How to interpret the survival and hazard curves. Although descriptive methods of data analysis provide a good overview of the data, they do little to reduce the complexity of the data and make it difficult to assess the relative likelihood of children from one service system making the transition to either of the other two. Another problem with the descriptive methods is that they only provide information on children who make a particular transition by the end of the two-year observation period. Nothing can be said about children who remain in the original service system at the end of the observation period. Children who are still at risk of transitioning to another service at the end of the observation period represent "censored" data. Censored data poses many problems for standard techniques of statistical analysis. We used survival analysis to deal with the problems associated with censored data while analyzing the likelihood of an event for the entire population at risk of experiencing the event.
In our analysis, it was important to distinguish different types of transitions among the human services. For example, we are interested in whether a child in AFDC is more likely to enter foster care or Medicaid or to receive system exit at all after leaving AFDC. Using survival analysis, we were able to do this using the method known as "competing risks". (Allison, p. 185). Survival analysis is most widely used in medical research, where it is often used to describe the risk of dying from a disease — thus "survival" analysis. A term often used in survival analyses is the hazard rate, or the risk (hazard) of experiencing something given that a person has not yet experienced it (or survived, in this case). In our case, one might say that the more time a child spends in AFDC, the hazard of moving into Medicaid decreases, that is, with every day spent receiving AFDC, the hazard rate goes down.
An example of a survival curve is found in Figure IL3. The curve shows the estimated more than X probability (y axis) that a child will "survive" (or not make the transition) for number of days (x axis). The "failure" — or the opposite of survival — is the estimated probability of making the transition prior to a specified time (again the x axis). Or, in figure 4, one could ask, what is the probability that a child has moved to "system exit" prior to 300 days. The failure rate is thus calculated by subtracting the survival rate from 1. (We speak mainly in terms of failure rates in the text, as in the following sentence: two years after entering AFDC, 20 percent had transitioned from AFDC to Medicaid-only while 50 percent had exited the system).
We also present the estimates of the hazard rate in a graph. By graphing the hazard rate of each of the "competing risks" (in this case each other possible transition, including the move to system exit) in a single graph, one can easily determine which type of transition was most likely over time. An example of the hazard is found in Figure IL2. In this figure, the hazard of exiting the system from AFDC is higher than the hazard of entering either Medicaid or foster care from AFDC. We can also see the declining likelihood of exiting the system or entering Medicaid from AFDC (or negative duration dependence) over time. In contrast, there is little changing in the hazard of entering foster care over time.
How to interpret the "competing risk Cox regression model". In order to understand the influence of individual and community characteristics on the hazard of a particular transition, we used Cox's proportional hazard models. The initial comparison looks at the likelihood of children to enter a selected service. In order to interpret the effect of a particular characteristic (age, race, poverty, gender) on the hazard of a transition, one looks at the risk ratio column. For example, Table NC1 shows that infants are more than 3 times (3.14) as likely than children age 15-17 (the comparison group is always indicated by 1.00) to move from AFDC to foster care, while controlling for the other variables (gender, race, poverty).
A second comparison tests whether or not the effect of a particular characteristic is the same across two different event types. For example, the column in Table IL1 labeled "comparison of coefficients" presents results of a test comparing the coefficients of the AFDC to foster care model with the coefficients of the AFDC to system exit model. If the number in the column meets the .05 significance level we can say that the effect of being an infant, for example, on the hazard of entering foster care is different than the effect on the hazard of entering system exit.
In addition to understanding the correlates of individual transitions, it is often of further interest to ask if, given departure from AFDC, are there factors that make individuals more or less likely to make one versus another transition. We know for example from Figure IL3 that in the aggregate, children who leave AFDC in Illinois are almost three times as likely to go to no services than to Medicaid only. For some individuals, this 3:1 ratio may differ. The 'comparison of coefficients' column presents results of a test comparing the coefficients of the AFDC to Medicaid model with the coefficients of the AFDC to No Services model. Consider the value of 12.1 for infants under the Medicaid only versus No Services column. As this number is positive and significant, we can say that infants are significantly more likely to enter Medicaid versus no services when compared with older children ages 15-17 after controlling for other variables. Note the coefficient does not tell us that infants are more likely to transition to Medicaid only than to No Services (infants may still for example be twice as likely to go to no services than to Medicaid only). We can only say that, given departure from AFDC, they are more likely than their older counterparts to transition to Medicaid-Only than to no services.