For some, dreams come true whilst for some, they don't. Take for example, my own self. As a child, I watched a procession of cinema actresses entering our house and leaving them. It was the sixties when my Aunt Ferdausi Begum was lending her voice to the films and all the ladies in the lead roles Shabnam, Kabori, Rosy, Sultana Zaman and Shabana (to name a few) frequented our house. I always thought that someday, I shall be a film star, but it didn't happen. However, I have one close friend for whom, his childhood dreams came true. This friend is Khozaima Ziauddin!
Khozaima was born in Dhaka, went to school and college in Dhaka and studied undergraduate in Shimer College, Chicago, USA (1983). As per his family tradition, he came back from USA and joined the family business. Ever since being in Shimer College, with concentrations in 19th century literature from Oxford. He wanted to pursue his Masters in English and maybe aim for PhD. When three of us flocked in a group, both of us, Refaat (now Justice Dr. Syed Refaat Ahmed) and I, went for higher studies aiming to be academics, Khozaima wanted to do the same. He always told us, "One day I shall wind up all my business and just go for my Masters and become a teacher like you." It took him many years to arrive at this point, but he did so!
After his graduation, Khozaima got married and had two lovely daughters, both grownups now. He looked after his parents, his siblings and finally when the time was right, he sold his business! He actually sold the business that he had inherited and expanded, downsized everything and got ready for being a student. Unbelievably and true to his word, Khozaima went back to university. He completed with flying colours, earning a Chancellor's Gold Medal in 2012.
Prof Imran Rahman (then VC of ULAB), who has the discerning eyes of the expert recruiter recruited him for his University of Liberal Arts, Bangladesh (ULAB). Khozaima Ziauddin teaches English there; his sections are always filled up on the day that they are announced. He is one of the most popular teachers. Other organizations hire him for a day or two to train the employees of various organizations (spoken English, conversational English). Khozaima loves his students and even looks after those who are in financial need. He knows his friends give zakat and from those funds, he helps some of the needy students at ULAB. In the middle of our friends', chit chat he gets calls from distressed students, whose mother is ill, or needs a ride or reference to the hospital or some other assistance, truly in line with the mentors that teachers are meant to be. It is so rare these days. It is such a feeling of fulfillment to watch him in this role, the students adore him and vice versa.
His determination and resolve to make his dream come true has been a source of inspiration to many. What I find most interesting and noteworthy is his resolve to become a teacher and give back to society. Many of us aspire to do so, how many of us can be over the moon?
Nashid Kamal is a Professor of Medical Demography, Nazrul exponent and translator.


text:
NBA Free Agency
Metta World Peace is a person.
That's all we know.
By Seth Jul 13, 2013, 9:07am EDT
Share All sharing options for: Metta World Peace is a person.
Doug Pensinger
Metta World Peace is a person who used to be called Ron Artest. The Lakers officially waived World Peace at 5 PM EST on Friday, July 12. He has 48 hours from that point to clear waivers. If he does clear waivers, he will be an unrestricted free agent and a potential target of the Knicks.
The above is what we know. Pretty much everything else is a convoluted web of rumors, which might be exactly how World Peace wants it. Observe:
When asked for his plans Friday, World Peace told reporters the following:
"I don't really want to play for anybody," World Peace said Friday at the Lakers' practice facility. "I don't want to go anywhere. I want to go to China, or coach or play arena football."
As for coming home, the Queensbridge product said: " I wanted to play in New York when I was in my prime and I was a young, fierce, lock-down [defender]. Madison Square Garden, that would have been sick. But right now, China is way more adventurous for me.''
Such intentions would, ya know, hurt the Knicks' chances of employing World Peace next season, since they are neither an arena football team nor a Chinese team. Then again, World Peace could be bluffing in an attempt to discourage teams under the cap from claiming him off waivers. His very own agent semi-acknowledged that as a possibility:
When told World Peace sounded serious, Davis told The Post, "Serious with Metta? You never know.''
So maybe World Peace does want to stay in the NBA, but hopes to pick his own team. In that case, what about the Knicks? Metta's dad likes New York's chances:
"I feel it's a big, big possibility,'' Artest Sr. told The Post Thursday night of his son joining the Knicks. "If the Knicks go after Ron, I can't see him turning them down.''
But Sam Amick's source suggests the Clippers-- who are probably the better team and wouldn't require relocation-- are the likeliest landing spot atop a long list of possibilities.
All that said, do the Knicks even want World Peace? Frank Isola hears that might not be the case, at least not solidly:
There are conflicting stories concerning the Knicks' interest in the 33-year-old forward from Queensbridge. A person familiar with the club's thinking says that the front office is more focused on adding another big man and a point guard. There were internal discussions regarding World Peace last week when rumors surfaced that he might become an amnesty casualty, but the Knicks' brain trust did not reach a conclusion about whether to pursue him.
So...yeah, I'm just gonna wait until Sunday at 5 PM and see what happens.
Knicks Free Agency 2013
And Justin Brownlee makes 20
Glen Grunwald and Andrea Bargnani spoke today
Game Thread: Knicks vs. Pelicans- 01/20/22 Win the game.
Game Preview: Knicks vs. Pelicans- 01/20/22 Seeking to snap the skid before the competition gets tougher.

redpajama_set_name: RedPajamaCommonCrawl


text:
<!DOCTYPE html>

    <meta http-equiv="content-type" content="text/html; charset=UTF-8">
	<meta charset="utf-8">
	<meta name="viewport" content="width=device-width, initial-scale=1">
	<title>Docs jQuery Mobile - Transitions</title>
	<link rel="stylesheet"  href="../../css/themes/default/jquery.mobile.css" />
	<link rel="stylesheet" href="../_assets/css/jqm-docs.css"/>

Transitions

		Accueil Recherche

Transitions de page

Le framework jQuery Mobile comprend un ensemble d'effets de transition à base de CSS qui peuvent être appliqués à n'importe quel lien de page ou de soumission de formulaire avec la navigation ajax :

fade

				dialog
				page

pop

				dialog
				page

flip

				dialog
				page

turn

				dialog
				page

flow

				dialog
				page

slidefade

				dialog
				page

slide

				dialog
				page

slideup

				dialog
				page

slidedown

				dialog
				page

none

				dialog
				page

Important : Certaines plateformes ont actuellement des problèmes avec les transitions. Nous travaillons sur une solution pour résoudre le problème pour tout le monde. Si vous rencontrez des scintillements et des flash pendant ou à la fin de la transition, nous suggérons la solution suivante. Notez que cette solution de contournement doit être minutieusement testé sur la plateforme cible avant le déploiement. Cette solution de contournement est connu pour causer des problèmes de performances et des plantages du navigateur sur certaines plateformes en particulier Android. Ajoutez le code suivant à votre CSS personnalisé.

                .ui-page { -webkit-backface-visibility: hidden; }

                Cette page est implémentée avec le contournement. Testez le soigneusement.

Seules les transitions fade sont visibles ? : Pour apercevoir tous les types de transition, vous devez être sur un navigateur qui supporte les transformations 3D. Par défaut, les dispositifs qui ne supportent pas la 3D (Comme Android 2.x), se replient sur "fade" pour tous les types de transition. Le comportement est configurable (voir ci-dessous).

Les transitions ont été initialement inspiré par jQTouch. Elles ont depuis été reconstruites, mais les accessoires de David  Kaneda et Jonathan Stark sont l'orientation initiale.

Définition d'une transition sur un lien ou à la soumission d'un formulaire

Par défaut, le framework applique la transition fade. Pour définir un effet de transition personnalisé, ajouter l'attribut data-transition au lien.

<a href="index.html" data-transition="pop">Je ferais Pop</a>

Lorsque le bouton Retour est pressé, le framework appliquera automatiquement la version inversée de la transition qui a été utilisée pour afficher la page. Pour spécifier que la version inversée de la transition doit être utilisée, ajouter l'attribut data-direction="reverse" au lien.

Configuration globale des transitions

Réglez l'option globale defaultPageTransition si vous préférez une transition par défaut différente. Les boîtes de dialogue ont une autre option appelée defaultDialogTransition qui peut également être configurée.

Navigateurs pris en charge et performance

Toutes les transitions sont construites avec des animations keyframe du CSS, qui comprennent -webkit, les règles fournisseur préfixés pour iOS, BlackBerry, Android et les navigateurs Safari et Chrome et -moz , les règles pour les navigateurs Firefox, et les règles préfixés pour Windows Phone 8 et IE10. Le support, pour les animations avec des keyframes et pour la fluidité de transition, est déterminé par la version du navigateur et du matériel. Si les animations ne sont pas supportées, le système se repliera sans risque vers une solution n'utilisant pas les transitions. Pour éviter de façon pro-active que la transition se retrouve dans des situations où les performances médiocres, nous excluons les navigateurs qui n'ont pas les transformations 3D, nous fournissons une transition de repli et appliquons une largeur maximum lorsque les transitions sont appliquées.

Définition de transitions de repli lorsque les transformations 3D ne sont pas supportées

Par défaut, toutes les transitions sauf fade ont besoin de la transformation 3D. Les appareils qui ne supportent pas la 3D, utiliseront la transition fade, indépendamment de la transition spécifiée. Nous faisons cela de manière pro-active sur les transitions avancées pour exclure les plateformes peu performantes comme Android 2.x et s'assurer ainsi une utilisation toujours fluide. Notez qu'il existe des plateformes telles que Android 3.0 qui supportent techniquement les transformations 3D, mais qui ont toujours une performance d'animation faible donc cela ne garantit pas que chaque navigateur sera à 100% sans scintillement, mais nous essayons de régler ce problème .

La transition de repli pour les navigateurs qui ne prennent pas en charge les transformations 3D peut être configurée pour chaque type de transition, mais par défaut, nous spécifions "fade" comme transition de repli. Par exemple, cela définira la transition de repli "none" pour la transition slideout :

		$.mobile.transitionFallbacks.slideout = "none"

Définition de largeur maximum du scroll pour les transitions

Par défaut, les transitions sont désactivées (définies à "none") lorsque vous venez DE ou allez VERS une page où la position du scroll est 3 fois plus haute que l'écran de l'appareil. Cette fonctionnalité a été ajoutée à cause de mauvais temps de réponse et la possibilité de planter le navigateur en cliquant sur un élément de la liste (ou tout autre élément de navigation) se trouvant loin dans une longue page, ce qui mène le navigateur à tenter d'animer massivement une grande page vers la position du scroll en haut de l'écran. C'est la position du scroll, et non celle de la hauteur totale de l'écran, qui est le facteur déterminant pour la performance. Ce point d'arrêt de la position du scroll est configurable via la nouvelle fonction getMaxScrollForTransition.

Définition de largeur maximum pour les transitions

Par défaut, les transitions peuvent être désactivées (définies à "none") lorsque la largeur de la fenêtre est plus grande qu'une certaine largeur de pixel. Cette fonctionnalité est utile car les transitions peuvent être gênantes ou peu performantes sur les grands écrans. Cette valeur est configurable via l'option globale $.mobile.maxTransitionWidth, qui par défaut est à false. L'option accepte n'importe quel nombre représentant une largeur en pixel ou la valeur false. Si elle n'est pas à false, le gestionnaire va utiliser une transition "none" lorsque la fenêtre est plus grande que la valeur spécifiée

Utilisation de la même transition de page

Les transitions vers la page active en cours sont ignorées par défaut, mais cela peut être activé en utilisant l'option  allowSamePageTransition de la méthode $.mobile.changePage. Notez que toutes les transitions fonctionneront comme prévu et cela peut aboutir à un résultat peu pratique.

Création de transitions personnalisées

jQuery Mobile permet l'ajout de  transitions personnalisées au dictionnaire $.mobile.transitionHandlers afin que vous puissiez étendre la sélection des transitions sur votre site ou application.

