Table 3.

Example of the three-state problem

	Health states for 124 patients with severe congestive heart failure who were good candidates for heart transplant
	Good candidate (gc)b	Transplant (t)	Dead (d)
Panel A: The transition probability matrixa
Good candidate (gc)	1 − pgct − pgcd	pgct	pgcd
Transplant (t)	0	1 − ptd	ptd
Dead (d)	0	0	1
Panel B: Study data and (incorrect) transition probabilities derived by the two-state formula for the first row of the transition matrix (panel A), for the study cohort of 124 people
Study outcomes at 3 years	9	92	23
3-year probability from study	0.0726	0.7419	0.1855
Annual probability, derived using the two-state formula	c	0.3633	0.0661

Projection end of year	Good candidates (A)	Transplants (B)	Cumulative transplants (C)	Dead (D)	Cumulative dead (E)	Sum of health states (A + C + E)
Panel C: Results of a Markov model using the annual two-state probabilities in panel B to project 3 years of outcomes for a cohort of 124 personsd
1	70.7494	45.0543	45.0543	8.1963	8.1963	124
2	40.3667	25.7061	70.7604	4.6765	12.8728	124
3	23.0316	14.6669	85.4273	2.6682	15.5411	124
Correct numbers	9		92		23	124

Source: Anguita 1993 [31], Fig. 1

^aThe rows represent the state a person can transition “from”; the columns represent the state a person can transition “to”

^bp_gct proportion of good candidates who receive a transplant during cycle t, p_gcd proportion of good candidates who die from all-cause mortality during cycle t, p_td proportion of transplanted patients who die during cycle t

^cThis probability is not derived from the formula. It is 1 minus the probabilities of transitioning to transplant or dead. See panel A, first row

^dApplying annual transition probabilities derived by the two-state formula results in incorrect numbers experiencing health-state transitions, as shown by the differences between the last two rows: 23 projected good candidates vs. the correct 9, 85.4 transplants vs. 92, and 15.5 deaths vs. 23 (see the values formatted in bold)