Title: Contingency tables as indicators of association
Course: UGQuants (SSPS) > Section: Associations Between 2 Variables

Contingency tables as indicators of association

Contingency tables have their limitations. Most fundamentally, we need to remember that they only reflect association based on the cases that are actually represented in the table. Consider the following points about tables 1-4:

 | The number of missing cases is not recorded. In fact, even though the Health Survey for England in 2010 questioned more than 14,000 individuals, only about a quarter of that total are represented in the tables. This is not necessarily a problem – it simply reflects the fact that only a random sub-group were asked the questions reflected in tables 1-4 – but it does emphasise the importance of making clear the number of cases represented in contingency tables. E.g. if this had not been shown, then anyone looking at these tables might assume that all 14,000 cases were included.
 | It is also evident – but worth emphasising – that we can only use the data in the tables as evidence of the association between smoking and level of education in England. The data do not necessarily tell us anything about the nature of this association in other parts of the UK (although we might judge it would likely be similar), let alone other parts of the world.
 | The tables do not tell us whether this association is consistent across all parts of England or whether it might also vary depending on the other social categories to which people belong. This is why we often include more than two variables in our analysis, and such approaches are addressed in other modules.
 | You might also think that even 14,000 cases (never mind 3,500), being such a tiny proportion of the population of England, might not give us a very reliable indication of association between variables in the population as a whole. But in fact, provided certain conditions are met, we often have very good grounds for thinking the opposite.

We have focused on the different ways of presenting data in a contingency table, but our key objective in producing such tables is to give us an indication of possible association between variables, or, if you prefer, provide initial evidence that one variable might influence the other. How do we do this?

Without conducting any further analysis, and provided we have followed convention and placed the independent variable in the columns, comparing column percentages across the columns by reading along each row can be quite an effective way of gaining an initial sense of the existence and degree of association. Since these percentages standardise the data by adjusting for the different numbers of cases in the marginals of the table, broadly speaking, the greater the difference between column percentages, the stronger the association. It can also be useful to compare each column percentage with the total column percentage value for all cases.

In later modules, we examine different ways of calculating the strength of association between variables, also known as correlation. But it is also useful at this point to explain that strength of association may also be explored in terms of this way of understanding data:

Data = fit + residual

To help us assess the degree of association reflected in the actual data we have, one approach we can take is to produce a distinctive type of ‘fit’ for the data: a model of no association. To do this, we create a cross-tabulation which shows the number of cases that would appear in each cell of the table if there were absolutely no association between our variables. Another way of describing these figures is the expected counts in our table. Or, as the expected cell percentage in each cell if there were no association.

Calculating these figures is not simply a matter of taking the total number of cases in the table and then dividing by the number of possible combinations of values. The reason this is so is because of our table marginals. E.g. in tables 1-4 it is evident that less than a quarter of cases overall have a degree and only about a third smoke. So we would not expect the number of cases in the cell representing people who both smoke and have a degree to represent a quarter of cases overall, but a much lower percentage than that.

We can then compare the expected counts with the actual (or ‘observed’) counts in each cell. This is done in table 5, with the expected counts in bold.

Table 5: Cigarette smoking in England by whether has university degree (observed and expected counts)

Currently smokes cigarettes? | Has university degree?
Degree | No Degree | All
Smoker
171

269
1010

912
1181
Non-Smoker
633

535
1717

1815
2350
Total | 804 | 2727 | 3531

Source: Health Survey for England, 2010

Essentially, table 5 shows us that fewer people with a degree are smokers, and more are non-smokers, than we would expect if there were no association between level of education and smoking. The opposite is true of people who do not have degrees. This is just another way of saying that people with degrees are less likely to be smokers than those who do not have degrees: essentially the same message we obtained by comparing column percentages in table 2. Another way of phrasing this is to say that people with degrees are under-represented among smokers.

Note also, however, that the association shown in tables 1-5 is very far from being a perfect association. In the example of smoking and level of education we would say that a perfect association existed if all cases with degrees were non-smokers and all cases without degrees were smokers. In practice, we rarely encounter such perfect associations in the social and political world – it is far more complex than that. People with degrees are less likely to smoke, but this is not the same as saying that whether or not one has a degree determines whether or not one will be a smoker.

If the expected counts are the ‘fit’ of our data, we can then think of the difference between the expected and observed count in each cell as the ‘residuals’ of our model of no association, just as we would use the difference between mean value and actual values as the basis for calculating standard deviation for ratio variables. In a model of no association for categorical variables, the larger the residuals, the greater the association.

Finally, if a key objective of this type of analysis is to identify associations then it is equally important that we do not just stop at this point. Our objective is to try to understand and explain the social and political world rather than only describe it. The finding that people with degrees are less likely to smoke might be interesting and valuable in itself, but the next step is to ask the question ‘Why might this be?’.

Is it because the typical university degree programme has a strong focus on the health risks of smoking cigarettes? This seems very unlikely. Perhaps more highly educated people are more likely to be aware of these risks? This seems more plausible, but still does not account for the fact that the health risks of smoking are very well-established and commonly known (in fact in some countries such as the UK they are highlighted on cigarette packets). Perhaps the association relates to differences in the social backgrounds of those who typically do and do not go to university, and the types of behaviour associated with those backgrounds?

This final suggestion is very significant: what it suggests is that smoking behaviour might not be directly influenced by whether or not someone has a degree. Instead we might need to explore other characteristics (i.e. variables) associated with levels of education and smoking behaviour. This observation also underpins one of the most important rules in quantitative analysis: correlation does not necessarily imply causation. This is just another way of saying that when there is an evident association between two variables, this does not necessarily mean that one variable is directly influencing the other.

Thinking through these kind of possibilities is often the most complex and difficult part of quantitative analysis, but also the most interesting and exciting because it requires us to engage what we might call our ‘social scientific imagination’. Sometimes we might do this before cross-tabulating two variables: we might use our previous knowledge to imagine how two variables might be associated before carrying out the analysis to confirm or refute our expectations.