# Introduction to Survival Analysis

## Introduction

Survival analysis is generally defined as a set of methods for analysing data where the outcome variable is the time until the occurrence of an event of interest. For example, if the event of interest is heart attack, then the survival time can be the time in years until a person develops a heart attack. For simplicity, we will adopt the terminology of survival analysis, referring to the event of interest as ‘death’ and to the waiting time as ‘survival’ time, but this technique has much wider applicability. The event can be death, occurrence of a disease, marriage, divorce, etc. The time to event or survival time can be measured in days, weeks, years, etc.

The specific difficulties relating to survival analysis arise largely from the fact that only some individuals have experienced the event and, subsequently, survival times will be unknown for a subset of the study group. This phenomenon is called censoring.

In longitudinal studies exact survival time is only known for those individuals who show the event of interest during the follow-up period. For others (those who are disease free at the end of the observation period or those that were lost) all we can say is that they did not show the event of interest during the follow-up period. These individuals are called censored observations. An attractive feature of survival analysis is that we are able to include the data contributed by censored observations right up until they are removed from the risk set.

## Survival and Hazard

T  –  a non-negative random variable representing the waiting time until the occurrence of an event.

The survival function, S(t), of an individual is the probability that they survive until at least time t, where t is a time of interest and T is the time of event.

The survival curve is non-increasing (the event may not reoccur for an individual) and is limited within [0,1].

F(t) – the probability that the event has occurred by duration t:

the probability density function (p.d.f.) f(t):

An alternative characterisation of the distribution of T is given by the hazard function, or instantaneous rate of occurrence of the event, defined as

The numerator of this expression is the conditional probability that the event will occur in the interval [t,t+dt] given that it has not occurred before, and the denominator is the width of the interval. Dividing one by the other we obtain a rate of event occurrence per unit of time. Taking the limit as the width of the interval goes down to zero, we obtain an instantaneous rate of occurrence.

Applying Bayes’ Rule

on the numerator of the hazard function:

Given that the event happened between time t to t+dt, the conditional probability of this event happening after time t is 1:

Dividing by dt and passing to the limit gives the useful result:

In words, the rate of occurrence of the event at duration t equals the density of events at t, divided by the probability of surviving to that duration without experiencing the event.

We will soon show that there is a one-to-one relation between the hazard and the survival function.

The derivative of S(t) is:

We will now show that the hazard function is the derivative of -log S(t):

If we now integrate from 0 to time t:

and introduce the boundary condition S(0) = 1 (since the event is sure not to have occurred by duration 0):

we can solve the above expression to obtain a formula for the probability of surviving to duration t as a function of the hazard at all durations up to t:

One approach to estimating the survival probabilities is to assume that the hazard function follow a specific mathematical distribution. Models with increasing hazard rates may arise when there is natural aging or wear. Decreasing hazard functions are much less common but find occasional use when there is a very early likelihood of failure, such as in certain types of electronic devices or in patients experiencing certain types of transplants. Most often, a bathtub-shaped hazard is appropriate in populations followed from birth.

The figure below hows the relationship between four parametrically specified hazards and the corresponding survival probabilities. It illustrates (a) a constant hazard rate over time (e.g. healthy persons) which is analogous to an exponential distribution of survival times, (b) strictly increasing (c) decreasing hazard rates based on a Weibull model, and (d) a combination of decreasing and increasing hazard rates using a log-Normal model. These curves are illustrative examples and other shapes are possible.

### Example

The simplest possible survival distribution is obtained by assuming a constant risk over time:

## Censoring and truncation

One of the distinguishing feature of the field of survival analysis is censoring: observations are called censored when the information about their survival time is incomplete; the most commonly encountered form is right censoring.

Right censoring occurs when a subject leaves the study before an event occurs, or the study ends before the event has occurred. For example, we consider patients in a clinical trial to study the effect of treatments on stroke occurrence. The study ends after 5 years. Those patients who have had no strokes by the end of the year are censored. Another example of right censoring is when a person drops out of the study before the end of the study observation time and did not experience the event. This person’s survival time is said to be censored, since we know that the event of interest did not happen while this person was under observation.

Left censoring is when the event of interest has already occurred before enrolment. This is very rarely encountered.

In a truncated sample, we do not even “pick up” observations that lie outside a certain range.

Unlike ordinary regression models, survival methods correctly incorporate information from both censored and uncensored observations in estimating important model parameters

## Non-parametric Models

The very simplest survival models are really just tables of event counts: non-parametric, easily computed and a good place to begin modelling to check assumptions, data quality and end-user requirements etc. When no event times are censored, a non-parametric estimator of S(t) is 1 − F(t), where F(t) is the empirical cumulative distribution function.

### Kaplan–Meier

When some observations are censored, we can estimate S(t) using the Kaplan-Meier product-limit estimator. An important advantage of the Kaplan–Meier curve is that the method can take into account some types of censored data, particularly right-censoring, which occurs if a patient withdraws from a study, is lost to follow-up, or is alive without event occurrence at last follow-up.

