RiskPaths: The underlying statistical models
General description
Events and parameter estimates
General description
Being a model for the study of childlessness, the main event of RiskPaths is the first pregnancy (which is always assumed to lead to birth). Pregnancy can occur at any point in time after the 15th birthday, with risks changing by both age and union status. The underlying statistical models are piecewise constant hazard regressions. With respect to fertility this implies the assumption of a constant pregnancy risk for a given age group (e.g. age 15-17.5) and union status (e.g. single with no prior unions).
For unions, we distinguish four possible state levels:
- single
- the first three years in a first union
- the following years in a first union
- all the years in a second union
(After the dissolution of a second union, women are assumed to stay single). Accordingly, we model five different union events:
- first union formation
- first union dissolution
- second union formation
- second union dissolution
- the change of union phase which occurs after three years in the first union.
The last event (change of union phase) is a clock event - it differs from other events in that its timing is not stochastic but predefined. (Another clock event in the model is the change of the age index every 2.5 years) Besides unions and fertility, we model mortality--a woman may die at any point in time. We stop the simulation of the pregnancy and union events either when a women dies, or at pregnancy (as we are only interested in studying childlessness), or at her 40th birthday (since later first pregnancies are very rare in Russia and Bulgaria and are thus ignored for this model).
At age fifteen a woman becomes subject to both pregnancy and union formation risks. These are competing risks. We draw random durations to first pregnancy and to first union formation. There are two additional competing events at this stage-mortality and change of age group. (As we assume that both pregnancy and union formation risks change with age, the risks underlying the random durations only apply for a given time period--2.5 years in our model--and have to be recalculated at that point in time.)
In other words, the 15th birthday will be followed by one of these four possible events:
- the woman dies
- she gets pregnant
- she enters a union
- she enters a new age group at age 17.5 because none of the first three events occurred before age 17.5
Death or pregnancy terminates the simulation. A change of age index requires that the waiting times for the competing events union formation and pregnancy be updated. The union formation event alters the risk of first pregnancy (making it much higher) and changes the set of competing risks. A woman is then no longer at risk of first union formation but becomes subject to union dissolution risk.
Events and parameter estimates
First pregnancy
As outlined above, first pregnancy is modeled by an age baseline hazard and relative risks dependent on union status and duration. The following Table 1 displays the parameter estimates for Bulgaria and Russia before and after the political and economical transition.
| Bulgaria | Russia | |
|---|---|---|
| 15-17.5 | 0.2869 | 0.2120 |
| 17.5-20 | 0.7591 | 0.7606 |
| 20-22.5 | 0.8458 | 0.8295 |
| 22.5-25 | 0.8167 | 0.6505 |
| 25-27.5 | 0.6727 | 0.5423 |
| 27.5-30 | 0.5105 | 0.5787 |
| 30-32.5 | 0.4882 | 0.4884 |
| 32.5-35 | 0.2562 | 0.3237 |
| 35-37.5 | 0.2597 | 0.3089 |
| 37.5-40 | 0.1542 | 0.0909 |
| before 1989 transition | 10 years after transition: 1999+ | |||
|---|---|---|---|---|
| Bulgaria | Russia | Bulgaria | Russia | |
| Not in union | 0.0648 | 0.0893 | 0.0316 | 0.0664 |
| First 3 years of first union | 1.0000 | 1.0000 | 0.4890 | 0.5067 |
| First union after three years | 0.2523 | 0.2767 | 0.2652 | 0.2746 |
| Second union | 0.8048 | 0.5250 | 0.2285 | 0.2698 |
The data from Table 1 is interpreted as follows in the model. As long as a woman has not entered a partnership, we have to multiply her age-dependent baseline risk of first pregnancy by the relative risk "not in a union". For example, the pregnancy risk of a 20 year old single woman of the pre-transition Bulgarian cohort can be calculated as 0.8458*0.0648 = 0.05481. At this rate of ? =0.05481:
- The expected mean waiting time to the pregnancy event is 1/ ? = 1/0.05481 = 18.25 years;
- The probability that a women does not experience pregnancy in the following 2.5 years (given that she stays single) is exp(-?t) = exp(-0.05481*2.5) = 87.2%.
Thus at her 20th birthday, we can draw a random duration to first pregnancy from a uniform distributed random number (a number that can obtain any value between 0 and 1 with the same probability) using the formula:
RandomDuration = -ln(RandomUniform) / (;
As we have calculated above, in 87.2% of the cases, no conception will take place in the next 2.5 years. Accordingly, if we draw a uniform distributed random number smaller than 0.872, the corresponding waiting time will be longer than 2.5 years, since
-ln(RandomUniform) / ( = -ln(0.872)/0.05481 = 2.5 years. A random draw greater than 0.872 will result in a waiting time smaller than 2.5 years-in this situation, if the woman does not enter a union before the pregnancy event, the pregnancy takes place in our simulation.
