Familywise error for multiple time-to-event endpoints in a group sequential design.

We investigate the familywise error rate (FWER) for time-to-event endpoints evaluated using a group sequential design with a hierarchical testing procedure for secondary endpoints. We show that, in this setup, the correlation between the log-rank test statistics at interim and at end of study is not congruent with the canonical correlation derived for normal-distributed endpoints. We show, both theoretically and by simulation, that the correlation also depends on the level of censoring, the hazard rates of the endpoints, and the hazard ratio. To optimize operating characteristics in this complex scenario, we propose a simulation-based method to assess the FWER which, better than the alpha-spending approach, can inform the choice of critical values for testing secondary endpoints.

Semaglutide and Cardiovascular Outcomes in Obesity without Diabetes.

BACKGROUND: Semaglutide, a glucagon-like peptide-1 receptor agonist, has been shown to reduce the risk of adverse cardiovascular events in patients with diabetes. Whether semaglutide can reduce cardiovascular risk associated with overweight and obesity in the absence of diabetes is unknown. METHODS: In a multicenter, double-blind, randomized, placebo-controlled, event-driven superiority trial, we enrolled patients 45 years of age or older who had preexisting cardiovascular disease and a body-mass index (the weight in kilograms divided by the square of the height in meters) of 27 or greater but no history of diabetes. Patients were randomly assigned in a 1:1 ratio to receive once-weekly subcutaneous semaglutide at a dose of 2.4 mg or placebo. The primary cardiovascular end point was a composite of death from cardiovascular causes, nonfatal myocardial infarction, or nonfatal stroke in a time-to-first-event analysis. Safety was also assessed. RESULTS: A total of 17,604 patients were enrolled; 8803 were assigned to receive semaglutide and 8801 to receive placebo. The mean (±SD) duration of exposure to semaglutide or placebo was 34.2±13.7 months, and the mean duration of follow-up was 39.8±9.4 months. A primary cardiovascular end-point event occurred in 569 of the 8803 patients (6.5%) in the semaglutide group and in 701 of the 8801 patients (8.0%) in the placebo group (hazard ratio, 0.80; 95% confidence interval, 0.72 to 0.90; P<0.001). Adverse events leading to permanent discontinuation of the trial product occurred in 1461 patients (16.6%) in the semaglutide group and 718 patients (8.2%) in the placebo group (P<0.001). CONCLUSIONS: In patients with preexisting cardiovascular disease and overweight or obesity but without diabetes, weekly subcutaneous semaglutide at a dose of 2.4 mg was superior to placebo in reducing the incidence of death from cardiovascular causes, nonfatal myocardial infarction, or nonfatal stroke at a mean follow-up of 39.8 months. (Funded by Novo Nordisk; SELECT ClinicalTrials.gov number, NCT03574597.).

Semaglutide for cardiovascular event reduction in people with overweight or obesity: SELECT study baseline characteristics.

OBJECTIVE: This paper describes the baseline characteristics of the Semaglutide Effects on Heart Disease and Stroke in Patients with Overweight or Obesity (SELECT) study, one of the largest cardiovascular (CV) outcome studies in the field of obesity, which evaluates the effect of semaglutide versus placebo on major CV events. METHODS: SELECT enrolled individuals with overweight or obesity without diabetes, with prior myocardial infarction, stroke, and/or peripheral artery disease. This study reports participants' baseline characteristics in the full study population and subgroups defined by baseline glycated hemoglobin (HbA1c ; <5.7%, ≥5.7 to <6.0%, ≥6.0 to <6.5%), baseline waist to height ratio tertile, and qualifying prior CV event or condition. RESULTS: The study enrolled 17,605 participants (72.5% male) with an average (SD) age of 61.6 (8.9) years and BMI of 33.34 (5.04) kg/m2 . The most common prior CV event was myocardial infarction (76.3% of participants), followed by stroke (23.3%) and peripheral artery disease (8.6%). Furthermore, 24.3% had a heart failure diagnosis. Two-thirds of participants (66%) had HbA1c in the prediabetes range (5.7%-6.4%). Across groups of increasing HbA1c , prevalence of all CV risk factors increased. CONCLUSIONS: The enrolled population in SELECT includes participants across a broad range of relevant risk categories. This will allow the study to garner information about the CV benefits of semaglutide across these relevant clinical subgroups.

