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The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As the non‐parametric model, additive hazards offer a very flexible way of modeling time‐varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper, we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non‐negative for all covariate values in their domain. We show that the maximum likelihood estimator may be obtained by separately maximizing the log‐likelihood contribution of each event time point, and we show that the maximizing problem is equivalent to fitting a series of Poisson regression models with an identity link under non‐negativity constraints. We derive an analytic solution to the maximum likelihood estimator. We contrast the maximum likelihood estimator with the ordinary least‐squares estimator in a simulation study and show that the maximum likelihood estimator has smaller mean squared error than the ordinary least‐squares estimator. An illustration with data on patients with carcinoma of the oropharynx is provided. [ABSTRACT FROM AUTHOR]

The elongation factor suppressor of Ty 5 homolog (Spt5) is a regulator of transcription and histone methylation. In humans, phosphorylation of SPT5 by P-TEFb, a protein kinase composed of Cyclin-dependent kinase 9 (CDK9) and cyclin T, interacts with the RNA polymerase II-associated factor1 (PAF1) complex. However, the mechanism of SPT5 phosphorylation is not well understood in plants. Here, we examine the function of SPT5 in Arabidopsis thaliana and find that spt5 mutant flowers early under long-day and short-day conditions. SPT5 interacts with the CDK-activating kinase 4 (CAK4; CDKD;2) and is specifically phosphorylated by CDKD;2 at threonines. The phosphorylated SPT5 binds VERNALIZATION INDEPENDENCE5 (VIP5), a subunit of the PAF1 complex. Genetic analysis showed that VIP5 acts downstream of SPT5 and CDKD;2. Loss of SPT5 or CDKD;2 function results in early flowering because of decreased amounts of FLOWERING LOCUS C (FLC) transcript. Importantly, CDKD;2 and SPT5 are required for the deposition of VIP5 and the enhancement of trimethylation of histone 3 lysine 4 in the chromatin of the FLC locus. Together, our results provide insight into the mechanism by which the Arabidopsis elongation factor SPT5 recruits the PAF1 complex via the posttranslational modification of proteins and suggest that the phosphorylation of SPT5 by CDKD;2 enables it to recruit VIP5 to regulate chromatin and transcription in Arabidopsis. [ABSTRACT FROM AUTHOR]