Estimating Companies’ Survival in Financial Crisis

Using the Cox Proportional Hazards Model

By: Niklas Andersson

Independent Thesis Advanced Level

Department of Statistics

Supervisor: Inger Persson

2014

i

Abstract

This master thesis is aimed towards answering the question What is the contribution from

a company’s sector with regards to its survival of a financial crisis?with the sub question

Can we use survival analysis on financial data to answer this?. Thus survival analysis

is used to answer our main question which is seldom used on financial data. This is

interesting since it will study how well survival analysis can be used on financial data at

the same time as it will evaluate if all companies experiences a financial crisis in the same

way. The dataset consists of all companies traded on the Swedish stock market during

2008. The results show that the survival method is very suitable the data that is used.

The sector a company operated in has a significant effect. However the power is to low

too give any indication of specific differences between the different sectors. Further on

it is found that the group of smallest companies had much better survival than larger

companies.

Keywords: Survival Analysis, Survival Data, Time to Event, 2008 Financial Crisis

and Swedish Stock Market.

Acknowledgments

First and foremost I would like to thank my family for their continuous support through-

out my academical studies. Secondly I wold like to thank my supervisor, Inger Persson,

for her support, inputs and dedication.

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Theory 5

2.1 Survival Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Survival Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Hazard Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.4 Estimations of Survival and Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.5 Test of Group Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.6 The Cox Proportional Hazards model . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.7 Assumptions, Goodness of Fit and Diagnostics . . . . . . . . . . . . . . . . . . . . 25

2.2 Key Performance Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.1 Liquidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.2 Solvency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Data 36

3.1 The Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 The Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 An Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Results 40

4.1 Survival Functions and Cumulative Hazards . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1.1 By Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.2 By Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Cox Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 Individual Study Start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.2 Day One Study Start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.3 Final Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Summary and Conclusion 63

6 Bibliography 65

7 Appendix 1

7.1 Descriptive Statistics for stratums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

7.2 Cummulative Hazards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

7.3 Companies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

7.4 Exampel Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

ii

Introduction

Below follows a short background to this thesis outlining the problem and environment

that we will work with. This is followed by a short presentation of the method that will

be used and the main question that the thesis will strive towards answering.

1.1 Background

Stock markets have always been susceptible to financial crisis and this in turn affects each

company with stocks traded in the market. Each company will do its best to survive a

financial crisis and keep the investors beveling in it. In extension the company is trying

to keep the price of its stock from falling too far.

The reasons for financial crises have been plenty during the history and although not

all financial crises are rooted in a stock market many have come to affect these markets.

Kindleberger and Aliber (2005, ch. 1) report of the first known financial crisis recorded,

the ”Tulipmania” that originated in overinflated prices on tulip bulbs. Kindleberger and

Aliber (2005, ch. 1) also tell of more recent financial crises like the 1920s stock price

bubble or even more close at heart the real estate and stock crisis during the 80s and the

beginning of the 90s that affected Sweden, Finland and Norway amongst other nations.

In times like these we might see stock prices falling rapidly and far with huge con-

sequences for companies and financial institutions. But we do not expect every stock

to experience the same price fall, after all no companies are the same. Therefore their

reaction and to what degree they are affected by a financial crises differ from company

to company. We also expect companies and the price of their stock to react differently

from crisis to crisis since in the same way as no company is the same as another there

are always parameters that make each crisis unique .

During 2008 the world’s stock markets went trough a recession leading to huge price

falls in stock prices. Sweden was no exception and the general price index of the OMX

1

1. INTRODUCTION 2

Stockholm exchange closed at a mere 50.2% of its notation at the beginning of the year.

The recession affected all stocks traded on the Stockholm market and no stock that was

traded in the beginning of the year closed on a positive note by the end of december.

However as we have stated stocks are not the same and there ought to be differences

between them. One could speculate that the size of the company and in what sector

it operates could make a difference in the stock’s sensitivity for a recession like the one

in 2008. There were most likely also other factors, such as Key Performance Indicators

(KPI), specific to each company that affected to what extent the price on their stock fell.

After all it is the traders that set the price and their confidence and belief in a company’s

stock determines what price they are wiling to pay. Figure: (1.1)below shows that there

is some difference in how the index price for different sectors have developed during the

time of interest in this thesis.

Figure 1.1: Section indexes during 2008 (base value 2008-01-01)

As of now we do not know if these differences are significant, if the sectors were

effected differently during this time period. The primary interest of this thesis will be

to determine if the different sectors played a roll in how well a stock resisted price fall

during 2008.

1. INTRODUCTION 3

1.2 Method

What we want to find out in this thesis is if there were differences in each company’s

stock’s resistance to price fall during the 2008 recession, specifically if there where dif-

ferences due to different sectors/industries that the companies operated in. In order to

do this we have chosen to use the statistical methods of survival analysis and the Cox

proportional hazards modeling.

We will be working with survival time data or Time to Event data where the depen-

dent variable is the time from when we start observing something until a specific event

occurs. When working with this kind of data the Cox proportional hazards model is the

by far most popular method. In fact according to Allison (2010, ch. 5) the original paper

written by Sir David Cox in 1972 where he presents the proportional hazards method

had been cited over 1000 times in 2009 making it the most cited paper in statistics and

earning it a place on the top 100 most cited papers of science. There are many reasons

for this popularity and the foremost is probably the fact that the model itself does not

need any information about some underlying distribution that we expect the survival

time to follow. This makes the Cox method semi-parametric and sets it aside from other

parametric models where you need to make a decision about the underlying distribution.

This makes the Cox proportional hazards model more robust. How this semi-parametric

model works we will see in Section: (2.1.6). There are also other reasons for the methods

popularity and amongst them is the fact that it is relative easy to include time depen-

dent covariates and it can use both discrete and continuous measurements of the Time

to Event. However this said, the Cox method is not a universal method and there are

times where a parametric model based on known distributions is preferred. Also using

a semi-parametric method will result in higher standard deviations while estimating pa-

rameters as we will see in Section: (2.1.4). (Allison, 2010, ch. 5)

In this thesis the Cox method will be used since the the underlying hazards is un-

known in our data and the dependent variable will be a Time to Event variable since we

are interested in effects that make a company survive longer in a financial crisis.

1. INTRODUCTION 4

In the research leading up to this paper I have found few cases of Time to Event data

and thus Cox proportional hazards modeling being used on financial data. The lack of

previous research suggests that it is not commonly used in this field however Ni (2009)

has successfully used this method in a paper focused on the effect from a number of share

(company stocks) related KPI:s. She defined a certain price fall in the companies’ stocks

(same for all stocks) to be the event of interest (essential in survival analysis) in order

to find each company’s survival time. In essence the method she used is the same as the

one used in this paper however the focus will be shifted towards the contribution of the

sector in which a company operates.

With the given background this thesis will strive towards answering the following

questions: What is the contribution from a company’s sector with regards to its survival

of a financial crisis? with the sub question Can we use survival analysis on financial

data to answer this?

As already mentioned all crises are not the same thus this thesis will be limited to

the crisis and following price fall that took place during 2008. In the same way not

all markets are the same and a second limitation will be that only stocks traded on the

OMX Stockholm exchange market will be included in the study. The dataset will initially

consist of stocks traded at the beginning of 2008 and they will be studied throughout the

entire year.

Theory

In this section we will look closer at the theories that will be used as a foundation in

this thesis. We will cover theory regarding survival analysis in general and the Cox

Proportional Hazards model as well as looking at financial theory regarding KPI:s that

can influence the investors’ decisions.

2.1 Survival Analysis

In survival analysis we are mainly concerned with studying time to event data. That

is, we have a specific event that either will or will not happen for the observations of

our study. This kind of data can occur in a wide number of fields such as medicine,

engineering and economics. A simple example of an event can be the death of a patient

or a specific diagnosis within the field of medicine while for an engineer it could mean the

breaking of a component in a machine. It is then the time to this specific event, survival

time that we will study in order to find the survival time of each observation. Of course

the event does not have to be something negative but could rather be a remission after

a treatment. (Klein and Moeschberger, 2005, ch. 2.1)

Throughout this theory section we will give examples on graphical and numerical re-

sults by applying the theories on a dataset created for this purpose. The event in this

dataset is, as it will be when we look at our main dataset, negative and is concerning

the time it takes for a mechanical component to brake. The dataset and its variables is

presented in depth in Appendix: (7.4).

5

2. THEORY 6

2.1.1 Survival Function

We will let the time to our event of interest occurs be T , then we can characterize the

distribution of T by using the Survival Function. This function tells us the probability

of an individual (observation) surviving beyond a specific time, t, or the probability of

experiencing the event after the time t (Klein and Moeschberger, 2005, ch. 2.2). We can

define the survival function as:

S(t) = Pr(T > t) (2.1)

The reason for the the survival function being Pr(T > t) rather than Pr(T = t) is that

some observations in the study will not experiencing the event during our study and thus

there time to event is unknown. These observations are called censored observations and

in Section: (2.1.3) we will lock closer at why and how censoring occur

The survival function itself is in fact the complement of the cumulative distribution

function [S(t) = 1− F (t)] or just as well the integral of the probability density function[S(t) =

∫∞tf(u)du

]. If T is continous then S(t) is also continous and a strictly decreasing

function. At time 0 the survival probability will always be 1 and as time goes towards in-

finity the probability of survival will go towards 0.(Klein and Moeschberger, 2005, ch. 2.2)

(Klein and Moeschberger, 2005, ch. 2.2) also report what happens when T is not

continuous. This is often the case in survival analysis due to lack of precision in the

measurement of time. A medical study might only measure for their event at check-ups

done at some time interval, such as once a year. In that case the researcher are only able

to determine that the event has happened between the last check-up and this check-up.

Then we have to tweak the definition of our survival function. As we will se in Section:

(3) this is the case in our study since we only can observe the event once a day. Assume

that time to event is discrete and can take on the values ti, i = 1, 2, 3, . . . then:

p(tj) = Pr(T = ti), i = 1, 2, 3, . . . where t1 < t2 < t3 < · · ·

which gives

S(t) = Pr(T > t) =∑ti>x

p(ti) (2.2)

2. THEORY 7

2.1.2 Hazard Function

Another way of looking at the survival time is trough the Hazard Function or Hazard

Rate [h(t)]. This function tells us the probability of experiencing the event in the next

instant conditioned on the fact that the event has not happened up to that point in

time. This in turn describes the probability of experiencing the event over time and how

that probability changes with time. The hazard rate is defined as (see e.g Klein and

Moeschberger (2005, ch. 2.3)):

h(t) = lim∆t→0

P [t ≤ T < t+ ∆t|T ≥ t]

∆t(2.3)

and if T is continuous then,

h(t) =f(t)

S(t)= −∂ln[S(t)]

δt(2.4)

Cumulative Hazard Function

Closely related to the hazard function is the Cumulative Hazard Function which as the

name suggests accumulate the hazard rates over time. This definition is also given by

Klein and Moeschberger (2005, ch. 2.3) for a continuous T :

H(t) =

∫ t

0

h(u)du = −ln[S(t)] (2.5)

where

S(t) = exp[−H(t)] = exp

[−∫ t

0

h(u)du

](2.6)

If we do not have continuous T then the cumulative hazard function is

H(t) =∑ti≤t

h(ti) (2.7)

2. THEORY 8

2.1.3 Censoring

When working with survival data one thing will almost always be a problem to consider,

censoring. Kleinbaum and Klein (2012, ch. 1.2) describe that censoring occurs when we

have some information about an observation’s survival time without knowing the exact

time. A simple example is when a patient in a study is no longer followed while the event

of interest has not yet happened. In this case we know that the patient ”survived” up

until the point where we stopped following that patient but not for how long afterwards.

Kleinbaum and Klein (2012, ch. 1.2) give three general reasons for censoring.

• The study ends without the event occurring for an observation

• The observation is lost to follow-up

• When working with people they can choose to withdraw from the study

In the end of a study all observations will either have experienced the event itself or

they will be censored. However not all cases of censoring are the same and it is common

to divide censoring into two major categories, right and left censoring which is determined

by how the data is collected.

Right Censoring

Right censoring is when the event is observed only if it occurs before a predetermined

time. This is the simplest form of right censoring and is often called Type I censoring.

An example of this is when a study starts off with a number of patients in whom we

await an event. However due to cost the study might end before all individuals have

experienced the event thus those individuals will be right censored. In this simplest form

all individuals that are censored when the study ends will have been observed for the

same length (from the start to the end of the study) and have the same censoring time.