Evaluating group-sequential non-inferiority clinical trials following interim stopping: The HIV Prevention Trials Network 083 trial.

BACKGROUND/AIMS: The HIV Prevention Trials Network 083 trial was a group-sequential non-inferiority trial designed to compare HIV incidence under a novel experimental regimen for HIV prevention, long-acting injectable cabotegravir, with an active-control regimen of daily oral tenofovir disoproxil fumarate/emtricitabine (brand name Truvada). In March of 2020, just as the trial had completed enrollment, the COVID-19 pandemic threatened to prevent trial participants from attending study visits and obtaining study medication, motivating the study team to update the interim monitoring plan. The Data and Safety Monitoring Board subsequently stopped the trial at the first interim review due to strong early evidence of efficacy. METHODS: Here we describe some unique aspects of the trial's design, monitoring, analysis, and interpretation. We illustrate the importance of computing point estimates, confidence intervals, and p values based on the sampling distribution induced by sequential monitoring. RESULTS: Accurate analysis, decision-making and interpretation of trial results rely on pre-specification of a stopping boundary, including the scale on which the stopping rule will be implemented, the specific test statistics to be calculated, and how the boundary will be adjusted if the available information fraction at interim review is different from planned. After appropriate adjustment for the sampling distribution and overrun, the HIV Prevention Trials Network 083 trial provided strong evidence that the experimental regimen was superior to the active control. CONCLUSIONS: For the HIV Prevention Trials Network 083 trial, the difference between corrected inferential statistics and naive results was quite small-as will often be the case-nevertheless, it is appropriate to report and publish the most accurate and unbiased statistical results.

Semaglutide Effects on Cardiovascular Outcomes in People With Overweight or Obesity (SELECT) rationale and design.

Cardiovascular disease (CVD) is a major cause of morbidity and mortality. Although it has been widely appreciated that obesity is a major risk factor for CVD, treatments that produce effective, durable weight loss and the impact of weight reduction in reducing cardiovascular risk have been elusive. Instead, progress in CVD risk reduction has been achieved through medications indicated for controlling lipids, hyperglycemia, blood pressure, heart failure, inflammation, and/or thrombosis. Obesity has been implicated as promoting all these issues, suggesting that sustained, effective weight loss may have independent cardiovascular benefit. GLP-1 receptor agonists (RAs) reduce weight, improve glycemia, decrease cardiovascular events in those with diabetes, and may have additional cardioprotective effects. The GLP-1 RA semaglutide is in phase 3 studies as a medication for obesity treatment at a dose of 2.4 mg subcutaneously (s.c.) once weekly. Semaglutide Effects on Heart Disease and Stroke in Patients with Overweight or Obesity (SELECT) is a randomized, double-blind, parallel-group trial testing if semaglutide 2.4 mg subcutaneously once weekly is superior to placebo when added to standard of care for preventing major adverse cardiovascular events in patients with established CVD and overweight or obesity but without diabetes. SELECT is the first cardiovascular outcomes trial to evaluate superiority in major adverse cardiovascular events reduction for an antiobesity medication in such a population. As such, SELECT has the potential for advancing new approaches to CVD risk reduction while targeting obesity.

Non-inferiority designs comparing placebo to a proven therapy for childhood pneumonia in low-resource settings.

BACKGROUND/AIMS: After a new treatment is recommended to be first-line treatment for a specific indication, outcome and population, it may be unethical to use placebo as a comparator in trials for that setting. Nevertheless, in specific circumstances, use of a placebo group might be warranted, for example, when it is believed that an active treatment may not be efficacious or cost-effective for a specific subpopulation. An example is antibiotic treatment for pneumonia, which may not be effective for many patients taking it due to the emergence of antibiotic-resistant strains or the high prevalence of viral and low prevalence of bacterial pneumonia. METHODS: We explore the applicability of different design options in cases where the benefit of an established treatment is questioned, with particular emphasis on issues that arise in a low-resource setting. Using the example of a clinical trial comparing the effectiveness of placebo versus amoxicillin in treating children 2-59 months of age with fast breathing pneumonia in Lilongwe, Malawi, we discuss the pros and cons of superiority versus non-inferiority designs, an intent-to-treat versus as-treated analysis and the use and interpretation of one- versus two-sided confidence intervals. RESULTS: We find that a non-inferiority design using an intent-to-treat analysis is the most appropriate design and analysis option. In addition, the presentation of one- versus two-sided confidence intervals can depend on the results but can maintain type I error. CONCLUSION: In the setting where the benefit of a previously established beneficial treatment is questioned, a non-inferiority design that includes placebo as the tested treatment option can be the most appropriate design option.

