Optimal weighted Bonferroni tests and their graphical extensions.

Regulatory guidelines mandate the strong control of the familywise error rate in confirmatory clinical trials with primary and secondary objectives. Bonferroni tests are one of the popular choices for multiple comparison procedures and are building blocks of more advanced procedures. It is usually of interest to find the optimal weighted Bonferroni split for multiple hypotheses. We consider two popular quantities as the optimization objectives, which are the disjunctive power and the conjunctive power. The former is the probability to reject at least one false hypothesis and the latter is the probability to reject all false hypotheses. We investigate the behavior of each of them as a function of different Bonferroni splits, given assumptions about the alternative hypotheses and correlations between test statistics. Under independent tests, unique optimal Bonferroni weights exist; under dependence, optimal Bonferroni weights may not be unique based on a fine grid search. In general, we propose an optimization algorithm based on constrained nonlinear optimization and multiple starting points. The proposed algorithm efficiently identifies optimal Bonferroni weights to maximize the disjunctive or conjunctive power. In addition, we apply the proposed algorithm to graphical approaches, which include many Bonferroni-based multiple comparison procedures. Utilizing the closed testing principle, we adopt a two-step approach to find optimal graphs using the disjunctive power. We also identify a class of closed test procedures that optimize the conjunctive power. We apply the proposed algorithm to a case study to illustrate the utility of optimal graphical approaches that reflect study objectives.

Metal- and light-free decarboxylative direct C-H alkylation of heteroarenes at room temperature.

This study reports a metal- and light-free decarboxylative C-H alkylation of heteroarenes at room temperature. The reaction generates various primary, secondary, and tertiary alkyl radicals and functionalizes seven different privileged scaffolds widely present in bioactive molecules. During this process, one equivalent of hypervalent iodine(III) carboxylates (HICs) plays dual roles as an alkyl radical precursor and an oxidant.

New strategies for confirmatory testing of secondary hypotheses on combined data from multiple trials.

BACKGROUND: Pivotal evidence of efficacy of a new drug is typically generated by (at least) two clinical trials which independently provide statistically significant and mutually corroborating evidence of efficacy based on a primary endpoint. In this situation, showing drug effects on clinically important secondary objectives can be demanding in terms of sample size requirements. Statistically efficient methods to power for such endpoints while controlling the Type I error are needed. METHODS: We review existing strategies for establishing claims on important but sample size-intense secondary endpoints. We present new strategies based on combined data from two independent, identically designed and concurrent trials, controlling the Type I error at the submission level. We explain the methodology and provide three case studies. RESULTS: Different strategies have been used for establishing secondary claims. One new strategy, involving a protocol planned analysis of combined data across trials, and controlling the Type I error at the submission level, is particularly efficient. It has already been successfully used in support of label claims. Regulatory views on this strategy differ. CONCLUSIONS: Inference on combined data across trials is a useful approach for generating pivotal evidence of efficacy for important but sample size-intense secondary endpoints. It requires careful preparation and regulatory discussion.

Disease severity and antiviral response in patients with chronic hepatitis B with non-obese NAFLD.

