Stratification #

Data analysis is often conducted in settings with extreme heterogeneity. Heterogeneity refers to a setting where the data being analyzed are obtained from analysis units (e.g. people) that are different in many ways, some of which we know about and that may be deeply related to the primary question being pursued, and many others of which we do not know about or are less relevant for addressing the question at-hand.

Heterogeneity is related to the notion of “dispersion” (or to its synonym “variation”) which we have already discussed, but is usually taken to have a slightly distinct connotation. Heterogeneity refers to dispersion that results from factors of secondary interest, or that is an unintended consequence of the way that the data were collected. As one example, suppose we are interested in the relationship between marital status and wealth. To assess this relationship, we collect data on marital status and wealth for people of different races who live in different states. The variation in marital status and in wealth are intrinsic to our research question – if there were no variation in these variables, we could not assess how they are related. But the variation by geographic location (i.e. state) or by race may be of secondary interest. In such a setting, these sources of variation introduce heterogeneity to our study data, and in most cases we would want to account for this somehow in the analysis.

As another example, let’s consider the setting of research with human subjects, and imagine that we are interested in understanding the risk of dying from cancer. We will use data from Japan for the year 2015, obtained from the World Health Organization (WHO). In Japan in 2015, a total of 370,322 people were reported to have died of cancer. The population of Japan in that year was 125,319,299. Therefore, the overall rate of cancer deaths in Japan was 296 deaths per 100,000 people. However, the rates of cancer death are very different among different subpopulations within Japan. A very basic way to partition a human population is based on age and biological sex. The table below shows the numbers of deaths, the population size, and the death rate per 100,000 people for eight “strata” of the population of Japan, defined by sex and age. We have collapsed the ages into four intervals for simplicity here.

The table shows us that the death rate of 296 per 100,000 is actually composed of a female-specific rate of 235 per 100,000, and a male-specific rate of 360 per 100,000. Or, if we focus on age, we see rates ranging from 2 per 100,000 for 0-19 year olds to 1,125 per 100,000 for people aged 70 and over. Looking across the eight strata, we see rates as low as 1.8 per 100,000 (for young females), and as high as 1,595 per 100,000 (for older males).

The purpose of this illustration is to show that over-summarizing data can hide potentially interesting findings. Although it is not incorrect to report the overall cancer death rate in Japan as 296 per 100,000, this is a heavily summarized result. Most individual people in Japan have a risk of dying of cancer in one year that is either much less than, or much greater than this number, depending on their sex and age.

The overall rate of death (296 per 100,000) may be referred to as the “marginal” cancer death rate. The other death rates are “stratified”, or conditional. The conditional rates can be conditioned on age (and marginal with respect to sex), or conditioned on sex (and marginal with respect to age), or conditioned on both age and sex. The stratum-level statistics may be referred to as disaggregated, while the statistics calculated for the overall population are aggregated.