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In some of the work I’m currently doing looking at households as derived from the Southwark patient register I wanted to go beyond a quantification of how many people lived in a households – a rudimentary household size, to looking at the composition of a household and hence what type of household it represented. In order to do this I looked at how types of household were generally reported in the UK Census, in European statistics, and in terms of social research on the life course, as well as in health literature itself. In terms of defining households, I found that although complex household typologies do exist, there exists a general set of likely household forms: as expected these revolve around the single, co-habiting, family, single parenthood, extended family etc models. As I have data on individuals I first decided to classify individuals into 5 broad categories that seem important in the literature and then look at the composition of these categories within households. The categories were:

1) Dependent Children (<18 yrs old)

2) Adult Male (18-65 yrs old)

3) Adult Female (18-60 yrs old)

4) Male Pensioner (65+ yrs old)

5) Female Pensioner (60+ yrs old)

Evidence suggests that these represent the coarsest categories that could usefully represent significant periods within the life course, as well as being relevant to changes in health status. In a sense, different type of household structure can be described by different combinations of these person classes for different household sizes.

I decided to test this by calculating all the possible combinations of these 5 classes for a 2 person household and then looking at their uptake in the actual household data I had derived from the Southwark patient register. It turned out that for a two person household there were 15 different ways in which you could combine the 5 person classes in order to create a unique household:

Children Only (Parents Unregistered); Single Parent Male and Child; Co-Habiting Men; Single Parent Female and Child; Single Parent Male Pensioner and Child; Co-Habiting Man and Woman; Co-Habiting Man and Male Pensioner; Co-Habiting Women; Single Parent Female Pensioner and Child; Cohabiting Woman and Male Pensioner; Cohabiting Man and Female Pensioner; Cohabiting Male Pensioners; Cohabiting Woman and Female Pensioner; Cohabiting Male and Female Pensioner; Cohabiting Female Pensioners.

Using this typology of 15 possible household types, I extracted the two person households from the larger dataset and wrote a Python script to classify these households. The result for 27,124 households was a follows:

What this graph seems to demonstrate is that roughly half of all 2 person households consist of a man and a woman (either adult or pensioner) cohabiting, and roughly a further 22% of same sex cohabitation. In this dataset for two person household, single parents only make up around 15% of households of which almost 13% is a single female parent (adult or pensioner) and a child. All other groups only make up around 13% of households, but crucially the only category in which no households were found to exist was the adult man cohabiting with a male pensioner category. Indeed many of the smaller categories can be interpreted as having inherently important social roles, the adult woman looking after a male or female pensioner for instance.

Essentially, the terrain of household type was a lot more nuanced and tricky than I’d at first though, made even more so by my realisation that as household size increases, the number of possible combinations of the person types within a  household increases dramatically. I wrote a python script to calculate the number of possible different sets of people for the household sizes 1 to 10:

This presents a difficult situation, even for reasonably small households. This is a problem known as “combinatorial mathematics” or “combinatorics“. I decided to see what I could learn about this distribution, so I looked for patterns in the sequence, as you are taught in pre-GCSE maths and soon found that the sequence had a constant fourth difference:

This constant fourth difference indicated that the sequence can be explained by a quartic function, of which is was easy to then calculate the form:

Sadly not one of those classically beautiful equations.

This all leads to the issue of how I now classify households, clearly the number of possible sets makes anything above around 4 people per household fairly intractable. I’ll experiment with 3 households and see whether I can account for most household types with a few set patterns and then look at households that fall outside of this remit.

Interesting none the less, I hadn’t expected to be doing much of this kind of maths!