Cardiovascular safety and lower severe hypoglycaemia of insulin degludec versus insulin glargine U100 in patients with type 2 diabetes aged 65 years or older: Results from DEVOTE (DEVOTE 7).

AIMS: The aim of this study was to describe the risks of cardiovascular (CV) events and severe hypoglycaemia with insulin degludec (degludec) vs insulin glargine 100 units/mL (glargine U100) in patients with type 2 diabetes (T2D) aged 65 years or older. MATERIALS AND METHODS: A total of 7637 patients in the DEVOTE trial, a treat-to-target, randomized, double-blind trial evaluating the CV safety of degludec vs glargine U100, were divided into three age groups (50-64 years, n = 3682; 65-74 years, n = 3136; ≥75 years, n = 819). Outcomes by overall age group and randomized treatment differences were analysed for major adverse cardiovascular events (MACE), all-cause mortality, severe hypoglycaemia and serious adverse events (SAEs). RESULTS: Patients with increasing age had higher risks of CV death, all-cause mortality and SAEs, and there were non-significant trends towards higher risks of MACE and severe hypoglycaemia. Treatment effects on the risk of MACE, all-cause mortality, severe hypoglycaemia and SAEs were consistent across age groups, based on the non-significant interactions between treatment and age with regard to these outcomes. CONCLUSIONS: There were higher risks of CV death, all-cause mortality and SAEs, and trends towards higher risks of MACE and severe hypoglycaemia with increasing age after adjusting for baseline differences. The effects across age groups of degludec vs glargine U100 on MACE, all-cause mortality and severe hypoglycaemia were comparable, suggesting that the risk of MACE, as well as all-cause mortality, is similar and the risk of severe hypoglycaemia is lower with degludec regardless of age. Evidence is conclusive only until 74 years of age.

Lower rates of cardiovascular events and mortality associated with liraglutide use in patients treated with basal insulin: A DEVOTE subanalysis (DEVOTE 10).

AIM: To compare the associations between concomitant liraglutide use versus no liraglutide use and the risk of major adverse cardiovascular events (MACE) and all-cause mortality among patients receiving basal insulin (either insulin degludec [degludec] or insulin glargine 100 units/mL [glargine U100]) in the Trial Comparing Cardiovascular Safety of Insulin Degludec versus Insulin Glargine in Patients with Type 2 Diabetes at High Risk of Cardiovascular Events (DEVOTE). MATERIALS AND METHODS: Patients with type 2 diabetes and high cardiovascular risk were randomized 1:1 to degludec or glargine U100. Hazard ratios for MACE/mortality were calculated using a Cox regression model adjusted for treatment and time-varying liraglutide use at any time during the trial, without interaction. Sensitivity analyses were adjusted for baseline covariates including, but not limited to, age, sex, smoking and prior cardiovascular disease. RESULTS: At baseline, 436/7637 (5.7%) patients were treated with liraglutide; after baseline, 187/7637 (2.4%) started and 210/7637 (2.7%) stopped liraglutide. Mean liraglutide exposure from randomization was 530.2 days. Liraglutide use versus no liraglutide use was associated with significantly lower hazard rates for MACE [0.62 (0.41; 0.92)95%CI ] and all-cause mortality [0.50 (0.29; 0.88)95%CI ]. There was no significant difference in the rate of severe hypoglycaemia with versus without liraglutide use. Multiple sensitivity analyses yielded similar results. CONCLUSIONS: Use of liraglutide was associated with significantly lower risk of MACE and death in patients with type 2 diabetes and high cardiovascular risk using basal insulin.

