4.2. Models
The time variable, duration of marriage in years, is categorized into five intervals: 0–1
, 1–2
, 2–3
, 3–6
, and 6
years. Primary level of education is used as baseline level and, hence, its corresponding relative hazard,
is set to
To make the proposed adjustment comparable to previous works on the same data set [
7,
8], the parameters from the common anticipatory approach and from a reduced model are also estimated. Using anticipatory analysis amounts to “back-dating” the times of highest educational achievement,
, for a number of individuals, while, in the reduced model, observations whose reported educational level was completed after marriage,
are discarded.
Neither the anticipatory manipulations nor the proposed adjustment change the observed marginal occurrences and exposures . In the reduced model, these marginals are reduced due to the reduction of the total number of respondents. The time at which the highest educational level is achieved is irrelevant as long as it precedes the time of marriage, but it is relevant for calculating exposure times whenever it occurs after the time of marriage.
The conditions required for using the Fisher information to construct confidence intervals may not be fulfilled, due to the nature of the problem at hand. Instead, the bootstrap method is used. The individuals are bootstrapped B times and the expected likelihood for each such sample is maximized. We then calculate empirical 95% confidence intervals for each parameter by computing the and percentiles of the bootstrap distributions. An alternative approach is to compute standard deviations and form normal approximation confidence intervals. But, this approach was considered problematic because of outliers in the bootstrap distributions. In the reduced and anticipatory cases, the number of bootstrap replications were 10,000. In the adjusted case, because estimation of the additional parameters and required heavy computation. In the bootstrap replications, integrals and derivatives were calculated in the same manner as in the estimation. The estimated parameters were taken as starting values for each bootstrap replication.
4.3. Numerical Considerations
Numerical integration was used to calculate the integrals for given parameter values. When maximizing the expected likelihood over the parameters, one problem may arise. All the are linear in the and, hence, for individuals with , the derivatives of may be non-zero for all possible . Hence, if all individuals would have , and, thus, conditioning would be unnecessary, the likelihood may be maximized at the boundary of the parameter space, or , for and These solutions might be considered unrealistic. For individuals with , the situation is less clear-cut, but if the majority of the individuals have , there is a risk that their contribution dominates the likelihood, and this seems to be the case in the data set used for our illustration.
Hence, the
parameters were fixed to the empirical proportions of men who did not continue to the next higher level (see
Table A1). We assigned
, obtained from
Table A1 as
and
obtained from
Table A1 as
. These are also almost identical to the Bayesian estimates obtained in [
7,
8].
Instead of considering the gamma distribution parameters , explicitly in the maximization, it was more convenient to use the transformed parameters and . These correspond to the expectations and standard deviations, respectively. We found that the likelihood has an asymptote as . To alleviate this problem, was fixed and maximization was carried over the other parameters. This turned out to work out well. Unlike the parameter, there was no natural choice of ad hoc value of . However, the maximum with respect to all other parameters, including and , was very robust to different choices. We decided to put , which seemed to be a reasonable value.
To obtain the maximum of the expected likelihood, several methods were combined. Each likelihood calculation was performed for a fixed set of the five gamma distribution parameters, , and , including a maximization with respect to the and parameters. Thus, in principle, the required task was to maximize over the gamma parameters. Initially, a random choice of parameter values was performed, and the ones that gave the largest value of the likelihood were selected. Then, using the Newton–Raphson maximization algorithm, the likelihood was maximized over the five gamma distribution parameters, one at a time, keeping the others fixed. Finally, we maximized over , and simultaneously, using the Newton–Raphson maximization algorithm in five dimensions.
To obtain approximate confidence intervals for the parameters, a bootstrap procedure was used to sample the individuals with replacement. We performed the maximization procedure described above for each bootstrap replication. While bootstrapping, the parameter drifted towards zero in about of the bootstrap replications during the Newton–Raphson iterations. As a negative value of is not reasonable, an extra condition, was imposed to overcome the problem. Further details on numerical issues, including Matlab codes, can be obtained from the authors upon request.