Aussi dans cette rubrique

Pages & Boîtes de dialogue

Anatomie de la page

Modèle de page unique

Modèle Multi-page

Titres des pages

Liaison de pages

Transitions de page

Widget de chargement de page

Boîtes de dialogue

Popups

Le préchargement & la mise en cache des pages

Ajax, hashes & historique

Injection dynamique des pages

Écriture des scripts des pages

Applications Backbone.js et Require.js

Application PhoneGap

Fonctionnalité de touchOverflow

Thématisation des pages

Copyright 2013 The jQuery Foundation

redpajama_set_name: RedPajamaGithub


text:
Q: Check which commit is ahead the origin/master On my master branch, when I run git status,  I got :
# Your branch is ahead of 'origin/master' by 1 commit.
#   (use "git push" to publish your local commits)
#

I would like to know what exactly is that commit which is ahead the remote master branch. How to achieve this?
I tried git log command, compared my local commitment history with the remote repo commitment history, they look exactly the same....

A: You can use the double dot annotation to find all commits which are reachable from one reference but not the other.
In your case you could use the following command:
git log origin/master..master

This effectively tells git to print all commits that are reachable from master but not from origin/master, so all commits which weren't pushed to the remote.

You can read more about the double dot annotation in the Revision Selection chapter of the gitpro book.
Or in the official documentation about git revisions.

redpajama_set_name: RedPajamaStackExchange


Temporomandibular disorder, or TMD, is often called "TMJ" by doctors, patients and insurance companies. However, the term TMJ actually refers only to the jaw joints themselves. TMD describes a group of diseases that can affect the jaw joints, the muscles that control jaw movement and/or the dental occlusion. TMDs are physical disorders that result from an imbalance in the delicate working relationship of the jaw and skull with the muscles that move the jaw, as well as the nervous system associated with these systems. This imbalance results in muscle fatigue, spasm and/or joint dysfunction, and even changes in the teeth, which in turn cause a variety of symptoms that are unique to each person.
The TM joint system is unique in many ways. The left and right joints must coordinate, working at the same time for the jaw to move. While the opening,lateral and forward movements of the jaw are controlled by the shape of the bones and are a function of muscles and ligaments, the closing end-point of the jaw movement is controlled by the coming together of teeth – the bite or occlusion. No other joint in the body has such a rigid end-point limit.
The proper, healthy function of the TMJ system requires normal structure and function of all the component parts, including muscles, nervous system, ligaments, joints (bones, discs and connecting tissues) and the dental occlusion.
Temporomandibular disorder, or TMD, is often called "TMJ" by doctors, patients and even insurance companies, although the term TMJ actually refers only to the jaw joints themselves. TMD describes a group of diseases that can involve the jaw joints, the muscles that control jaw movement and the dental occlusion. TMDs are physical disorders arising from an imbalance in the delicate working relationship of the jaw and skull with the muscles that move the jaw, as will as the nervous system associated with these systems. This imbalance results in muscle fatigue, spasm and/or joint dysfunction, and even changes in the teeth, which in turn cause a variety of symptoms, unique for each person.
The symptoms of TMD, which affect millions of adults and children, can gradually appear with no apparent, specific cause. They can also occur after severe trauma, such as a traffic accident or a blow to the facial area. Even subtle repeated traumas, such as excessive teeth clenching, gum chewing or nail biting can cause TMD. These conditions can also be caused by a bad bite (malocclusion), which is characterized by unhealthy muscle and joint function. Systemic diseases and developmental abnormalities can also cause TMD.
Stress and tension may awaken a quiet, asymptomatic TMD or aggravate an existing temporomandibular condition. However, stress alone does not cause TMD if a patient has healthy dental occlusion and muscle and jaw function.
All too often patients are classified as "chronic pain patients", as a result of their initial complaints being wrongly identified as stress or psychological disorder. These patients may be relegated to treatment focusing only on medication and counseling, with no treatment of the physical cause of their illness. A thorough analysis of each patient, as well as early and appropriate treatment of TMD may avoid the progression to a chronic pain state.


Created by the designer Francisco Providência, the museography is based on thematic rooms focusing on such subjects as: pre-monetary currency; money throughout the world and its history over the centuries; banknote and coin production; and also personal testimonies about the role of money in the life of ordinary citizens.
This museum is aware of its social role and also of the symbolic presence that it enjoys in the city as well as in the country; it is a place where contact with diverse audiences takes priority, and one that invests in educational and cultural programming, preserving strong ties to both the community and the city of Lisbon.
The Money Museum guarantees accessibility and satisfaction to its visitors at the social, physical, and cognitive levels. It is projected as a place of sociability that provides a variety of services and communication interfaces with the public: cafeteria, museum shop, conference room, library, welcome areas, workshop rooms.
As it believes in the value of cooperation, the Money Museum aims to establish partnerships with other institutions and socio-cultural spaces in Lisbon. These partnerships will allow it to promote its activities; exchange ideas, collections, and projects; consolidate contact networks; or even facilitate the flow of audiences and of its collection. The underlying aim is to enrich the museum, secure visitor loyalty and, naturally, the participation of the museum itself in the society that not only hosts but also transforms it.


Field Dependence/Independence Cognitive Styles: Are They Significant At Different Levels Of Vocabulary Knowledge?
Cognitive styles influence the performance of language learners and can predict their success in the process of language learning. Considering field dependence/independence cognitive styles, this study aims at determining if they are significant in English vocabulary knowledge. A number of EFL university students took part in the study. The investigation was done through using Vocabulary Size Test (VST) (Nation, 2007) and the Group Embedded Figures Test (GEFT) (Witkin, Oltman, Raskin, and Karp, 1971). Using the Vocabulary Size Test (VST), the participants were divided into three groups of high, mid, and low. Moreover, with respect to the Group Embedded Figures Test (GEFT), they were divided into two groups, field dependents and field independents. Mean score comparison revealed there was a credible and meaningful relationship between field dependence/independence cognitive styles and total vocabulary knowledge. It was also indicated there was a significant relationship between field dependence/independence cognitive styles and vocabulary knowledge in the high and mid groups. Finally, based on the findings, teachers should take learners' individual differences into consideration so that they could adopt and apply teaching methods in line with the learners' various cognitive styles.


text:
Jibou (; veraltet Jibău oder Șibău; , ) ist eine Kleinstadt im Kreis Sălaj im Kreischgebiet in Rumänien.

Geographische Lage
Jibou liegt am Fluss Someș, etwa 25 Kilometer nordöstlich der Kreishauptstadt Zalău, mit der sie über den drum național 1H und der Kreisstraße (drum județean) DJ 108A verbunden ist.

Persönlichkeiten

 Nikolaus Wesselényi (1796–1850), Großgrundbesitzer, Politiker

Städtepartnerschaften
Die Partnerstadt von Jibou ist seit August 1992 Glauchau in Sachsen.

Siehe auch
 Liste deutscher und ungarischer Bezeichnungen rumänischer Orte

Weblinks

Einzelnachweise

redpajama_set_name: RedPajamaWikipedia


text:
videos News>
agriculture News>
Crop insurance scheme designed to profit insurance firms, says P Chidambaram
videos | November 14, 2018, 10:32 PM IST
By Archana Shukla | Nov 15, 2018, 08:23 AM IST (Updated)
Former finance minister and senior Congress leader P Chidambaram attacked the government over its crop insurance scheme on Wednesday, alleging that it was designed to profit insurance companies and 84 lakh farmers, including those from Rajasthan and Madhya Pradesh, had withdrawn from it.
Taking to Twitter, Chidambaram claimed that insurance companies had made profits of Rs 6,460 crore in 2016-17 and were set to make profits to the tune of Rs 8,000-9,000 crore after "robbing the farmers".
"As we had cautioned, PM's Fasal Bima Yojana (crop insurance scheme) was designed to profit the insurance companies at the cost of the farmers.
CNBC-TV18's Archana Shukla caught up with PP Kapoor an RTI Activist to understand the government's crop insurance scheme.
With inputs from PTI.
First Published: Nov 14, 2018, 10:32 PM IST
crop insurance scheme
Fasal Bima Yojana


Want take a look new pictures naked celebrity? means you in the right place, we often publish selfie celebrity and their Fappening, as well as the bra in iCloud. On the site you see full collection video fusion on the internet. We occasionally improve update our content.


text:
Gonzalez, Marlins reach deal
Free agent outfielder Luis Gonzalez has reached a preliminary agreement on a $2 million, one-year contract. He must pass a physical. The contract for the 40-year-old Gonzalez includes $1 million in performance bonuses, according to www.foxsports.com.
Johnny Estrada agreed to a $1.25 million, one-year contract, giving Washington an experienced catcher to back up Paul Lo Duca, who is recovering from knee surgery. Estrada, a switch-hitter, played for Milwaukee in 2007, batting .278 with 10 homers and 54 RBI.
Free-agent third baseman Pedro Feliz finalized an $8.5 million, two-year contract. With bonuses, the contract could be worth up to $15 million over three years. Feliz hit .253 with 20 homers and 72 RBI for San Francisco last season.
Right-hander Matt Belisle agreed to a one-year, $1.25 million contract. Belisle went 8-9 with a 5.32 ERA in 30 starts last season.
Brad Wilkerson agreed to a one-year contract, a move that could signal Seattle is close to acquiring pitcher Erik Bedard from the Orioles. The Mariners have discussed trading outfielder Adam Jones to Baltimore along with others for Bedard.
Morgan Ensberg agreed Thursday to a minor league contract with the New York Yankees. Ensberg would get a $1.75 million, one-year contract if added to the 40-man roster.


text:
\section{Introduction}

The fundamental concept in survival analysis is the hazard rate. There are several regression models describing the hazard rate, such as multiplicative risk models and additive risk models. The dominant hazard model in survival analysis is the multiplicative proportional hazards model~\citep{cox1972regression}. In many cases, however, the proportional hazards assumption is not met. In such cases, developing models that adequately describe the non-proportional effect of the covariate(s) is not straightforward \citep{gore1984regression, schemper1992cox, perperoglou2006reduced, vHP12} and different models should be considered.

In this article, we focus on the non-parametric additive hazards model proposed by~\cite{Aalen1980, Aalen1989}, extensively studied in~\citet{martinussen2007dynamic}. The additive model defines the hazard rate as a linear form of the vector of covariates. Unconventionally, it does not naturally force the hazard rate to be positive. However, it has several useful properties, as set out by~\citet{aalen2008survival}. Firstly, the additive hazards model is internally consistent: it retains its additive structure if covariates are measured with uncertainty or are dropped from the linear expression. This contrasts with the proportional hazards model, which loses its proportional hazards property when covariates are omitted~\citep{strutherskalbfleisch1986, schumacher1987, bretagnolle1988}. This drawback of proportional hazards has triggered a recent debate about the danger of using the hazard ratio as a causal effect measure in survival analysis~\citep{hernan2010hazards, aalen2015does}. A second advantage of additive hazards is that it allows implemention of dynamic structures~\citep{martinussen2000nonparametric}, such as self-exciting processes, which are impossible to incorporate in most other nonlinear regression models.

The usual way to estimate the parameters in the additive hazards model is by Aalen's method, which uses ordinary least squares to estimate the cumulative effect of the covariates. Aalen's ordinary least squares method is straightforward to implement, but it has some disadvantages. The most important disadvantage is that it does not guarantee that the model-based hazards are positive for all time points for all subjects in the data, but only for the subjects that experience an event, and only at their event times. Furthermore, the ordinary least squares estimator is defined as a method-of-moments estimator and does not have the support of likelihood theory.

One method that generally yields efficient estimators is maximum likelihood. To the best of our knowledge, the maximum likelihood method has not been considered in any detail in the context of the additive hazards model, since no analytical expression for the maximum likelihood estimator was available. The objective of this paper is to derive the maximum likelihood estimator for the additive hazards model and to determine its advantages and limitations and in comparison to Aalen's ordinary least squares method. We find that there is no maximum solution to the survival likelihood in the unconstrained parameter space. However, we may find the solution if we choose a proper domain. A natural choice for such a domain is one that guarantees positivity of the hazard rate at all time points for all subjects in the data.