DEVOTE 3: temporal relationships between severe hypoglycaemia, cardiovascular outcomes and mortality.

AIMS/HYPOTHESIS: The double-blind Trial Comparing Cardiovascular Safety of Insulin Degludec vs Insulin Glargine in Patients with Type 2 Diabetes at High Risk of Cardiovascular Events (DEVOTE) assessed the cardiovascular safety of insulin degludec. The incidence and rates of adjudicated severe hypoglycaemia, and all-cause mortality were also determined. This paper reports a secondary analysis investigating associations of severe hypoglycaemia with cardiovascular outcomes and mortality. METHODS: In DEVOTE, patients with type 2 diabetes were randomised to receive either insulin degludec or insulin glargine U100 (100 units/ml) once daily (between dinner and bedtime) in an event-driven, double-blind, treat-to-target cardiovascular outcomes trial. The primary outcome was the first occurrence of an adjudicated major adverse cardiovascular event (MACE; cardiovascular death, non-fatal myocardial infarction or non-fatal stroke). Adjudicated severe hypoglycaemia was the pre-specified secondary outcome. In the present analysis, the associations of severe hypoglycaemia with both MACE and all-cause mortality was evaluated in the pooled trial population using time-to-event analyses, with severe hypoglycaemia as a time-dependent variable and randomised treatment as a fixed factor. An investigation with interaction terms indicated that the effect of severe hypoglycaemia on the risk of MACE and all-cause mortality were the same for both treatment arms, and so the temporal association for severe hypoglycaemia with subsequent MACE and all-cause mortality is reported for the pooled population. RESULTS: There was a non-significant difference in the risk of MACE for individuals who had vs those who had not experienced severe hypoglycaemia during the trial (HR 1.38, 95% CI 0.96, 1.96; p = 0.080) and therefore there was no temporal relationship between severe hypoglycaemia and MACE. There was a significantly higher risk of all-cause mortality for patients who had vs those who had not experienced severe hypoglycaemia during the trial (HR 2.51, 95% CI 1.79, 3.50; p < 0.001). There was a higher risk of all-cause mortality 15, 30, 60, 90, 180 and 365 days after experiencing severe hypoglycaemia compared with not experiencing severe hypoglycaemia in the same time interval. The association between severe hypoglycaemia and all-cause mortality was maintained after adjustment for the following baseline characteristics: age, sex, HbA1c, BMI, diabetes duration, insulin regimen, hepatic impairment, renal status and cardiovascular risk group. CONCLUSIONS/INTERPRETATION: The results from these analyses demonstrate an association between severe hypoglycaemia and all-cause mortality. Furthermore, they indicate that patients who experienced severe hypoglycaemia were particularly at greater risk of death in the short term after the hypoglycaemic episode. These findings indicate that severe hypoglycaemia is associated with higher subsequent mortality; however, they cannot answer the question as to whether severe hypoglycaemia serves as a risk marker for adverse outcomes or whether there is a direct causal effect. TRIAL REGISTRATION: ClinicalTrials.gov NCT01959529.

Day-to-day fasting glycaemic variability in DEVOTE: associations with severe hypoglycaemia and cardiovascular outcomes (DEVOTE 2).

