# Department of Mathematics and Statistics Faculty Publications

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Item Penalized Likelihood Ratio Test for a Biomarker Threshold Effect in Clinical Trials Based on Generalized Linear Models(Wiley, 2022-02-24) Gavanji, Parisa; Jiang, Wenyu; Chen, BingshuIn a clinical trial, the responses to the new treatment may vary among patient subsets with different characteristics in a biomarker. It is often necessary to examine if there is a cutpoint for the biomarker that divides the patients into two subsets of those with more-favourable and less-favourable responses. More generally, we approach this problem as a test of homogeneity in the effects of a set of covariates in generalized linear regression models. The unknown cutpoint results in a model with nonidentifiability and a nonsmooth likelihood function to which the ordinary likelihood methods do not apply. We first use a smooth continuous function to approximate the indicator function defining the patient subsets. We then propose a penalized likelihood ratio test to overcome the model irregularities. Under the null hypothesis, we prove that the asymptotic distribution of the proposed test statistic is a mixture of chi-squared distributions. Our method is based on established asymptotic theory, is simple to use, and works in a general framework that includes logistic, Poisson, and linear regression models. In extensive simulation studies, we find that the proposed test works well in terms of size and power. We further demonstrate the use of the proposed method by applying it to clinical trial data from the Digitalis Investigation Group (DIG) on heart failure.Item High-Q Spectral Peaks and Nonstationarity in the Deep Ocean Infragravity Wave Band: Tidal Harmonics and Solar Normal Modes(Wiley, 2019-02-20) Chave, Alan D.; Luther, Douglas S.; Thomson, David J.Infragravity waves have received the least study of any class of waves in the deep ocean. This paper analyzes a 389-day-long deep ocean pressure record from the Hawaii Ocean Mixing Experiment for the presence of narrowband (≲2 μHz) components and nonstationarity over 400–4,000 μHz using a combination of fitting a mixture noncentral/central χ2 model to spectral estimates, high-resolution multitaper spectral estimation, and computation of the offset coherence between distinct frequencies for a given data segment. In the frequency band 400–1,000 μHz there is a noncentral fraction of 0.67 ± 0.07 that decreases with increasing frequency. Evidence is presented for the presence of tidal harmonics in the data over the 400- to 1,400-μHz bands. Above 2,000 μHz the noncentral fraction rises with frequency, comprising about one third of the spectral estimates over 3,000–4,000 μHz. The power spectrum exhibits frequent narrowband peaks at 6–11 standard deviations above the noise level. The widths of the peaks correspond to a Q of at least 1,000, vastly exceeding that of any oceanic or atmospheric process. The offset coherence shows that the spectral peaks have substantial (p = 0.99–0.9999) interfrequency correlation, both locally and between distinct peaks within a given analysis band. Many of the peak frequencies correspond to the known values for solar pressure modes that have previously been observed in solar wind and terrestrial data, while others are the result of nonstationarity that distributes power across frequency. Overall, this paper documents the existence of two previously unrecognized sources of infragravity wave variability in the deep ocean.Item Nonparametric Testing Methods for Treatment-Biomarker Interaction based on Local Partial-Likelihood(Wiley, 2015-06-17) Liu, Yicong; Jiang, Wenyu; Chen, BingshuIn clinical trials, patients with different biomarker features may respond differently to the new treatments or drugs. In personalized medicine, it is important to study the interaction between treatment and biomarkers in order to clearly identify patients that benefit from the treatment. With the local partial likelihood estimation (LPLE) method proposed by Fan et al. (2006), the treatment effect can be modeled as a flexible function of the biomarker. In this paper, we propose a bootstrap test method for survival outcome data based on the LPLE, for assessing whether the treatment effect is a constant among all patients or varies as a function of the biomarker. The test method is called local partial likelihood bootstrap (LPLB) and is developed by bootstrapping the martingale residuals. The test statistic measures the amount of changes in treatment effects across the entire range of the biomarker and is derived based on asymptotic the- ories for martingales. The LPLB method is nonparametric, and is shown in simulations and data analysis examples to be flexible to identify treatment effects of any form in any biomarker defined subsets, and more powerful to detect treatment-biomarker interaction of complex forms than the Cox regression model with a simple interaction. We use data from a breast cancer and a prostate cancer clinical trial to illustrate the proposed LPLB test.Item Why is sterility virulence most common in sexually transmitted infections? Examining the role of epidemiology(2019-03-12) McLeod, David V.; Day, TroySterility virulence, or the reduction in host fecundity due to infection, occurs in many host–pathogen systems. Notably, sterility virulence is more common for sexually transmitted infections (STIs) than for directly transmitted pathogens, while other forms of virulence tend to be limited in STIs. This has led to the suggestion that sterility virulence may have an adaptive explanation. By focusing upon finite population models, we show that the observed patterns of sterility virulence can be explained by consideration of the epidemiological differences between STIs and directly transmitted pathogens. In particular, when pathogen transmission is predominantly density invariant (as for STIs), and mortality is density dependent, sterility virulence can be favored by demographic stochasticity, whereas if pathogen transmission is predominantly density dependent, as is common for most directly transmitted pathogens, sterility virulence is disfavored. We show these conclusions can hold even if there is a weak selective advantage to sterilizing.Item Prime divisors of sparse values of cyclotomic polynomials and Wieferich primes(2019-03-20) Murty, M. Ram; Séguin, FrançoisBang (1886), Zsigmondy (1892) and Birkhoff and Vandiver (1904) initiated the study of the largest prime divisors of sequences of the form an−bn, denoted P(an−bn), by essentially proving that for integers a>b>0, P(an−bn)≥n+1 for every n>2. Since then, the problem of finding bounds on the largest prime factor of Lehmer sequences, Lucas sequences or special cases thereof has been studied by many, most notably by Schinzel (1962), and Stewart (1975, 2013). In 2002, Murty and Wong proved, conditionally upon the abc conjecture, that P(an−bn)≫n2−ϵ for any ϵ>0. In this article, we improve this result for the specific case where b=1. Specifically, we obtain a more precise result, and one that is dependent on a condition we believe to be weaker than the abc conjecture. Our result actually concerns the largest prime factor of the nth cyclotomic polynomial evaluated at a fixed integer a, P(Φn(a)), as we let n grow. We additionally prove some results related to the prime factorization of Φn(a). We also present a connection to Wieferich primes, as well as show that the finiteness of a particular subset of Wieferich primes is a sufficient condition for the infinitude of non-Wieferich primes. Finally, we use the technique used in the proof of the aforementioned results to show an improvement on average of estimates due to Erdős for certain sums.