We show that the maximum likelihood estimator of the cumulative baseline hazard and cumulative covariate effects is a step function changing values only at the event time points, and that the maximum of the log-likelihood can be found by separately maximising the log-likelihood contributions corresponding to each of the event time points, as with Aalen's ordinary least squares solution. We define the constraint domain, and show that the maximisation problem is equivalent to fitting a series of identity-link Poisson regression models under non-negativity constraints. This problem has been studied before by~\citet{Marschner2010} and \citet{MARSCHNER2012}, who proposed an EM algorithm to obtain maximum likelihood estimators. In contrast, our solution is analytical, and computation time at each time point is linear in the sample size and in the number of covariates per time point. We also discuss the new method's connections to the ordinary least squares estimator. In particular, when only a single binary covariate is considered, we will show that maximum likelihood and ordinary least squares are equivalent. In Section~\ref{sec:simulation} we report on the results of a simulation study comparing maximum likelihood with ordinary least squares. Section~\ref{sec:application} illustrates our methods on data from a randomised clinical trial on patients with carcinoma of the oropharynx.

\section{Notation and model definition}
\label{sub:notation}

We use bold letters to indicate vectors and matrices. Define $T^*$ to be the time to event, $C$ to be the time to censoring, and let $T=\min(T^*, C)$, and $\Delta = I(T^* \leq C)$ the event indicator. We observe $(t_i, \delta_i, \mathbf{x}_i^*)$, $i=1,\ldots,n$, with $\mathbf{x}_i^* = (x_{i1},\ldots,x_{ip})^\top$ a $p$-vector of covariates. We assume that the covariates $x_{ij}$ are restricted to the interval $[0, 1]$. This looks like a severe restriction, but it is not, because for any covariate with a minimum of $a$ and a maximum of $b$ in the observed data, we can rescale the covariate to be $(x-a)/(b-a)$, which takes values in $[0, 1]$. Extend $\mathbf{x}_i^*$ with the constant $x_{i0} = 1$, obtaining $\mathbf{x}_i = (1, x_{i1},\ldots,x_{ip})^\top$. Define $\mathcal{E}$ to be the set of event time points $t_i$, with corresponding $\delta_i=1$ and let $|\mathcal E| = K $. We assume there are no ties, and define the ordered sequence of event time points $t_1^* < \ldots < t_K^*$. Finally, define $Y_i(t) =\sum_{i=1}^n I(t_i\geq t)$
as the at risk indicator of subject $i$, taking the value $1$ if subject $i$ is at risk for the event of interest just before time $t$ and the value $0$ otherwise.

The additive hazards model~\citep{Aalen1980, Aalen1989} assumes the hazard rate to be of the form
\[
    h(t \,|\, \mathbf{x}_i^*) = \mathbf x^\top_i \bm \beta(t) =  \beta_0(t) + \beta_1(t)x_{i1} + \cdots + \beta_p(t)x_{ip}.
\]
The parameters $\beta_j(t)$ allow effects of the covariates to change over time, thus,  the additive  model is fully non-parametric.

\section{Aalen's method}
\label{sub:OLS}

To estimate the parameters $\bm{\beta}(t)$, the most commonly used approach is Aalen's ordinary least squares method. Let us define the counting process $N_i(t)$ as the number of events experienced by subject $i$ before or at time $t$. Then the intensity $\lambda_i(t)$ of the counting process has the form
\[
        \lambda_i(t) =  Y_i(t)(\beta_0(t) + \beta_1(t)x_{i1} + \cdots + \beta_p(t)x_{ip})
            = Y_i(t) h(t_i \,|\, \mathbf{x}_i^*).
\]
The formula $dN_i(t) = \lambda_i(t)dt + dM_i(t)$ gives equations
\[
    d\mathbf{N}(t) = \mathbf{X}(t)d\mathbf{B}(t) + d\mathbf{M}(t),
\]
where $\mathbf{N}(t) = (N_1(t),N_2(t),\ldots,N_n(t))^\top$ is the vector of counting processes, $\mathbf{X}(t)$ is the matrix of covariates multiplied with $Y_i(t)$ at each $i$th row, $\mathbf{B}(t) = (B_0(t),B_1(t),\ldots,B_p(t))^\top$ is the cumulative beta defined as $B_j(t) = \int_0^t \beta_j(s)ds$ and $\mathbf{M}(t) = (M_1(t), \ldots, M_n(t))^\top$, where $M_i(t)$ is the $i$th martingale error term. Ordinary least square regression thus gives Aalen's estimator of the $\bm{\beta}(t)$ if $X(t)$ is of full rank:
\[
    d\widehat{\mathbf{B}}(t) = (\mathbf{X}(t)^\top\mathbf{X}(t))^{-1} \mathbf{X}(t)^\top d\mathbf{N}(t).
\]
When $X(t)$ is not of full rank, we set $d\widehat{\mathbf{B}}(t) = 0$~\citep{aalen2008survival}.

\section{Maximum likelihood estimation}
\label{sub:ML}

In this section, we will derive an analytic solution for the maximum likelihood estimator under the natural constraint that the hazard is non-negative at each time point for all covariate values within the domain.

\subsection{The shape of the likelihood}

To begin with, let us consider the likelihood function for the additive hazards model. The log likelihood contribution of subject $i$ with an event or censored at time $t_i$ is given by
\[
    \ell_i = \delta_i \log\{h(t_i \,|\, \mathbf{x}_i^*)\}+ \log\{ S(t_i \,|\, \mathbf{x}_i^*)\},
\]
where $S(t \,|\, \mathbf{x}_i^*) = P(T^* > t \,|\, \mathbf{x}_i^*) = \exp(- H(t \,|\, \mathbf{x}_i^*))$ is the survival probability and $H(t \,|\, \mathbf{x}_i^*)$ the cumulative hazard of subject $i$. The log-likelihood is therefore given by~\citep{aalen2008survival}

\begin{equation}
    \label{eq:logliki}
     \ell = \sum_{i=1}^n \{\delta_i \log \mathbf \mathbf{x}_i^\top \bm{\beta}(t_i) - \mathbf{x}_i^\top \mathbf B(t_i)\}.
\end{equation}

Our first result is
\begin{proposition}
The likelihood function $\ell$ is unbounded.
\end{proposition}

\noindent%
We therefore introduce the natural constraint that the hazard of each possible covariate value of our data is non-negative at each time point, that is, we require that for all $t \geq 0$,
\begin{equation}
    \label{eq:hpos}
 h(t \,|\, \mathbf{x}^*) \geq 0,\ \textrm{for}\ \mathbf{x}^* \in \{0, 1\}^p,
\end{equation}
which implies positivity of the hazard for $\mathbf{x}^* \in [0, 1]^p$. Our objective is to maximise the total log-likelihood given in~\eqref{eq:logliki}, subject to the constraint~\eqref{eq:hpos}.

Let us assume $\hat{\mathbf{B}}(t)$ is the function which maximises the likelihood function~\eqref{eq:logliki}, subject to the constraint~\eqref{eq:hpos}. Since $\hat{\mathbf{B}}(t)$ is an estimator of cumulative beta, which is generally the negative logarithm of the survival function, we may assume that it is right continuous with left limits (cadlag). Thus, we may decompose $\hat{\mathbf{B}}(t)$ as the summation of a continuous function $\hat{\mathbf{B}}_c(t)$ and a step function $\hat{\mathbf{B}}_s(t)$.

\begin{lemma}
\label{prop:beta0}
Under the constraint $h(t \,|\, \mathbf{x}^*) \geq 0$ for $\mathbf{x}^* \in \{0, 1\}^p$, the log-likelihood can only achieve a maximum if $\hat{\mathbf{B}}_c(t) \equiv 0$ and $\hat{\mathbf{B}}_s(t)$ is a step function with jumps only at the event time points in $\mathcal{E}$.
\end{lemma}

\noindent%
Lemma~\ref{prop:beta0} implies that the maximisation problem is the same as maximising the total log-likelihood with respect to the jumps of $\mathbf{B}(t)$ at the event time points. Denoting the jump of $B_j(t)$ at the $k^{\textrm{th}}$ event time point $t_k^*$ as $\beta_{kj}$, $k=1,\ldots,K$, $j=0,\ldots,p$, this leads to $B_j(t) = \sum_{k: t_k^* \leq t} \beta_{kj}$.

The total log likelihood $\ell$ from~\eqref{eq:logliki} can now be rewritten in terms of the $\beta_{kj}$ as
\[
    \ell = \sum_{i=1}^n \Biggl\{ \delta_i \log \Bigl(\sum_{j=0}^p x_{ij} \beta_{k(i),j}\Bigr) - \sum_{j=0}^p x_{ij} \sum_{k: t_k^* \leq t_i} \beta_{kj} \Biggr\},
\]
where $k(i)$ is the index of the event time points $t_k^*$ corresponding to $t_i$, i.e.~$t_{k(i)}^* = t_i$ (in case $\delta_i=0$, $k(i)$ can be chosen as an arbitrary index in $\{1,\ldots,n\}$). We reorder this sum over subjects as a sum over the distinct event time points $t_k^*$, obtaining
\[
    \ell = \sum_{k=1}^{K} \Biggl\{ \log \Bigl(\sum_{j=0}^p x_{i(k),j} \beta_{kj}\Bigr) - \sum_{j=0}^p \beta_{kj} \sum_{l: t_l \geq t_k^*} x_{lj} \Biggr\} = \sum_{k=1}^{K} \ell_k^*,
\]
with
\begin{equation}\label{eq:ellk}
    \ell_k^* = \log \Bigl(\sum_{j=0}^p x_{i(k),j} \beta_{kj}\Bigr) - \sum_{j=0}^p \beta_{kj} s_{kj},
\end{equation}
$s_{kj} = \sum_{l: t_l \geq t_k^*} x_{lj}$,
and where $i(k)$ is the subject index corresponding to the $k^{\textrm{th}}$ event time point, i.e.~$t_{i(k)} = t_k^*$.

It is easy to see that each term $\ell_k^*$ is a function only of the variables $\beta_{kj}$, $j = 0,\ldots,p$. Since the terms $\ell_k^*$ do not have parameters in common, we may reduce the problem by separately maximising each $\ell_k^*$ as a function of $\bm{\beta}_k = (\beta_{k0}, \beta_{k1}, \ldots, \beta_{kp})$.

For the remainder of this section, we will fix (any) one of the time points $t_k^*$. To simplify notation, we are going to suppress dependence on $k$, and consider maximisation of
\begin{equation}\label{eq:ell}
    \ell^* = \log \Bigl( \sum_{j=0}^p x_j \beta_{j} \Bigr) - \sum_{j=0}^p s_j \beta_j =
    \log \Bigl( \mathbf{x}^\top \bm{\beta} \Bigr) - \mathbf{s}^\top \bm{\beta}
\end{equation}
with respect to $\bm{\beta} = (\beta_0, \beta_1, \ldots, \beta_p)$, where $\mathbf{s} = (s_0, s_1, \ldots, s_p)^\top$, $x_j$ stands for the $j^{\textrm{th}}$ element of the covariate vector of the subject that failed at time $t_k^*$, and $x_0=1$.

\subsection{Maximum likelihood with the full constraint matrix}

\noindent%
From now on, we will give an alternative description of the constraint~(\ref{eq:hpos}) using a matrix form.  Let $\bm{\beta} = (\beta_0, \beta_1, \ldots, \beta_p)^\top$ be the vector of parameters of the function $\ell^*$. We can restate the constraint~(\ref{eq:hpos})   by the matrix $M_\mathbf{D}$ through the inequalities $M_\mathbf{D} \bm{\beta} \geq \bm 0$. We construct the matrix $M_\mathbf{D}$ such that the rows consist of all the $2^p$ possible $(p+1)$-tuples $(m_0, m_1, \ldots, m_p)$ in $\{0,1\}^{p+1}$ with $m_0 =1$.  So $M_\mathbf{D}$ is a $2^p \times (p+1)$ matrix.

