Biological rhythms are recognized as critical aspects of cellular, physiological, organismal, and ecological function. Circadian rhythms are 24 hour (h) physiological rhythms driven by photoperiod and temperature yet maintained in their absence, and occur throughout all kingdoms of life. Disruptions in circadian rhythms are both causes and consequences of many neurological and metabolic diseases, making their study relevant from both a basic science and a biomedical perspective. In the past two decades, high-throughput molecular methods have been applied to probe circadian regulation and behavior of RNA and protein expression levels throughout various organisms, tissues, and conditions. The high-dimensional data that result are sparse, noisy, and contain high levels of null results. Adequately and efficiently detecting rhythms in these data and identifying differences across datasets are ongoing challenges for which many statistical methods have been developed. I introduce three new methods for analyzing these data. In Chapter 2, I improve upon a popular rhythm detection method by more accurately calculating its p-values, which allows its further improvement to increase its sensitivity to asymmetric rhythmic time series; my new method is called empirical JTK CYCLE (eJTK). In Chapter 3, I discuss incorrect assumptions regarding the independence of p-values in two leading rhythm detection methods, RAIN and MetaCycle, and suggest approaches to correct and improve those methods. In Chapter 4, I combine eJTK with empirical Bayes-based bootstrap replicates of experimental time series, adding sensitivity for time series with low relative uncertainty in their time point measurements; my new method is called Bootstrap eJTK (BooteJTK). I also enhance the efficiency of calculating accurate p-values for the method and identify two areas in p-value calculation in other methods in the field. In Chapter 5, I build upon BooteJTK to create a method that rigorously compares two time series from different conditions to determine if they have significantly different rhythmicity and phase. My methods allow for greater sensitivity of rhythmic time series detection while simultaneously providing improved rigor with regard to noisy time series and identification of differences across datasets.