Learning curves are a useful way of representing the rate of learning over time. Features include an index of baseline performance (y-intercept), the efficiency of learning over time (slope parameter) and the maximal theoretical performance achievable (upper asymptote). Each of these parameters can be statistically modelled on an individual and group basis with the resulting estimates being useful to both learners and educators for feedback and educational quality improvement. In this primer, we review various descriptive and modelling techniques appropriate to learning curves including smoothing, regression modelling and application of the Thurstone model. Using an example dataset we demonstrate each technique as it specifically applies to learning curves and point out limitations.

Vasectomy has been implicated as a risk factor for prostate cancer in multiple epidemiologic studies over the past 25 years. Whether this relationship is causal remains unclear. This study examines the association between vasectomy and prostate cancer in the Prostate, Lung, Colorectal and Ovarian (PLCO) Cancer Screening Trial, which randomized men to usual care or annual prostate cancer screening. We performed a retrospective analysis of 13-year screening and outcomes data from the PLCO trial. Multivariable Cox proportional hazards regression stratified by study arm and age at vasectomy was performed. There was an increased risk of prostate cancer in men who had undergone a vasectomy and were randomized to the usual care arm of the study (adjusted HR, 1.11; 95% confidence interval, 1.03-1.20; = 0.008). There was no association between vasectomy and diagnosis of prostate cancer in men randomized to the prostate cancer screening arm. Only men undergoing vasectomy at an older age in the usual care arm of the study, but not the prostate cancer screening arm, were at increased risk of being diagnosed with prostate cancer. Vasectomy was not associated with prostate cancer risk among men who were screened for prostate cancer as part of a clinical trial, but was associated with prostate cancer detection in men receiving usual care. The positive association between vasectomy and prostate cancer is likely related to increased detection of prostate cancer based on patterns of care rather than a biological effect of vasectomy on prostate cancer development. .

Summary We introduce a novel Bayesian estimator for the class proportion in an unlabeled dataset, based on the targeted learning framework. The procedure requires the specification of a prior (and outputs a posterior) only for the target of inference, and yields a tightly concentrated posterior. When the scientific question can be characterized by a low-dimensional parameter functional, this focus on target prior and posterior distributions perfectly aligns with Bayesian subjectivism. We prove a Bernstein–von Mises-type result for our proposed Bayesian procedure, which guarantees that the posterior distribution converges to the distribution of an efficient, asymptotically linear estimator. In particular, the posterior is Gaussian, doubly robust, and efficient in the limit, under the only assumption that certain nuisance parameters are estimated at slower-than-parametric rates. We perform numerical studies illustrating the frequentist properties of the method. We also illustrate their use in a motivating application to estimate the proportion of embolic strokes of undetermined source arising from occult cardiac sources or large-artery atherosclerotic lesions. Though we focus on the motivating example of the proportion of cases in an unlabeled dataset, the procedure is general and can be adapted to estimate any pathwise differentiable parameter in a non-parametric model.

<div>Abstract<p><b>Background:</b> Vasectomy has been implicated as a risk factor for prostate cancer in multiple epidemiologic studies over the past 25 years. Whether this relationship is causal remains unclear. This study examines the association between vasectomy and prostate cancer in the Prostate, Lung, Colorectal and Ovarian (PLCO) Cancer Screening Trial, which randomized men to usual care or annual prostate cancer screening.</p><p><b>Methods:</b> We performed a retrospective analysis of 13-year screening and outcomes data from the PLCO trial. Multivariable Cox proportional hazards regression stratified by study arm and age at vasectomy was performed.</p><p><b>Results:</b> There was an increased risk of prostate cancer in men who had undergone a vasectomy and were randomized to the usual care arm of the study (adjusted HR, 1.11; 95% confidence interval, 1.03–1.20; <i>P</i> = 0.008). There was no association between vasectomy and diagnosis of prostate cancer in men randomized to the prostate cancer screening arm. Only men undergoing vasectomy at an older age in the usual care arm of the study, but not the prostate cancer screening arm, were at increased risk of being diagnosed with prostate cancer.</p><p><b>Conclusions:</b> Vasectomy was not associated with prostate cancer risk among men who were screened for prostate cancer as part of a clinical trial, but was associated with prostate cancer detection in men receiving usual care.</p><p><b>Impact:</b> The positive association between vasectomy and prostate cancer is likely related to increased detection of prostate cancer based on patterns of care rather than a biological effect of vasectomy on prostate cancer development. <i>Cancer Epidemiol Biomarkers Prev; 26(11); 1653–9. ©2017 AACR</i>.</p></div>

<p>Supplemental Table 2. Risk of prostate cancer by vasectomy status and trial arm using multiple imputation.</p>