Multiple testing of composite null hypotheses for discrete data using randomized p-values.

P-values that are derived from continuously distributed test statistics are typically uniformly distributed on (0,1) under least favorable parameter configurations (LFCs) in the null hypothesis. Conservativeness of a p-value P (meaning that P is under the null hypothesis stochastically larger than uniform on (0,1)) can occur if the test statistic from which P is derived is discrete, or if the true parameter value under the null is not an LFC. To deal with both of these sources of conservativeness, we present two approaches utilizing randomized p-values. We illustrate their effectiveness for testing a composite null hypothesis under a binomial model. We also give an example of how the proposed p-values can be used to test a composite null in group testing designs. We find that the proposed randomized p-values are less conservative compared to nonrandomized p-values under the null hypothesis, but that they are stochastically not smaller under the alternative. The problem of establishing the validity of randomized p-values has received attention in previous literature. We show that our proposed randomized p-values are valid under various discrete statistical models, which are such that the distribution of the corresponding test statistic belongs to an exponential family. The behavior of the power function for the tests based on the proposed randomized p-values as a function of the sample size is also investigated. Simulations and a real data example are used to compare the different considered p-values.

Environmental pollution mitigation through utilization of carbon dioxide by microalgae.

Anthropogenic emissions of CO2 have reached a critical level and the global surface temperature is expected to rise by 1.5 °C between 2030 and 2050. To ameliorate the current global warming scenario, the research community has been struggling to find more economical and innovative solutions for carbon sequestration. Among such techniques, the use of microalgal species such as Chlorella sp., Dunaliella tertiolecta, Spirulina platensis, Desmodesmus sp., and Nannochloropsis sp., among others have shown high carbon tolerance capacity (10-100%) for establishing carbon capture, utilization and storage systems. To make microalgal-based carbon capture more economical, the microalgal biomass (∼2 g/L) can be converted biofuels, pharmaceuticals and nutraceuticals through biorefinery approach with product yield in the range of 60-99.5%. Further, CRISPR-Cas9 has enabled the knockout of specific genes in microalgal species that can be used to generate low pH tolerant strains with high lipid production. Inspite of the emerging developments in pollution control by microalgae, only limited investigations are available on its economic aspects which indicate a production cost of ∼$ 0.5-15/kg microalgal biomass. This review intends to summarize the advancements in different carbon sequestration techniques while highlighting their mechanisms and major research areas that need attention for economical microalgae-based carbon sequestration.

Randomized p$p$ -values for multiple testing and their application in replicability analysis.

We are concerned with testing replicability hypotheses for many endpoints simultaneously. This constitutes a multiple test problem with composite null hypotheses. Traditional p$p$ -values, which are computed under least favorable parameter configurations (LFCs), are over-conservative in the case of composite null hypotheses. As demonstrated in prior work, this poses severe challenges in the multiple testing context, especially when one goal of the statistical analysis is to estimate the proportion π0$\pi _0$ of true null hypotheses. Randomized p$p$ -values have been proposed to remedy this issue. In the present work, we discuss the application of randomized p$p$ -values in replicability analysis. In particular, we introduce a general class of statistical models for which valid, randomized p$p$ -values can be calculated easily. By means of computer simulations, we demonstrate that their usage typically leads to a much more accurate estimation of π0$\pi _0$ than the LFC-based approach. Finally, we apply our proposed methodology to a real data example from genomics.