1. A Multi-State Generalization of the HMD Methodology, Applied to Fertility by Parity in the United States, John R. Wilmoth Department of Demography University of California, Berkeley
Presentation for the 1st HMD Symposium “Human Mortality over Age, Time, Sex, and Place” Rostock, Germany, June 2004

2. Lexis Diagram
Age
Lifeline for a single individual
Time

3. Three Shapes for 1x1 Lexis Region
Age
Age-cohort
Age-period
Period-cohort
Time

4. Lexis Model – Definition
A complete description of a population over age and time, including detailed estimates of population size (and perhaps characteristics), as well as the transition rates (and events) associated with changes in population size (and characteristics), as a function of both age and time.

5. Lexis Model – Mortality example
Population surface: matrix of estimated population sizes, P(x, t), for 1-year age intervals (from age x to x+1) on January 1st of each year, t.
Mortality surface: matrix of death rates, m(x, t), for 1-year intervals of both age and time (i.e., in 1x1 Lexis squares).

6. Lexis Model – Applications
Lexis models can be used to describe:
The actual life history of cohorts
The hypothetical life history of cohorts (for example, mortality/longevity of a cohort in the absence of migration)
The hypothetical life history of synthetic cohorts (for example, mortality/longevity during some time period)

7. Single- vs. Multi-State Models of Demographic Processes
In a single-state model, individuals are either in or out of the population. Exit can occur through mortality or emigration, and entry can occur through birth or immigration.
In a multi-state model, members of a population are classified according to certain states (i.e., characteristics or statuses), which may change over the life course.

8. Single-State Model of Mortality and Migration, with Transition Ratesb
Alive
Birth
b birth rate
m mortality rate
i immigration rate
e emigration rate i
m+e
Death / Migration

9. Three Types of Multi-State Models
Closed population (no mortality or migration)
Open population with invariant rates of exit/entry by state
Open population with variable rates of exit/entry by state

10. Multi-State Fertility Model for Closed Population
fn + f1 f2
fn -2
fn -1
Parity 0
Parity 1 …
Parity n - 2
Parity (n -1) +

11. Multi-State Fertility Model for Open Population with Invariant Rates of Exit/ Entry by Parity
fn + f1 f2
fn -2
fn -1
Parity 0
Parity 1 …
Parity n - 2
Parity (n -1) +
m+e i
m+e i
m+e i b i
m+e
Death / Migration
Birth

12. Multi-State Fertility Model for Open Population with Variable Rates of Exit/Entry by Parity
fn + f1 f2
fn -2
fn -1
Parity 0
Parity 1 …
Parity n - 2
Parity (n -1) +
i(n-1)+
m0+e0
m1+e1
in-2
mn-2+en-2 i1 b
m(n-1)++e(n-1)+ i0
Death / Migration
Birth

13. Two Kinds of Fertility Rates
Transition rates (Type I)
Accrual rates (Type II)

14. Multi-State versus Multiple-Decrement Models
Multi-state models describe variation in characteristics of a population
Multiple-decrement (or increment) models describe variation in modes of transition (e.g., competing causes of death, causes of migration, causes of pregnancy)

15. Lifelines in Single-State Lexis Model
Age a
x + 2 d
· Out migration
x + 1 o b x
o In migration x x
x Death e c
x - 1 o
t - 1 t
t + 1
t + 2
Time

16. Lifelines in Multi-State Lexis Model
Age a z3
x + 2 d
· Out migration
x + 1 o b
o In migration x z2 x z3 x
x Death e z1 c
z1 z2 z3 etc. State Transitions z1
x - 1 z2 o
t - 1 t
t + 1
t + 2
Time

17. Population Changes in Single-State Model (mortality only)
Age
x + 1 P2 DL x DU
for k = 1, 2, … , n+ P1
x - 1 t
t + 1
Time

