlier in the literature, between-treatment covariance is modelled through random coefficients but the residuals are assumed to be independent identically distributed (i.i.d.). Finally, we consider a mixed model with saturated covariance structure. We investigate the precision and robustness of those models by fitting them to a large group of real data sets from thorough QT studies. Our findings suggest: (i) Point estimates of treatment contrasts from all five models are similar. (ii) The random coefficients model with i.i.d. residuals is not robust; the model potentially leads to both under- and overestimation of standard errors of treatment contrasts and therefore cannot be recommended for the analysis of conditional QT prolongation. (iii) The combined random effects/random coefficients model does not always converge; in the cases where it converges, its precision is generally inferior to the other models considered. (iv) Both the random effects and the random coefficients model are robust. (v) The random effects, the random coefficients, and the saturated model have similar precision and all three models are suitable for the one-step assessment of conditional QT prolongation.

Comparative evaluation of the Roche Cobas AmpliPrep/Cobas TaqMan HIV-1 version 2 test using the TaqMan 48 analyzer and the Abbott RealTime HIV-1 assay.

Acceptable precision was achieved in a comparison study of the Abbott RealTime (RT) and Roche CAP/CTM-48 V2 HIV-1 assays, but viral load quantification was under- and overestimated, respectively, compared to the 2nd HIV-1 WHO International Standard. The same quantification patterns were observed for patient cohorts from Africa and the United States.

Residuals and outliers in replicate design crossover studies.

Outliers in bioequivalence trials may arise through various mechanisms, requiring different interpretation and handling of such data points. For example, regulatory authorities might permit exclusion from analysis of outliers caused by product or process failure, while exclusion of outliers caused by subject-by-treatment interaction generally is not acceptable. In standard 2 x 2 crossover studies it is not possible to distinguish between relevant types of outliers based on statistical criteria alone. However, in replicate design (2-treatment, 4-period) crossover studies three types of outliers can be distinguished: (i) Subject outliers are usually unproblematic, at least regarding the analysis of bioequivalence, and may require no further action; (ii) Subject-by-formulation outliers may affect the outcome of the bioequivalence test but generally cannot simply be removed from analysis; and (iii) Removal of single-data-point outliers from analysis may be justified in certain cases. As a very simple but effective diagnostic tool for the identification and classification of outliers in replicate design crossover studies we propose to calculate and plot three types of residual corresponding to the three different types of outliers that can be distinguished. The residuals are obtained from four mutually orthogonal linear contrasts of the four data points associated with each subject. If preferred, outlier tests can be applied to the resulting sets of residuals after suitable standardization.