BACKGROUND: The burden of nonalcoholic fatty liver disease (NAFLD) is growing in patients with chronic hepatitis B (CHB). NAFLD is typically associated with obesity, however, it is increasingly being identified in non-obese patients. This study aimed to investigate disease severity and antiviral response in non-obese patients with CHB with NAFLD (CHB + NAFLD). METHODS: A total of 809 patients with CHB + NAFLD were prospectively recruited and followed up for 3 years. NAFLD was diagnosed by transient elastography and defined as controlled attenuation parameter ≥248 dB/m, in the absence of excessive alcohol intake. Obesity status was defined by the Asian body mass index (BMI) cutoff of 25 kg/m2. Metabolic abnormality was defined by the presence of dyslipidemia, hypertension or diabetes. Fibrosis staging was defined according to the EASL-ALEH guidelines, with fibrosis progression defined as ≥1-stage increment. RESULTS: In the total cohort (median age 40 years, 59.0% antiviral-treated), 33.3% were non-obese. Non-obese patients were less metabolically abnormal than obese patients (60.2% vs 72.0%, P = 0.003). After 3-year follow up, the rate of fibrosis progression was comparable between non-obese and obese patients (17.5% vs 21.9% in the total cohort, P = 0.145; 15.7% vs 14.6% in antiviral-treated cohort with persistent viral suppression, P = 0.795). No significant differences in virological and biochemical responses were observed between non-obese and obese patients (P >0.05 for all). CONCLUSION: Approximately one third of CHB + NAFLD patients were non-obese. Non-obese patients, while less metabolically abnormal, had a similar risk for fibrosis progression as obese patients. Obesity status did not impact the efficiency of antiviral therapy.

Covariate adjustment and estimation of difference in proportions in randomized clinical trials.

Difference in proportions is frequently used to measure treatment effect for binary outcomes in randomized clinical trials. The estimation of difference in proportions can be assisted by adjusting for prognostic baseline covariates to enhance precision and bolster statistical power. Standardization or g-computation is a widely used method for covariate adjustment in estimating unconditional difference in proportions, because of its robustness to model misspecification. Various inference methods have been proposed to quantify the uncertainty and confidence intervals based on large-sample theories. However, their performances under small sample sizes and model misspecification have not been comprehensively evaluated. We propose an alternative approach to estimate the unconditional variance of the standardization estimator based on the robust sandwich estimator to further enhance the finite sample performance. Extensive simulations are provided to demonstrate the performances of the proposed method, spanning a wide range of sample sizes, randomization ratios, and model specification. We apply the proposed method in a real data example to illustrate the practical utility.

CITIES: Clinical trials with intercurrent events simulator.

Although clinical trials are often designed with randomization and well-controlled protocols, complications will inevitably arise in the presence of intercurrent events (ICEs) such as treatment discontinuation. These can lead to missing outcome data and possibly confounding causal inference when the missingness is a function of a latent stratification of patients defined by intermediate outcomes. The pharmaceutical industry has been focused on developing new methods that can yield pertinent causal inferences in trials with ICEs. However, it is difficult to compare the properties of different methods developed in this endeavor as real-life clinical trial data cannot be easily shared to provide benchmark data sets. Furthermore, different methods consider distinct assumptions for the underlying data-generating mechanisms, and simulation studies often are customized to specific situations or methods. We develop a novel, general simulation model and corresponding Shiny application in R for clinical trials with ICEs, aptly named the Clinical Trials with Intercurrent Events Simulator (CITIES). It is formulated under the Rubin Causal Model where the considered treatment effects account for ICEs in clinical trials with repeated measures. CITIES facilitates the effective generation of data that resemble real-life clinical trials with respect to their reported summary statistics, without requiring the use of the original trial data. We illustrate the utility of CITIES via two case studies involving real-life clinical trials that demonstrate how CITIES provides a comprehensive tool for practitioners in the pharmaceutical industry to compare methods for the analysis of clinical trials with ICEs on identical, benchmark settings that resemble real-life trials.

Weighted gene coexpression network analysis and machine learning reveal oncogenome associated microbiome plays an important role in tumor immunity and prognosis in pan-cancer.

