A gender quota for candidates was imposed in the 2016 Irish election (see e.g., http://www.thejournal.ie/readme/gender-quota). The question arises whether it had an impact. As a first pass consider the following table:¹

           |        gender
      dail |         f          m |     Total
-----------+----------------------+----------
      2011 |        21        145 |       166 
           |     12.65      87.35 |    100.00 
-----------+----------------------+----------
      2016 |        32        116 |       148 
           |     21.62      78.38 |    100.00 
-----------+----------------------+----------
     Total |        53        261 |       314 
           |     16.88      83.12 |    100.00

This is clearly a big rise. Is it statistically significant?

. tab dail gender, chi exact

           |        gender
      dail |         f          m |     Total
-----------+----------------------+----------
      2011 |        21        145 |       166 
      2016 |        32        116 |       148 
-----------+----------------------+----------
     Total |        53        261 |       314 

          Pearson chi2(1) =   4.4881   Pr = 0.034
           Fisher's exact =                 0.036
   1-sided Fisher's exact =                 0.025

Both Pearson’s \(\chi^2\) test and Fisher’s exact test suggest it is relatively (but not hugely) unlikely that both cohorts have the same underlying gender proportions. In other words, there is evidence of change.

There are two different difficulties here:

• The gender quota isn’t the only thing that changed between 2011 and 2016
• The two subsamples (electoral cohorts) are not independent: a lot of incumbents held their seats (and didn’t change gender).

The former is a general problem with observational data: an association that is consistent with a particular hypothesis is not necessarily proof that it is true. The increase in women TDs might be due to social change over the five years, and the quota could possibly have had no effect. The statistical test allows us to say the apparent association is likely real, not due to chance, but not to what it should be attributed.

The latter difficulty is statistical. The \(\chi^2\) test requires the rows and columns to be independent: for instance, the 2011 and 2016 rows should be independent draws from two different populations (or the same population at two different times, as here).² If the rows indicated different regions or countries, for instance, this would be satisfied. But in this case it is not true: a signficant number of TDs present at time 1 are also present at time 2, i.e., are incumbents who got re-elected. This means the two rows are much more alike than they would be if they were separate draws from the population. In this case, the consequence is that the difference over time is understated by this table.

We can try to deal with this by excluding the retained incumbents, and just deal with the change: those who fail to get re-elected and those newly elected. But that probably varies too much in the opposite direction: the TDs that hold their seats are also part of the story (for instance, a gender quota might make it harder – or easier – for an existing female TD to hold her seat). If instead of pooling the data so we have one observation per seat per Dáil, we do so such that we have one observation per TD, whether present in one or two cohorts, we can include everyone, and assess the evidence for gender change in a better way:

           |        gender
    tdtype |         f          m |     Total
-----------+----------------------+----------
    Exited |         9         68 |        77 
           |     11.69      88.31 |    100.00 
-----------+----------------------+----------
   Entered |        19         40 |        59 
           |     32.20      67.80 |    100.00 
-----------+----------------------+----------
 Stayed in |        12         77 |        89 
           |     13.48      86.52 |    100.00 
-----------+----------------------+----------
     Total |        40        185 |       225 
           |     17.78      82.22 |    100.00 

          Pearson chi2(2) =  11.4759   Pr = 0.003
           Fisher's exact =                 0.005

This shows much stronger evidence of a change, with a p-value well below 0.01. While exiting incumbents seem to be slightly more male than those who held their seats (but not signficantly), the clear picture is that almost a third of new entrants are female, way above the 18% average for the whole group.

This counts as quite strong evidence of change, just from looking at the seats filled: there are statistically significantly more female TDs than there were. Whether this can be attributed to the gender quota, however, is a question that these stats can’t quite answer. The 2011–2016 difference is completely consistent with the intention of the quota, but we can’t know the counterfactual of what would have happened in 2016 without the quota.