(Klein and Moeschberger, 2005, ch. 3.2)

Similar to the Type I censoring is the generalized Type I censoring which as in the

basic Type I case has a predetermined end date of the study. In this case though the

starting date of all individuals are not the same, rather they enter the study at individual

times. As a consequence each observation will have a different fixed censoring time and

2. THEORY 9

when the study ends those observations that are time censored will not have the same

study time. (Klein and Moeschberger, 2005, ch. 3.2)

Another common type of right censoring that is often used within the field of tech-

nology according to Klein and Moeschberger (2005, ch. 3.2) is the Type II censoring. In

this type of censoring a number of observations (n) is entered in the study at the same

time but instead of a pre-determined time at which the study ends a number of events

(r) is chosen. We have the condition that r < n in this case and when r amount of the

observations have experienced the event the study ends. One property of this kind of

censoring is that the data will consist of the r smallest event times from a random sample

n.

A pseudo case of right censoring that is also common in survival analysis which can

coexist with many other types of censoring is Random Censoring. Random censoring oc-

cur when we, for some reason, no longer can study our observation/individual although

we know that it has not yet experienced the event. Random censoring implies that the

censoring time and the event time are independent. This might come about due to a

patient in a medical study moving outside of the area that hospital performing the study

operated in an thus the patient is lost to further follow ups which in turn leads to the

event never being observed although the study itself is still ongoing. These cases are

censored at the time when they are lost and not, if for example type I censoring is used,

at the end of the study. (Klein and Moeschberger, 2005, ch. 3.2)

In our study random censoring will be present. This might be due to a number of

things such as merges, buy ups, or termination of trade. Other types of censoring exist

such as Left Censoring and Truncation but these will not be precent in this thesis. For

information about other types of censoring se i.g Klein and Moeschberger (2005, ch 3) or

Kleinbaum and Klein (2012, ch 1)

2. THEORY 10

2.1.4 Estimations of Survival and Hazard

In the next two sections we will look closer at estimations of survival time and hazard

rates. There are two different methods that we will cover which are both applicable when

working with right censured data. The focus on right censured data is due to the fact

that this is the kind of censoring present in our dataset. The dataset itself is outlined in

detail in Section: (3). The two methods that we will look at are the Kaplan-Meier (K-M)

or Product-Limit estimator and the Nelson-Aalen (N-A) estimator. As we will see both

these methods can be used to derive the survival function and the cumulative hazard

function. Since our dataset will be right censored we will only focus on these methods.

When presenting and working with the methods we will use the following notations and

assumptions. We will have a total of n individuals or observations and each of those will

hold information about their time to event ti as well as if they experienced an event or were

censored. The data will be discrete thus allowing for possible ties and the event/censoring

can only be observed at specific time points thus t1 < t2 < t3 < · · · < tD. t1 represent the

first time at which we can observe an event, ti the ith time point and tD the time point at

which the last event is observed. At a given time point we know the number of individuals

that are risking to experience the event Yi and we can observe di number of events. If

di > 1 we will have a tie at time point ti. The ratio diYi

can thus be interpreted as the

conditional probability of experiencing the event at time i if you have not experienced it

at any previous times. This quantity will be used in both methods in order to estimate

the survival function S(t) and the cumulative hazard function H(t). These notations are

equivalent to those used by Klein and Moeschberger (2005) in their presentation of these

estimates with the assumption that the censoring time is unrelated to the event time

(there is no information about the time to the event for a censored observation).

Kaplan–Meier

The Product-Limit estimator of the survival function was introduced by Kaplan and

Meier (1958) and thus is also called the Kaplan-Meier estimator. This estimator has a

fault that we need to be aware of, it is only well defined for the time interval that has

been observed in the data, t1 − tD. In other words beyond the largest observed time the

estimations are not reliable (Klein and Moeschberger, 2005, ch. 4.2). The K-M estimator

2. THEORY 11

is defined as (see e.g Klein and Moeschberger (2005, ch. 4.2)):

S(t) =∏ti≤t

[1− di

Yi

], for t1 ≤ t (2.8)

If the condition of t1 ≤ t is not met then S(t) = 1 , that is the probability of survival

before the first observation is 1. The variance of these estimates can be derived trough

the Greenwood’s formula (see e.g Klein and Moeschberger (2005, ch. 4.2)):

V [S(t)] = S(t)2∑ti≤t

diYi(Yi − di)

(2.9)

Earlier we have pointed out the connection between the Survival function and Hazard

Rates trough which we can use the K-M estimator to estimate the cumulative hazard

function as well as the survival function. By taking −ln to the estimated survival func-

tion at each time point we can acquire an estimation of the cumulative hazard ratio for

that time point, −ln[S(t)] = H(t).

Example: In Figure: (2.1) the survival function for the example data is estimated

using the Kaplan-Meier estimator. The time to event data is discreet and the survival

function have a step or stair shaped downward slope. The survival function is as discussed

in Section: (2.1.1) strictly decreasing. The last observation (in time) is censored thus the

survival function ends in a horizontal line.

Figure 2.1: Estimated survival function on the exaple data

2. THEORY 12

Nelson–Aalen

Although the K-M estimator can be used to estimate the cumulative hazard rate the

Nelson-Aalen estimator of the cumulative hazard is considered to have better low sample

properties and will therefore be used in this thesis (Klein and Moeschberger, 2005, ch. 4.2).

The estimator was developed by Nelson (1972) and Aalen (1978) where Aalen in the same

publication gave the expression for the variance of this estimator that is given in (2.11).

The estimator itself is given below in (2.10) as it is presented by Klein and Moeschberger

(2005, ch. 4.2).

S(t) =∑ti≤t

[1− di

Yi

], for t1 ≤ t (2.10)

If the condition of t1 ≤ t is not met then H(t) = 0 , that is the cumulative hazard

before the first observation is 0.

σ2H(t) =

∑ti≤t

diY 2i

(2.11)

As with the K-M estimator this estimator is well defined up to the largest observation

of t. Just as we could transform S(t) to be an estimator of H(t) trough −ln[S(t)] = H(t)

we can transform the N-A estimator to be an alternative estimator of the survival func-

tion, S(t) = −exp[H(t)] . (Klein and Moeschberger, 2005, ch. 4.2)

Example: Below in Figure: (2.2) the cumulative hazards are estimated on the exam-

ple data using the Nelson-Aalen estimator. The cumulative hazards is strictly increasing

since it is the cumulative of the estimated hazard at each point in time. The last obser-

vation (in time) is censored thus the cumulative hazards ends in a horizontal line in the

same way as the survival function in Figure: (2.1).

2. THEORY 13

Figure 2.2: Estimated cumulative hazards on the exaple data

2.1.5 Test of Group Differences

When we are studying a variable that can be divided into two or more groups we might

be interested in testing if there is a difference between these groups regarding their sur-

vival/hazard. Example of these kinds of studies where different survival between groups

might be of interest is in the field of medicine where you can have three different groups

of patients all suffering of the same disease but treated with different medicines. Thus the

primary interest of that study might be to determine if there is a difference in survival

between the groups and in extension possibly also between the medicines.In the same

way an engineer might desire to test if there is a difference between how long a com-

ponent lasts before breaking in a machine depending on which manufacturer made that

component. In order to keep the notation equivalent to that of Klein and Moeschberger

(2005) we will let K represent the groups where Kj represents the jth group. Also τ will

represent the last point in time where there is one observation still at risk for all groups.

The test itself focuses on the hazard at each time point where there is an observed

event and determines if there is a differenc between the groups at that time point. A weak-

ness with this kinds of multigroup tests that is pointed out by Klein and Moeschberger

(2005, ch. 7.3) is that it tests if at least one population differs from the others at any

2. THEORY 14

point in time. In other words while it can detect if there is a significant difference between

the groups, it only tells us if there is a difference between one, not which, group and the

rest. The hypothesis is:

H0 : h1(t) = h2(t) = h3(t) = · · · = hK(t), for all t ≤ τ

H1 : at least one of hj(t) differ for one, or more, t ≤ τ(2.12)

with the test function

Zj(τ) =D∑i=1

Wj(ti)

[dijYij− diYi

], J = 1, . . . , K. (2.13)

This test function will be a corner peace in creating the test statistic. In (2.13) Wj(ti)

is representing a weight that would/could be different for each group. However this is not

the case in the most common variations of this test, some of which we will cover below,

thus (2.13) can be simplified. Those tests that will concern us use a weighting function

where Wj(ti) = W (ti)Yij thus W (ti) is the same for all groups and Yij is the numbers at

risk in the jth group at the tthi time. Using this we can simplify (2.13) to (2.14). (Klein

and Moeschberger, 2005, ch. 7.3)

Zj(τ) =D∑i=1

W (ti)

[dij − Yij

(diYi

)]j = 1, . . . , K. (2.14)

Further on the variance and covariances of Zj(τ) from (2.14) are given by:

σjj =D∑i=1

W (ti)2YijYi

(1− Yij

Y i

)(Yi − diYi − 1

)di, j = 1, . . . , K (2.15)

and

σjg = −D∑i=1

W (ti)2YijYi

YigYi

(Yi − diYi − 1

)di, g 6= j (2.16)

In both 2.15 and 2.16 we find the term(Yi−diYi−1

)which will be equal to one in all cases

except when two observations have the same time to event. Thus this term corrects for

possible ties. All possible Zj(τ) are linearly dependent and the test statistic that we

will use is produced by excluding any one of the Z ′js. From this choice we can get the

estimated (K − 1)(K − 1) variance-covariance matrix Σ. This matrix will then be used

2. THEORY 15

in the test statistic (2.17). The test statistic is on quadratic form and when the null

hypothesis is true and we have a large sample it is chi-square distributed with K − 1

degrees of freedom. That means that when using significance level α to test H0 the test

will be rejected when χ2 is larger than the αth upper percentage of a χ2K−1 distribution.

(Klein and Moeschberger, 2005, ch. 7.3)

χ2 = (Z1(τ), . . . , Zk−1(τ))Σ−1(Z1(τ), . . . , Zk−1(τ))t (2.17)

In the special case where K = 2 the test statistic in (2.17) simplifies to:

Z =

D∑i=1

W (ti)[di1 − Yi1

(diYi

)]D∑i=1

W (ti)2 Yi1Yi

(1− Yi1

Y i

) (Yi−diYi−1

)di

=Z1(τ)

σ1(2.18)

Where Z is standard normally distributed.

There is a multitude of possible variation on this test due to the weight function and

altering this weight function will give the test different properties. The arguably most

simple weight is 1 where all time points have the same importance. This example is

commonly known as the Log-Rank test. Below follows a list of some variations on the test

presented by Klein and Moeschberger (2005, ch. 7.3) and Kleinbaum and Klein (2012,

ch. 2.5-2.6).

Log-Rank test

W (t) = 1: All time points have the same weight which makes it ideal when the hazard

rates in all groups are proportional.

Gehan’s or Wilcoxon Test

W (ti) = Yi: More weight is laid at time points where more people are in the risk group.

That is, weighted towards earlier failures. This test is best if we have reasons to suspect

that ”treatment” effect is strongest early on in the research. Focus is on differences early

in the study..

2. THEORY 16

Tarone-Ware test

W (ti) = f(Yi) where f(y) =√y: This test puts most weight to differences between the

observed and expected events at time points where there is the most data.

In Section: (4.1) both the Log-Rank and Wilcoxon tests will be presented and looked

at when determining significant difference between different sectors and company sizes.

This in order to find both general differences throughout time with the unweighted Log-

Rank test at the same time as possible differences early on can be detected using the

Wilcoxon test. In our study we will have the most data early on thus the Tarone-Ware

test wild also put emphasis on early differences and it will not be used in favor of the

Wilcoxon test.

Example: Now lets implement this group test on our example data. The Factor vari-

able consist of two different entries Factor1 and Factor2 which we can stratify the survival

time by. First in order to get a graphical image of the differences Figure: (2.3) shows the

estimated survival function stratified by the Factor variable.

Figure 2.3: Estimated survival functions on the exaple data stratifyed by Factory

2. THEORY 17

Since the red line for Factor2 is consistently below the line for Factor1 we can suspect

that Factor1 produces components with longer survival than Factor2. However we need

to look at the test results in Table: (2.1) to confirm this. Looking at Table: (2.1) we se

that nether of the test statistics are significant on the 5% level thus there are (so far) no

significant differences between the two factories.

Table 2.1: Tests of equality in the example data stratifyed by Factory

Test Chi-Square DF Pr> Chi-Square

Log-Rank 0.4644 1 0.4956

Wilcoxon 1.2141 1 0.2705

2.1.6 The Cox Proportional Hazards model

Sometime there is more than just some group variable that differentiate the individuals

in a study. In such a case the method presented in Section: 2.1.5 would not suffice. What

we need is a more complex model that can take multiple variables into account in the

same way as an ordinary linear model does. The proportional hazards model is such a

model and it was presented by Cox (1972). Thus the model is often referred to as the

Cox proportional hazards model.

