# PORTIONS OF APPLICATION OF DURATION MODELS IN POLLUTING INDUSTRIES ON DISEASE INCIDENCES AND IN UNEMPLOYMENT

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```					PORTIONS OF APPLICATION OF DURATION MODELS IN POLLUTING INDUSTRIES ON DISEASE INCIDENCES AND
IN UNEMPLOYMENT SITUATION

PATRICK MAZICK (RESOURCE ECONOMIST)

pat_mazick@rocketmail.com

(00265) 999 365 952

UNIVERSITY OF MALAWI

BUNDA COLLEGE

DEPARTMENT OF AGRICULTURAL AND APPLED ECONOMICS

BOX 219

LILONGWE

MALAWI

1
First portion

Modeling survival analysis out of polluting factory
Introduction to survival analysis (survival and hazard function)

The term "survival analysis" pertains to a statistical approach designed to take into account the
amount of time an experimental unit contributes to a study. That is, it is the study of time
between entry into observation and a subsequent event. Originally, the event of interest was
death hence the term, "survival analysis." The analysis consisted of following the subject until
death. The uses in the survival analysis of today vary quite a bit. Applications now include time
until onset of disease, time until stock market crash, time until equipment failure, time until
earthquake, and so on. The best way to define such events is simply to realize that these events
are a transition from one discrete state to another at an instantaneous moment in time. Of course,
the term "instantaneous", which may be years, months, days, minutes, or seconds, is relative and
has only the boundaries set by the analyst.

In the modeling of survival and hazard, there are several tools that are of importance, thus;
probability mass function, cumulative distribution function, hazard and survival functions. To
start with PMF measures the probability of separate discrete value. For example, in a list of
integers from 1 up to 100, PMF will account for p(1), p(2) up to p(100) whereas the CDF will
consider the P(1,2,3,100) or P(20,10,24 and 70). The PMF is denoted by the lower case f(t) while
the CDF is denoted by the upper case F(t). In modeling the survival and hazard functions we
need these probability functions.

Survival and Hazard models

Since we are told that there is a polluting factory and in the course of its operations, there are
visible signs of an outbreak probably due to pollution. So, we label this as a public bad in the
language of environmental economics.

We assume that the outbreak is because of pollution and nothing else to ease the analysis. We
further hypothesize that after the operation of the factory, the outbreak will cease even though
the effects will remain.

To model this situation, we consider carefully, the time on which an individual is seen having the
visible signs and the time of death (Or the time of starting operations and observance of visible
signs of outbreak: in both cases the modeling is the same provided the t and T have been
properly defined) because timing the start of operation and observation of outbreak provides
difficulties in defining who a victim is or not. At the same time, extending the analysis to assume
that the outbreak effects are fatal, just to simplify the modeling and demonstrate the practicality.

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Let t be the time of first visible signs, ̂ be average time of sickness and T be the terminal period
which we expect the individual to die. From the theory of survival analysis in modeling survival
cases, we know that an individual found before T, thus; t<T is called hazard (hazard model) and
after T, thus; t>T, is called survival and we need survival model.

Again, we assume that t is discrete random variable not continuous. Further than that, we also
assume that T≥0.

Given the probability mass function, f(t) and cumulative distribution function F(t) and T≥0, then
survival function will look as follows

=       <      ……………………………………………………………………………….(1)

=     >      =1−         <     ……………………………………………………………..(2)

The survival function gives the probability of surviving or being event-free beyond time t.
Because S(t) is a probability, it is positive and ranges from 0 to 1.

Let T be a discrete duration random variable accounting for the values 1, 2, 3 …., n taking on the
PMF;        =       = …………………………………………………………..…………….(3)

Where t=1, 2, 3,

CDF;        =       ≤    =      1 +    2 +    3 +⋯+            …………………………………(4)

From the above equations we can derive survival and hazard models

The exit rate or hazard rate:

ℎ    =       = ∣     ≥     =          ……………………………………………………………..(5)

ℎ    =              ………………………………………………………………………..…….(6)

When t>1, equations (5) and (6) above imply;

ℎ    =                              ………………………………………………………………(7)
⋯

This implies that

ℎ    =              ………………………………………………………………………………(8)

In all cases ℎ 2 =      2 =     2

Since we know from theory that          +ℎ      = 1 identity equation, then by computing the model
for hazard rate, survival rate will be generated from the identity equation.

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Therefore;      =1−                     ⋯
=1−               =1−ℎ        ……………..(9)

Conclusions: The most important consideration in modeling this situation is looking at given the
time in which range do we survive or die as hazards. Once you get either of the function,
subtracting it from 1 you get the other. Besides, the definition of the victim is very crucial in
terms of the time range when modeling hazards.

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Second portion

As major differences involved in modeling unemployment from disease incidences may be
involved in defining the time range and dynamics involved in unemployment.