 The domain $\mathbf D\subset \mathbb R^{p+1}$ defined by constraint~\eqref{eq:hpos} is a polytope enclosed by the flat boundaries defined by $\gamma_i = M_i^\top \bm{\beta} = 0$, $i=1,\ldots,2^p$, where $M_i$ is the $i$th row of $M_{\mathbf D}$.  Now, we can show that the maximum point lies on the boundary of this polytope.

\begin{lemma} \label{prop:maximumpointReverse}
If the maximum point of $\ell^*$ in the domain $\mathbf D$ exists, then the maximum point  lies in the intersection of $p$ flat boundaries given by $\gamma_{i_j} = M_{i_j}^\top \bm \beta= 0$, $j = 1 \ldots, p$.
\end{lemma}

\noindent%
In Lemma~\ref{prop:maximumpointReverse}, we have given the geometric property of the maximum point of the $\ell^*$. The intersection of $p$ flat boundaries is a one-dimensional line. So, the maximum point always lies on a one-dimensional line edge on the boundary of the domain. The following lemma defines all the one-dimensional line edges on the boundary of the domain. There are $2p$ such line edges.

\begin{lemma}\label{prop:geometryofedge}
The one-dimensional line edges on the boundary of the domain $M_\mathbf{D}\bm \beta \geq 0$ are given by $(0,\ldots,0,l,0,\ldots,0)$, with the position of $l$ ranging from $2$ to $p+1$, and $(l,\ldots,0,-l,0,\ldots,0)$, with the position of $-l$ ranging from $2$ to $p+1$.
\end{lemma}

\noindent%
Let us denote by $\bm e_j$ the $(p+1)$-vector $(0, 0, \ldots, 0, 1, 0, \ldots, 0)^\top$, with the 1 at position $j+1$, and by $\bm f_j$ the $(p+1)$-vector $(1, 0, \ldots, 0, -1, 0, \ldots, 0)^\top$, with the -1 at position $j+1$, and define the sets $\mathcal{A}^+ = \{ \bm e_1, \ldots, \bm e_p \}$ and $\mathcal{A}^- = \{ \bm f_1, \ldots, \bm f_p \}$. We may call the set $\mathcal{A} = \mathcal{A}^+ \cup \mathcal{A}^-$, a finite set of cardinality $2p$, the set of admissible directions. Lemma's~\ref{prop:maximumpointReverse} and~\ref{prop:geometryofedge} imply that the maximum of $\ell^*$ lies on one of the edges ${\bm v} \cdot l$, with ${\bm v} \in \mathcal{A}$. We now come to our main theorem which describes the structure of the maximum likelihood estimator of the additive hazards model under the constraint of non-negativity of the hazards. Let $i(k)$ denote the index of the subject that fails at $t_k^*$, the maximum likelihood estimator $\hat{ \bm \beta}_k$ at the event time point $t_k^*$ can be determined as follows:

\begin{algorithm}[H]
\caption{Algorithm to find the maximum point within the domain $\mathbf D$.\label{alg:s1}}

\begin{itemize}
    \item Calculate $\mathbf s_{k} = \sum_{l: t_l \geq t_k^*} \mathbf x_{l}$;
    \item Calculate the values of ratio  $\bm v_j = x_{i(k), j} / s_{k,j}$ for $1 \leq j \leq p$;
    \item Calculate the values of ratio  $\bm v_{p + j} = (x_{i(k), 0} - x_{i(k), j}) / (s_{k,0} - s_{k,j})$ for $1 \leq j \leq p$;
    \item Find the set $M$ of indices $m = 1, \ldots, 2p$, for which $\bm v_m$ is maximised;
    \item For any index $m \in M$, calculate the value of $\hat{\bm \beta}_{k}^{(m)}$ by
\begin{equation*}\hat{\bm \beta}_{k}^{(m)} =  \begin{cases}
            \bm e_m / s_{k,m}, & \hbox{if $1 \leq m \leq p$;} \\
            \bm f_{m-p} / (s_{k,0} - s_{k,(m-p)}),& \hbox{if $p + 1 \leq m \leq 2p$;}
          \end{cases}
         \end{equation*}
    \item Return $\hat{\bm \beta}_{k}^{(m)}$, $m \in M$.
\end{itemize}

\end{algorithm}

\begin{theorem}\label{thm:maintheorem}
For an additive hazards model $h(t \,|\, \mathbf{x}_i^*) = \bm \beta(t)\mathbf{x}_i$ with constraints  $h(t \,|\, \mathbf{x}^*) \geq 0,\ \textrm{with }\ \mathbf{x}^* \in [0, 1]^p$, assuming no ties,  the values returned by Algorithm \ref{alg:s1} are maximum likelihood estimators.
\end{theorem}

\noindent%
Theorem \ref{thm:maintheorem} offers some insight into the maximum likelihood estimator. There are only two possible patterns of the jump $\bm \beta(t_k^*)$. Either there is an increment of the coefficient $\beta_j(t_k^*)$ for one covariate or there is a decrease of $\beta_j(t_k^*)$ for one covariate which is compensated by an increment of the intercept. So, this estimator is sparse in the change of $\bm \beta(t)$.

Theorem \ref{thm:maintheorem} also implies a condition for the existence and uniqueness of the likelihood estimator. The likelihood estimator exists if and only if $s_{k,m} \neq 0$ and $s_{k,0}- s_{k,(m-p)} \neq 0$. which correspond to the cases all $x_i$'s are zeros or all $x_i$'s are ones, $i \geq i(k)$. Uniqueness is not guaranteed by Theorem \ref{thm:maintheorem}. In fact,  all choices of the ratios $x_{i(k), j} / s_{k,j}$ and/or $(x_{i(k), 0} - x_{i(k), j}) / (s_{k,0} - s_{k,j})$ found by Algorithm \ref{alg:s1} result in the same maximum. In that case any ${\bm v} \in \mathcal{A}$ corresponding to such a maximum is allowed, as well as any convex combination, as stated in the following theorem.

\begin{theorem}\label{thm:mutisolution}
If $\hat{\bm\beta}_{k}^{(m_1)}$ and $\hat{\bm\beta}_{k}^{(m_2)}$ are solutions given by Algorithm \ref{alg:s1} which maximise the likelihood $\ell$, then any convex combination $\lambda \bm \hat{\bm\beta}_{k}^{(m_1)} + (1 - \lambda) \hat{\bm\beta}_{k}^{(m_2)}$, with $0 \leq \lambda \leq 1$, attains the same maximised likelihood.
\end{theorem}

In practice, we suggest an average of all solutions given by Theorem~\ref{thm:maintheorem}. By Theorem~\ref{thm:mutisolution}, this is also a solution which maximises the likelihood.

With our result on the maximum likelihood estimator, we also have the following result which reveals the connection between the maximum likelihood estimator and Aalen's estimator.

\begin{proposition}
For an additive hazards model with one binary covariate, the maximum likelihood estimator and the Aalen estimator coincide.
\end{proposition}

This can also be explained intuitively by Theorem \ref{thm:maintheorem} as follows. For one binary covariate, Aalen's ordinary least squares method can be viewed as selecting one of two competing trends at each event time. Theorem~\ref{thm:maintheorem} shows that the nature of maximum likelihood estimator is similar; the jump of the coefficient takes place for only one covariate, i.e., only one trend is selected at each event time.

\subsection{Example}

We illustrate Theorem~\ref{thm:maintheorem} using a simple example with two covariates. Let us assume that at a certain event time point $t^*$ a subject fails with covariate values $x_1=0$ and $x_2=1$, so we have $\mathbf{x}^* = (0,1)$, and $\mathbf{x} = (1,0,1)$. Suppose that $\mathbf{s} = (8,5,6)^\top$. Then the log-likelihood is given by
\begin{equation}\label{eq:twocovml}
    \ell^* = \log (\beta_0 + \beta_2)- 8 \beta_0 -5\beta_1 -6 \beta_2.
\end{equation}

In this case, the domain is defined by the matrix $\mathbf{M_D}$ through the boundary conditions $M_{\mathbf D} \bm \beta \geq 0$, where
\[ M_{\mathbf D} = \begin{pmatrix}
    1 & 0 & 0 \\
    1 & 0 & 1 \\
    1 & 1 & 0 \\
    1 & 1 & 1 \\
\end{pmatrix}.\]

Theorem~\ref{thm:maintheorem} directs us to compare the values of $x_1/s_1$, $x_2/s_2$, $(x_0-x_1)/(s_0-s_1)$, and $(x_0-x_2)/(s_0-s_2)$, namely $0$, $\frac{1}{6}$, $\frac{1}{3}$, $0$, corresponding to the one-dimensional line edges ${\bm v} \cdot l$, with $\bm v = (0,1,0)^\top, (0,0,1)^\top, (1,-1,0)^\top$ and $(1,0,-1)^\top$, respectively, in the set of admissible directions $\mathcal{A}$. The third of these numbers is largest, so the one-dimensional line ${\bm v} \cdot l$, with $\bm v = (1,-1,0)^\top$ is the edge where the maximum point lies. Maximising over $l$ gives $l = \frac{1}{\mathbf{s}^{\top} \bm{v}} = \frac{1}{3}$, $\hat{\bm \beta}(t) = (\frac{1}{3}, -\frac{1}{3},0)$ and the maximised log-likelihood $\ell^* = \log(\frac{1}{3})- 1$.

\subsection{Maximum likelihood with a partial constraint matrix}
\label{sub:MLEfind}

\noindent%
In previous sections, we show that the maximum point of the likelihood function lies in one of the $2p$ one-dimensional line edges on the boundary. This result is derived for the case where the domain is given by the matrix $M_{\mathbf D}$ whose rows are all the combinations of $1$ and $0$ in the last $p$ columns. Sometimes we need more flexible domains, for example when dummy covariates are introduced or when it is known that some coefficients are positive. In this more general case Theorem \ref{thm:maintheorem} does not apply, and numerical solutions are needed.

We consider a general domain defined by $M_D \bm \beta \geq 0$ for some matrix $M_D$, where the matrix $M_D$ can be an arbitrary matrix of $p+1 $ columns. We assume that the rank of $M_D$ is $p+1$. We assert that Lemma \ref{prop:maximumpointReverse} still applies for this constraint matrix $M_D$. The maximum point lies on a one-dimensional edge on the boundary, defined as the intersection of $p$ flat boundaries. So, any local maximum point of a   one-dimensional edge is a potential solution of the maximum likelihood of the domain $D$.  Given any one-dimensional edge, we have $p$ equations $\gamma_i = M_{i} \bm \beta = 0$. Combined with the first order condition which is also a linear equation, we have $p+1$ linear equations whose solution is the maximum point on this edge. By testing its derivatives and constraint conditions, we can check if it is a maximum point of the domain $D$.  However, it is difficult to find the right edge on which the maximum point lies. We introduce three methods to find the right edge: the na\"{\i}ve method, the ascending method and the descending method.

The na\"{\i}ve method loops through all the possible combinations of $p$ rows of $M_D$. Every combination of $p$ rows defines a one-dimensional edge and gives a possible solution of the maximum point $\bm \beta^*$. We can test its derivatives and constraint conditions to check if it is the maximum point we want to find. This na\"{\i}ve method is easy to understand, but not efficient, since the complexity grows approximately with the $p$th power of number of the rows of $M_D$.

sThe ascending method searches for a series of $\bm \beta^*$ with increasing likelihood. More specifically, the algorithm  starts from a $p$-combination of rows, whose maximum point is $\bm \beta^*_0$. We assume this edge is inside the domain. If  $\bm \beta^*_0$ is not the maximum point of the domain $D$, its gradient points to the inside of the domain. Following this direction, we can find an adjacent one-dimensional edge whose maximum point $\bm \beta^*_1$ has a larger likelihood.  Since this algorithm starts from an edge already on the domain, as a result, it is quite efficient when a good starting edge of the domain is known. To find a new edge, we use the gradient of the log likelihood, constrained to the adjacent hyperplanes. This method works well if the geometry of the domain is not too complicated.