Sticking with the notations of Klein and Moeschberger (2005) we will let Tj denote

the time individual j has been/was in the study, δj is indicating if the individual has

experienced the event (δj = 1 if the event has occurred) and Zj(t) is a vector consisting

of the k = 1, . . . , p observed covariates for individual j at time point t. In total there are

n individuals (j = 1, . . . , n). For simplistic reasons and due to the fact that there will be

no time dependent variable in this study we can let Zj(t) = Zj. Using these inputs we

can create the model that was suggested by Cox (1972) where h(t|Z) is the hazard rate

at time t and Z is the covariate vector.

h(t|Z) = h0(t)c(βtZ) (2.19)

2. THEORY 18

In this equation h0(t) represents a baseline hazard, that is, ignoring all other variables

there will still be some hazard attributed toward experiencing the event. The fact that

this proportional hazard model allows for the actual distribution of the survival to be

unknown, or unspecified, is a key feature that is part of what has made the model very

popular. This is also what makes the model semiparametric. By specifying h0(t) one

could acquire other linear models such as the exponential or Weibull models. (Allison,

2010, p. 126-127)

In Equation: (2.19) c(βtZ) is a known function and a requirement is that h(t|Z) must

be positive. According to Klein and Moeschberger (2005, ch. 8.1) a commonly chosen

function for c(βtZ) is:

c(βtZ) = exp(βtZ) (2.20)

This is also the function that Cox (1972) uses. In this equation,

βt =(β1 β2 β3 . . . βp

)and Z =

Z1

Z2

Z3

...

Zp

(2.21)

thus

βtZ = β1Z1 + β2Z2 + β3Z3 + · · ·+ βpZp =

p∑k=1

βkZk (2.22)

and we can rewrite Equation: 2.19 to

h(t|Z) = h0(t) exp(βtZ) = h0(t) exp

(p∑

k=1

βkZk

). (2.23)

2. THEORY 19

Hazard rate

So why is it called “proportional hazards”? This is due to the fact that we can use the

Cox Proportion Hazards model to look at the difference in hazards (of experiencing the

event) between two individuals with different covariate values via the hazard rate: (Klein

and Moeschberger, 2005, ch. 8.1)

h(t|Z)

h(t|Z∗)=

h0(t) exp

(p∑

k=1

βkZk

)h0(t) exp

(p∑

k=1

βkZ∗k

) = exp

[p∑

k=1

βk(Zk − Z∗k)

](2.24)

This is a ratio of the two different individuals’ hazards where Z and Z∗ are the

sets of covariates for the first and second individual respectively. This ratio will be

a constant which means that the hazard rates are proportional over time between two

different individuals. The proportionality of the hazards between to individuals is the key

assumption of this model. The easiest example of describing how to interpret this ratio

is when only one covariate differs between the two individuals. Klein and Moeschberger

(2005, ch. 8.1) use the case of one individual receiving a medicine (Z1 = 1) while the

other individual receives a placebo (Z1 = 0) and all other covariates (Z2 − Zp) have the

same value. In this case Equation: 2.24 would result in,

h(t|Z)

h(t|Z∗)= exp(β1) (2.25)

where exp(β1) is the risk of the event occurring if an individual have received the

medicine, in comparison with placebo. If the medicine improves the survival then the

ratio will be less than 1.

Estimating β using Partial Maximum Likelihood

We can estimate the values of β when there are no ties in the time to event by the partial

likelihood in Equation: 2.26. In the next section we will explore the method when ties are

pressent. The partial likelihood function was also introduced by D. R. Cox in his article

from 1972 and it is called partial due to the fact that the baseline hazard (h0(t)) can be

unknown and is left out of the covariate estimation. Due to this lack of information in

the likelihood function the standard errors will be larger than if the whole model had

been used to estimate the coefficients, however this is not a huge problem. The gain is

2. THEORY 20

that the estimates still will have good properties without any knowledge of the baseline

hazard and they will be consistent and asymptotically normal for large samples. (Allison,

2010, p. 126-129)

This likelihood can be derived from Equation: 2.23 as presented by Cox (1972):

L(β) =D∏i=1

exp

(p∑

k=1

βkZ(i)k

)∑

j∈R(ti)exp

(p∑

k=1

βkZjk

) (2.26)

In this equation Z(i)k is the kth covariate associated with the individual that have

failure time ti. The nominator only consists of the individual that experience the event

at time ti and the denominator is defined as the conditional set of individuals who were

still in the study just before time ti (the risk set R(ti)). Taking the log of the likelihood

in 2.26 gives:

logL(β) =D∑i=1

(p∑

k=1

βkZ(i)k

)−

D∑i=1

ln

∑j∈R(ti)

exp

(p∑

k=1

βkZjk

) (2.27)

The maximum likelihood (or partial maximum likelihood in this case) estimates can

be found by maximizing this equation. This is done by solving the derivative of 2.27

(known as the score function) with respect to the β. The score function is given by Cox

(1972) and presented in our notation by Klein and Moeschberger (2005, ch. 8.1):

Uh(β) =∂ logL(β)

∂βh=

D∑i=1

Z(i)h −D∑i=1

∑j∈R(ti)

Zjh exp

(p∑

k=1

βkZjk

)∑

j∈R(ti)exp

(p∑

k=1

βkZjk

) (2.28)

The estimates themselves are then found by solving Uh(β) = 0 for each h = 1, . . . , p.

Calculating these estimates can be done numerically using some kind of iterative process,

such as the Newton-Raphson metode, but in this thesis we will rely on the results that

are given to us by our software.

2. THEORY 21

In the next section we will look at ways to test the global hypothesis regarding the

estimations and in those tests we will need the information matrix. This matrix is the

negative of the 2nd derivative of Equation: 2.27. The information matrix will be de-

noted by I(β) which will be a p × p matrix where the (g, h)th element is: (Klein and

Moeschberger, 2005, ch. 8.1)

Igh(β) =D∑i=1

∑j∈R(ti)

ZjgZjh exp

(p∑

k=1βkZjk

)∑

j∈R(ti)exp

(p∑

k=1βkZjk

)

−D∑i=1

∑j∈R(ti)

Zjg exp

(p∑

k=1βkZjk

)∑

j∈R(ti)exp

(p∑

k=1βkZjk

)

×D∑i=1

∑j∈R(ti)

Zjh exp

(p∑

k=1βkZjk

)∑

j∈R(ti)exp

(p∑

k=1βkZjk

)

(2.29)

Ties

Estimating the β values using the Partial Maximum Likelihood method described above

is fine as long as there are no ties in the survival time. If any two events occur at the same

point in time then the estimation of β must be adjusted. There are a number of methods

to handle and set up the partial maximum likelihood when ties are present and Klein

and Moeschberger (2005, ch. 8.4) outline three classical methods, the Breslow, Efron and

Cox or Discrete methods.

Allison (2010, ch. 5) describes the Discreate method and another method in depth,

the Exact method. The Discrete method assumes that time is discrete hence if two events

happen at the same time there is no underlying order, both events really happened at

the same time. In most cases this is highly unlikely and tied events are often due to the

fact that we can not measure time exact enough. The Exact method assumes that there

is an underlying order in the events but since we can only observe event at time intervals

we can not determine which of the events occurred first if two events are recorded at the

same time point, ti. It is this exact method that will be relevant in our case since our

time data will be on a daily interval, thus two events can happen on the same day but

we will not be able to tell which occurred earlier in the day. We will shortly go trough

the theory of the Exact method next.

2. THEORY 22

The partial maximum likelihood derivation of β described in the previous section

needed all events to be individually ordered in Equation: (2.26) thus ties will pose a

problem when using this method. As explained the exact method assumes that ties are

due to inexact measurement of time and that there are a true underlying order. Using

this assumption and some basic probability theory the Exact method can estimate the

likelihood at those points in time where ties are present using all possible ways of ordering

the events. If we assume that 3 events are tied at the 5th point in time where events are

observed, t5, then there are 3! = 6 different ways these events could have been ordered

in reality. If we let each of these 6 set of orders be denoted by Ai and since each Ai is

mutually exclusive the sum of all Ai will be the union of these events. The likelihood at

the 5th point in time will then be L5 =6∑i=1

Pr(Ai).

Global test

The global hypothesis that we will be concerned with is H0 : β = β0 vs H1 : β 6= β0.

What we test is if our model with the covariates is significantly different from a reduced

model. This reduced model β0 can be seen as the model without the variables of interest

while β represent the model including the variables we want to test. β is called the full

model. (Kleinbaum and Klein, 2012, p. 103)

We will let b = (b1, b2, b3, . . . , bp)′ represent the estimated coefficients of β derived

using the maximum likelihood method presented in the previous section. There are three

different tests that we will look at in this section and use later on in our analysis in Sec-

tion: 4.2. All of these use this hypothesis but with slightly different test statistics. The

three tests presented below are those that will be reported by SAS when we run our model.

The Wald test uses the fact that b has a p-variate normal distribution for large

samples with mean β and variance-covariance matrix I−1(b). The test statistic is then χ2

distributed with p degrees of freedom where the test statistic is: (Klein and Moeschberger,

2005, p. 254)

χ2W = (b− β0)tI(b)(b− β0) (2.30)

2. THEORY 23

The Likelihood Ratio test is also χ2 distributed with p degrees of freedom for large

samples of n but uses the following test statistic: Klein and Moeschberger (2005, p. 254)

χ2LR = 2[logL(b)− logL(β0)] (2.31)

In this test statistic logL(b) represent Equation: 2.27 using the estimated values of

β while logL(β0) is Equation: 2.27 with the reduced model. According to Kleinbaum

and Klein (2012, p. 104) the Likelihood ratio test has better statistical properties than

the Wald test but in general, at least for large samples, they produce fairly similar test

statistics and rejections.

The last test that we will look at is the Score test which as the name suggests uses

the score function. In this test U(β) = (U1(β), . . . , Up(β))t and U(β) is asymptotically

p-variate normal with mean 0 and covariance matrix I(β). The test statistic is χ2

distributed with p degrees of freedom for large samples and is on the following form:

Klein and Moeschberger (2005, ch. 8.3)

χ2S = U(β0)

tI−1(β0)U(β0) (2.32)

In Section: (4.2) we will mainly focus on the Likelihood Ratio statistics due to the

better statistical properties but all of these statistics will be presented when a model has

been estimated. We will also perform some testes on a subset of variables from the model

and in those cases only the Wald statistic since it is readily available in SAS for these tests.

Example: In Table: (2.2) the result from an fitted Cox model is presented. In this

model all available covariates are used to estimate the survival of our components. All

covariates are significant with the largest P-value being 0.0155. Since the hazard ratio

for Usage is smaller than 1, 0.815, each extra step of 1 in the usage lowers the hazard

of breakage with a ratio of 0.815. A higher grade awarded in the tensile strength test

increases the hazard of the event occurring (lower test score is better). Factor1 is a

dummy for the Factor covariate and reference is Factor2. This means that the estimated

hazard ratio for Factor1 is the hazard for Factor1 components in comparison to those

from Factor2. In comparison to the result in Table: (2.1) we now have significant effect

from the Factor variable.

2. THEORY 24

Table 2.2: Estimated Cox model using example Data

Variable DF Parameter Standard Chi-Square Pr> Hazard Lower 95% Hazard Ratio Upper 95% Hazard Ratio

Estimate Error Chi-Square Ratio Confidence Limit Confidence Limit

Usage 1 -0.20406 0.05470 13.9180 0.0002* 0.815 0.729 0.905

Grade 1 0.08821 0.01872 22.2083 < 0.0001* 1.092 1.054 1.135

Factory1 1 -1.138955 0.47061 5.8573 0.0155* 0.320 0.122 0.783

∗=significant on the 5% level

The global test of the fitted Cox model is presented in Table: (2.3) and from the

results we can deduct that the model itself is significant since all three test statistics have

a P-value of less than 0.0001. In this table we can also se the Akaike information criterion

which we will look closer at in the next section.

Table 2.3: Tests of global hypothesis, β = 0, using example data

Test Chi-Square DF Pr> Chi-Square

Likelihood Ratio 41.9147 3 < 0.0001*

Score 40.3968 3 < 0.0001*

Wald 31.3107 3 < 0.0001*

Akaike information criterion (AIC) 134.695

∗=significant on the 5% level

2. THEORY 25

2.1.7 Assumptions, Goodness of Fit and Diagnostics

In this section we will look at methods to check the vital proportional hazards assump-

tions and evaluate how good the model is. We will begin by looking at two methods of

checking for proportional hazards then we will look at some information criteria before

we will examine some residuals.