BACKGROUND: For many years, the role of the microbiome in tumor progression, particularly the tumor microbiome, was largely overlooked. The connection between the tumor microbiome and the tumor genome still requires further investigation. METHODS: The TCGA microbiome and genome data were obtained from Haziza et al.'s article and UCSC Xena database, respectively. Separate WGCNA networks were constructed for the tumor microbiome and genomic data after filtering the datasets. Correlation analysis between the microbial and mRNA modules was conducted to identify oncogenome associated microbiome module (OAM) modules, with three microbial modules selected for each tumor type. Reactome analysis was used to enrich biological processes. Machine learning techniques were implemented to explore the tumor type-specific enrichment and prognostic value of OAM, as well as the ability of the tumor microbiome to differentiate TP53 mutations. RESULTS: We constructed a total of 182 tumor microbiome and 570 mRNA WGCNA modules. Our results show that there is a correlation between tumor microbiome and tumor genome. Gene enrichment analysis results suggest that the genes in the mRNA module with the highest correlation with the tumor microbiome group are mainly enriched in infection, transcriptional regulation by TP53 and antigen presentation. The correlation analysis of OAM with CD8+ T cells or TAM1 cells suggests the existence of many microbiota that may be involved in tumor immune suppression or promotion, such as Williamsia in breast cancer, Biostraticola in stomach cancer, Megasphaera in cervical cancer and Lottiidibacillus in ovarian cancer. In addition, the results show that the microbiome-genome prognostic model has good predictive value for short-term prognosis. The analysis of tumor TP53 mutations shows that tumor microbiota has a certain ability to distinguish TP53 mutations, with an AUROC value of 0.755. The tumor microbiota with high importance scores are Corallococcus, Bacillus and Saezia. Finally, we identified a potential anti-cancer microbiota, Tissierella, which has been shown to be associated with improved prognosis in tumors including breast cancer, lung adenocarcinoma and gastric cancer. CONCLUSION: There is an association between the tumor microbiome and the tumor genome, and the existence of this association is not accidental and could change the landscape of tumor research.

Multiplicity adjustments for the Dunnett procedure under heterokcedasticity.

We give a simulation-based method for computing the multiplicity adjusted p-values and critical constants for the Dunnett procedure for comparing treatments with a control under heteroskedasticity. The Welch-Satterthwaite test statistics used in this procedure do not have a simple multivariate t-distribution because their denominators are mixtures of chi-squares and are correlated because of the common control treatment sample variance present in all denominators. The joint distribution of the denominators of the test statistics is approximated by correlated chi-square variables and is generated using a novel algorithm proposed in this paper. This approximation is used to derive critical constants or adjusted p-values. The familywise error rate (FWER) of the proposed method is compared with some existing methods via simulation under different heteroskedastic scenarios. The results show that our proposed method controls the FWER most accurately, whereas other methods are either too conservative or liberal or control the FWER less accurately. The different methods considered are illustrated on a real data set.

Two calcium-dependent protein kinases enhance maize drought tolerance by activating anion channel ZmSLAC1 in guard cells.

Stomatal closure is an important process to prevent water loss in plants response to drought stress, which is finely modulated by ion channels together with their regulators in guard cells, especially the S-type anion channel AtSLAC1 in Arabidopsis. However, the functional characterization and regulation analyses of anion channels in gramineous crops, such as in maize guard cells are still limited. In this study, we identified an S-type anion channel ZmSLAC1 that was preferentially expressed in maize guard cells and involved in stomatal closure under drought stress. We found that two Ca2+ -dependent protein kinases ZmCPK35 and ZmCPK37 were expressed in maize guard cells and localized on the plasma membrane. Lesion of ZmCPK37 resulted in drought-sensitive phenotypes. Mutation of ZmSLAC1 and ZmCPK37 impaired ABA-activated S-type anion currents in maize guard cells, while the S-type anion currents were increased in the guard cells of ZmCPK35- and ZmCPK37-overexpression lines. Electrophysiological characterization in maize guard cells and Xenopus oocytes indicated that ZmCPK35 and ZmCPK37 could activate ZmSLAC1-mediated Cl- and NO3- currents. The maize inbred and hybrid lines overexpressing ZmCPK35 and ZmCPK37 exhibited enhanced tolerance and increased yield under drought conditions. In conclusion, our results demonstrate that ZmSLAC1 plays crucial roles in stomatal closure in maize, whose activity is regulated by ZmCPK35 and ZmCPK37. Elevation of ZmCPK35 and ZmCPK37 expression levels is a feasible way to improve maize drought tolerance as well as reduce yield loss under drought stress.

