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Let $X$ be an $(n-2)$-connected $2n$-dimensional Poincar\'e complex with torsion-free homology, where $n\geq 4$. We prove that $X$ can be decomposed into a connected sum of two Poincar\'e complexes: one being $(n-1)$-connected, while the other having trivial $n$th homology group. Under the additional assumption that $H_n(X)=0$ and $Sq^2:H^{n-1}(X;\mathbb{Z}_2)\to H^{n+1}(X;\mathbb{Z}_2)$ is trivial, we can prove that $X$ can be further decomposed into connected sums of Poincar\'e complexes whose $(n-1)$th homology is isomorphic to $\mathbb{Z}$. As an application of this result, we classify the homotopy types of such $2$-connected $8$-dimensional Poincar\'e complexes., Comment: 18 pages

A stepped wedge design is a unidirectional crossover design where clusters are randomized to distinct treatment sequences. While model-based analysis of stepped wedge designs is standard practice to evaluate treatment effects accounting for clustering and adjusting for covariates, their properties under misspecification have not been systematically explored. In this article, we focus on model-based methods, including linear mixed models and generalized estimating equations with an independence, simple exchangeable, or nested exchangeable working correlation structure. We study when a potentially misspecified working model can offer consistent estimation of the marginal treatment effect estimands, which are defined nonparametrically with potential outcomes and may be functions of calendar time and/or exposure time. We prove a central result that consistency for nonparametric estimands usually requires a correctly specified treatment effect structure, but generally not the remaining aspects of the working model (functional form of covariates, random effects, and residual distribution), and valid inference is obtained via the sandwich variance estimator. Furthermore, an additional g-computation step is required to achieve model-robust inference under non-identity link functions or for ratio estimands. The theoretical results are illustrated via several simulation experiments and re-analysis of a completed stepped wedge cluster randomized trial.

In-Context Learning (ICL) combined with pre-trained large language models has achieved promising results on various NLP tasks. However, ICL requires high-quality annotated demonstrations which might not be available in real-world scenarios. To overcome this limitation, we propose \textbf{D}ata \textbf{A}ugmentation for \textbf{I}n-Context \textbf{L}earning (\textbf{DAIL}). DAIL leverages the intuition that large language models are more familiar with the content generated by themselves. It first utilizes the language model to generate paraphrases of the test sample and employs majority voting to determine the final result based on individual predictions. Our extensive empirical evaluation shows that DAIL outperforms the standard ICL method and other ensemble-based methods in the low-resource scenario. Additionally, we explore the use of voting consistency as a confidence score of the model when the logits of predictions are inaccessible. We believe our work will stimulate further research on ICL in low-resource settings., Comment: Course project for DSC 253 (Advanced Data-Driven Text Mining) at UCSD

Cluster randomized trials (CRTs) frequently recruit a small number of clusters, therefore necessitating the application of small-sample corrections for valid inference. A recent systematic review indicated that CRTs reporting right-censored, time-to-event outcomes are not uncommon, and that the marginal Cox proportional hazards model is one of the common approaches used for primary analysis. While small-sample corrections have been studied under marginal models with continuous, binary and count outcomes, no prior research has been devoted to the development and evaluation of bias-corrected sandwich variance estimators when clustered time-to-event outcomes are analyzed by the marginal Cox model. To improve current practice, we propose 9 bias-corrected sandwich variance estimators for the analysis of CRTs using the marginal Cox model, and report on a simulation study to evaluate their small-sample properties. Our results indicate that the optimal choice of bias-corrected sandwich variance estimator for CRTs with survival outcomes can depend on the variability of cluster sizes, and can also slightly differ whether it is evaluated according to relative bias or type I error rate. Finally, we illustrate the new variance estimators in a real-world CRT where the conclusion about intervention effectiveness differs depending on the use of small-sample bias corrections. The proposed sandwich variance estimators are implemented in an R package CoxBcv.

In this article, we develop methods for sample size and power calculations in four-level intervention studies when intervention assignment is carried out at any level, with a particular focus on cluster randomized trials (CRTs). CRTs involving four levels are becoming popular in health care research, where the effects are measured, for example, from evaluations (level 1) within participants (level 2) in divisions (level 3) that are nested in clusters (level 4). In such multi-level CRTs, we consider three types of intraclass correlations between different evaluations to account for such clustering: that of the same participant, that of different participants from the same division, and that of different participants from different divisions in the same cluster. Assuming arbitrary link and variance functions, with the proposed correlation structure as the true correlation structure, closed-form sample size formulas for randomization carried out at any level (including individually randomized trials within a four-level clustered structure) are derived based on the generalized estimating equations approach using the model-based variance and using the sandwich variance with an independence working correlation matrix. We demonstrate that empirical power corresponds well with that predicted by the proposed method for as few as 8 clusters, when data are analyzed using the matrix-adjusted estimating equations for the correlation parameters with a bias-corrected sandwich variance estimator, under both balanced and unbalanced designs.