The descending method is the reverse of the ascending method. Assuming that we find a maximum point $\bm \beta^*$ on a one-dimensional edge, it is not necessary that this $\bm \beta^*$ satisfies all the constraint conditions $M_D \bm \beta^* \geq 0$. However, it may satisfy some of the constraint conditions, i.e., we have $\gamma_i = M_i \bm \beta^* \geq 0$ for some row vector $M_i$. So, for a submatrix $M_D'$ consisting of these  rows, we have $M_D' \bm \beta \geq 0$. In many cases, this means we find a maximum point of the domain $D'$ defined by $M_D' \bm \beta \geq 0$. This new domain $D'$ is larger than the domain $ D$ and we have  $D' \supset D$. So the maximum likelihood of domain $D'$ is larger than the maximum likelihood of domain $D$. Starting from this domain $D'$, we can add the constraints step by step, which gives a smaller and smaller domains and finally reaches the domain $D$ that we want. Correspondingly, we have a series of decreasing maximum likelihood. This gives the name of descending method. The descending method also requires a starting edge whose maximum point is the maximum point of some larger domain $D' \supset D$. We can apply the na\"{\i}ve method to find such an edge and this is the time consuming part of the algorithm.

Details and pseudo-code of all three methods are given in the Supplemental Information.

\section{Simulation}
\label{sec:simulation}

The purpose of the simulation study is to show feasibility of our maximum likelihood estimator and compare its performance with the ordinary least squares estimator. We assume $\bm \beta(t) = (0.05,0.02,0.04,0.06,0.08)t$, hence the true hazard $h$ is given by:
\[
    h(t \,|\, \mathbf{x}) = \sum_i \beta_i x_i t= (0.05 + 0.02\  x_1 + 0.04\  x_2 + 0.06 \  x_3 +0.08\  x_4)t.
\]
This corresponds to a Weibull distribution with shape $b=2$ and rate $a = (0.05 + 0.02 x_1 + 0.04 x_2 + 0.06 x_3 + 0.08 x_4) / 2$, using the parametrisation $h(t; a, b) = a b t^{b-1}$ for the Weibull hazard.

We randomly generate covariate data $\mathbf{x}_i$, for $i=1,\ldots, n$, with $n=500$, with independent uniform distributions on $[0, 1]$. For each subject an independent censoring time $C_i$ is generated with uniform distribution over $(2.5,7.5)$, giving a censoring rate of about $22\%$. For Aalen's ordinary least squares method we use the implemention in the \textbf{timereg} package~\citep{martinussen2007dynamic}. We use the maximum likelihood estimator with full constraint matrix as introduced in Section~\ref{sub:ML}.

We choose a subject with data of values $x = (0.4,0.6,0.4,0.6)$.
Figure~\ref{fig:expsurCCM} shows the true survival curve and a single realisation of the estimated survival curves by the two methods. The survival curves given by both estimators are similar and approximate the true survival curve well.

\begin{comment}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\textwidth]{SurC_simu_CM.png}
\caption{Estimated survival curves by ordinary least squares (OLS) and by maximum likelihood (MLE)}
\label{fig:expsurCCM}
\end{figure}
\end{comment}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{SurC_simu_TG.pdf}
\caption{Estimated survival curves by ordinary least squares (OLS) and by maximum likelihood (MLE)}
\label{fig:expsurCCM}
\end{figure}

We repeat the same simulation set-up for $1000$ replications with the same subject. We check the estimated cumulative hazards at the three quartiles of the true survival function for that subject, at $t=1.93$, $3.00$ and $4.24$. Figure~\ref{fig:hist} shows histograms of the estimated cumulative hazards for the two methods at the median. The true value of the cumulative hazard is $0.693$, shown by the red vertical line. The ordinary least squares estimate is quite symmetric around the true value, while the maximum likelihood estimator is slightly skewed to the left. However, the variability of the maximum likelihood estimation is smaller.
\begin{figure}[H]
\centering
\includegraphics[width=0.95\textwidth]{Hist_simu_time3.pdf}
\caption{Histograms of estimated cumulative hazards at time $t=3$ (median) by ordinary least squares (OLS) and maximum likelihood (ML).}\label{fig:hist}
\end{figure}

This behaviour is also seen in Table~\ref{tab:cumulativebeta}, which shows that for all three time points, the maximum likelihood estimator has more bias but smaller variance. As a result, the root mean squared error is about 10 to 16 percent smaller for the maximum likelihood estimator, compared to ordinary least squares.
\begin{center}
\begin{tabular}{p{1.5cm}<{\centering} c p{2cm}<{\centering} c p{2cm}<{\centering} c c c  }
\hline
True survival probability & Time point & True cumulative hazard & Method & Estimated cumulative hazard & Bias & Empirical SE & RMSE\\
\hline
\multirow{2}*{$0.75$}    & \multirow{2}*{$1.93$} & \multirow{2}*{$0.288$} & OLS & $0.286$ & $-0.002$ & $0.031$ & $0.031$\\
             &     &  & MLE & $0.280$  & $- 0.007$ & $0.025$ & $0.026$\\
\hline
  \multirow{2}*{$0.5$}    & \multirow{2}*{$3.00$} & \multirow{2}*{$0.693$} & OLS & $0.691$ & $-0.002$ & $0.056$ & $0.056$\\
             &     &  & MLE & $0.676$  & $- 0.017$ & $0.046$ & $0.049$\\
\hline
   \multirow{2}*{$0.25$}    & \multirow{2}*{$4.24$} & \multirow{2}*{$1.386$} & OLS & $1.384$ & $-0.002$ & $0.107$ & $0.107$\\
             &     &  & MLE & $1.353$  & $- 0.034$ & $0.089$ & $0.095$\\
\hline
\end{tabular}
\captionof{table}{Bias (mean estimate minus true value), empirical SE (the standard deviation of the estimates around their mean), and RMSE (root mean squared error, square root of the mean squared difference between estimate and truth) of cumulative hazards estimators by ML and OLS.}
\label{tab:cumulativebeta}
\end{center}

\section{Application}
\label{sec:application}

We apply our methods to data from a clinical trial with 195 patients with carcinoma of the oropharynx by the Radiation Therapy Oncology Group in the United States. The data are introduced by~\citet[Section 1.1.2, Appendix A]{KalbfleishPrentice2012} and used there for illustration of additive hazards models in~\citet{aalen2008survival}. Patients were randomised into two treatment groups ("standard" and "experimental" treatment), and survival times were measured in days from diagnosis. Seven covariates were included in the data: Sex, Treatment, Grade, Age, Condition, T-stage and N-stage. Following~\citet{aalen2008survival}, we take all the covariates as continuous. We re-scale them to $[0,1]$.

The cumulative beta's of the baseline and covariates sex, treatment group, condition, T-stage and N-stage are shown in Figure~\ref{fig:cbeta_orapha}. The ordinary least squares baseline hazard estimate decreases over the first nine months and between 2 and 2.5 years, which would correspond to a negative hazard for subjects with all covariates 0, and an increasing survival curve. The maximum likelihood estimator of the baseline hazard is monotone increasing over time, as desired. For treatment effect, condition, T-stage and N-stage factors the two estimated curves of the cumulative beta are very close to each other in theses cases. For sex, the two methods show opposite trends. The differences in estimates between the two methods may partly be explained by the large number of covariates in the model, in relation to the modest sample size, leading to highly variable estimates for both methods.

\begin{figure}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{CBeta_orapha.pdf}
\caption{Estimated cumulative beta's of factors by maximum likelihood (MLE) and ordinary least squares (OLS)}\label{fig:cbeta_orapha}
\end{figure}

To illustrate the difference in the behavior of the survival curves between maximum likelihood and ordinary least squares, we choose a subject who is female of age $55$ in the treatment group, of cancer grade moderate differentiated, of condition restricted worked, of second T-stage and first N-stage. The covariates of the subject are given by $\mathbf{x} = (1, 1, 0, 1, -0.5, 1, 1, 1)$. The estimated survival curves according to the ordinary least squares and maximum likelihood models are shown in Figure~\ref{fig:estsurv}. The two estimators give  similar results in the first half year when the risk set is large. After that, the ordinary least squares survival curve starts to fluctuate wildly, while the maximum likelihood estimate shows a steady decrease.

\begin{comment}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\textwidth]{SurC_orapha_CM.jpeg}
\caption{Estimated survival curves by the two methods}
\end{figure}
\end{comment}
\begin{figure}[H]
\centering
\includegraphics[width=0.95\textwidth]{SurC_orapha_TG.pdf}
\caption{Estimated survival curves by the two methods maximum likelihood (MLE) and ordinary least squares (OLS).}\label{fig:estsurv}
\end{figure}

\begin{comment}
\section{Discussion}

\section*{acknowledgements}
No acknowledgements, I think.
\end{comment}

\section*{Acknowledgements}

The project is sponsored by the Leiden University Fund / Hakkenberg $\alpha$ $\beta$ $\gamma$-integratie Fonds, www.luf.nl. We would like to thank and acknowledge their support.\vspace*{-8pt}

\section*{conflict of interest}
No conflict of interest.

\section{Introduction}

The fundamental concept in survival analysis is the hazard rate. There are several regression models describing the hazard rate, such as multiplicative risk models and additive risk models. The dominant hazard model in survival analysis is the multiplicative proportional hazards model~\citep{cox1972regression}. In many cases, however, the proportional hazards assumption is not met. In such cases, developing models that adequately describe the non-proportional effect of the covariate(s) is not straightforward \citep{gore1984regression, schemper1992cox, perperoglou2006reduced, vHP12} and different models should be considered.

In this article, we focus on the non-parametric additive hazards model proposed by~\cite{Aalen1980, Aalen1989}, extensively studied in~\citet{martinussen2007dynamic}. The additive model defines the hazard rate as a linear form of the vector of covariates. Unconventionally, it does not naturally force the hazard rate to be positive. However, it has several useful properties, as set out by~\citet{aalen2008survival}. Firstly, the additive hazards model is internally consistent: it retains its additive structure if covariates are measured with uncertainty or are dropped from the linear expression. This contrasts with the proportional hazards model, which loses its proportional hazards property when covariates are omitted~\citep{strutherskalbfleisch1986, schumacher1987, bretagnolle1988}. This drawback of proportional hazards has triggered a recent debate about the danger of using the hazard ratio as a causal effect measure in survival analysis~\citep{hernan2010hazards, aalen2015does}. A second advantage of additive hazards is that it allows implemention of dynamic structures~\citep{martinussen2000nonparametric}, such as self-exciting processes, which are impossible to incorporate in most other nonlinear regression models.

The usual way to estimate the parameters in the additive hazards model is by Aalen's method, which uses ordinary least squares to estimate the cumulative effect of the covariates. Aalen's ordinary least squares method is straightforward to implement, but it has some disadvantages. The most important disadvantage is that it does not guarantee that the model-based hazards are positive for all time points for all subjects in the data, but only for the subjects that experience an event, and only at their event times. Furthermore, the ordinary least squares estimator is defined as a method-of-moments estimator and does not have the support of likelihood theory.

One method that generally yields efficient estimators is maximum likelihood. To the best of our knowledge, the maximum likelihood method has not been considered in any detail in the context of the additive hazards model, since no analytical expression for the maximum likelihood estimator was available. The objective of this paper is to derive the maximum likelihood estimator for the additive hazards model and to determine its advantages and limitations and in comparison to Aalen's ordinary least squares method. We find that there is no maximum solution to the survival likelihood in the unconstrained parameter space. However, we may find the solution if we choose a proper domain. A natural choice for such a domain is one that guarantees positivity of the hazard rate at all time points for all subjects in the data.

We show that the maximum likelihood estimator of the cumulative baseline hazard and cumulative covariate effects is a step function changing values only at the event time points, and that the maximum of the log-likelihood can be found by separately maximising the log-likelihood contributions corresponding to each of the event time points, as with Aalen's ordinary least squares solution. We define the constraint domain, and show that the maximisation problem is equivalent to fitting a series of identity-link Poisson regression models under non-negativity constraints. This problem has been studied before by~\citet{Marschner2010} and \citet{MARSCHNER2012}, who proposed an EM algorithm to obtain maximum likelihood estimators. In contrast, our solution is analytical, and computation time at each time point is linear in the sample size and in the number of covariates per time point. We also discuss the new method's connections to the ordinary least squares estimator. In particular, when only a single binary covariate is considered, we will show that maximum likelihood and ordinary least squares are equivalent. In Section~\ref{sec:simulation} we report on the results of a simulation study comparing maximum likelihood with ordinary least squares. Section~\ref{sec:application} illustrates our methods on data from a randomised clinical trial on patients with carcinoma of the oropharynx.