Hepatic exosomes with declined MiR-27b-3p trigger RIG-I/TBK1 signal pathway in macrophages.

BACKGROUND AND AIMS: Evidence suggests that interferon alpha (IFNα) plays an essential role in decreasing the HBsAg quantification and elevating the rate of clinical cure in chronic hepatitis B (CHB). However, the mechanisms underlying the effects of the exosomes on the expression of host genes in IFNα treatment remain unclear. METHODS: CHB patients with IFNα treatment were divided into responders and non-responders according to the degree of HBsAg decline. Through microRNA sequencing and a series of molecular biology methods, the key microRNAs in serum exosomes associated with clinical antiviral response of Peg-IFNα treatment in nucleotide analogue-treated CHB patients were investigated. The roles of exosomal miRNAs on the IFNα signal pathway were explored in macrophages. RESULTS: MicroRNA sequencing and RT-qPCR assays confirmed six distinctly declined miRNAs in serum exosomes of responders at week 12 compared with levels at baseline. Exosomes with declined miR27b-3p in the serum of Peg-IFNα-treated responders activated phosphorylation of interferon regulatory factor 3/7 (IRF3/7) in IFNα synthesis pathway in macrophages. However, miR27b-3p overexpression in HepAD38 cells suppressed IFNα synthesis in macrophages, resulting in insufficient ability to eliminate HBV, whereas the inhibitory effect could be blocked by inhibitors of exosomes release. Luciferase assay showed miR-27b-3p directly suppressed retinoic acid-inducible gene I (RIG-I) and TANK-binding kinase 1 (TBK1) expressions, and these effects could be abrogated in mutation experiments. CONCLUSIONS: In IFNα treatment, exosomes with declined miR-27b-3p triggered activation of RIG-I/TBK1 signalling in macrophages against HBV. Serum exosomal miR-27-3p might represent a potential biomarker for patients with CHB.

Causal discoveries for high dimensional mixed data.

Causal relationships are of crucial importance for biological and medical research. Algorithms have been proposed for causal structure learning with graphical visualizations. While much of the literature focuses on biological studies where data often follow the same distribution, for example, the normal distribution for all variables, challenges emerge from epidemiological and clinical studies where data are often mixed with continuous, binary, and ordinal variables. We propose to use a mixed latent Gaussian copula model to estimate the underlying correlation structure via the rank correlation for mixed data. This correlation structure is then incorporated into a popular causal discovery algorithm, the PC algorithm, to identify causal structures. The proposed algorithm, called the latent-PC algorithm, is able to discover the true causal structure consistently under mild conditions in high dimensional settings. From simulation studies, the latent-PC algorithm delivers a competitive performance in terms of a similar or higher true positive rate and a similar or lower false positive rate, compared with other variants of the PC algorithm. In the high dimensional settings where the number of variables is more than the number of observations, the causal graphs identified by the latent-PC algorithm are closer to the true causal structures, compared to other competing algorithms. Further, we demonstrate the utility of the latent-PC algorithm in a real dataset for hepatocellular carcinoma. Causal structures for patient survival are visualized and connected with clinical interpretations in the literature.

Interferon-Induced Macrophage-Derived Exosomes Mediate Antiviral Activity Against Hepatitis B Virus Through miR-574-5p.

