Question

20.(5 ps) Construct the weighted directional graph form of a Markov model consisting of the following 3 states: Wel Sick dead. O well Sick dead 21. (5 pts) construct the matrix form of the above Markov model D Focus 1
i really need both done

Answer 1

Markov-state diagram. Each circle represents a Markov state. Arrows indicate allowed transitions.

times spent in the individual states to arrive at an expected survival for the process.

where ts is the time spent in state s. Usually, however, the quality of survival is considered important. Each state is associated with a quality factor representing the quality of life in that state relative to perfect health. The utility that is associated with spending one cycle in a particular state is referred to as the incremental utility. Consider the Markov process depicted in figure 3. If the incremental utility of the sick state is 0.7, then spending the cycle in the sick state contributes 0.7 quality-adjusted cycles to the expected utility. Utility accrued for the entire Markov process is the total number of cycles spent in each state, each multiplied by the incremental utility for that state. n Expected utility = ~ ts X Us s=l i Let us assume that the DEAD state has an incremental utility of zero,* and that the WELL state has an incremental utility of 1.0. This means that for every cycle spent in the WELL state the patient is credited with a quantity of utility equal to the duration of a single Markov cycle. If the patient spends, on average, 2.5 cycles in the WELL state and 1.25 cycles in the sick state before entering the DEAD state, the utility assigned would be (2.5 X 1) + (1.25 X 0.7), or 3.9 quality-adjusted cycles. This number is the quality-adjusted life expectancy of the patient.

When performing cost-effectiveness analyses, a separate incremental utility may be specified for each state, representing the financial cost of being in that state for one cycle. The model is evaluated separately for cost and survival. Cost-effectiveness ratios are calculated as for a standard decision tree.10,11 TYPES OF MARKOV PROCESSES Markov processes are categorized according to whether the state-transition probabilities are constant over time or not. In the most general type of Markov process, the transition probabilities may change over time. For example, the transition probability for the transition from WELL to DEAD consists of two components. The first component is the probability of dying from unrelated causes. In general, this probability changes over time because, as the patient gets older, the probability of dying from unrelated causes will increase continuously. The second component is the probability of suffering a fatal hemorrhage or embolus during the cycle. This may or may not be constant over time. A special type of Markov process in which the transition probabilities are constant over time is called a Markov chain. If it has an absorbing state, its behavior over time can be determined as an exact solution by simple matrix algebra, as discussed below. The DEALE can be used to derive the constant mortality rates needed to implement a Markov chain. However, the availability of specialized software to evaluate Markov processes and the greater accuracy afforded by agespecific mortality rates have resulted in greater reliance on Markov processes with time-variant probabilities. The net probability of making a transition from one state to another during a single cycle is called a transition probability. The Markov process is completely defined by the probability distribution among the starting states and the probabilities for the individual allowed transitions. For a Markov model of n states, there will be n2 transition probabilities. When these probabilities are constant with respect to time, they can be represented by an n x n matrix, as shown in table 1. Probabilities representing disallowed transitions will, of course, be zero. This matrix, called the P matrix, forms the basis for the fundamental matrix solution of Markov chains described in detail by Beck and Pauker.

Το DEAD WELL 0.6 0 0 WELL DISABLED DEAD DISABLED 0.2 0.6 0 From 0.2 0.4 1