Suppose that 100 subjects of a certain type were tracked over a period of time to determine how many survived for one year, two years, three years, and so forth. If all the subjects remained accessible throughout the entire length of the study, the estimation of year-by-year survival probabilities for subjects of this type in general would be an easy matter. The survival of 87 subjects at the end of the first year would give a one-year survival probability estimate of 87/100=0.87; the survival of 76 subjects at the end of the second year would yield a two-year estimate of 76/100=0.76; and so forth.

But in real-life longitudinal research it rarely works out this neatly. Typically there are subjects lost along the way (censored) for reasons unrelated to the focus of the study.

Suppose that 100 subjects of a certain type were tracked over a period of two years determine how many survived for one year and for two years. Of the 100 subjects who are “at risk” at the beginning of the study, 3 become unavailable (censored) during the first year and 3 are known to have died by the end of the first year. Another 2 become unavailable during the second year and another 10 are known to have died by the end of the second year.

Kaplan and Meier proposed that subjects who become unavailable during a given time period be counted among those who survive through the end of that period, but then deleted from the number who are at risk for the next time period.

The table below shows how these conventions would work out for the present example. Of the 100 subjects who are at risk at the beginning of the study, 3 become unavailable during the first year and 3 die. The number surviving the first year (Year 1) is therefore 100 (at risk) – 3 (died) = 97 and the number at risk at the beginning of the second year (Year 2) is 100 (at risk) – 3 (died) – 3 (unavailable) = 94. Another 2 subjects become unavailable during the second year and another 10 die. So the number surviving Year 2 is 94 (at risk) – 10 (died) = 84.

As illustrated in the next table, the Kaplan-Meier procedure then calculates the survival probability estimate for each of the t time periods, except the first, as a compound conditional probability.

The estimate for surviving through Year 1 is simply 97/100=0.97. And if one does survive through Year 1, the conditional probability of then surviving through Year 2 is 84/94=0.8936. The estimated probability of surviving through both Year 1 and Year 2 is therefore (97/100) x (84/94)=0.8668.

## Incorporating covariates: proportional hazards models

Up to now we have not had information for each individual other than the survival time and censoring status ie. we have not considered information such as the weight, age, or smoking status of individuals, for example. These are referred to as covariates or explanatory variables.

### Cox Proportional Hazards Modelling

The most interesting survival-analysis research examines the relationship between survival — typically in the form of the hazard function — and one or more explanatory variables (or covariates).

where λ0(t) is the non-parametric baseline hazard function and βx is a linear parametric model using features of the individuals, transformed by an exponential function. The baseline hazard function λ0(t) does not need to be specified for the Cox model, making it semi-parametric. The baseline hazard function is appropriately named because it describes the risk at a certain time when x = 0, which is when the features are not incorporated. The hazard function describes the relationship between the baseline hazard and features of a specific sample to quantify the hazard or risk at a certain time.

The model only needs to satisfy the proportional hazard assumption, which is that the hazard of one sample is proportional to the hazard of another sample. Two samples xi and xj satisfy this assumption when the ratio is not dependent on time as shown below:

The parameters can be estimated by maximizing the partial likelihood.

# Changing the Game with Data and Insights – Data Science Singapore

Another great Data Science Singapore (DSSG) event! Hong Cao from McLaren Applied Technologies shared his insights on applications of data science at McLaren.

The first project is using economic sensors for continuous human conditions monitoring, including sleep quality, gait and activities, perceived stress and cognitive performance.

Gait outlier analysis provides unique insight on fatigue levels while exercising, probability of injury and post surgery performance and recovery.

A related study looks into how biotelemetry assist in patient treatment such as ALS (Amyotrophic Lateral Sclerosis) disease progression monitoring. The prototype tools collect heart rate, activity and speech data to analyse disease progression.

HRV (Heart Rate Variability) features are extracted from both the time and from the frequency domains.

Activity score is derived from the three-axis accelerometer data.

The second project was a predictive failure POC, to help determine the condition of Haul Trucks in order to predict when a failure might happen. The cost of having an excavator go down in the field is $5 million a day, while the cost of losing a haul truck is$1.8 million per day. If you can prevent it from going down in the field, that makes a huge difference

# Data Scientists, With Great Power Comes Great Responsibility

It is a good time to be a data scientist.

In 2012 the Harvard Business Review hailed the role of data scientist “The sexiest job of the 21st century”. Data scientists are working at both start-ups and well-established companies like Twitter, Facebook, LinkedIn and Google receiving a total average salary of $98k ($144k for US respondents only) .

Data – and the insights it provides – gives the business the upper hand to better understand the clients, prospects and the overall operation. Till recently, it was not uncommon for million- and -billion- dollar deals to be accepted or rejected based on the intuition & instinct. Data scientists add value to the business by leading to informed and timely decision-making process using quantifiable, data driven evidence and by translating the data into actionable insights.