Test of Proportional Hazards

Although we have not considered time dependent variables so far due to the fact that we

will not have any such variables in this thesis there is still a usage of them that we will

mention. Klein and Moeschberger (2005, ch. 9.2) describe how time dependent covariates

can be used to test the critical assumption of proportional hazards for our covariates that

is needed when modeling with the Cox proportional hazards model. In order to test if a

covariate, Z1, that is not time dependent violates the proportional hazards assumption

we first create a new covariate using Z1 which is artificially time dependent. Let this new

variable be Z2(t) = Z1 × g(t) where g(t) is a function of time such as g(t) = ln t. The

hazard rate at time t (Equation: 2.19) would be,

h(t|Z1) = h0(t) exp[β1Z1 + β2(Z1 × g(t))] (2.33)

and the hazard ratio between two individuals with different values on Z1 is

h[t|Z1]

h[t|Z∗1 ]= exp[β1(Z1 − Z∗1) + β2g(t)(Z1 − Z∗1)]. (2.34)

It is clear from this hazard ratio that it will only be dependent on the time through

g(t) if β2 6= 0. Thus by testing the hypothesis H0 β2 = 0 we will test if the proportional

hazards assumption holds for the covariate in mind. A rejection of H0 would mean that

the assumption is violated. What this means in practice is that we look at the proba-

bility of the estimated parameter for the new time dependent variable when estimating

the model. If that parameter is significant (different from zero) then the proportional

hazards assumption does not hold for that variable. In our case we will also test the as-

sumption on dummy variables and since all dummies have to be tested together a linear

test assuming that all the created time dependent dummy variables are zero together will

2. THEORY 26

be used.

Persson and Khamis (2008) present and test the statistical properties and powers for

a number of g(t) choices such as√t, et and ln(t). They found that both

√t and ln(t)

are good choices and since ln(t) also is the choice that Klein and Moeschberger (2005,

ch. 9.2) propose we will use g(t)=ln(t) when we test the proportional hazards assumption

in Section: (4.2).

There are different solutions to use if the proportional hazard assumptions fails when

testing it in this manner. One solution is to include the created time dependent variable in

the estimated model however that variable will be hart to interpret. For further discussion

on this se e.g. Klein and Moeschberger (2005, ch. 9.2).

Graphical Evaluation of Proportional Hazards, Arjas Plot

One way to graphically check for the proportional hazards assumption is trough the Ar-

jas plot first presented by Arjas (1988). The Arjas plot is not limited to checking the

assumption of proportional hazards, it can also be used to check the overall fit of a pro-

portional hazards regression model such as the Cox model. Let us assume that Z∗ is a

set of covariates included in the model and we are considering adding a new covariate, Z1.

Using the Arjas plot we can evaluate if Z1 should be included and if Z1 has proportional

hazards adjusted for the existing covariates. (Klein and Moeschberger, 2005, ch. 11.4)

There exists a number of methods to check for proportional hazards graphically as

described by Persson and Khamis (2007) in their comparison of different methods. They

recommend that the Arjas plot/method should be the preferred method of assessing

the proportional hazards assumption graphically except for the special case where the

hazard is strictly increasing. Thus this method will be used together with the test of pro-

portional hazard using a created function of time variable (described in previous section).

Klein and Moeschberger (2005, ch. 11.4) outline what is needed in order to produce

the Arjas plot and first we will need to estimate the proportional hazards model using

the covariate set Z∗. If Z1 is continuous we will need to group it into K levels making

2. THEORY 27

it discrete. For each of these levels (or the existing levels of a categorical variable) and

at every event time, ti, the Total time On Test (TOT) is calculated from the estimated

cumulative hazard rate of the model and the total number of observed events up to that

time point, N . The calculations of these two statistics are given next:

TOTg(ti) =∑

Z1, j=g

H(min(ti, Tj)|Z∗j) (2.35)

Ng(ti) =∑

Z1, j=g

δjI(Tj ≤ ti) (2.36)

What this means is that TOTg(ti) is the sum of the estimated cumulative hazards

for the the model using the covariate set Z∗ for all individuals in the group g where

g = 1, . . . , K up to time ti, or the largest event time in the group Tj. Ng(ti) is simply the

number of experienced events in the same group up to the same point in time. If Z1 is

redundant and not needed for the model’s fit a plot of Ng versus TOTg, the Arjas plot,

will result in a 45◦ line trough the origin. If we plot Ng versus TOTg for all the groups

this will result in K lines and if they are linear but differ from the 45◦ line Z1 should

be included in the model. Finally if the lines produced are not linear it indicates a vi-

olation of the proportional hazards assumption. (Klein and Moeschberger, 2005, ch. 11.4)

Example: We will look at an example of the Arjas plot using our example data. In

Figure: (2.4) the Factor variable is checked for proportional hazards. Since it consist of

Factory1 and Factory2 this grouping is insinuative to use in the Arjas test and looking

at Figure: (2.4) nether of the two factories seam to have proportional hazards, both lines

are curving.

2. THEORY 28

Figure 2.4: Arjas plot of the example data by Factory

Comparing Models using Information Criteria

In order to evaluate if one model is better than another, to evaluate the goodness of fit,

we can use so called information criteria. These information criteria only give an informal

indication of which model has the best fit, the difference between two criteria can not be

tested . (Allison, 2010, p. 74)

All three of these criterions use the log-likelihood, in specific they use −2 × logL or

−2× [Equation : (2.27)]. While you can use just −2× logL as a goodness of fit statistic

we will focus on three variations of this statistic which adjust for the number of covariates

in the respective model. To be specific they penalise a model with more covariates. Below

these statistics are presented as Allison (2010, p. 74-74) present them (remember that we

have a total of p covariates and n observations in our notation):

2. THEORY 29

AIC = −2 logL+ 2k

BIC = −2 logL+ k log n

AICC = −2 logL+ 2k + 2k(k+1)n−k1

(2.37)

Akaike’s information criterion (AIC) and Bayesian information criterion (BIC) both

penalise for additional covariates where (BIC) in most applications penalise the most.

The Akaike’s information criterion corrected (AICC) is a slight alteration of the (AIC)

statistic which takes the number of observations into account and thus may behave better

in small samples. (Allison, 2010, p. 74-75) 1 When we estimate our models in Section:

(4.2) we will use the AIC value to compare models.

Residuals

Martingale residuals

While we might know which covariates we want to use in our Cox proportional haz-

ards model we might be uncertain what form of that variable would explain its effect

on survival best. It might be that, Z2, logZ or some other transformation explains the

variable’s contribution to survival better than the plain form Z. Further on we will have

continuous variables in this thesis and it might be appropriate to discretisize one or more

of those variables in order to better estimate their influence on the model. A modification

of the Cox-Snell residuals called Martingale residuals can be used to find an appropriate

functional form of a variate and whether it should be discretisized. The Martingale resid-

ual, M , for right-censored data and time independent covariates is defined in Equation:

2.38. It has the property ofn∑j=1

Mj = 0 and for large samples each Mj is uncorrelated

with population mean zero. These residuals can be seen as the difference between ob-

served number of events, δj, and the expected number of events, rj. Thus the Martingale

residuals represent the excess number of events in the data that is not predicted by the

Cox proportional hazard model. (Klein and Moeschberger, 2005, ch. 11.3)

Mj = δj − rj (2.38)

1Important to remember is that when comparing the goodness of fit statistics for different models

you can not compare (AIC) for one model with (BIC) for another and so on. Only the same criterion

can be compared between different models.

2. THEORY 30

The method for finding an appropriate functional form for the variable of interest is

to exclude that variable when estimating the Cox proportional hazards model and then

calculate the Martingale residuals. Let Z1 represent the residual of interest. Next we

plot the residuals Mj against Z1 for each jth observation. Usually a smoothed fit of the

diagram is used to get a clear sense of the “linearity” in the scatter plot through time.

It is this fitted curve that gives us an indication of what functional form of Z1 should

be used. The aim is to get a fitted (smoothed) curve that is linear. If the line contains

a clear break in it when Z1 is continuous this indicates that discretisizing Z1 whould be

apropriate. (Klein and Moeschberger, 2005, ch. 11.3)

Example: In the example data we have a variable called Grade. This variable could

illustrate a test score given to the component in a tensile strength test and thus it might

tell if a component will break. In Figure: (2.5) the Martingale residuals for Grade are

plotted with a fitted curve. There is a clear break in this fitted curve which indicates

that the Grade variable which is continuous should be discretized. Locking at the figure

suggests that all test scores below 60-65 could be grouped into one group and all above

into another.

Figure 2.5: Plot of the Martingale residuals for the Grade variable in the example data

2. THEORY 31

Cox-Snell residuals

While AIC, BIC and AICC can be used to compare the goodness of fit for different

models they do not tell if a specific model has a good fit. One way of testing if an estimated

Cox proportional hazards model has good fit is through the Cox-Snell residuals. Lee and

Wang (2003, ch. 8.4) present the Cox-Snell residuals as,

rj = − log S(tj) (2.39)

where tj is the observed survival time, censored or uncensored for individual j. S(t) is

the estimated survival function based on the estimated covariate. If the observation at

tj is censored then the corresponding rj will be treated as an censored observation. This

means that when plotting these residuals they will be represented by a step like line just

as the survival function and cumulative hazards function.

If the residuals of a fitted Cox proportional hazards model are good, that is if the

model fits the data, then the residuals should line up on a 45◦ line when plotted versus

the estimated cumulative hazard rate for these residuals. If a model fits the dataset

then the Cox-Snell residuals will follow the unit exponential distribution. If we let SR(r)

represent the Kaplan-Meier estimates with respect to the residuals then − log SR will be

the estimated cumulative hazard and for each individual − log SR(rj) = rj if the model

is appropriate. (Lee and Wang, 2003, ch. 8.4)

While the Cox-Snell residuals are useful when determining the over all goodnes of fit

for a model Klein and Moeschberger (2005, p. 358) point out that this method gives no

indicator why a model does not fit well in such cases. Further on assuming exponential

distribution for the residuals only truly holds when the actual covariate values are used

rather than the estimated values and since we want to check the goodness of fit for an

estimated model these values will be estimated. Thus departure from the exponential

distribution (which will be observed as departure from the 45◦ line in the figure) can be

due to uncertainty in the estimations of β and the cumulative hazard. This effect will be

largest in the right tail of the distribution (and in the right end of the figure) for small

samples. (Klein and Moeschberger, 2005, p. 359)

2. THEORY 32

Example: In Table: (2.2) we estimated a Cox model using our example data. Now we

will look at the goodness of fit for that model using the Cox-Snell residuals. In Figure:

(2.6) the Cox-Snell plot is presented and the residuals deviates somewhat from the 45◦

line. This model seams to not fit the data perfectly.

Figure 2.6: Plot of Cox-Snell residuals for the estimated model (all covariates) of the

example data

2. THEORY 33

2.2 Key Performance Indicators

There is a very large amount of Key Performance Indicators (KPI:s) or financial ratios

used to analyse all sorts of aspects of a company’s financial health and well being. Using

these an investor can decide if he find the company worthy of investment. To include

them all would be, close to, if not impossible. In this thesis we will rather focus on two

kinds of KPI:s aimed specifically towards the companies’ ability to pay off loans.

Penman (2010, ch. 19) defines two types of ratios that are of importance when

analysing a company’s ability to pay off its loans and avoid defaulting (going bankrupt

and being terminated or sold in order to give back what is possible to the lenders/banks).

These two main types of ratios are Liquidity Ratio and Solvency Ratio.

2.2.1 Liquidity

The liquidity Ratio is concerned with the short term papers (loans and debts that will

have to be repaid within one year). As this might imply liquidity gives an indication of

how well a company will succeed with paying of loans and dept that are due in a near

future. It means that these ratios are constructed using assets (things that can/will be

turned into cash) and liabilities (debts and loans) that are, for assets, going to result in

cash within a year, and for liabilities, are coming due within a year. To differ these assets

and liabilities from those with a longer time frame they are called current. (Penman,

2010, ch. 19)

So why is this of interest? Brealey et al. (2011, ch. 28.7) makes the comparison to a

household and a typical financial situation you can find yourself in. If you for some reason

are facing a large unexpected bill you need to use capital easily accessible in order to pay

it quickly. Savings and capital invested in stocks are examples of quick, easy money. But

if those do not suffice then you might have trouble with realizing assets such as a car or

a vacation house into cash quickly enough to meat the bill. In the same way companies

have assets that are easy to realize as well as those that will take a significant amount of

time before they are turned into cash.