BACKGROUND: Interferon alfa (IFN-α) has been proved effective in treating chronic hepatitis B (CHB), owing to its ability to suppress hepatitis B surface antigen and hepatitis B virus (HBV) covalently closed circular DNA. However, the underlying mechanisms are unclear. METHODS: We investigated the antiviral activities of exosomes from responders and nonresponders to pegylated IFN-α (PegIFN-α) as well as the supernatants of IFN-α-treated macrophages derived from THP-1 (the human leukemia monocyte cell line). Then the expression profiles of exosomal microRNAs (miRNAs) were analyzed using miRNA sequencing. The luciferase reporter assay was used to locate the binding position of HBV genomic sequence targeted by the identified miRNA. RESULTS: Exosomes from PegIFN-α-treated patients, particularly responders, as well as the supernatants of IFN-α-treated macrophages exhibited anti-HBV activities, as manifested by the suppression of hepatitis B surface antigen, hepatitis B e antigen, HBV DNA, and covalently closed circular DNA levels in HBV-related cell lines. PegIFN-α treatment up-regulated exosomal hsa-miR-193a-5p, hsa-miR-25-5p, and hsa-miR-574-5p, which could partially inhibit HBV replication and transcription, and hsa-miR-574-5p reduced pregenomic RNA and polymerase messenger RNA levels by binding to the 2750-2757 position of the HBV genomic sequence. CONCLUSIONS: Exosomes can transfer IFN-α-related miRNAs from macrophages to HBV-infected hepatocytes, and they exhibit antiviral activities against HBV replication and expression.

lncRNA ZNF667-AS1 (NR_036521.1) inhibits the progression of colorectal cancer via regulating ANK2/JAK2 expression.

Long noncoding RNAs (lncRNAs) participate in many biological processes by affecting gene expression at the posttranscriptional level. lncRNAs are dysregulated in colorectal cancer (CRC) and this dysregulation is closely related to tumorigenesis, metastasis, and prognosis. Although many lncRNAs have been identified in CRC, the relation between ZNF667 antisense RNA 1 (head to head; ZNF667-AS1, accession: NR_036521.1) and CRC remains unclear. In this study, a total of 2,218 differentially expressed genes and 428 differentially expressed lncRNAs were identified between tumor and pericarcinous tissues. They were mainly enriched in cancer pathways, chemokine signaling, phosphoinositide 3-kinase-protein kinase B signaling pathway, and others. Key lncRNAs, including ZNF667-AS1, and their corresponding genes, such as ankyrin 2 (ANK2), were downregulated in CRC tumor tissues. In addition, downregulated ZNF667-AS1 (NR_036521.1) expression is associated with poor prognosis and disease progression. Overexpression of ZNF667-AS1 (NR_036521.1) inhibited the proliferation, migration, and invasion of VOLO cells both in vitro and in vivo. Moreover, Janus kinase 2 (JAK2) and ANK2 were significantly down- and upregulated in the overexpressed ZNF667-AS1 VOLO cells compared to those in the negative-control group. Knockdown of ANK2 or overexpression of JAK2 significantly counteracted the inhibitory effects of overexpression of ZNF667-AS1 on LOVO cell proliferation and migration. Taken together, it is indicated in our research that ZNF667-AS1 interaction with ANK2/JAK2 maybe important in CRC progression. Overexpression of ZNF667-AS1 could inhibit the proliferation, migration, and invasion of CRC cells, which may be related with the high ANK2 and low JAK2 levels.

Group sequential Holm and Hochberg procedures.

The problem of testing multiple hypotheses using a group sequential procedure often arises in clinical trials. We review several group sequential Holm (GSHM) type procedures proposed in the literature and clarify the relationships between them. In particular, we show which procedures are equivalent or, if different, which are more powerful and what are their pros and cons. We propose a step-up group sequential Hochberg (GSHC) procedure as a reverse application of a particular step-down GSHM procedure. We conducted an extensive simulation study to evaluate the familywise error rate (FWER) and power properties of that GSHM procedure and the GSHC procedure and found that the GSHC procedure controls FWER more closely and is more powerful. All procedures are illustrated with a common numerical example, the data for which are chosen to bring out the differences between them. A real case study is also presented to illustrate application of these procedures. R programs for applying the proposed procedures, additional simulation results, and the proof of the FWER control of the GSHC procedure in a special case are provided in Supplementary Material.

