τ$\tau$‐Inflated Beta Regression Model for Estimating τ$\tau$‐Restricted Means and Event‐Free Probabilities for Censored Time‐to‐Event Data

In this research, we propose analysis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -restricted censored time-to-event data via a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -inflated beta regression ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -IBR) model. The outcome of interest is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>min</mml:mi> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:annotation>${\rm min}(\tau,T)$</mml:annotation></mml:semantics> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>T</mml:mi> <mml:annotation>$T$</mml:annotation></mml:semantics> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> are the time-to-event and follow-up duration, respectively. Our analysis goals include estimation and inference related to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -restricted mean survival time ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -RMST) values and event-free probabilities at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> that address the censored nature of the data. In this setting, it is common to observe many individuals with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>min</mml:mi> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>τ</mml:mi></mml:mrow> <mml:annotation>${\rm min}(\tau,T)=\tau$</mml:annotation></mml:semantics> </mml:math> , a point mass that is typically overlooked in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -restricted event-time analyses. Our proposed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -IBR model is based on a decomposition of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>min</mml:mi> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:annotation>${\rm min}(\tau,T)$</mml:annotation></mml:semantics> </mml:math> into <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>τ</mml:mi> <mml:mo>[</mml:mo> <mml:mi>I</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>/</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> <mml:mi>I</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo><</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> <mml:mo>]</mml:mo></mml:mrow> <mml:annotation>$\tau [I(T \ge \tau) +(T/\tau) I(T &lt;\tau)]$</mml:annotation></mml:semantics> </mml:math> . We model the mean of this latter expression using joint logistic and beta regression models that are fit using an expectation-maximization algorithm. An alternative multiple imputation (MI) algorithm for fitting the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -IBR model has the additional advantage of producing uncensored datasets for analysis. Simulations indicate excellent performance of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -IBR model(s), and corresponding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -RMST estimates, in independent and dependent censoring settings. We apply our method to the Azithromycin for Prevention of Chronic Obstructive Pulmonary Disease (COPD) Exacerbations Trial. In addition to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -IBR model results providing a nuanced understanding of the treatment effect, visually appealing heatmaps of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:semantics><mml:mi>τ</mml:mi> <mml:annotation>$\tau$</mml:annotation></mml:semantics> </mml:math> -restricted event times based on our MI datasets are given, a visualization not typically available for censored time-to-event data.