To continue this example, let us assume that the first event that happens in our simulation is a union formation at age 20.5. We now have to update the pregnancy risk. While the baseline risk still stays the same for the next two years (i.e. 0.8458), the relative risk is now 1.0000 (as per the reference category in Table 1) because the woman is in the first three years of a union. The new hazard rate for pregnancy (applicable for the next two years, until age 22.5) is considerably higher now at 0.8458*1.0000 = 0.8458. The average waiting time at this rate is thus only 1/0.8458 = 1.18 years and for any random number greater than exp(-0.8458*2)=0.1842 the simulated waiting time would be smaller than two years. That is, 81.6% (1 - 0.1842) of women will experience a first pregnancy within the first two years of a first union or partnership that begins at age 20.5.
First union formation
Risks are given as piecewise constant rates changing with age. Again we model age intervals of 2.5 years. These are the rates for women prior to any conception, as such an event would stop our simulation.
| before 1989 transition | 10 years after transition: 1999+ | |||
|---|---|---|---|---|
| Bulgaria | Russia | Bulgaria | Russia | |
| 15-17.5 | 0.0309 | 0.0297 | 0.0173 | 0.0303 |
| 17.5-20 | 0.1341 | 0.1342 | 0.0751 | 0.1369 |
| 20-22.5 | 0.1672 | 0.1889 | 0.0936 | 0.1926 |
| 22.5-25 | 0.1656 | 0.1724 | 0.0927 | 0.1758 |
| 25-27.5 | 0.1474 | 0.1208 | 0.0825 | 0.1232 |
| 27.5-30 | 0.1085 | 0.1086 | 0.0607 | 0.1108 |
| 30-32.5 | 0.0804 | 0.0838 | 0.0450 | 0.0855 |
| 32.5-35 | 0.0339 | 0.0862 | 0.0190 | 0.0879 |
| 35-37.5 | 0.0455 | 0.0388 | 0.0255 | 0.0396 |
| 37.5-40 | 0.0400 | 0.0324 | 0.0224 | 0.0330 |
The parameterization example given in Table 2 has the following interpretation: the first union formation hazard of Bulgarian women of the first cohort is 0 until the 15th birthday; afterwards it changes in time steps of 2.5 years from 0.0309 to 0.1341, then from 0.1341 to 0.1672, and so on. The risk is highest for the age group 20-22.5--at a rate of 0.1672, the expected time to union formation is 1/0.1672=6 years. A women who is single on her 20th birthday has a 34% probability of experiencing a first union formation in the following 2.5 years (p=1-exp(-0.1672*2.5)).
Second union formation
A woman becomes exposed to the second union formation risk if and when her first union dissolves. As a difference to the first union formation which is based on age, this process does not start at a fixed point in time but is triggered by another event (first union dissolution). Accordingly, the time intervals of the estimated piecewise constant hazard rates refer to the time since first union dissolution.
| before 1989 transition | 10 years after transition: 1999+ | |||
|---|---|---|---|---|
| Bulgaria | Russia | Bulgaria | Russia | |
| <2 years after dissolution | 0.1996 | 0.2554 | 0.1457 | 0.2247 |
| 2-6 years after dissolution | 0.1353 | 0.1695 | 0.0988 | 0.1492 |
| 6-10 years after dissolution | 0.1099 | 0.1354 | 0.0802 | 0.1191 |
| 10-15 years after dissolution | 0.0261 | 0.1126 | 0.0191 | 0.0991 |
| >5 years after dissolution | 0.0457 | 0.0217 | 0.0334 | 0.0191 |
Union dissolution
Both first and second unions can dissolve, with such processes starting at the first and second union formations, respectively. As the sample size is very small for the modeling of the second union dissolution event we do not distinguish the before and after transition cohorts for this event.
| before 1989 transition | 10 years after transition: 1999+ | |||
|---|---|---|---|---|
| Bulgaria | Russia | Bulgaria | Russia | |
| First year of union | 0.0096 | 0.0380 | 0.0121 | 0.0601 |
| Union duration 1-5 | 0.0200 | 0.0601 | 0.0252 | 0.0949 |
| Union duration 5-9 | 0.0213 | 0.0476 | 0.0269 | 0.0752 |
| Union duration 9-13 | 0.0151 | 0.0408 | 0.0190 | 0.0645 |
| Union duration >13 | 0.0111 | 0.0282 | 0.0140 | 0.0445 |
| Bulgaria | Russia | |
|---|---|---|
| First 3 years of union | 0.0371 | 0.0810 |
| Union duration 3-9 | 0.0128 | 0.0744 |
| Union duration 9+ | 0.0661 | 0.0632 |
Mortality
In this sample model, we leave it to the model user to either set death probabilities by age or to "switch off" mortality allowing the study of fertility without interference from mortality. In the latter case, all women reach the maximum age of 100 years. If the user chooses to simulate mortality, the specified probabilities are internally converted to piecewise constant hazard rates (based on the formula -ln(1-p) for p<1) so that death can happen at any time in a year. If a probability is set to 1 (as is the case when age=100), immediate death is assumed.
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