2. THEORY 34

There are in total six different types of liquidity measures presented by Penman (2010,

ch. 19) and we will focus on those that qre also presented by Brealey et al. (2011, ch. 28.7).

Those that indicate the ability of a company’s current assets to pay for the current

liabilities. The three that we will not focus on are concerned with how well different

types of cash flow can cover liabilities and expenditures.

Current Ratio =Current Assets

Current Liabilities(2.40)

Quick (or Acid Test) Ratio =Cash + Short-Term Investments + Receivables

Current Liabilities2 (2.41)

Cash Ratio =Cash + Short-Term Investments

Current Liabilities(2.42)

These three ratios in general explain the same thing but there are slight differences

between them in the numerator. The Current Ratio is the most general one based on all

current assets while the Quick Ratio excludes Inventories due to the fact that these are

a bit slower to turn into cash. Cash Ratio as the name suggests only takes into account

the Cash and those investment that can be liquified almost immediately. (Penman, 2010,

ch. 19)

One thing to keep in mind is that not all companies are the same and since they have

different businesses what might be considered a good current ratio for one company could

be bad for another in the eyes of an investor. A current ratio of 1:1 could be good for a

company which quickly sells and restocks its inventory while for a manufacturing com-

pany with a slow process from inventory to cash a ratio of 2:1 might just be acceptable.

This slowness from inventory to cash might also bee seen as an overoptimistic view of

the situation and hence the quick ratio could bee seen as harsher since it excludes the

inventories. (Melville, 2011, p. 364)

2Receivables is debt that other companies’ or customers own the company

2. THEORY 35

Later we will use the Quick Ratio as an good middle ground between three kind of

ratios to represent liquidity in our study

2.2.2 Solvency

While liquidity gives the investors a picture of the short-term situation investors might be

interested in how well a company is equipped to pay off long-term debts. Thus investors

look at Solvency Ratios in order to estimate a companies ability to cover debts in a more

distant future. (Penman, 2010, ch. 19)

Below are presented three different types of solvency ratios that are of interest. There

are some differences between them but in general they are rather alike. As before these

three ratios are gathered from Penman (2010, ch. 19).

Debt to Total Assets =Total Debt

Total Assets3 (2.43)

Debt to Equity4 =Total Debt

Total Equity(2.44)

Long-Term Debt Ratio =Long-Term Debt

Long-Term Debt + Total Equity(2.45)

It is interesting to note what Brealey et al. (2011, ch. 28.6) mention regarding these

ratios. They use the book (or accounting) value of a company, the equity, instead of

the market value. While in a default situation the market value is what determines if

debt holders get their money back it also includes things that the investors assume to be

positive for the future value of the shares. Assets such as research and development are

included in the mark value during good times but these values might disappear if times

become bad, thus the market value is often ignored by lenders.

The Debt to Equity ratio will be used as a solvency ratio later on which in fact is

closely related to another KPI commonly used in Sweden, the Solidity (Soliditet).

3Total Assets=Liabilities + Total Equity4Known in Sweden as Skuldsattningsgrad

Data

In this part of the thesis we will focus on the data set gathered in order to answer our

questions from Section: (1.2). We will present a summary of how the dataset and the

variables were acquired and how the observations (company stocks) were selected. We

will also look closer at how our event is defined.

3.1 The Observations

In Section: (1.2) it was briefly mentioned that this study would span the year of 2008 and

contain stocks listed on the main Swedish stock market, the OMX Stockholm exchange

market. Thus an initial list of stocks traded on the first day of trade, January 2nd, 2008

were gathered from the newspaper Dagens Industri (DI). The January 3rd edition was

used since it contains the closing prices of all stocks traded in Sweden during the first day

of trade. The actual time series data were then collected using the software Datastream

Professional from Thomson Reuter. Due to missing information/data a total of 293 stocks

from the original list of 301 companies and their daily closing prices during 2008 were

gathered from Datastream. In Appendix: (7.3) a list of all 293 stocks is presented.

3.2 The Variables

Apart from the necessary event time variable that will be presented in Section (3.3) a

number of covariates will be used in our study. What sector a company operated in will

of course be a vital variable since the primary interest in this thesis is the effect on stock

price attributed to the sectors during 2008. During 2008 stocks in Sweden were typically

divided into 9 different sectors and the information about which sector a particular com-

pany operated in were gathered from the January 2nd edition of DI.

Another variable that is also collected from DI is company size. It is logical to include

size in this study since the size of a company very well could contribute to how well it

survives a financial crisis like the one in 2008. By DI, and in general, companies are

36

3. DATA 37

sorted into three size groups, Small Cap, Mid Cap and Large Cap. What size a company

is sorted to is determined by the accumulated value of its stocks (number of stocks ×

price of one stock). If the stock value of a company is larger than 1 billion euro it is

sorted to Large Cap. If the value is between 150 million euro and 1 billion euro it is a

Mid Cap company and finally companies with less that 150 million in accumulated stock

value is regarded to be a Small Cap company.

In Section: (2.2) we have discussed the usage of Liquidity and Solvency ratios to eval-

uate the capacity off a company to pay of its loans. We will use these two measurements

or ratios in our model in order to further, apart from size, adjust for the fact that not all

companies are the same.

These ratios are presented by the companies for each fiscal year (12 month interval

of financial reporting, typically Jan-Dec) and presented in the annual report presenting

figures for that year. This means that, in general, last years ratios will be available 3-5

months into the new year. When our study starts in january 2008, harshly speaking, it

was only the KPI:s from the financial report regarding the year 2006 (presented during

the spring of 2007) that were available to the traders. This would imply that it is the

Liquidity and Solvency ratios from 2006 that should be used in the study since these are

the latest figures available at the beginning of the study. However we vill use the KPI:s

that were presented in the financial reports for 2007, thus available to the traders during

the spring of 2008, rather than those from 2006 in order to get figures that are a bit more

up to date.

The two ratios that will be used in this study are the Quick Ratio as an indicator of

Liquidity and Debt to Equity as a solvency ratio. As with the time-series data for each

stock these KPI:s are gathered from Datastream Professional where unfortunatly some

missing values regarding debt, liability and assets resulted in a total of 201 observation

with information about both their Quick Ratio and Debt to Equity. As we will see in Sec-

tion: (4.2) this will mean that when including these variables will have less observations

in the study.

3. DATA 38

3.3 An Event

In order to find the time to event for each of our stocks we need to define what an event

is. Since we are interested in finding differences that determine how well a company can

withstand a recession the death or termination of a company would be a suitable event.

However since few of the companies traded in the beginning of 2008 were terminated

during the year this would lead to few events. Thus this thesis will use the same method

as Ni (2009) where the event is defined as a specific amount of fall in a stock’s price.

Looking at the OMX Stockholm share price index, which is an index compiled of the

prices for all stocks traded on the OMX Stockholm exchange, we can determine that the

index lost almost 50% of its value. In fact the index was at its all year high notation on

the first day of trade, january 2nd. Using that as the index base (value 100) its lowest

notation of 50.18 were measured on the 21st of November. Since the index is a weight of

all companies some will have a higher value at a specific point in time and some will have

a lower value than the index at that point. This means that although some companies

also fell to 50% of their initial value during this year others will not have done so.

In this study we will work with two types of event times. One where the study starts

at the beginning of the year for all stocks and ends on the last day of trade. This would be

Type I censoring as described in Section: (2.1.3). Since the price index fell to a minimum

of 50% of its initial value at the beginning of the study the arbitrary point of a price fall

to below 60% of the initial value of the stock will be regarded as an event. This point is

chosen such that most observations (stocks) will have experienced the event and avoid a

huge amount of censored observations (choosing the event to be a price fall to 50% of the

initial stock value whould result in more censored cases). To separate this time to event

from the next one we will refer to this as ’Day One’ start.

The second time to event variable that will be used is individual for each stock. This

time to event is going to use the same window in time for the study however the study of

each observation will start at its respective year high notation. That is for some stocks

the time to event might be counted from the first day of the study, january 2nd, while

other stocks will have their highest notation later in the year. In either case the study

3. DATA 39

will end at the end of the year. Thus we are still are working with right censoring but

in this case Type I generalized censoring. Once again we will use the price index to find

a suitable threshold for an event. Since the index has its highest notation on the first

day of trade in the study the maximum price fall from the highest notation and the first

notation in the year will be the same. This means that we can use a price fall to below

60% of each stock’s highest notation as an event as well. This method is the one used by

Ni (2009) and this will be referred to as individual time to event.

A difference that is noteworthy between these two methods is that in the first case,

’Day One’, all censored observations (except for random censoring events) will be at the

same time point (the maximum possible study time, one year or 261 days of trade). The

censoring time for observations that have not experienced the event will differ when we

use individual start since each observation is going to be observed for a different amount

of time.

A third method of defining the start of our study could be to determine the start

of the financial crisis of 2008. One such definition of start time could be the Lehman

Brothers collapse of 2008 however other starts could be defined as well. Attempting to

of define the start of the 2008 financial crisis is left out of this study and we will only use

the two time to event variables presented above.

In this dataset there are some cases where we can not follow the stock’s price to the

end of the study while the event still has not occurred at the point where the observation

is lost. Thus there are some cases where stocks are lost to random censoring (se Section:

(2.1.3)). As explained these observations can have experienced a number of things such

as merges, buy ups or termination of trade which have resulted in the stock no longer

being traded. For both types of time to event data , ’Day One’and individual, there are a

total of seven cases were the stocks have been lost to the study. These have been censored

at that time.

Results

Before we look at the different variables and their effect on the survival of the stocks we

will take a general look at the estimated survival function and cumulative hazards unad-

justed for any covariates. As explained in the previous section we will use two different

ways of measuring the survival time, the individual start where each stock is observed

after their respective highest notation and the ’Day One’ start where all stocks are ob-

served from the 2nd of January.

4.1 Survival Functions and Cumulative Hazards

First let’s look at the estimated survival function for the two different event variables using

the K-M or Product-Limit estimator. This method was outlined in Section: (2.1.4). The

estimated survival curves are presented in Figures: (4.1) and (4.2) below.

Figure 4.1: Estimated survival finction with individual start

40

4. RESULTS 41

Figure 4.2: Estimated survival finction with ’Day One’ start

Looking at Figure: (4.1), the one with individual start, the events are rather evenly

distributed throughout the study. However in the ’Day One’ figure, Figure: (4.2), the

events are mostly clumped together , specifically around the day 200. Since all stocks

have the same starting day we can deduct that day 200 represents the 6th of October

which, if we look back to Figure: (1.1) is around the time where the different indices have

the highest price losses. Another difference is the censored cases. With individual start

we have 35 censored cases with very different event times. In the ’Day One’ study there

are 63 censored cases and all except seven of these are censored at the last day of the

study, day 261. The seven that are not censored at day 261 are those observations that

are lost due to competing risks.

Looking at the cumulative hazards functions in Figures: (4.3) and (4.4) we see that

the same patterns as we observed for the survival functions are present for the cumulative

hazards. When we observe each stock with individual start the cumulative hazard rises

more evenly than for the ’Day One’ method. As discussed in Section: (2.1.4) the Nelson-

Aalen estimation is used to estimator these cumulative hazards.

4. RESULTS 42

Figure 4.3: Estimated cumulative hazard with individual start

Figure 4.4: Estimated cumulative hazard with ’Day One’ start

4. RESULTS 43

4.1.1 By Sectors

Now we will group our two ordinal variables, Sector and Size and re-estimate the survival

functions and the cumulative hazards. We will start with our main variable of interest,

the Sector variable1. For the survival functions the pattern from the previous section

where we noticed that the events are more evenly spread over time in the individual

start study continues when we have grouped sectors but there are differences between the

different sectors. The survival functions are presented in Figures: (4.5) and (4.6). One

sector that appears to have low survival time in both time methods is the Energy sector

however if we have a look at the Log-Rank and Wilcoxon test statistics in Tables: (4.1)

and (4.2) these statistics are simultaneously significant only for the ’Day One’ method

on the 5% level. While the Log-Rank test, which as we described in Section: (2.1.5) puts

equal weight over all time points is not significant for the individual method (P-value

0.1032). This could indicate that the differences between at least one of the sectors and

the rest are clearer when we use the ’Day One’ time to event. Since the Wilcoxon test

is significant (P-value 0.0234) in Tables: (4.1) this indicates that for the individual start

data there are differences between the sector in an early stage. The cumulative hazards

figures of this stratification are found in Appendix: (7.2).