Expression and prognostic significance of the DNA damage response pathway and autophagy markers in gastric cancer.

Gastric cancer (GC) is a leading cause of mortality and morbidity worldwide. We assessed the expression patterns of DNA damage response (DDR)-related markers, including ATM, CHK2, p-p53 (S15), Rad51, and BRCA2 and autophagy-related proteins including p62 and Beclin-1 in 153 GC specimens using immunohistochemistry staining. GC tissues showed lower levels of ATM, CHK2, p-p53, BRCA2, and higher levels of Rad51 compared to adjacent normal tissues. The autophagy-related protein p62 was upregulated, whereas Beclin-1 was downregulated in human GC groups. Additionally, different statuses of DDR pathways and autophagy characterized by protein expression were associated with overall survival. Our results indicated that the impairment of DNA damage and autophagy may be implicated in gastric cancer progression and its clinical prognosis.

CALCIUM-DEPENDENT PROTEIN KINASE 32-mediated phosphorylation is essential for the ammonium transport activity of AMT1;1 in Arabidopsis roots.

Protein kinase-mediated phosphorylation modulates the absorption of many nutrients in plants. CALCIUM-DEPENDENT PROTEIN KINASES (CPKs) are key players in plant signaling to translate calcium signals into diverse physiological responses. However, the regulatory role of CPKs in ammonium uptake remains largely unknown. Here, using methylammonium (MeA) toxicity screening, CPK32 was identified as a positive regulator of ammonium uptake in roots. CPK32 specifically interacted with AMMONIUM TRANSPORTER 1;1 (AMT1;1) and phosphorylated AMT1;1 at the non-conserved serine residue Ser450 in the C-terminal domain. Functional analysis in Xenopus oocytes showed that co-expression of CPK32 and AMT1;1 significantly enhanced the AMT1;1-mediated inward ammonium currents. In transgenic plants, the phosphomimic variant AMT1;1S450E, but not the non-phosphorylatable variant AMT1;1S450A, fully complemented the MeA insensitivity and restored high-affinity 15NH4+ uptake in both amt1;1 and cpk32 mutants. Moreover, in the CPK32 knockout background, AMT1;1 lost its ammonium transport activity entirely. These results indicate that CPK32 is a crucial positive regulator of ammonium uptake in roots and the ammonium transport activity of AMT1;1 is dependent on CPK32-mediated phosphorylation.

Parthenolide ameliorates intracerebral hemorrhage-induced brain injury in rats.

Neuroinflammation and oxidative stress are key contributors to intracranial hemorrhage (ICH)-induced brain injury. Parthenolide (PN) is a sesquiterpene lactone that has been observed to have antioxidative, anti-inflammatory, and neuroprotective potentials. However, the role of PN in ICH remains unclear. Therefore, we investigated the neuroprotective effects and underlying mechanisms of PN on an experimental model of ICH in rats. Our results showed that PN treatment improved neurological deficit and brain edema in ICH rats. The ipsilateral hemispheres of the brain were separated and homogenized. The concentrations of TNF-α, interleukin (IL)-6, and IL-17 in the homogenates were detected by enzyme-linked immunosorbent assay. We found that PN inhibited the production of proinflammatory cytokines in an ICH rat model. The ROS and glutathione (GSH) levels, as well as the activity of superoxide dismutase (SOD) in the homogenates were measured. ICH caused an increase in ROS level, and the decreases in GSH level and SOD activity were mitigated by PN treatment. Furthermore, PN significantly suppressed the expressions of active caspase-3 and Bax in ipsilateral hemispheres of the brain at Day 3 after ICH, as well as increased the surviving neurons. Finally, the ICH-induced activation of TLR4/NF-κB pathway was suppressed by PN treatment. These findings suggested that PN could be beneficial in the therapeutic strategy for ICH treatment.

Interference of miR-943-3p with secreted frizzled-related proteins4 (SFRP4) in an asthma mouse model.