AIMS/HYPOTHESIS: The Trial Comparing Cardiovascular Safety of Insulin Degludec vs Insulin Glargine in Patients with Type 2 Diabetes at High Risk of Cardiovascular Events (DEVOTE) was a double-blind, randomised, event-driven, treat-to-target prospective trial comparing the cardiovascular safety of insulin degludec with that of insulin glargine U100 (100 units/ml) in patients with type 2 diabetes at high risk of cardiovascular events. This paper reports a secondary analysis investigating associations of day-to-day fasting glycaemic variability (pre-breakfast self-measured blood glucose [SMBG]) with severe hypoglycaemia and cardiovascular outcomes. METHODS: In DEVOTE, patients with type 2 diabetes were randomised to receive insulin degludec or insulin glargine U100 once daily. The primary outcome was the first occurrence of an adjudicated major adverse cardiovascular event (MACE). Adjudicated severe hypoglycaemia was the pre-specified secondary outcome. In this article, day-to-day fasting glycaemic variability was based on the standard deviation of the pre-breakfast SMBG measurements. The variability measure was calculated as follows. Each month, only the three pre-breakfast SMBG measurements recorded before contact with the site were used to determine a day-to-day fasting glycaemic variability measure for each patient. For each patient, the variance of the three log-transformed pre-breakfast SMBG measurements each month was determined. The standard deviation was determined as the square root of the mean of these monthly variances and was defined as day-to-day fasting glycaemic variability. The associations between day-to-day fasting glycaemic variability and severe hypoglycaemia, MACE and all-cause mortality were analysed for the pooled trial population with Cox proportional hazards models. Several sensitivity analyses were conducted, including adjustments for baseline characteristics and most recent HbA1c. RESULTS: Day-to-day fasting glycaemic variability was significantly associated with severe hypoglycaemia (HR 4.11, 95% CI 3.15, 5.35), MACE (HR 1.36, 95% CI 1.12, 1.65) and all-cause mortality (HR 1.58, 95% CI 1.23, 2.03) before adjustments. The increased risks of severe hypoglycaemia, MACE and all-cause mortality translate into 2.7-, 1.2- and 1.4-fold risk, respectively, when a patient's day-to-day fasting glycaemic variability measure is doubled. The significant relationships of day-to-day fasting glycaemic variability with severe hypoglycaemia and all-cause mortality were maintained after adjustments. However, the significant association with MACE was not maintained following adjustment for baseline characteristics with either baseline HbA1c (HR 1.19, 95% CI 0.96, 1.47) or the most recent HbA1c measurement throughout the trial (HR 1.21, 95% CI 0.98, 1.49). CONCLUSIONS/INTERPRETATION: Higher day-to-day fasting glycaemic variability is associated with increased risks of severe hypoglycaemia and all-cause mortality. TRIAL REGISTRATION: ClinicalTrials.gov NCT01959529.

Efficacy and Safety of Degludec versus Glargine in Type 2 Diabetes.

BACKGROUND: Degludec is an ultralong-acting, once-daily basal insulin that is approved for use in adults, adolescents, and children with diabetes. Previous open-label studies have shown lower day-to-day variability in the glucose-lowering effect and lower rates of hypoglycemia among patients who received degludec than among those who received basal insulin glargine. However, data are lacking on the cardiovascular safety of degludec. METHODS: We randomly assigned 7637 patients with type 2 diabetes to receive either insulin degludec (3818 patients) or insulin glargine U100 (3819 patients) once daily between dinner and bedtime in a double-blind, treat-to-target, event-driven cardiovascular outcomes trial. The primary composite outcome in the time-to-event analysis was the first occurrence of an adjudicated major cardiovascular event (death from cardiovascular causes, nonfatal myocardial infarction, or nonfatal stroke) with a prespecified noninferiority margin of 1.3. Adjudicated severe hypoglycemia, as defined by the American Diabetes Association, was the prespecified, multiplicity-adjusted secondary outcome. RESULTS: Of the patients who underwent randomization, 6509 (85.2%) had established cardiovascular disease, chronic kidney disease, or both. At baseline, the mean age was 65.0 years, the mean duration of diabetes was 16.4 years, and the mean (±SD) glycated hemoglobin level was 8.4±1.7%; 83.9% of the patients were receiving insulin. The primary outcome occurred in 325 patients (8.5%) in the degludec group and in 356 (9.3%) in the glargine group (hazard ratio, 0.91; 95% confidence interval, 0.78 to 1.06; P<0.001 for noninferiority). At 24 months, the mean glycated hemoglobin level was 7.5±1.2% in each group, whereas the mean fasting plasma glucose level was significantly lower in the degludec group than in the glargine group (128±56 vs. 136±57 mg per deciliter, P<0.001). Prespecified adjudicated severe hypoglycemia occurred in 187 patients (4.9%) in the degludec group and in 252 (6.6%) in the glargine group, for an absolute difference of 1.7 percentage points (rate ratio, 0.60; P<0.001 for superiority; odds ratio, 0.73; P<0.001 for superiority). Rates of adverse events did not differ between the two groups. CONCLUSIONS: Among patients with type 2 diabetes at high risk for cardiovascular events, degludec was noninferior to glargine with respect to the incidence of major cardiovascular events. (Funded by Novo Nordisk and others; DEVOTE ClinicalTrials.gov number, NCT01959529 .).