18. Demographic Accounting in Single-State Model (mortality only)

19. Population Changes in Multi-State Model (both fertility and mortality)
Age
x + 1 DL
P2(k)
BL (k) x
where k = 1, 2, … , n for births
but k = 0, 1, … , (n-1) for women
BU (k)
P1(k) DU
x - 1 t
t + 1
Time

20. Demographic Accounting, Multi-State Model (both fertility and mortality)
for k = 1, 2, …, n - 2

21. Demographic Accounting, Multi-State Model (cont.)

22. Necessary or Simplifying Assumptions (for purpose of estimating models)
For both single- and multi-state models, it is necessary to choose between:
Uniform distribution of events
Constant transition rates
over some age-time interval (e.g., Lexis triangle)
For multi-state model, it is convenient to assume that rates of death and migration are equal across parity states

23. Demographic Accounting in Two Directions
Time moving forward
versus
Time moving backward

24. Available Fertility Data for the U.S.
Age 50 14
1917
1940
2001
Time

25. Elementary Formulas for Estimating Single- and Multi-State Lexis Models

26. Some Results Age patterns of fertility rates
Time trends in fertility rates
Summary measures of lifetime fertility (for both periods and cohorts)
Distributions of women by parity

27. Age Patterns of Fertility Rates

28. Parity-Specific Fertility Rates over Age by Decade, U. S
Parity-Specific Fertility Rates over Age by Decade, U.S., Ages a) Birth Order k = 1

29. Parity-Specific Fertility Rates over Age by Decade, U. S
Parity-Specific Fertility Rates over Age by Decade, U.S., Ages b) Birth Order k = 2

30. Parity-Specific Fertility Rates over Age by Decade, U. S
Parity-Specific Fertility Rates over Age by Decade, U.S., Ages c) Birth Order k = 3

31. Parity-Specific Fertility Rates over Age by Decade, U. S
Parity-Specific Fertility Rates over Age by Decade, U.S., Ages d) Birth Order k = 4+

32. Time Trends in Fertility Rates

33. Parity-Specific Fertility Rates over Time by Age Group, U. S
Parity-Specific Fertility Rates over Time by Age Group, U.S., a) Birth Order k = 1

34. Parity-Specific Fertility Rates over Time by Age Group, U. S
Parity-Specific Fertility Rates over Time by Age Group, U.S., b) Birth Order k = 2

35. Parity-Specific Fertility Rates over Time by Age Group, U. S
Parity-Specific Fertility Rates over Time by Age Group, U.S., c) Birth Order k = 3

36. Parity-Specific Fertility Rates over Time by Age Group, U. S
Parity-Specific Fertility Rates over Time by Age Group, U.S., d) Birth Order k = 4+

37. Summary Measures of Lifetime Fertility

38. Total Fertility Rate (TFR), 1917-2000, and Completed Fertility Rate (CFR), 1903-1950, U.S.

39. Distributions of Women by Parity

40. Observed and Hypothetical Distributions of Women by Parity, U. S
Observed and Hypothetical Distributions of Women by Parity, U.S., a) Age 30

41. Observed and Hypothetical Distributions of Women by Parity, U. S
Observed and Hypothetical Distributions of Women by Parity, U.S., b) Age 50

42. Possible Applications for Multi-State Lexis Model
Fertility status (parity 0, 1, 2, 3, etc.)
Marital status (single, married, divorced, widowed, remarried, etc.)
Health status (healthy, mildly disabled, severely disabled, etc.)
Criminal status (unrestricted, incarcerated, parole, probation, etc.)

43. Possible Applications (cont.)
Immigration status (citizen, permanent resident, temporary resident, illegal, etc.)
Residency status (United States, Canada, Mexico, etc.)
Employment status (employed full-time, employed part-time, unemployed, etc.)
Educational status (full-time student, part-time student, graduate, dropout, etc.)

44. Simplifying Characteristics of Multi-State Fertility Model
Data are reliable and readily available (at least for some countries)
States are well defined
No reverse transitions