\section{Notation and model definition}
\label{sub:notation}

We use bold letters to indicate vectors and matrices. Define $T^*$ to be the time to event, $C$ to be the time to censoring, and let $T=\min(T^*, C)$, and $\Delta = I(T^* \leq C)$ the event indicator. We observe $(t_i, \delta_i, \mathbf{x}_i^*)$, $i=1,\ldots,n$, with $\mathbf{x}_i^* = (x_{i1},\ldots,x_{ip})^\top$ a $p$-vector of covariates. We assume that the covariates $x_{ij}$ are restricted to the interval $[0, 1]$. This looks like a severe restriction, but it is not, because for any covariate with a minimum of $a$ and a maximum of $b$ in the observed data, we can rescale the covariate to be $(x-a)/(b-a)$, which takes values in $[0, 1]$. Extend $\mathbf{x}_i^*$ with the constant $x_{i0} = 1$, obtaining $\mathbf{x}_i = (1, x_{i1},\ldots,x_{ip})^\top$. Define $\mathcal{E}$ to be the set of event time points $t_i$, with corresponding $\delta_i=1$ and let $|\mathcal E| = K $. We assume there are no ties, and define the ordered sequence of event time points $t_1^* < \ldots < t_K^*$. Finally, define $Y_i(t) =\sum_{i=1}^n I(t_i\geq t)$
as the at risk indicator of subject $i$, taking the value $1$ if subject $i$ is at risk for the event of interest just before time $t$ and the value $0$ otherwise.

The additive hazards model~\citep{Aalen1980, Aalen1989} assumes the hazard rate to be of the form
\[
    h(t \,|\, \mathbf{x}_i^*) = \mathbf x^\top_i \bm \beta(t) =  \beta_0(t) + \beta_1(t)x_{i1} + \cdots + \beta_p(t)x_{ip}.
\]
The parameters $\beta_j(t)$ allow effects of the covariates to change over time, thus,  the additive  model is fully non-parametric.

\section{Aalen's method}
\label{sub:OLS}

To estimate the parameters $\bm{\beta}(t)$, the most commonly used approach is Aalen's ordinary least squares method. Let us define the counting process $N_i(t)$ as the number of events experienced by subject $i$ before or at time $t$. Then the intensity $\lambda_i(t)$ of the counting process has the form
\[
        \lambda_i(t) =  Y_i(t)(\beta_0(t) + \beta_1(t)x_{i1} + \cdots + \beta_p(t)x_{ip})
            = Y_i(t) h(t_i \,|\, \mathbf{x}_i^*).
\]
The formula $dN_i(t) = \lambda_i(t)dt + dM_i(t)$ gives equations
\[
    d\mathbf{N}(t) = \mathbf{X}(t)d\mathbf{B}(t) + d\mathbf{M}(t),
\]
where $\mathbf{N}(t) = (N_1(t),N_2(t),\ldots,N_n(t))^\top$ is the vector of counting processes, $\mathbf{X}(t)$ is the matrix of covariates multiplied with $Y_i(t)$ at each $i$th row, $\mathbf{B}(t) = (B_0(t),B_1(t),\ldots,B_p(t))^\top$ is the cumulative beta defined as $B_j(t) = \int_0^t \beta_j(s)ds$ and $\mathbf{M}(t) = (M_1(t), \ldots, M_n(t))^\top$, where $M_i(t)$ is the $i$th martingale error term. Ordinary least square regression thus gives Aalen's estimator of the $\bm{\beta}(t)$ if $X(t)$ is of full rank:
\[
    d\widehat{\mathbf{B}}(t) = (\mathbf{X}(t)^\top\mathbf{X}(t))^{-1} \mathbf{X}(t)^\top d\mathbf{N}(t).
\]
When $X(t)$ is not of full rank, we set $d\widehat{\mathbf{B}}(t) = 0$~\citep{aalen2008survival}.

\section{Maximum likelihood estimation}
\label{sub:ML}

In this section, we will derive an analytic solution for the maximum likelihood estimator under the natural constraint that the hazard is non-negative at each time point for all covariate values within the domain.

\subsection{The shape of the likelihood}

To begin with, let us consider the likelihood function for the additive hazards model. The log likelihood contribution of subject $i$ with an event or censored at time $t_i$ is given by
\[
    \ell_i = \delta_i \log\{h(t_i \,|\, \mathbf{x}_i^*)\}+ \log\{ S(t_i \,|\, \mathbf{x}_i^*)\},
\]
where $S(t \,|\, \mathbf{x}_i^*) = P(T^* > t \,|\, \mathbf{x}_i^*) = \exp(- H(t \,|\, \mathbf{x}_i^*))$ is the survival probability and $H(t \,|\, \mathbf{x}_i^*)$ the cumulative hazard of subject $i$. The log-likelihood is therefore given by~\citep{aalen2008survival}

\begin{equation}
    \label{eq:logliki}
     \ell = \sum_{i=1}^n \{\delta_i \log \mathbf \mathbf{x}_i^\top \bm{\beta}(t_i) - \mathbf{x}_i^\top \mathbf B(t_i)\}.
\end{equation}

Our first result is
\begin{proposition}
The likelihood function $\ell$ is unbounded.
\end{proposition}

\noindent%
We therefore introduce the natural constraint that the hazard of each possible covariate value of our data is non-negative at each time point, that is, we require that for all $t \geq 0$,
\begin{equation}
    \label{eq:hpos}
 h(t \,|\, \mathbf{x}^*) \geq 0,\ \textrm{for}\ \mathbf{x}^* \in \{0, 1\}^p,
\end{equation}
which implies positivity of the hazard for $\mathbf{x}^* \in [0, 1]^p$. Our objective is to maximise the total log-likelihood given in~\eqref{eq:logliki}, subject to the constraint~\eqref{eq:hpos}.

Let us assume $\hat{\mathbf{B}}(t)$ is the function which maximises the likelihood function~\eqref{eq:logliki}, subject to the constraint~\eqref{eq:hpos}. Since $\hat{\mathbf{B}}(t)$ is an estimator of cumulative beta, which is generally the negative logarithm of the survival function, we may assume that it is right continuous with left limits (cadlag). Thus, we may decompose $\hat{\mathbf{B}}(t)$ as the summation of a continuous function $\hat{\mathbf{B}}_c(t)$ and a step function $\hat{\mathbf{B}}_s(t)$.

\begin{lemma}
\label{prop:beta0}
Under the constraint $h(t \,|\, \mathbf{x}^*) \geq 0$ for $\mathbf{x}^* \in \{0, 1\}^p$, the log-likelihood can only achieve a maximum if $\hat{\mathbf{B}}_c(t) \equiv 0$ and $\hat{\mathbf{B}}_s(t)$ is a step function with jumps only at the event time points in $\mathcal{E}$.
\end{lemma}

\noindent%
Lemma~\ref{prop:beta0} implies that the maximisation problem is the same as maximising the total log-likelihood with respect to the jumps of $\mathbf{B}(t)$ at the event time points. Denoting the jump of $B_j(t)$ at the $k^{\textrm{th}}$ event time point $t_k^*$ as $\beta_{kj}$, $k=1,\ldots,K$, $j=0,\ldots,p$, this leads to $B_j(t) = \sum_{k: t_k^* \leq t} \beta_{kj}$.

The total log likelihood $\ell$ from~\eqref{eq:logliki} can now be rewritten in terms of the $\beta_{kj}$ as
\[
    \ell = \sum_{i=1}^n \Biggl\{ \delta_i \log \Bigl(\sum_{j=0}^p x_{ij} \beta_{k(i),j}\Bigr) - \sum_{j=0}^p x_{ij} \sum_{k: t_k^* \leq t_i} \beta_{kj} \Biggr\},
\]
where $k(i)$ is the index of the event time points $t_k^*$ corresponding to $t_i$, i.e.~$t_{k(i)}^* = t_i$ (in case $\delta_i=0$, $k(i)$ can be chosen as an arbitrary index in $\{1,\ldots,n\}$). We reorder this sum over subjects as a sum over the distinct event time points $t_k^*$, obtaining
\[
    \ell = \sum_{k=1}^{K} \Biggl\{ \log \Bigl(\sum_{j=0}^p x_{i(k),j} \beta_{kj}\Bigr) - \sum_{j=0}^p \beta_{kj} \sum_{l: t_l \geq t_k^*} x_{lj} \Biggr\} = \sum_{k=1}^{K} \ell_k^*,
\]
with
\begin{equation}\label{eq:ellk}
    \ell_k^* = \log \Bigl(\sum_{j=0}^p x_{i(k),j} \beta_{kj}\Bigr) - \sum_{j=0}^p \beta_{kj} s_{kj},
\end{equation}
$s_{kj} = \sum_{l: t_l \geq t_k^*} x_{lj}$,
and where $i(k)$ is the subject index corresponding to the $k^{\textrm{th}}$ event time point, i.e.~$t_{i(k)} = t_k^*$.

It is easy to see that each term $\ell_k^*$ is a function only of the variables $\beta_{kj}$, $j = 0,\ldots,p$. Since the terms $\ell_k^*$ do not have parameters in common, we may reduce the problem by separately maximising each $\ell_k^*$ as a function of $\bm{\beta}_k = (\beta_{k0}, \beta_{k1}, \ldots, \beta_{kp})$.

For the remainder of this section, we will fix (any) one of the time points $t_k^*$. To simplify notation, we are going to suppress dependence on $k$, and consider maximisation of
\begin{equation}\label{eq:ell}
    \ell^* = \log \Bigl( \sum_{j=0}^p x_j \beta_{j} \Bigr) - \sum_{j=0}^p s_j \beta_j =
    \log \Bigl( \mathbf{x}^\top \bm{\beta} \Bigr) - \mathbf{s}^\top \bm{\beta}
\end{equation}
with respect to $\bm{\beta} = (\beta_0, \beta_1, \ldots, \beta_p)$, where $\mathbf{s} = (s_0, s_1, \ldots, s_p)^\top$, $x_j$ stands for the $j^{\textrm{th}}$ element of the covariate vector of the subject that failed at time $t_k^*$, and $x_0=1$.

\subsection{Maximum likelihood with the full constraint matrix}

\noindent%
From now on, we will give an alternative description of the constraint~(\ref{eq:hpos}) using a matrix form.  Let $\bm{\beta} = (\beta_0, \beta_1, \ldots, \beta_p)^\top$ be the vector of parameters of the function $\ell^*$. We can restate the constraint~(\ref{eq:hpos})   by the matrix $M_\mathbf{D}$ through the inequalities $M_\mathbf{D} \bm{\beta} \geq \bm 0$. We construct the matrix $M_\mathbf{D}$ such that the rows consist of all the $2^p$ possible $(p+1)$-tuples $(m_0, m_1, \ldots, m_p)$ in $\{0,1\}^{p+1}$ with $m_0 =1$.  So $M_\mathbf{D}$ is a $2^p \times (p+1)$ matrix.