More data, less information? Potential for nonmonotonic information growth using GEE.

Statistical intuition suggests that increasing the total number of observations available for analysis should increase the precision with which parameters can be estimated. Such monotonic growth of statistical information is of particular importance when data are analyzed sequentially, such as in confirmatory clinical trials. However, monotonic information growth is not always guaranteed, even when using a valid, but inefficient estimator. In this article, we demonstrate the theoretical possibility of nonmonotonic information growth when using generalized estimating equations (GEE) to estimate a slope and provide intuition for why this possibility exists. We use theoretical and simulation-based results to characterize situations that may result in nonmonotonic information growth. Nonmonotonic information growth is most likely to occur when (1) accrual is fast relative to follow-up on each individual, (2) correlation among measurements from the same individual is high, and (3) measurements are becoming more variable further from randomization. In situations that may lead to nonmonotonic information growth, study designers should plan interim analyses to avoid situations most likely to result in nonmonotonic information growth.

Design of DEVOTE (Trial Comparing Cardiovascular Safety of Insulin Degludec vs Insulin Glargine in Patients With Type 2 Diabetes at High Risk of Cardiovascular Events) - DEVOTE 1.

DEVOTE was designed to evaluate the cardiovascular safety of insulin degludec (IDeg) vs insulin glargine U100 (IGlar) in patients with T2D at high risk of cardiovascular events. DEVOTE is a phase 3b, multicenter, international, randomized, double-blind, active comparator-controlled trial, designed as an event-driven trial that would continue until 633 positively adjudicated primary events were accrued. The primary end point was the time from randomization to a composite outcome consisting of the first occurrence of cardiovascular death, nonfatal myocardial infarction, or nonfatal stroke. Patients with T2D at high risk of cardiovascular complications were randomized 1:1 to receive either IDeg or IGlar, each added to background therapies. This trial was designed to demonstrate statistical noninferiority of IDeg vs IGlar for the primary end point. DEVOTE enrolled 7,637 patients between October 2013 and November 2014 at 436 sites in 20 countries. Of these, 6,506 patients had prior cardiovascular disease or chronic kidney disease, and the remainder had multiple cardiovascular risk factors. DEVOTE was designed to provide conclusive evidence regarding the cardiovascular safety of IDeg relative to IGlar in a high-risk population of patients with T2D.

Violations of the independent increment assumption when using generalized estimating equation in longitudinal group sequential trials.

In phase 3 clinical trials, ethical and financial concerns motivate sequential analyses in which the data are analyzed prior to completion of the entire planned study. Existing group sequential software accounts for the effects of these interim analyses on the sampling density by assuming that the contribution of subsequent increments is independent of the contribution from previous data. This independent increment assumption is satisfied in many common circumstances, including when using the efficient estimator. However, certain circumstances may dictate using an inefficient estimator, and the independent increment assumption may then be violated. Consequences of assuming independent increments in a setting where the assumption does not hold have not been previously explored. One important setting in which independent increments may not hold is the setting of longitudinal clinical trials. This paper considers dependent increments that arise because of heteroscedastic and correlated data in the context of longitudinal clinical trials that use a generalized estimating equation (GEE) approach. Both heteroscedasticity over time and correlation of observations within subjects may lead to departures from the independent increment assumption when using GEE. We characterize situations leading to greater departures in this paper. Despite violations of the independent increment assumption, simulation results suggest that operating characteristics of sequential designs are largely maintained for typically observed patterns of accrual, correlation, and heteroscedasticity even when using analyses that use standard software that depends on an independent increment structure. More extreme scenarios may require greater care to avoid departures from the nominal type I error rate and power.

An evaluation of inferential procedures for adaptive clinical trial designs with pre-specified rules for modifying the sample size.