In the first place, assumptions about t are the same in all cases but a hazard individual in the
model above is the survivor in the unemployment model. For instance, if t>T, this is a survivor
in above model but he is automatically the hazard in unemployment model. Before we come to

Since we see students graduating from college joining the job market almost each and every
year, one thing which is so real is that the absorptive capacity of the job market is less than the
flow of graduates into the job market. As such, in trying to have an understanding about when
and individual gets into the job market and if possible starts benefiting from safety nets for
unemployed folks and when gets the job, duration data models are called for in this analysis. In
reality, some people stay long before securing a job while others do not.

Let T be allowable time to stay benefiting from the safety nets, t the time an individual stays in
the job market. For those where t ≤ T are survivors (they have secured a job before terminal
time) whereas those in which t ≥ T are hazards (haven’t secured a job and are also kicked out of
the safety nets.

Please! Observe differences coming in t ≤ or t ≥ T. the opposites of what we are saying here
are the actualities in the disease incidence model in question 1 above.

Now, after establishing the basics, let’s go to actual modeling;

Given the equations (3) and (4) as PMF and CDF above,

ℎ    =       = ∣     ≤     ……………………………………………………………………(10)

ℎ    =        …………………………………………………………………………………(11)

ℎ    =             ……………………………………………………………………………...(12)

ℎ    =                                  ……………………………………………………….(13)
⋯

ℎ    =              The same as equation (8)

So, this gives the probability of exit over the surviving population at each time. It represents the
probability of not securing a job within the given period and h (t) subtracting it from 1 gives
survivor rate representing people who secure the job with the given period.

5
Conclusions: the main focus is how we set ℎ         =      = ∣ ≥ for disease incidence model
as hazard function and opposite being the survivor function of the same whereas ℎ   =     =
∣ ≤ is the hazard function and the opposite being the survivor model of the same. These
differences in setting direction are very crucial in the analysis.

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Third portion

The interpretation of the hazards and spells of the given output is as follows: In general, the
hazards in the table represent the probability whereas spells represent the period of exiting or
getting employed whereas list represents the serial for an individual. So, when interpreting the
hazard column, we will be talking about the given figure in the column representing the
probability of the so much of getting employed or exiting the spell whereas for the spell we will
be talking about the duration between for instance 2and 3 weeks when an individual(s) will
secure a job.

Individuals 1, 10 and 20: There is about 20% chance or probability of 0.20107399 that these
individuals will exit the spell (or in other words get employed) between 5 and 6 weeks. In short
the spell is denoted λ(5) representing between 5 and 6 weeks.

Individual 2: There is 18.5% chance or there is 0.18467583 probability that individual 2 will
exit the spell between 13 and 14 weeks. In other words, the probability that an individual will
exit the spell between 13 and 14 weeks (λ(13)) is 0.18467583.

Individual 3: The probability that individual will exit the spell between 21 and 22 weeks (λ(21))
is 0.24770642 or about 24.8% chance of exiting the spell within λ(21).

Individuals 4, 8 and 14: The probability that individuals 4, 8 and 14 will exit the spell between
3 and 4 weeks or λ(3) is 0.1826988 or about 18.3% chance of exiting the spell in λ(3).

Individual 5: there is about 15% chance or probability of 0.1544813 that an individual will exit
the spell (being employed) is between 9 and 10 weeks or λ(9). In other words, the probability
that an individual will exit the spell between 9 and 10 weeks or λ(9) is 0.1544813.

Individual 6: there is about 15.6 % chance or probability of 0.15629742 that an individual will
become employed (exit the spell) is between 11 and 12 weeks or λ(11). In other words, the
probability that an individual will exit the spell between 11 and 12 weeks or λ(11) is
0.15629742.

Individuals 7, 15, 18 and 19: there is about 16.2% chance or there is a probability of
0.16153156 that individuals 7, 15, 18 and 19 will exit the spell (being employed) between 1 and
2 weeks time period or λ(1). In other words, the probability that individuals 7, 15, 18 and 19 will
exit the spell between 1 and 2 weeks or λ(1) is 0.16153156.

Individuals 9 and 11: there is about 21.5% chance or there is a probability of 0.21489967 that
individuals will exit the spell (being employed) between 7 and 8 weeks or λ(7). Or the
probability that individuals 9 and 11 will exit the spell between 7 and 8 weeks is 0.21489967.

Individual 12: the probability that an individual 12 will exit the spell between 15 and 16 weeks
or λ(15) is 0.18971061.

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Individuals 13 and 16: the probability that individuals 13 and 16 will exit the spell between 2
and 3 weeks or λ(2) is 0.17195862.

Individual 17: there is about 25% chance or there is a probability of 0.25060241 that an
individual 17 will exit the spell (being employed) between 14 and 15 weeks or λ(14).

8

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