 The domain $\mathbf D\subset \mathbb R^{p+1}$ defined by constraint~\eqref{eq:hpos} is a polytope enclosed by the flat boundaries defined by $\gamma_i = M_i^\top \bm{\beta} = 0$, $i=1,\ldots,2^p$, where $M_i$ is the $i$th row of $M_{\mathbf D}$.  Now, we can show that the maximum point lies on the boundary of this polytope.

\begin{lemma} \label{prop:maximumpointReverse}
If the maximum point of $\ell^*$ in the domain $\mathbf D$ exists, then the maximum point  lies in the intersection of $p$ flat boundaries given by $\gamma_{i_j} = M_{i_j}^\top \bm \beta= 0$, $j = 1 \ldots, p$.
\end{lemma}

\noindent%
In Lemma~\ref{prop:maximumpointReverse}, we have given the geometric property of the maximum point of the $\ell^*$. The intersection of $p$ flat boundaries is a one-dimensional line. So, the maximum point always lies on a one-dimensional line edge on the boundary of the domain. The following lemma defines all the one-dimensional line edges on the boundary of the domain. There are $2p$ such line edges.

\begin{lemma}\label{prop:geometryofedge}
The one-dimensional line edges on the boundary of the domain $M_\mathbf{D}\bm \beta \geq 0$ are given by $(0,\ldots,0,l,0,\ldots,0)$, with the position of $l$ ranging from $2$ to $p+1$, and $(l,\ldots,0,-l,0,\ldots,0)$, with the position of $-l$ ranging from $2$ to $p+1$.
\end{lemma}

\noindent%
Let us denote by $\bm e_j$ the $(p+1)$-vector $(0, 0, \ldots, 0, 1, 0, \ldots, 0)^\top$, with the 1 at position $j+1$, and by $\bm f_j$ the $(p+1)$-vector $(1, 0, \ldots, 0, -1, 0, \ldots, 0)^\top$, with the -1 at position $j+1$, and define the sets $\mathcal{A}^+ = \{ \bm e_1, \ldots, \bm e_p \}$ and $\mathcal{A}^- = \{ \bm f_1, \ldots, \bm f_p \}$. We may call the set $\mathcal{A} = \mathcal{A}^+ \cup \mathcal{A}^-$, a finite set of cardinality $2p$, the set of admissible directions. Lemma's~\ref{prop:maximumpointReverse} and~\ref{prop:geometryofedge} imply that the maximum of $\ell^*$ lies on one of the edges ${\bm v} \cdot l$, with ${\bm v} \in \mathcal{A}$. We now come to our main theorem which describes the structure of the maximum likelihood estimator of the additive hazards model under the constraint of non-negativity of the hazards. Let $i(k)$ denote the index of the subject that fails at $t_k^*$, the maximum likelihood estimator $\hat{ \bm \beta}_k$ at the event time point $t_k^*$ can be determined as follows:

\begin{algorithm}[H]
\caption{Algorithm to find the maximum point within the domain $\mathbf D$.\label{alg:s1}}

\begin{itemize}
    \item Calculate $\mathbf s_{k} = \sum_{l: t_l \geq t_k^*} \mathbf x_{l}$;
    \item Calculate the values of ratio  $\bm v_j = x_{i(k), j} / s_{k,j}$ for $1 \leq j \leq p$;
    \item Calculate the values of ratio  $\bm v_{p + j} = (x_{i(k), 0} - x_{i(k), j}) / (s_{k,0} - s_{k,j})$ for $1 \leq j \leq p$;
    \item Find the set $M$ of indices $m = 1, \ldots, 2p$, for which $\bm v_m$ is maximised;
    \item For any index $m \in M$, calculate the value of $\hat{\bm \beta}_{k}^{(m)}$ by
\begin{equation*}\hat{\bm \beta}_{k}^{(m)} =  \begin{cases}
            \bm e_m / s_{k,m}, & \hbox{if $1 \leq m \leq p$;} \\
            \bm f_{m-p} / (s_{k,0} - s_{k,(m-p)}),& \hbox{if $p + 1 \leq m \leq 2p$;}
          \end{cases}
         \end{equation*}
    \item Return $\hat{\bm \beta}_{k}^{(m)}$, $m \in M$.
\end{itemize}

\end{algorithm}

\begin{theorem}\label{thm:maintheorem}
For an additive hazards model $h(t \,|\, \mathbf{x}_i^*) = \bm \beta(t)\mathbf{x}_i$ with constraints  $h(t \,|\, \mathbf{x}^*) \geq 0,\ \textrm{with }\ \mathbf{x}^* \in [0, 1]^p$, assuming no ties,  the values returned by Algorithm \ref{alg:s1} are maximum likelihood estimators.
\end{theorem}

\noindent%
Theorem \ref{thm:maintheorem} offers some insight into the maximum likelihood estimator. There are only two possible patterns of the jump $\bm \beta(t_k^*)$. Either there is an increment of the coefficient $\beta_j(t_k^*)$ for one covariate or there is a decrease of $\beta_j(t_k^*)$ for one covariate which is compensated by an increment of the intercept. So, this estimator is sparse in the change of $\bm \beta(t)$.

Theorem \ref{thm:maintheorem} also implies a condition for the existence and uniqueness of the likelihood estimator. The likelihood estimator exists if and only if $s_{k,m} \neq 0$ and $s_{k,0}- s_{k,(m-p)} \neq 0$. which correspond to the cases all $x_i$'s are zeros or all $x_i$'s are ones, $i \geq i(k)$. Uniqueness is not guaranteed by Theorem \ref{thm:maintheorem}. In fact,  all choices of the ratios $x_{i(k), j} / s_{k,j}$ and/or $(x_{i(k), 0} - x_{i(k), j}) / (s_{k,0} - s_{k,j})$ found by Algorithm \ref{alg:s1} result in the same maximum. In that case any ${\bm v} \in \mathcal{A}$ corresponding to such a maximum is allowed, as well as any convex combination, as stated in the following theorem.

\begin{theorem}\label{thm:mutisolution}
If $\hat{\bm\beta}_{k}^{(m_1)}$ and $\hat{\bm\beta}_{k}^{(m_2)}$ are solutions given by Algorithm \ref{alg:s1} which maximise the likelihood $\ell$, then any convex combination $\lambda \bm \hat{\bm\beta}_{k}^{(m_1)} + (1 - \lambda) \hat{\bm\beta}_{k}^{(m_2)}$, with $0 \leq \lambda \leq 1$, attains the same maximised likelihood.
\end{theorem}

In practice, we suggest an average of all solutions given by Theorem~\ref{thm:maintheorem}. By Theorem~\ref{thm:mutisolution}, this is also a solution which maximises the likelihood.

With our result on the maximum likelihood estimator, we also have the following result which reveals the connection between the maximum likelihood estimator and Aalen's estimator.

\begin{proposition}
For an additive hazards model with one binary covariate, the maximum likelihood estimator and the Aalen estimator coincide.
\end{proposition}

This can also be explained intuitively by Theorem \ref{thm:maintheorem} as follows. For one binary covariate, Aalen's ordinary least squares method can be viewed as selecting one of two competing trends at each event time. Theorem~\ref{thm:maintheorem} shows that the nature of maximum likelihood estimator is similar; the jump of the coefficient takes place for only one covariate, i.e., only one trend is selected at each event time.

\subsection{Example}

We illustrate Theorem~\ref{thm:maintheorem} using a simple example with two covariates. Let us assume that at a certain event time point $t^*$ a subject fails with covariate values $x_1=0$ and $x_2=1$, so we have $\mathbf{x}^* = (0,1)$, and $\mathbf{x} = (1,0,1)$. Suppose that $\mathbf{s} = (8,5,6)^\top$. Then the log-likelihood is given by
\begin{equation}\label{eq:twocovml}
    \ell^* = \log (\beta_0 + \beta_2)- 8 \beta_0 -5\beta_1 -6 \beta_2.
\end{equation}

In this case, the domain is defined by the matrix $\mathbf{M_D}$ through the boundary conditions $M_{\mathbf D} \bm \beta \geq 0$, where
\[ M_{\mathbf D} = \begin{pmatrix}
    1 & 0 & 0 \\
    1 & 0 & 1 \\
    1 & 1 & 0 \\
    1 & 1 & 1 \\
\end{pmatrix}.\]

Theorem~\ref{thm:maintheorem} directs us to compare the values of $x_1/s_1$, $x_2/s_2$, $(x_0-x_1)/(s_0-s_1)$, and $(x_0-x_2)/(s_0-s_2)$, namely $0$, $\frac{1}{6}$, $\frac{1}{3}$, $0$, corresponding to the one-dimensional line edges ${\bm v} \cdot l$, with $\bm v = (0,1,0)^\top, (0,0,1)^\top, (1,-1,0)^\top$ and $(1,0,-1)^\top$, respectively, in the set of admissible directions $\mathcal{A}$. The third of these numbers is largest, so the one-dimensional line ${\bm v} \cdot l$, with $\bm v = (1,-1,0)^\top$ is the edge where the maximum point lies. Maximising over $l$ gives $l = \frac{1}{\mathbf{s}^{\top} \bm{v}} = \frac{1}{3}$, $\hat{\bm \beta}(t) = (\frac{1}{3}, -\frac{1}{3},0)$ and the maximised log-likelihood $\ell^* = \log(\frac{1}{3})- 1$.

\subsection{Maximum likelihood with a partial constraint matrix}
\label{sub:MLEfind}

\noindent%
In previous sections, we show that the maximum point of the likelihood function lies in one of the $2p$ one-dimensional line edges on the boundary. This result is derived for the case where the domain is given by the matrix $M_{\mathbf D}$ whose rows are all the combinations of $1$ and $0$ in the last $p$ columns. Sometimes we need more flexible domains, for example when dummy covariates are introduced or when it is known that some coefficients are positive. In this more general case Theorem \ref{thm:maintheorem} does not apply, and numerical solutions are needed.

We consider a general domain defined by $M_D \bm \beta \geq 0$ for some matrix $M_D$, where the matrix $M_D$ can be an arbitrary matrix of $p+1 $ columns. We assume that the rank of $M_D$ is $p+1$. We assert that Lemma \ref{prop:maximumpointReverse} still applies for this constraint matrix $M_D$. The maximum point lies on a one-dimensional edge on the boundary, defined as the intersection of $p$ flat boundaries. So, any local maximum point of a   one-dimensional edge is a potential solution of the maximum likelihood of the domain $D$.  Given any one-dimensional edge, we have $p$ equations $\gamma_i = M_{i} \bm \beta = 0$. Combined with the first order condition which is also a linear equation, we have $p+1$ linear equations whose solution is the maximum point on this edge. By testing its derivatives and constraint conditions, we can check if it is a maximum point of the domain $D$.  However, it is difficult to find the right edge on which the maximum point lies. We introduce three methods to find the right edge: the na\"{\i}ve method, the ascending method and the descending method.

The na\"{\i}ve method loops through all the possible combinations of $p$ rows of $M_D$. Every combination of $p$ rows defines a one-dimensional edge and gives a possible solution of the maximum point $\bm \beta^*$. We can test its derivatives and constraint conditions to check if it is the maximum point we want to find. This na\"{\i}ve method is easy to understand, but not efficient, since the complexity grows approximately with the $p$th power of number of the rows of $M_D$.

sThe ascending method searches for a series of $\bm \beta^*$ with increasing likelihood. More specifically, the algorithm  starts from a $p$-combination of rows, whose maximum point is $\bm \beta^*_0$. We assume this edge is inside the domain. If  $\bm \beta^*_0$ is not the maximum point of the domain $D$, its gradient points to the inside of the domain. Following this direction, we can find an adjacent one-dimensional edge whose maximum point $\bm \beta^*_1$ has a larger likelihood.  Since this algorithm starts from an edge already on the domain, as a result, it is quite efficient when a good starting edge of the domain is known. To find a new edge, we use the gradient of the log likelihood, constrained to the adjacent hyperplanes. This method works well if the geometry of the domain is not too complicated.

The descending method is the reverse of the ascending method. Assuming that we find a maximum point $\bm \beta^*$ on a one-dimensional edge, it is not necessary that this $\bm \beta^*$ satisfies all the constraint conditions $M_D \bm \beta^* \geq 0$. However, it may satisfy some of the constraint conditions, i.e., we have $\gamma_i = M_i \bm \beta^* \geq 0$ for some row vector $M_i$. So, for a submatrix $M_D'$ consisting of these  rows, we have $M_D' \bm \beta \geq 0$. In many cases, this means we find a maximum point of the domain $D'$ defined by $M_D' \bm \beta \geq 0$. This new domain $D'$ is larger than the domain $ D$ and we have  $D' \supset D$. So the maximum likelihood of domain $D'$ is larger than the maximum likelihood of domain $D$. Starting from this domain $D'$, we can add the constraints step by step, which gives a smaller and smaller domains and finally reaches the domain $D$ that we want. Correspondingly, we have a series of decreasing maximum likelihood. This gives the name of descending method. The descending method also requires a starting edge whose maximum point is the maximum point of some larger domain $D' \supset D$. We can apply the na\"{\i}ve method to find such an edge and this is the time consuming part of the algorithm.

