Several biostatistical techniques are available for relating the outcome of a chronic disease to individual patient characteristics.The stochastic process approach updates the information available at the time of initial presentation and makes it responsive to the individual clinical course.At any point in a patient’s history the most recently occupied state contains all of the information which is relevant to the patient’s future course. This is called the Markov Property.Mantel-Haenszel (MH) procedures proved quite useful, in estimating relative risks of death for the various states by using the strata defined by the time intervals.

1. By Carol Hargreaves

2. 


While some changes reflect only statistical fluctuations of the
measuring instruments, others are real and of sufficient
magnitude to reflect a beneficial treatment effect or a
worsening of prognosis.
Several biostatistical techniques are available for relating the
outcome of a chronic disease to individual patient
characteristics.
Dynamic aspects of the disease are most easily investigated
in a stochastic process framework, which explicitly displays
the time dependence.

3. 


A stochastic process analysis to clinical and laboratory
data on advanced prostate cancer can model the
probabilities of changes in patient status that occur
during treatment and follow-up.
The stochastic process approach updates the
information available at the time of initial presentation
and makes it responsive to the individual clinical
course.
This approach is, thus, an adaptive one which enables
the physician to base his decisions on current
probability estimates, specific to his individual patient.

4. 

Prognostic information is not limited to survival analysis but
may be used to estimate the probabilities that the patient will
improve or have a tumour regression in a given amount of
time.
It is hoped that this presentation serves to familiarise my
audience with the notion and terminology of a stochastic
process, through an application to advanced prostate cancer.

5. 


The initial data analysis used the proportional hazards model
to express survival as a function of baseline measurements in
order to determine which variables had prognostic
significance.
Serum alkaline and acid phosphate (AP and AcP) emerged as
the variables most highly associated with survival, with lower
enzyme levels predicting longer survival.
A stochastic process model was developed which
incorporates both the effect of these variables on survival and
changes in the variables over time.

6. 
A Stochastic Process represents the patient at any point in time as being
in one of a number of states.

Time is a discrete variable in the model and is measured in units of 90
days. Time 0 is the study entry, time 1 is 90 days later etc.

The state of each patient at time 0 was determined from baseline data.
The patient was observed over 3 years.

A patients state at a later time point was determined from the last
available measurement of each variable in the preceding 90 day interval.


If the patient was alive and at least one of the two variables was not
measured during a given 90 day interval, then state 6 (missing data) was
assigned.
88 Patients were included in the study.

7. 
1. Dead

2. AP > 120 and AcP > 2

3. AP > 120 and AcP ≤ 2

4. AP ≤ 120 and AcP > 2

5. AP ≤ 120 and AcP ≤ 2

6. Missing data

8. 


Suppose patient 33’s sequence of states is
22421 from study entry to death at 3 years.
The likelihood of undergoing a particular
change of state, given the preceding
experience of the patient, is called a
transition probability.
These probabilities are assumed to be the
same for any two patients with the same
sequence of states.

9. 


At any point in a patient’s history the most recently occupied
state contains all of the information which is relevant to the
patient’s future course. This is called the Markov Property.
The Markov Property would imply that in the example
involving paths 321 and 221, the transition from state 2 to
death (state 1) had the same probability in each case, and
was not affected by the differences at study entry.
The second assumption is that the transition probabilities are
stationary, i.e., transition probabilities depend only on the
states of departure and arrival, and not on the time of
departure.

10. 


For example, patients no.5 (path 21) and no.8
(path 221) represent stationary transition
probabilities.
If the process is Markov with stationary transition
probabilities, then the transitions to death were
equally likely.
The 88 patients analyzed survived an average of
almost 5 intervals and supplied 349 observations
on the transition law of the process.

11. 


It is possible to compare various states with
respect to mortality independent of the
stationary assumption.
With time intervals indexed by 0, 1, 2, and
3+, the MH estimated relative risk of death for
state 2 relative to state 5 is 8.1 (p < 0.001).
For pooled live states other than state 5, relative
to state 5, the MH estimated risk is 8.6 (p <
0.001).

12. State
State at time
T
1
Dead
1 Dead
63(1.00)
2 Both
unfavourable
36 (0.24)
at
time
T+1
2
Both
unfavourable
3
Only acid
favourable
4
Only
alkaline
favourable
5
Both
favourable
0 (0.00)
0 (0.00)
0 (0.00)
0 (0.00)
63
4 (0.03)
3 (0.02)
2 (0.01)
10 (0.07)
148
0 (0.00)
93 (0.63)
6
Missing
data
Total
3 Only acid
favourable
9 (0.16)
10 (0.17)
24 (0.41)
1 (0.02)
11 (0.19)
3 (0.05)
58
4 Only
alkaline
favourable
6 (0.21)
5 (0.17)
1 (0.03)
14 (0.48)
1 (0.03)
2 (0.07)
29
5 Both
favourable
3 (0.03)
2 (0.02)
8 (0.09)
5 (0.06)
72 (0.80)
0 (0.00)
90
6 Missing
data
9 (0.37)
4 (0.17)
1 (0.04)
0 (0.00)
0 (0.00)
10 (0.42)
24

13. 
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The most important analysis involves comparisons of the onestep transition probabilities to death.
Table 1 shows that 3/90=0.033 visits to state 5
(AP≤120, AcP≤2) resulted in a death within 3months, while
36/148 =0.243 to state 2
(AP > 120,AcP>2) resulted in death within 3 months.
These two states account for over 2/3 of the occupancies
(238/349), and provide a good prognostic discrimination.

14. 
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The relative risk of death to within 3 months is greater than
7 for state 2 relative to state 5 (0.243/0.033) =7.3. Fisher’s
exact two-tailed p=0.0001.
State 3 (high alkaline, low acid) and 4 (low alkaline, high acid)
had estimated death frequencies (0.155 and
0.206, respectively) between those of state 2 and 5.
Pairwise comparisons at the 0.05 level showed state 3
differed from state 5 and 6, and state 4 differed from state 5.

15. 
From state 3 (high alkaline, low acid) there appears to be a
significant chance
(11/58 =0.189) of improving to state 5.
However, there seems a little chance
(1/29=0.034) of improving to state 5 from
state 4 (high acid, low alkaline).
For the most favourable state, the prognosis is relatively
good; a patient remains in state 5 with estimated probability
72/90 =0.80

16. 


The use of statistical stochastic process models for displaying
and analyzing any (medical) dataset containing repeated
measurements taken at different points in time.
Mantel-Haenszel (MH) procedures proved quite useful, in
estimating relative risks of death for the various states by
using the strata defined by the time intervals.
The relative risk of death within 3 months of all states other
than state 5, relative to state 5, was estimated as 8.6
(p<0.001).