1In Appendix:(7.1) some descriptive statistics are presented for this grouping.

4. RESULTS 44

Figure 4.5: Estimated survival function stratified by sector, individual start

Figure 4.6: Estimated survival function stratified by sector, ’Day One’ start

4. RESULTS 45

Table 4.1: Tests of equality stratifyed by sector, individual start

Test Chi-Square DF Pr> Chi-Square

Log-Rank 13.2618 8 0.1032

Wilcoxon 17.7207 8 0.0234*

∗=significant on the 5% level

Table 4.2: Tests of equality stratifyed by sector, ’Day One’ start

Test Chi-Square DF Pr> Chi-Square

Log-Rank 25.3026 8 0.0014*

Wilcoxon 27.7024 8 0.0005*

∗=significant on the 5% level

4. RESULTS 46

4.1.2 By Size

Although size is not the main variable of interest in our study the variable is easily strat-

ified into the three groups and we will quickly have a look at the survival function for our

data grouped by size 2. The estimated survival functions are presented in Figures:(4.7)

and (4.8).

Figure 4.7: Estimated survival finction stratified by size, individual start

Figure 4.8: Estimated survival finction stratified by size, ’Day One’ start

2In Appendix: (7.1) some descriptive statistics are presented

4. RESULTS 47

It is clear from Figure: (4.7) that there is very little difference between the differ-

ent sized companies’ survival time when we use the individual start. However when we

follow every company from the first day of trade it looks like there are some differences

between the sizes. In Figure: (4.8) the survival function for Small Cap stocks are above

those for Mid and Large Cap for most of the time. Looking at the test statistics for

equality between the different sizes in Tables: (4.3) and (4.4) we find that no test is sig-

nificant (far from) for the individual time but both the Log-Rank and Wilcoxon tests are

highly significant for the ’Day One’ time to event data with P-values 0.0002 and less than

0.0001. The cumulative hazards figures of this stratification are found in Appendix: (7.2).

Table 4.3: Tests of equality stratifyed by size, individual start

Test Chi-Square DF Pr> Chi-Square

Log-Rank 1.5523 2 0.4602

Wilcoxon 0.3442 2 0.8419

Table 4.4: Tests of equality stratifyed by size, ’Day One’ start

Test Chi-Square DF Pr> Chi-Square

Log-Rank 16.7187 2 0.0002*

Wilcoxon 19.0375 2 < 0.0001*

∗=significant on the 5% level

4. RESULTS 48

4.2 Cox Modeling

We will now move on from evaluating the survival functions to looking at some estimated

Cox proportional hazards models. Since ties are present we will use the Exact method

from here on as outlined in Section: (2.1.6) and in general a significance level of 5% will

be used to determine significances.

4.2.1 Individual Study Start

First we will use the time to event data that uses individual start and since our main

interest is to find significant effects in the sector dummies we will begin with estimating

the model using only the sector variable. In Table: (4.5) the significance of this variable

is tested. It is a global test where H0 : βMaterial = βIndustry = . . . = βTele = 0 thus it tests

if the “Sector” itself is significant and if, as in Table: (4.5), this test is not significant we

should not estimate the model using the sector as a covariate. In this test we will use the

Wald test since it is the one that SAS gives. In Section: (2.1.6) we mentioned that the

Likelihood Ratio statistic have better statistical properties for low samples but the Wald

statistic should suffice for our purpose. With a P-value of 0.1201 the Sector variable is

not significant. The global test itself is also insignificant as seen in Table: (4.6) where the

P-value for the Likelihood Ratio is 0.1395. In this case since Sector is the only variable in

the model the Global test is the same as the test of the Sector significance (Same Wald

Chi-Square value).

Table 4.5: Tests of variable significance, idividual start

Variables Wald Chi-Square DF Pr> Chi-Square

Sector 12.7671 8 0.1201

4. RESULTS 49

Table 4.6: Tests of global hypothesis, β = 0, individual start, Sector

Test Chi-Square DF Pr> Chi-Square

Likelihood Ratio 12.2726 8 0.1395

Score 13.2064 8 0.1049

Wald 12.7671 8 0.1201

In Table: (4.7) the Size variable is included in the model however this does not im-

prove the model. This time both the Size and Sector variable fail their individual global

tests with p-values of 0.117 and 0.488. In Table: (4.8) we can also see that the model in

general is insignificant with a p-value of 0.1882 for the Likelihood Ratio.

Table 4.7: Tests of variable significanse, idividual start

Variables Wald Chi-Square DF Pr> Chi-Square

Sector 12.8479 8 0.1172

Size 1.4337 2 0.4883

Table 4.8: Tests of global hypothesis, β = 0, individual start, Sector and Size

Test Chi-Square DF Pr> Chi-Square

Likelihood Ratio 13.6782 10 0.1882

Score 14.6315 10 0.1461

Wald 14.1759 10 0.1651

Adjusting for companies’ different KPI values by including the Quick Ratios and Debt

to Equity ratios from the financial year of 2007 into the model in Table: (4.10) results

in yet another insignificant model. The whole model is insignificant with P-value 0.1325

for the Likelihood Ratio test. Also in Table: (4.9) none of the variables are significant.

Sector has a P-value of 0.14 which is far from the P-value of 0.05 that is our threshold

for significance. At this point we will abandon a model using the individual start data

4. RESULTS 50

and move our focus to the time to event data were all stocks are observed from the first

day of trade in our study.

Table 4.9: Tests of variable significance, idividual start

Variables Wald Chi-Square DF Pr> Chi-Square

Sector 12.2571 8 0.1401

Size 1.1709 2 0.5569

QR 07 1.3094 1 0.2525

D E 07 1.0549 1 0.3044

Table 4.10: Tests of global hypothesis, β = 0 individual start, Sector, Size and KPI:s

Test Chi-Square DF Pr> Chi-Square

Likelihood Ratio 17.4770 12 0.1325

Score 16.4837 12 0.1701

Wald 15.8733 12 0.1971

4.2.2 Day One Study Start

As in the previous section we will start by estimating a Cox proportional hazards model

using only the Sector variable. In Table: (4.11) we test the significance of the Sector

variable and find that it is significant (p-value=0.0025) which means that which means

that it would be valuable to include the sector dummies in the model. The SAS output

from this estimation is shown in Table: (4.12). The sector dummies are not significant

but if we were to interpret their estimated Hazard Ratios we would find that all sectors

except for the Commodity and Health sectors have higher hazard of experiencing the

event than the base sector. In this case (and throughout) the reference sector is Energy.

Since the Sector variable was significant in Table: (4.11) we can conclude that there is

a simultaneous effect from the Sectors but it is too weak to give any significant sector

dummies. In Section: (4.1.1) we suspected that the Material sector had among the worst

survivals and alltough not significant the hasard ratio for Material in Table: (4.12) is

4. RESULTS 51

the second highest which is in line with that suspision. In Table: (4.13) we see that the

model itself is significant (Likelihood Ratio P-value 0.0011) which is as expected since

Sector were significant and it is the only variable included. The AIC value of this model

is 1927 as presented in Table: (4.13).

Table 4.11: Tests of variable significance, ’Day One’ start

Variables Wald Chi-Square DF Pr> Chi-Square

Sector 23.7271 8 0.0025*

∗=significant on the 5% level

Table 4.12: Estimated Cox model using ’Day One’ start with Sector as covariate (repre-

sented by dummies, reference sector is Energy)

Variable DF Parameter Standard Chi-Square Pr> Hazard Lower 95% Hazard Ratio Upper 95% Hazard Ratio

Estimate Error Chi-Square Ratio Confidence Limit Confidence Limit

Commodity 1 -0.45377 0.60568 0.5613 0.4537 0.635 0.183 2.110

Finance 1 0.08029 0.43313 0.0344 0.8529 1.084 0.502 2.828

Health 1 -0.74116 0.49404 2.2506 0.1336 0.477 0.188 1.357

IT 1 0.00124 0.43533 0.0000 0.9977 1.001 0.461 2.621

Industry 1 0.18257 0.42798 0.1820 0.6697 1.200 0.563 3.109

Material 1 0.59217 0.48345 1.5003 0.2206 1.808 0.734 5.073

Rare Commodity 1 0.55293 0.44401 1.5508 0.2130 1.738 0.783 4.609

Tele 1 0.82165 0.60858 1.8228 0.1770 2.274 0.652 7.592

Table 4.13: Tests of global hypothesis, β = 0, ’Day One’ start, Sector

Test Chi-Square DF Pr> Chi-Square

Likelihood Ratio 25.8037 8 0.0011*

Score 25.2700 8 0.0014*

Wald 23.7271 8 0.0025*

Akaike information criterion (AIC) 1927

∗=significant on the 5% level

4. RESULTS 52

When we studied the survival function of this data stratified by size in Figure: (4.8)

and Table: (7.4) we found that this stratification was highly significant and that the

one survival curve that stood out was the on for Small Cap companies. If we look at

Table: (4.14) we can see that both Sector (P-value=0.0145) and Size (P-value=0.0026)

are significant. This means that we can include the dummies in the model. Including the

size dummies in our Cox model in Table: (4.15) below the trend continues. The dummy

for Small Cap companies is highly significant with a P-value of 0.0016. The estimated

Hazard Ratio for Small Cap is 0.570 (Large Cap is the reference) which means that the

hazard of experiencing the event is almost half for Small Cap companies in comparison

to Large Cap companies. No other variable in the Cox model is significant but as we can

see in Table: (4.16) the model as a whole is highly significant with P-values of 0.0001 and

less than 0.0001 for the different test statistics. The AIC value of this model has gone

down from 1927 to 1909 in this model (Table: (4.16)).

Table 4.14: Tests of variable significance, ’Day One’ start

Variables Wald Chi-Square DF Pr> Chi-Square

Sector 19.0697 8 0.0145*

Size 11.9238 2 0.0026*

∗=significant on the 5% level

Table 4.15: Estimated Cox model using ’Day One’ start with Sector and Size as covariate

(represented by dummies, reference sectors are Energy and Larg Cap)

Variable DF Parameter Standard Chi-Square Pr> Hazard Lower 95% Hazard Ratio Upper 95% Hazard Ratio

Estimate Error Chi-Square Ratio Confidence Limit Confidence Limit

Commodity 1 -0.44101 0.60846 0.5253 0.4686 0.643 0.184 2.148

Finance 1 0.17605 0.44007 0.1600 0.6891 1.192 0.543 3.145

Health 1 -0.47106 0.50346 0.8754 0.3495 0.624 0.242 1.806

IT 1 0.35851 0.44969 0.6356 0.4253 1.431 0.637 3.830

Industry 1 0.33156 0.43179 0.5896 0.4426 1.393 0.648 3.629

Material 1 0.58308 0.48874 1.4233 0.2329 1.792 0.719 5.070

Rare Commodity 1 0.67495 0.44780 2.2718 0.1317 1.964 0.877 5.237

Tele 1 0.86611 0.62125 1.9436 0.1633 2.378 0.667 8.131

Mid Cap 1 -0.07074 0.17387 0.1655 0.6841 0.932 0.662 1.311

Small Cap 1 -0.56165 0.17807 9.9482 0.0016* 0.570 0.402 0.809

∗=significant on the 5% level

4. RESULTS 53

Table 4.16: Tests of global hypothesis, β = 0, ’Day One’ start, Sector and Size

Test Chi-Square DF Pr> Chi-Square

Likelihood Ratio 38.1300 10 < 0.0001*

Score 37.4240 10 < 0.0001*

Wald 35.5295 10 0.0001*

Akaike information criterion (AIC) 1909

∗=significant on the 5% level

Finally we will include the KPI:s into the model in order to adjust for the difference

in financial status in the companies. Table: (4.17) shows that when including the KPI:s

Sector and Size are still significant which means that we again can move on with estimat-

ing a Cox model including the dummies for Sector and Size together with the KPI:s. This

model’s output is presented in Table: (4.18). Neither of the added continuous variables

are significant (Quick Ratio P-value is 0.2821 and Debt to Equity P-value is 0.9351). The

Debt to Equity value in fact has an estimated Hazard Ratio of 1 which means that a

one point increase or decrease has no impact on the survival in this model. Again the

global test is highly significant which we can see in Table:(4.19) were the Likelihood Ratio

P-value is less than 0.0001. The AIC value of 1262 in Table: (4.19) is the best for any of

the models using ’Day One’ data and the available variables which means that this model

will be selected as the ”best” model, even though the KPI:s don’t have a significant effect.

3

3Since there weere some missing values in the collection of these KPI:s this model is estimated using

201 observations with 38 censored cases.