Details and pseudo-code of all three methods are given in the Supplemental Information.

\section{Simulation}
\label{sec:simulation}

The purpose of the simulation study is to show feasibility of our maximum likelihood estimator and compare its performance with the ordinary least squares estimator. We assume $\bm \beta(t) = (0.05,0.02,0.04,0.06,0.08)t$, hence the true hazard $h$ is given by:
\[
    h(t \,|\, \mathbf{x}) = \sum_i \beta_i x_i t= (0.05 + 0.02\  x_1 + 0.04\  x_2 + 0.06 \  x_3 +0.08\  x_4)t.
\]
This corresponds to a Weibull distribution with shape $b=2$ and rate $a = (0.05 + 0.02 x_1 + 0.04 x_2 + 0.06 x_3 + 0.08 x_4) / 2$, using the parametrisation $h(t; a, b) = a b t^{b-1}$ for the Weibull hazard.

We randomly generate covariate data $\mathbf{x}_i$, for $i=1,\ldots, n$, with $n=500$, with independent uniform distributions on $[0, 1]$. For each subject an independent censoring time $C_i$ is generated with uniform distribution over $(2.5,7.5)$, giving a censoring rate of about $22\%$. For Aalen's ordinary least squares method we use the implemention in the \textbf{timereg} package~\citep{martinussen2007dynamic}. We use the maximum likelihood estimator with full constraint matrix as introduced in Section~\ref{sub:ML}.

We choose a subject with data of values $x = (0.4,0.6,0.4,0.6)$.
Figure~\ref{fig:expsurCCM} shows the true survival curve and a single realisation of the estimated survival curves by the two methods. The survival curves given by both estimators are similar and approximate the true survival curve well.

\begin{comment}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\textwidth]{SurC_simu_CM.png}
\caption{Estimated survival curves by ordinary least squares (OLS) and by maximum likelihood (MLE)}
\label{fig:expsurCCM}
\end{figure}
\end{comment}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{SurC_simu_TG.pdf}
\caption{Estimated survival curves by ordinary least squares (OLS) and by maximum likelihood (MLE)}
\label{fig:expsurCCM}
\end{figure}

We repeat the same simulation set-up for $1000$ replications with the same subject. We check the estimated cumulative hazards at the three quartiles of the true survival function for that subject, at $t=1.93$, $3.00$ and $4.24$. Figure~\ref{fig:hist} shows histograms of the estimated cumulative hazards for the two methods at the median. The true value of the cumulative hazard is $0.693$, shown by the red vertical line. The ordinary least squares estimate is quite symmetric around the true value, while the maximum likelihood estimator is slightly skewed to the left. However, the variability of the maximum likelihood estimation is smaller.
\begin{figure}[H]
\centering
\includegraphics[width=0.95\textwidth]{Hist_simu_time3.pdf}
\caption{Histograms of estimated cumulative hazards at time $t=3$ (median) by ordinary least squares (OLS) and maximum likelihood (ML).}\label{fig:hist}
\end{figure}

This behaviour is also seen in Table~\ref{tab:cumulativebeta}, which shows that for all three time points, the maximum likelihood estimator has more bias but smaller variance. As a result, the root mean squared error is about 10 to 16 percent smaller for the maximum likelihood estimator, compared to ordinary least squares.
\begin{center}
\begin{tabular}{p{1.5cm}<{\centering} c p{2cm}<{\centering} c p{2cm}<{\centering} c c c  }
\hline
True survival probability & Time point & True cumulative hazard & Method & Estimated cumulative hazard & Bias & Empirical SE & RMSE\\
\hline
\multirow{2}*{$0.75$}    & \multirow{2}*{$1.93$} & \multirow{2}*{$0.288$} & OLS & $0.286$ & $-0.002$ & $0.031$ & $0.031$\\
             &     &  & MLE & $0.280$  & $- 0.007$ & $0.025$ & $0.026$\\
\hline
  \multirow{2}*{$0.5$}    & \multirow{2}*{$3.00$} & \multirow{2}*{$0.693$} & OLS & $0.691$ & $-0.002$ & $0.056$ & $0.056$\\
             &     &  & MLE & $0.676$  & $- 0.017$ & $0.046$ & $0.049$\\
\hline
   \multirow{2}*{$0.25$}    & \multirow{2}*{$4.24$} & \multirow{2}*{$1.386$} & OLS & $1.384$ & $-0.002$ & $0.107$ & $0.107$\\
             &     &  & MLE & $1.353$  & $- 0.034$ & $0.089$ & $0.095$\\
\hline
\end{tabular}
\captionof{table}{Bias (mean estimate minus true value), empirical SE (the standard deviation of the estimates around their mean), and RMSE (root mean squared error, square root of the mean squared difference between estimate and truth) of cumulative hazards estimators by ML and OLS.}
\label{tab:cumulativebeta}
\end{center}

\section{Application}
\label{sec:application}

We apply our methods to data from a clinical trial with 195 patients with carcinoma of the oropharynx by the Radiation Therapy Oncology Group in the United States. The data are introduced by~\citet[Section 1.1.2, Appendix A]{KalbfleishPrentice2012} and used there for illustration of additive hazards models in~\citet{aalen2008survival}. Patients were randomised into two treatment groups ("standard" and "experimental" treatment), and survival times were measured in days from diagnosis. Seven covariates were included in the data: Sex, Treatment, Grade, Age, Condition, T-stage and N-stage. Following~\citet{aalen2008survival}, we take all the covariates as continuous. We re-scale them to $[0,1]$.

The cumulative beta's of the baseline and covariates sex, treatment group, condition, T-stage and N-stage are shown in Figure~\ref{fig:cbeta_orapha}. The ordinary least squares baseline hazard estimate decreases over the first nine months and between 2 and 2.5 years, which would correspond to a negative hazard for subjects with all covariates 0, and an increasing survival curve. The maximum likelihood estimator of the baseline hazard is monotone increasing over time, as desired. For treatment effect, condition, T-stage and N-stage factors the two estimated curves of the cumulative beta are very close to each other in theses cases. For sex, the two methods show opposite trends. The differences in estimates between the two methods may partly be explained by the large number of covariates in the model, in relation to the modest sample size, leading to highly variable estimates for both methods.

\begin{figure}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{CBeta_orapha.pdf}
\caption{Estimated cumulative beta's of factors by maximum likelihood (MLE) and ordinary least squares (OLS)}\label{fig:cbeta_orapha}
\end{figure}

To illustrate the difference in the behavior of the survival curves between maximum likelihood and ordinary least squares, we choose a subject who is female of age $55$ in the treatment group, of cancer grade moderate differentiated, of condition restricted worked, of second T-stage and first N-stage. The covariates of the subject are given by $\mathbf{x} = (1, 1, 0, 1, -0.5, 1, 1, 1)$. The estimated survival curves according to the ordinary least squares and maximum likelihood models are shown in Figure~\ref{fig:estsurv}. The two estimators give  similar results in the first half year when the risk set is large. After that, the ordinary least squares survival curve starts to fluctuate wildly, while the maximum likelihood estimate shows a steady decrease.

\begin{comment}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\textwidth]{SurC_orapha_CM.jpeg}
\caption{Estimated survival curves by the two methods}
\end{figure}
\end{comment}
\begin{figure}[H]
\centering
\includegraphics[width=0.95\textwidth]{SurC_orapha_TG.pdf}
\caption{Estimated survival curves by the two methods maximum likelihood (MLE) and ordinary least squares (OLS).}\label{fig:estsurv}
\end{figure}

\begin{comment}
\section{Discussion}

\section*{acknowledgements}
No acknowledgements, I think.
\end{comment}

\section*{Acknowledgements}

The project is sponsored by the Leiden University Fund / Hakkenberg $\alpha$ $\beta$ $\gamma$-integratie Fonds, www.luf.nl. We would like to thank and acknowledge their support.\vspace*{-8pt}

\section*{conflict of interest}
No conflict of interest.

redpajama_set_name: RedPajamaArXiv


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South African Transport and Allied Workers' Union (SATAWU) has completed the third day of negotiations at the Road Accident Fund (RAF) with an undertaking to take the latest salary increase offer to members.
SATAWU is at this point not disclosing the percentage as negotiations are at a sensitive stage. However, both parties agreed to labour consulting members on whether to accept the latest offer. Consultations will take place early next with the final feedback planned for mid-week.
Salary negotiations at RAF were set for three days starting Wednesday. SATAWU is the only union representing 1 250 of the total 2 200 workers at the fund since NUMSA was derecognised early this month after it was found to have attained its recognition status unduly. An attempt to reserve the situation was dismissed by the Labour Court yesterday with costs.


text:
Henry B. Plant Museum
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Music and Meaning: Spirituals at the Tampa Bay Casino
View a performance of a spiritual sung at the Tampa Bay Casino and learn about the significance of this musical form.
Dirty Laundry: True Tales of Women Workers at the Tampa Bay Hotel
This exhibit showcases the lives of the housekeepers, maids, and laundresses who labored at the Tampa Bay Hotel.
Minstrelsy in Black and White
Explore minstrel performances at the Tampa Bay Casino.
Lucky Charms: An Opera, a Steamship, and Tampa's Civic Identity
Enjoy a performance from a 19th century opera and learn about the connections between the work and the City of Tampa.
The Sportin' Life
Dive into the sporting lifestyle of America's grand resorts from the 1880s to the 1930s.
Ann Lowe, the First Internationally Recognized Black Fashion Designer
Learn about Ann Lowe and three of her gowns designed for Gasparilla celebration - including 3D scans of the gowns.
Strolling in a Perfect Paradise
Tour Henry Plant Park through a series of historic postcards.
Red Cross Nursing and the War of 1898: The Tampa Connection
Follow the path that led dozens of young women from New York to Tampa and Cuba in 1898.
Mr. Skinner, at Your Service: The Labor of Fine Dining at the Tampa Bay Hotel
Peek behind the scenes of the Tampa Bay Hotel's dining room and meet the men and women responsible for its epicurean delights.
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From young gun to vet, Dellavedova aims to get Cavs clicking
Matthew Dellavedova's first game in his second run with the Cleveland Cavaliers was about what you'd expect — lots of hustle, lots of leadership running the offense and just general scrappiness.
Still, things were a little clunky, or maybe even a lot clunky, as the Cavs (6-21) were never really in it and dropped a 108-92 decision to the host Milwaukee Bucks on Monday.
"I was just trying to call the few plays that I know and keep us organized," Dellavedova said, via Chris Fedor of Cleveland.com. "I was just trying to keep talking to the guys and string some stops together in the paint. We probably did a better job of that in the second half."
Dellavedova finished with 11 points on 4-of-7 shooting, with four assists, in 16 minutes.
Once one of the Cavs' young energy guys, his job is now to push Collin Sexton in practice and show the rookie how to run the offense like a pro.
"He's going to pick up 94 feet (defensively) and if you leave him open he can shoot the three," center Tristan Thompson said. "In the pick-and-roll he's going to make the right play, whether it's the floater, the lob or the cross-court pass for a three. So he's good for the second unit."
Thompson is now out for two-to-four weeks with a sprained ankle, an injury he suffered in the loss. Before exiting Monday, he finished with 12 points on 6-of-7 shooting.
Thompson and Dellavedova have always had a special on-the-court connection, Delly often driving to the middle of the lane and lobbing a pass toward the rim for Thompson to finish.
"I'm happy he's here," Thompson said.
That appears to be true of all the Cavs, as Dellavedova has been in this situation already. While the Cavs may not win much this season, players such as Dellavedova will help with important aspects such as energy and culture.
And it took all of one game to see that.


After growing sharply throughout the 1990s, participation rates in 401(k)s leveled off during the next decade, especially during the period between 2005 and 2009, according to trend analysis from the Employee Benefit Research Institute.
After growing sharply throughout the 1990s, participation rates in 401(k)s leveled off during the next decade, especially during the period between 2005 and 2009, according to trend analysis from the Employee Benefit Research Institute. In the case of IRA ownership, the rates declined slightly for the same period.