Many papers have introduced adaptive clinical trial methods that allow modifications to the sample size based on interim estimates of treatment effect. There has been extensive commentary on type I error control and efficiency considerations, but little research on estimation after an adaptive hypothesis test. We evaluate the reliability and precision of different inferential procedures in the presence of an adaptive design with pre-specified rules for modifying the sampling plan. We extend group sequential orderings of the outcome space based on the stage at stopping, likelihood ratio statistic, and sample mean to the adaptive setting in order to compute median-unbiased point estimates, exact confidence intervals, and P-values uniformly distributed under the null hypothesis. The likelihood ratio ordering is found to average shorter confidence intervals and produce higher probabilities of P-values below important thresholds than alternative approaches. The bias adjusted mean demonstrates the lowest mean squared error among candidate point estimates. A conditional error-based approach in the literature has the benefit of being the only method that accommodates unplanned adaptations. We compare the performance of this and other methods in order to quantify the cost of failing to plan ahead in settings where adaptations could realistically be pre-specified at the design stage. We find the cost to be meaningful for all designs and treatment effects considered, and to be substantial for designs frequently proposed in the literature.

Detecting differential gene expression in subgroups of a disease population.

In many disease settings, it is likely that only a subset of the disease population will exhibit certain genetic or phenotypic differences from the healthy population. Therefore, when seeking to identify genes or other explanatory factors that might be related to the disease state, we might expect a mixture distribution of the variable of interest in the disease group. A number of methods have been proposed for performing tests to identify situations for which only a subgroup of samples or patients exhibit differential expression levels. Our discussion here focuses on how inattention to standard statistical theory can lead to approaches that exhibit some serious drawbacks. We present and discuss several approaches motivated by theoretical derivations and compare to an ad hoc approach based upon identification of outliers. We find that the outlier-sum statistic proposed by Tibshirani and Hastie offers little benefit over a t-test even in the most idealized scenarios and suffers from a number of limitations including difficulty of calibration, lack of robustness to underlying distributions, high false positive rates owing to its asymmetric treatment of groups, and poor power or discriminatory ability under many alternatives.

Adaptive clinical trial designs with pre-specified rules for modifying the sample size: understanding efficient types of adaptation.

Adaptive clinical trial design has been proposed as a promising new approach that may improve the drug discovery process. Proponents of adaptive sample size re-estimation promote its ability to avoid 'up-front' commitment of resources, better address the complicated decisions faced by data monitoring committees, and minimize accrual to studies having delayed ascertainment of outcomes. We investigate aspects of adaptation rules, such as timing of the adaptation analysis and magnitude of sample size adjustment, that lead to greater or lesser statistical efficiency. Owing in part to the recent Food and Drug Administration guidance that promotes the use of pre-specified sampling plans, we evaluate alternative approaches in the context of well-defined, pre-specified adaptation. We quantify the relative costs and benefits of fixed sample, group sequential, and pre-specified adaptive designs with respect to standard operating characteristics such as type I error, maximal sample size, power, and expected sample size under a range of alternatives. Our results build on others' prior research by demonstrating in realistic settings that simple and easily implemented pre-specified adaptive designs provide only very small efficiency gains over group sequential designs with the same number of analyses. In addition, we describe optimal rules for modifying the sample size, providing efficient adaptation boundaries on a variety of scales for the interim test statistic for adaptation analyses occurring at several different stages of the trial. We thus provide insight into what are good and bad choices of adaptive sampling plans when the added flexibility of adaptive designs is desired.

Manipulation of the coccyx with anesthesia for the management of coccydynia.

Coccydynia is pain in and around the coccyx, with trauma being the most common etiologic factor. The authors describe the case of a 60-year-old woman who was injured in an automobile accident several months before consultation. After the physician recorded the patient's history, performed a physical examination, and reviewed imaging reports, lumbar radiculopathy and somatic dysfunction of the lumbar and sacral vertebrae were diagnosed as the cause of her coccydynia. During a 6-week period, the patient underwent a series of 3 epidural steroid injections, osteopathic mobilization of the lumbar and sacral spine, and manual manipulation of the coccyx with anesthesia. After each treatment session, the patient reported a substantial improvement in radicular pain as well as a subjective increase in her lower extremity range of motion.