4. RESULTS 54

Table 4.17: Tests of variable significanse, ’Day One’ start

Variables Wald Chi-Square DF Pr> Chi-Square

Sector 19.6852 8 0.0116*

Size 14.9865 2 0.0006*

Qick Ratio 1.1569 1 0.2821

Debt to Equity 0.0066 1 0.9351

∗=significant on the 5% level

Table 4.18: Estimated Cox model using ’Day One’ start with Sector, Size KPI:s as

covariate (Sector and Size are represented by dummies, reference sectors are Energy and

Larg Cap)

Variable DF Parameter Standard Chi-Square Pr> Hazard Lower 95% Hazard Ratio Upper 95% Hazard Ratio

Estimate Error Chi-Square Ratio Confidence Limit Confidence Limit

Commodity 1 -0.19476 0.73496 0.0702 0.7910 0.823 0.184 3.632

Finance 1 0.44098 0.60601 0.5295 0.4668 1.554 0.505 5.797

Health 1 -0.47210 0.62681 0.5673 0.4513 0.624 0.191 2.391

IT 1 0.71883 0.54566 1.7354 0.1877 2.052 0.785 7.045

Industry 1 0.39899 0.52148 0.5854 0.4442 1.490 0.606 4.945

Material 1 0.48143 0.58875 0.6687 0.4135 1.618 0.550 5.892

Rare Commodity 1 1.02829 0.54297 3.5865 0.0583 2.796 1.075 9.559

Tele 1 0.89225 0.73077 1.4908 0.2221 2.441 0.554 10.770

Mid Cap 1 -0.07209 0.21328 0.1142 0.7354 0.930 0.613 1.417

Small Cap 1 -0.77934 0.22589 11.9035 0.0006* 0.459 0.294 0.715

Quick Ratio 1 -0.10458 0.09723 1.1569 0.2821 0.901 0.727 1.070

Debt to Equity 1 0.0000552 0.0006771 0.0066 0.9351 1.000 0.999 1.001

∗=significant on the 5% level

Table 4.19: Tests of global hypothesis, β = 0, ’Day One’ start, Sector, Size and KPI:s

Test Chi-Square DF Pr> Chi-Square

Likelihood Ratio 42.8679 12 < 0.0001*

Score 39.6337 12 < 0.0001*

Wald 37.0490 12 0.0002*

Akaike information criterion (AIC) 1262

∗=significant on the 5% level

4. RESULTS 55

Residuals and Assumptions

Now we know that our final model will use the ’Day One’ data set and we have seen

that the best model has all available variables included. The sector and size variables

are dummies and we can’t do much about them but the two KPI:s included in the

model are on a continuous scale and might either be better suited as a discrete variables

or transformed in some form in order to explain their effect on survival the best. As

explained in Section: (2.1.7) the Martingale residuals can be used to find a good form

for these covariates. We will first plot the Martingale residuals for the two KPI:s with

no transformation and evaluate their form. These plots are shown in Figures: (4.9) and

(4.10) below.

Figure 4.9: Martingale residuals for the Quick Ratio

4. RESULTS 56

Figure 4.10: Martingale residuals for the Debt to Equity

None of these figures have a straight fitted line which we are looking for but they

also have no clear breaking points suggesting points that could be used to make them

discrete (in comparison to the example in Figure: (2.5)). There are different ways to

transform a variable such as squaring it in order to make the fitted line straighter and

thus improving the variable’s influence in the Cox model. In our case since we have some

large observations regarding the ratios but mostly lower values we will test transforming

the variables using the logarithm. By taking the logarithm of these variables we will

decrease the influence of large values and make the difference larger between the smaller

values. This might improve our martingale plots, make the fitted line straighter, and we

will test it next. The resulting Martingale plots are shown in Figure: (4.11) and (4.12),

both variables have improved slightly. The fitted curves are still not perfectly straight

but they do look much smoother this time and we will use these transformations of the

two KPI variables.

4. RESULTS 57

Figure 4.11: Martingale residuals for log(Quick Ratio)

Figure 4.12: Martingale residuals for log(Debt to Equity)

4. RESULTS 58

Next we will check if the assumption of proportional hazards is valid. There were two

methods of checking the assumption in Section: (2.1.7). Since we will keep the Quick

Ratio and Debt to Equity measurements continuous we will not use the Arjas plot to

check the proportional hazard assumption for these variables but we will use it on our

categorical variables.

In the Arjas plot we want the lines for each group to be straight with a slope different

from 1 as explained in Section: (2.1.7). In Figure: (4.13) we can see that some of the

lines are not straight. Especially the line for Industry which starts off with a larger slope

than the 45◦ however it curves downwards as the number of failures accumulate and in

the end it is below the 45◦ line. The lines for Rare Commodities and material have the

same tendencies to curve in the end of each respective line.

Figure 4.13: Arjas plot for Sector

In Figure: (4.14) the Arjas plot for the Size groups are presented and these look much

better than those in Figure: (4.13). There is a slight upwards curve on the Mid Cap line

but in general all three groups look very good.

4. RESULTS 59

Figure 4.14: Arjas plot for Size

Next we will create an artificially time dependent variable of the variables we are

testing and look at the estimated parameters for that variable to evaluate if the assump-

tion holds as outlined in Section: (2.1.7). Since Sector and Size are two sets of dummy

variables all dummies in the sets will have to be tested together and in those cases it is

the hypothesis that all estimated parameters for the time dependent covariates have a

value of zero that we will evaluate just as when we test if the variables should be included

in the Cox model.

Table 4.20: Test of proportional hazards for each of the covariates

Tested Variable Chi-Square DF PR>Chi-Square

Sector 12.914 7 0.074

Size 1.755 1 0.185

log(Quick Ratio) 0.043 1 0.836

log(Debt to Equity) 1.003 1 0.317

4. RESULTS 60

Table: (4.20) consist of the results from four estimated Cox models each one estimated

with only one covariate and its artificial time dependent variable. In Table: (4.20) we

can se that although the sector variable set is close to rejection all test are not rejected

on the 5% level. This result matches the one we found from the Arjas plot. In Figure:

(4.13) some of the groups were not perfectly straight however the proportional hazards

assumption is not rejected for any of the included variables in our study and there is no

need to adjust our model.

4.2.3 Final Model

Having concluded that the proportional hazards assumption is not rejected for any of the

covariates and having deducted on what form we want the continuous KPI variables we

can now model the final Cox proportional hazards model. Checking in Table: (4.21) it is

clear that although we have changed the KPI variables slightly by taking the logarithm

of them the categorical variables are still significant (P-value of Sector is 0.0153 and Size

is 0.0008). The estimated Cox model output is presented in Table: (4.22) where still

the only variable that is individually significant is the Small Cap dummy with P-value of

0.0011. The Sector variable with lowest P-value (0.0773) is Rare Commodities. Although

not significant it appears that the Rare Commodities sector has the worst survival with a

hazard ratio of 2.621, the Health sector has the lowest hazard rate (0.669) and seems to

have the best survival. The transformation of the Debt to Equity has not made a huge

difference on the estimated Hazard Ratio. In Table: (4.23) we can see that all global tests

reported are highly significant with P-values of 0.0001 or less. The AIC value 1260 for

this model is slightly better than for the full model with the untransformed KPI variables

(Table: (4.23)).

4. RESULTS 61

Table 4.21: Tests of variable significanse, Final Model

Variables Wald Chi-Square DF Pr> Chi-Square

Sector 18.9253 8 0.0153*

Size 14.2516 2 0.0008*

log(Quick Ratio) 2.3325 1 0.1267

log(Debt to Equity) 0.6711 1 0.4127

∗=significant on the 5% level

Table 4.22: Estimated Cox model using ’Day One’ start with the final set of covariates

Variable DF Parameter Standard Chi-Square Pr> Hazard Lower 95% Hazard Ratio Upper 95% Hazard Ratio

Estimate Error Chi-Square Ratio Confidence Limit Confidence Limit

Commodity 1 -0.28487 0.68822 0.1713 0.6789 0.752 0.193 3.128

Finance 1 0.46301 0.60650 0.5828 0.4452 1.589 0.515 5.929

Health 1 -0.40215 0.62720 0.4111 0.5214 0.669 0.204 2.566

IT 1 0.80682 0.54844 2.1642 0.1413 2.241 0.852 7.726

Industry 1 0.40552 0.52124 0.6053 0.4366 1.500 0.610 4.976

Material 1 0.45956 0.58806 0.6107 0.4345 1.583 0.539 5.757

Rare Commodity 1 0.96359 0.54553 3.1200 0.0773 2.621 1.001 8.993

Tele 1 0.93390 0.73115 1.6315 0.2015 2.544 0.577 11.235

Mid Cap 1 -0.01327 0.21558 0.0038 0.9509 0.987 0.647 1.510

Small Cap 1 -0.73423 0.22518 10.6313 0.0011* 0.480 0.308 0.746

log(Quick Ratio) 1 -0.25932 0.16980 2.3325 0.1267 0.772 0.553 1.075

log(Debt to Equity) 1 0.06412 0.07826 0.6711 0.4127 1.066 0.917 1.247

∗=significant on the 5% level

Table 4.23: Tests of global hypothesis, β = 0, Final Model

Test Chi-Square DF Pr> Chi-Square

Likelihood Ratio 45.6193 12 < 0.0001*

Score 41.6097 12 < 0.0001*

Wald 39.0325 12 0.0001*

Akaike information criterion (AIC) 1260

∗=significant on the 5% level

4. RESULTS 62

Now we have ended up with a model that is highly significant and we will end with

a quick evaluation of how good this model fits the data. In Section: (2.1.7) we described

how the Cox-Snell residuals can be used to evaluate the goodness of fit for an estimated

model. When plotting the Cox-Snell residuals against the estimated cumulative hazard

rates of the model we want them to line up on a 45◦ line. Remember that the goodness

of fit “test” is strongest in the left end of the plot since it contains more observations

(residuals). The Cox-Snell plot is shown in Figure: (4.15).

Figure 4.15: Cox-Snell residuals plotted against cumulative hazards of he final model

The models seems to fit the data rather well. Initially the residuals follow the 45◦ line

quite closely only to deviate from it in the top-right part where the numbers of residuals

are fewer.

Summary and Conclusion

Throughout this thesis we have looked at the theory behind survival analysis and the

Cox proportional hazards model. We applied this theory on Time to Event data from

the Swedish stock market and the financial crisis of 2008. Our aim was to answer the

questions: What is the contribution from a company’s sector with regards to its survival

of a financial crisis? with the sub question Can we use survival analysis on financial

data to answer this?.

Beginning with the sub question we have found that using the Cox proportional haz-

ards model on financial data is viable. What you need to make sure is that the dependent

variable is on the Time to Event form meaning that it should be a measurement of time

from a starting point to the time at which a specific event occurs. If your dependent

variable is on this form and you aim to find covariates that effect the time for the event

to occur then this method should work well. In Table: (4.23) we could see that our final

model was highly significant and from looking at Figure: (4.15) we can deduct that our

estimated model seems to have rather good fit.

In contradiction to the result that Ni (2009) obtained our KPI variables were insignif-

icant however this does not mean that KPI:s are irrelevant to this kind of study. Survival

analysis could very well be used successfully to deduct the effect of different financial

ratios on the development of stock prices. Remember that the event does not need to be

the loss of stock price.

Moving on to our main question the results are divided. In Table: (4.22) none of

the dummies regarding different sectors were significant. This gives no support for a

contribution to stocks’ survival during 2008. However in Table: (4.21) the simultaneous

test of the Sector variable is significant. This indicates that there in fact is a significant

difference between the sectors during this time period (the model is fitted towards the

event data following all stocks from the first day of trade). However the power of the

Sector variable is not strong enough to make the individual dummies significant but still

63

5. SUMMARY AND CONCLUSION 64

there was an effect from the different sectors on the survival of a stock during 2008.

A variable included in the model that we put no theoretical background to except

that it would help sort for different types of companies was the Size variable. In our final

model we found that the Small Cap companies had a better survival time in comparison

to the Large Cap companies. We have not looked at any theory regarding if small or large

companies survive financial crises best thus we will not say if this result was expected or

not but it is interesting to notice that Small Cap companies have a Hazard Rate of 0.48

(Table: (4.22)) when compared to Large Cap companies.

I would recommend using the survival analysis method when working with financial

data and there are a lot of possible implementations of this method. Possible future

studies could move the focus from company sector to size in order to find why the Small

Cap variable is significant. Another possible application and further study is to do as Ni

(2009) who focused on the financial ratios. This study could also be tweaked in regards

to the time to event variables. We mentioned briefly in Section: (3.3) that the start of

the study could be chosen to be the start of the crisis itself. Defining the actual start of

the 2008 crisis and use it as the start of a time to event study was left out in this thesis

but could very well be part of future research which could yield different results.