The aim of this study is to investigate the potential roles of miR-943-3p and its target gene secreted frizzled-related proteins4 (SFRP4) in allergic asthma and elucidate its underlying mechanism, which may prompt a new clue about developing novel treatments of this disease. An allergic asthma mouse model was generated by challenging with ovalbumin (OVA); lung pathological features of mice were viewed using H&E staining; thickness of subepithelial fibrosis and smooth muscle was measured using Masson's trichrome staining. Inflammatory cells from bronchoalveolar lavage fluid (BALF) were counted based on Diff-Quik staining and morphometric analysis. Expressions of miR-943-3p, SFRP4 and Wnt signal pathway-associated proteins were detected using RT-PCR or immunoblotting, respectively. SFRP4 was downregulated in the bronchial biopsies of allergic asthma patients and represented a unique intersection between differentially expressed genes (DEGs) and genes in the Wnt signal pathway. Both miR-943-3p upregulation and SFRP4 downregulation were detected in allergic asthma patients and OVA-induced mice. Besides, OVA-induced mice possessed more inflammatory cells in BALF including macrophage (mac), eosinophil (eos), lymphocyte (lym) and neutrophil (neu), higher expression of collagen, ß-catenin and c-Myc as well as thicker subepithelial fibrosis and smooth muscle in lung than control mice. In vivo delivery of miR-943-3p agomir worsened these symptoms, while both miR-943-3p antagomir and Ad-SFRP4 administration effectively alleviated this disease. Taken together, miR-943-3p accelerated the progression of airway inflammation and remodeling in allergic asthma via suppressing the activity of SFRP4 through Wnt signaling pathway in asthma patients and OVA-induced mice.

Symmetric graphs for equally weighted tests, with application to the Hochberg procedure.

The graphical approach to multiple testing provides a convenient tool for designing, visualizing, and performing multiplicity adjustments in confirmatory clinical trials while controlling the familywise error rate. It assigns a set of weights to each intersection null hypothesis within the closed test framework. These weights form the basis for intersection tests using weighted individual p-values, such as the weighted Bonferroni test. In this paper, we extend the graphical approach to intersection tests that assume equal weights for the elementary null hypotheses associated with any intersection hypothesis, including the Hochberg procedure as well as omnibus tests such as Fisher's combination, O'Brien's, and F tests. More specifically, we introduce symmetric graphs that generate sets of equal weights so that the aforementioned tests can be applied with the graphical approach. In addition, we visualize the Hochberg and the truncated Hochberg procedures in serial and parallel gatekeeping settings using symmetric component graphs. We illustrate the method with two clinical trial examples.

An additive boundary for group sequential designs with connection to conditional error.

Group sequential designs allow stopping a clinical trial for meeting its efficacy objectives based on interim evaluation of the accumulating data. Various methods to determine group sequential boundaries that control the probability of crossing the boundary at an interim or the final analysis have been proposed. To monitor trials with uncertainty in group sizes at each analysis, error spending functions are often used to derive stopping boundaries. Although flexible, most spending functions are generic increasing functions with parameters that are difficult to interpret. They are often selected arbitrarily, sometimes using trial and error, so that the corresponding boundaries approximate the desired behavior numerically. Lan and DeMets proposed a spending function that approximates in a natural way the O'Brien-Fleming boundary based on the Brownian motion process. We extend this approach to a general family that has an additive boundary for the Brownian motion process. The spending function and the group sequential boundary share a common parameter that regulates how fast the error is spent. Three subfamilies are considered with different additive terms. In the first subfamily, the parameter has an interpretation as the conditional error rate, which is the conditional probability to reject the null hypothesis at the final analysis. This parameter also provides a connection between group sequential and adaptive design methodology. More choices of designs are allowed in the other two subfamilies. Numerical results are provided to illustrate flexibility and interpretability of the proposed procedures. A clinical trial is described to illustrate the utility of conditional error in boundary determination.