So you have a rewarding corporate day job, how about doing data science for social good?

You have been endowed with tremendous data science and leadership powers and the world needs them! Mission-driven organizations are tackling huge social issues like poverty, global warming and public health. Many have tons of unexplored data that could help them make a bigger impact, but don’t have the time or skills to leverage it. Data science has the power to move the needle on critical issues but organizations need access to data superheroes like you to use it

DataKind Blog

There are a few of programs that exist specifically to facilitate this, the United Nations #VisualizeChange challenge is the one I’ve just taken.

As the Chief Information Technology Officer, I invite the global community of data scientists to partner with the United Nations in our mandate to harness the power of data analytics and visualization to uncover new knowledge about UN related topics such as human rights, environmental issues, and political affairs.

Ms. Atefeh Riazi – Chief Information Technology Officer at United Nations

The United Nations UNITE IDEAS published a number of data visualization challenges. For the latest challenge, #VisualizeChange: A World Humanitarian Summit Data Challenge , we were provided with unstructured information from nearly 500 documents that the consultation process has generated as per July 2015. The qualitative data is categorized in emerging themes and sub-themes that have been identified according to a developed taxonomy. The challenge was to process the consultation data in order to develop an original and thought provoking illustration of information collected through the consultation process.

Over the weekend I’ve built an interactive visualization using open-source tools (R and Shiny) to help and identify innovative ideas and innovative technologies in humanitarian action, especially on communication and IT technology. By making it to the top 10 finalists, the solution is showcased here, as well as on the Unite Ideas platform and other related events worldwide, so I hope that this visualization will be used to uncover new knowledge.

#VisualizeChange Top 10 Visualizations

Opening these challenges to the public helps raising awareness – during the process of analysing the data and designing the visualization I’ve learned on some of most pressing humanitarian needs such as Damage and Need Assessment, Communication, Online Payment and more and on the most promising technologies such as Mobile, Data Analytics, Social Media, Crowdsourcing and more.

#VisualizeChange Innovative Ideas and Technologies

Kaggle is another great platform where you can apply your data science skills for social good. How about applying image classification algorithms to automate the right whale recognition process using a dataset of aerial photographs of individual whale? With fewer than 500 North Atlantic right whales left in the world’s oceans, knowing the health and status of each whale is integral to the efforts of researchers working to protect the species from extinction.

Right Whale Recognition

There are other excellent programs.

The DSSG program ran by the University of Chicago, where aspiring data scientists take on real-world problems in education, health, energy, transportation, economic development, international development and work for three months on data mining, machine learning, big data, and data science projects with social impact.

DataKind bring together top data scientists with leading social change organizations to collaborate on cutting-edge analytics and advanced algorithms to maximize social impact.

Bayes Impact  is a group of practical idealists who believe that applied properly, data can be used to solve the world’s biggest problems.

Are you aware of any other organizations and platforms doing data science for social good? Feel free to share.

Tools & Technologies

R for analysis & visualization
Shiny.io for hosting the interactive R script
The complete source code and the data is hosted here

# How Individualized Medical Geographic Information Systems and Big Data will Transform Healthcare

The modern healthcare system is experiencing a significant disruptive change consistent with the technological shifts that have altered the communications, publishing, travel, and banking industries. The roots of this transformation can be found across many topics including the Quantified Self movement, mobile technology platforms, wearable computing, and rapid advances in genomic and precision medicine.

Here are some conclusions and thoughts following a seminar held last week at the Institute of Systems Science at the National University of Singapore (NUS).

Dr. Steven Tucker, MD, shared his view of how individualised medical Geographic Information Systems (GIS) will transform medicine:

“This is a medical Geographical Information Systems, compared to a Google map of the individual. We can do this with biology and health”.

There’s much more to it than just collecting data points from your mobile phone or from a wearable device:

“Your DNA plus your bacteria and your epigenome (a record of the chemical changes to the DNA and histone proteins of an organism) exposures, protein and unique appearance go to making you a unique biological individual. We are not in a standard distribution anymore… That is transformative” said Dr. Tucker.

These singular, individual data and information set up a remarkable and unprecedented opportunity to improve medical treatment and develop preventive strategies to preserve health.

But how is this related to Big Data?

Michael Snyder of Stanford University was one of the first humans to have such a construct made of himself. Snyder had not only gene expression analysed, but also proteomic and metabalomic sequencing as well, as described in Topol’s recent article in Cell, “Individualized Medicine from Prewomb to Tomb” (March 27, 2014).

The procedure required one Terabyte of storage for the DNA sequence, two Terabytes for the epigenomic data, and three Terabytes for the microbiome, the article said. Storage requirements grow quickly to one Petaybytes (1,000 Terabytes) for 100 people, so do the math of how many Petabytes of storage are required for storing the individual data of millions of people.

“The longer you can follow a person, the more you’ll learn about their health states, the more you can do to help them stay healthy. That’s the way it should be.” (Michael Snyder, Making It Personal).