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R. A. Brealey, S. C. Myers, and F. Allen. Principles of Coporate Finance. McGraw-

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6. BIBLIOGRAPHY 0

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Appendix

7.1 Descriptive Statistics for stratums

Table 7.1: Number of observed stocks, events, and censored cases by sector, individual

start

Sector Total Events Censored

Commodity 8 6 2

Energy 8 6 2

Financials 58 51 7

Health Care 27 21 6

IT 59 52 7

Industry 73 67 6

Materials 17 15 2

Rare Commodity 38 35 3

T/CM 5 5 0

Total 293 258 35

1

7. APPENDIX 2

Table 7.2: Number of observed stocks, events, and censored cases by sector, ’Day One’

start

Sector Total Events Censored

Commodity 8 5 3

Energy 8 6 2

Financials 58 48 10

Health Care 27 13 14

IT 59 44 15

Industry 73 61 12

Materials 17 15 2

Rare Commodity 38 33 5

T/CM 5 5 0

Total 293 230 63

Table 7.3: Number of observed stocks, events, and censored cases by size, individual start

Size Total Events Censored

Large Cap 88 78 10

Mid Cap 83 75 8

Small Cap 122 105 17

Total 293 258 35

Table 7.4: Number of observed stocks, events, and censored cases by size, ’Day One’ start

Size Total Events Censored

Large Cap 88 74 14

Mid Cap 83 70 13

Small Cap 122 86 36

Total 293 230 63

7. APPENDIX 3

7.2 Cummulative Hazards

Figure 7.1: Estimated cumulative hazard stratified by sector, individual start

Figure 7.2: Estimated cumulative hazard stratified by sector, ’Day One’ start

7. APPENDIX 4

Figure 7.3: Estimated cumulative hazard stratified by size, individual start

Figure 7.4: Estimated cumulative hazard stratified by size, ’Day One’ start

7. APPENDIX 5

7.3 Companies

In the following table all included companies/stocks in this study is presented. The names

are those used in 2014 and all financial data are collected from Datastream Professional

by Thomson Reuters.

7. APPENDIX 6

Table 7.5: Stocks included in the study

ASTRAZENECA (OME) ELECTROLUX ’A’ LUNDIN PETROLEUM SAGAX

A-COM ELECTROLUX ’B’ LUXONEN SDB SAGAX PREFERRED

AARHUSKARLSHAMN ELEKTA ’B’ MALMBERGS ELEKTRISKA ’B’ SANDVIK

ABB (OME) ELOS ’B’ MEDA ’A’ SAS

ACANDO ’B’ ENEA MEDIVIR ’B’ SCA ’A’

ACAP INVEST ’A’ ENIRO MEKONOMEN SCA ’B’

ACAP INVEST ’B’ ERICSSON ’A’ MELKER SCHORLING SCANIA ’A’

ACTIVE BIOTECH ERICSSON ’B’ METRO INTL.SDB ’A’ SCANIA ’B’

ADDNODE ’B’ FABEGE METRO INTL.SDB ’B’ SEB ’A’

ADDTECH ’B’ FAGERHULT MICRONIC MYDATA SEB ’C’

AEROCRINE ’B’ FAST PARTNER MIDSONA ’A’ SECO TOOLS ’B’

AF ’B’ FASTIGHETS BALDER ’B’ MIDSONA ’B’ SECTRA ’B’

AFFARSSTRATEGERNA ’B’ FAZER KONFEKTYR SERVICE MIDWAY HOLDINGS ’A’ SECURITAS ’B’

ALFA LAVAL FEELGOOD SVENSKA MIDWAY HOLDINGS ’B’ SEMCON

ALLENEX FENIX OUTDOOR ’B’ MILLICOM INTL.CELU.SDR SENSYS TRAFFIC

ALLIANCE OIL SDR (OTC) FINGERPRINT CARDS ’B’ MODERN TIMES GP.MTG ’A’ SIGMA B

ANOTO GROUP G & L BEIJER MODERN TIMES GP.MTG ’B’ SINTERCAST

ARTIMPLANT GETINGE MODUL 1 DATA SKANDITEK INDRI.FRV.

ASPIRO GEVEKO ’B’ MSC KONSULT ’B’ SKANSKA ’B’

ASSA ABLOY ’B’ GUNNEBO MULTIQ INTERNATIONAL SKF ’A’

ATLAS COPCO ’A’ GUNNEBO INDUSTRIER MUNTERS SKF ’B’

ATLAS COPCO ’B’ HALDEX NCC ’A’ SKISTAR ’B’

ATRIUM LJUNGBERG ’B’ HAVSFRUN INVESTMENT ’B’ NCC ’B’ SOFTRONIC ’B’

AUDIODEV ’B’ HEBA ’B’ NEDERMAN HOLDING SSAB ’A’

AUTOLIV SDB HEMTEX NEONET SSAB ’B’

AVANZA BANK HOLDING HENNES & MAURITZ ’B’ NET INSIGHT ’B’ STOCKWIK FORVALTNING

AXFOOD HEXAGON ’B’ NETONNET STORA ENSO ’A’

AXIS HIQ INTERNATIONAL NEW WAVE GROUP ’B’ STORA ENSO ’R’

B&B TOOLS ’B’ HL DISPLAY ’B’ NIBE INDUSTRIER ’B’ STUDSVIK

BALLINGSLOV INTL. HMS NETWORKS NILORNGRUPPEN ’B’ SVEDBERGS I DALSTORP ’B’

BE GROUP HOGANAS ’B’ NOBEL BIOCARE (OME) SVENSKA HANDBKN.’A’

BEIJER ALMA ’B’ HOLMEN ’A’ NOBIA SVENSKA HANDBKN.’B’

BEIJER ELECTRONICS HOLMEN ’B’ NOKIA (OME) SVITHOID TANKERS ’B’

BERGS TIMBER ’B’ HOME PROPERTIES NOLATO ’B’ SVOLDER ’A’

BETSSON ’B’ HQ NORDEA BANK SVOLDER ’B’

BILIA ’A’ HUFVUDSTADEN ’A’ NORDNET ’B’ SWECO ’A’

BILLERUD KORSNAS HUFVUDSTADEN ’C’ NOTE SWECO ’B’

BIOGAIA ’B’ HUMAN CARE H C NOVACAST TECHS.’B’ SWEDBANK ’A’

BIOINVENT INTL. HUSQVARNA ’A’ NOVESTRA SWEDISH MATCH

BIOLIN SCIENTIFIC HUSQVARNA ’B’ NOVOTEK ’B’ SWEDISH ORPHAN BIOVITRUM

BIOPHAUSIA ’A’ I A R SYSTEMS GROUP OEM INTERNATIONAL ’B’ SYSTEMAIR

BIOTAGE IBS ’B’ OPCON TANGANYIKA OIL SDB

BJORN BORG ICA GRUPPEN ORC GROUP TECHNOLOGY NEXUS

BOLIDEN IMAGE SYSTEMS ORESUND INVESTMENT TELE2 ’A’

BONG INDL.& FINL.SYS.’A’ OREXO TELE2 ’B’

BORAS WAFVERI ’B’ INDL.& FINL.SYS.’B’ ORIFLAME COSMETICS SDR TELECA ’B’

BOSS MEDIA INDUSTRIVARDEN ’A’ ORTIVUS ’A’ TELIASONERA

BRINOVA FASTIGHETER INDUSTRIVARDEN ’C’ ORTIVUS ’B’ TELIGENT

BRIO ’B’ INDUTRADE OXIGENE (OME) TICKET TRAVEL

BROSTROM INTELLECTA ’B’ PA RESOURCES ’B’ TIETO CORPORATION (OME)

BTS GROUP INTRUM JUSTITIA PARTNERTECH TILGIN

BURE EQUITY INVESTOR ’A’ PEAB ’B’ TRACTION ’B’

CARDO INVESTOR ’B’ PEAB INDUSTRI ’B’ TRADEDOUBLER

CARL LAMM JEEVES INFO.SYSTEMS PHONERA TRANSCOM WWD.SDB.A

CASHGUARD ’B’ JM POOLIA ’B’ TRANSCOM WWD.SDB.B

CASTELLUM KABE HUSVAGNAR ’B’ PRECISE BIOMETRICS TRELLEBORG ’B’

CATELLA ’A’ KAPPAHL PREVAS ’B’ TRICORONA

CATELLA ’B’ KARO BIO PRICER ’B’ UNIBET GROUP SDB

CATENA KAUPTHING BANK PROACT IT GROUP UNIFLEX ’B’

CISION KINNEVIK ’A’ PROBI VBG GROUP

CLAS OHLSON ’B’ KINNEVIK ’B’ PROFFICE ’B’ VENUE RETAIL GROUP ’B’

CONCORDIA MARITIME ’B’ KLOVERN PROFILGRUPPEN ’B’ VITROLIFE

CONNECTA KNOW IT Q-MED VOLVO ’A’

CONSILIUM ’B’ KUNGSLEDEN RATOS ’A’ VOLVO ’B’

CTT SYSTEMS LAGERCRANTZ GROUP ’B’ RATOS ’B’ VOSTOK GAS SDB

CYBERCOM GROUP EUROPE LAMMHULTS DESIGN GROUP RAYSEARCH LABS.’B’ VOSTOK NAFTA INV.SDR

DAGON LATOUR INVESTMENT ’A’ READSOFT ’B’ WALLENSTAM ’B’

DIN BOSTAD SVERIGE LATOUR INVESTMENT ’B’ REDERI AB TNSAT.’B’ WIHLBORGS FASTIGHETER

DIOS FASTIGHETER LAWSON SOFTWARE REJLERS B XANO INDUSTRI ’B’

DORO LBI INTERNATIONAL REZIDOR HOTEL GROUP XPONCARD

DUNI LEDSTIERNAN ’B’ RNB RETAIL AND BRANDS ZODIAK TELEVISION ’B’

DUROC ’B’ LINDAB INTERNATIONAL RORVIK TIMBER

EAST CAPITAL EXPLORER LUNDBERGFORETAGEN ’B’ ROTTNEROS

ELANDERS ’B’ LUNDIN MINING SDB SAAB ’B’

7. APPENDIX 7

7.4 Exampel Data

In Table: (7.6) the example dataset that is used to create examples is presented. This

dataset is generated in order to give good examples and is in no way made up of real

data. We can pretend that we are studying how long a mechanical component last before

it breaks thus the individuals will be components. Time to Event and Event is the time

the component have been studied before it brakes (Event=1 and time is in months).

If the component have not broken before time 30 it is time censored. There are two

different Factories that produces the component, Factory1 and Factory2. We also know

how much the component is used, Usage could be the amount of weight that is put on

the component or the number of times it is used. Finally we have a Grade variable which

could be the result from a tensile strength test on the batch from which the component

comes from (lower is better).

7. APPENDIX 8

Tab

le7.

6:E

xam

ple

Dat

a

Indiv

idual

Tim

eto

Event

Event

Fact

ory

Usa

ge

Gra

de

Indiv

idual

Tim

eto

Event

Event

Fact

ory

Usa

ge

Gra

de

14

1F

acto

ry2

1464

2610

1F

acto

ry2

1580

222

1F

acto

ry2

2282

278

1F

acto

ry2

1780

315

1F

acto

ry1

787

2811

1F

acto

ry2

968

414

0F

acto

ry1

1665

2921

1F

acto

ry2

1159

59

1F

acto

ry2

766

3019

1F

acto

ry2

1769

613

1F

acto

ry2

1166

3130

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acto

ry1

2048

717

1F

acto

ry2

1792

3230

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ry1

1357

819

1F

acto

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1380

3330

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acto

ry2

1558

910

0F

acto

ry2

1169

3430

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acto

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2354

1023

0F

acto

ry1

1569

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1158

117

1F

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1464

3630

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1848

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1F

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2255

1315

1F

acto

ry1

787

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1740

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1659

169

1F

acto

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169

4230

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acto

ry1

2157

1817

1F

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1186

4330

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acto

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1757

1925

1F

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ry1

1769

4430

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1258

2013

1F

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1266

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acto

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2054

2118

0F

acto

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1775

4630

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1758

2216

0F

acto

ry1

1364

4730

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1554

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1F

acto

ry1

1587

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acto

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2154

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1F

acto

ry1

1677

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acto

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1753

2518

1F

acto

ry1

1378

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acto

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1249