Unobserved Heterogeneity

An Example

Consider the ages at marriage in a rural area of Bangladesh (Figure 1). The heavy line shows the distribution that arises when gender is not considered (or left unmeasured). Suppose a lognormal distribution is believed to be a proper model for age at marriage. Fitting the observations to a lognormal distribution (without reference to sex) yields parameters that give a MEAN age (STANDARD DEVIATION) of 23.3 (5.9) years. Failure to consider gender masks substantial subgroup differences. In fact, a gender-specific analysis yields a mean age of marriage of 27.3 (5.5) years for males and 19.5 (3.4) years for females. The subgroup differences are so large that parameter estimates for any other measured covariate (say, the effect of education on age at marriage) are likely to be BIASED by the lack of information on gender.

Figure 1 Ages at Marriage in Matlab, Bangladesh, in 1985

Data are from the International Centre for Diarrhoeal Disease Research, Bangladesh (1992).

With no information on gender, one solution is to estimate parameters for a finite mixture model. A four-parameter mixture model can be estimated assuming equal numbers of males and females and yields estimates of 25.4 (7.1) years for one subgroup and 21.4 (4.0) years for a second subgroup. The subgroups correspond to males and females, respectively.

“Devious Dynamics” in Hazards Models

Failure to account for unobserved heterogeneity can lead to serious problems of interpretation for survival analysis and related methods that deal with durations to events (hazards models or EVENT HISTORY models). For these models, unobserved heterogeneity refers to unmeasured differences among the units of observation in their risk of “failure” (i.e., risk of experiencing the event under investigation). Ignoring unmeasured heterogeneity in risk of failure leads to a downward bias in risk with time.

Consider mortality risk in infants, which is initially very high and declines rapidly over the first years of life. Taken literally, this observation implies that infants undergo reverse senescence: Individuals “get better” with age. In part, this pattern reflects known biological processes such as maturation of the immune system. On the other hand, the same “infant mortality” pattern is seen for other complex systems such as computers and automobiles—it is less plausible to believe [Page 1161]that individual computers experience a decline in risk of failure during early service. The apparent reverse senescence is an artifact of the changing composition of the sample over time, a product of the “devious dynamics of aging cohorts” (Vaupel & Yashin, 1985a, 1985b).

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