Published on February 16, 2014
By Carol Hargreaves
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.
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.
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.
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.
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.
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
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.
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.
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.
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).
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
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.
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.
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
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).
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