Decomposing mean changes

Suppose we observe a mean \(P\) at two discrete times. A mean can be expressed as a numerator \(n\) divided by a denominator \(d\). We can define the change in the mean as follows:

\[ P_{\Delta} = P_2 - P_1 = \frac{n_2}{d_2} - \frac{n_1}{d_1} \\ \]

For the rest of this paper, we talk about the example of a population proportion, but this decomposition would work for any ratio statistic.

Now let’s suppose all changes in the numerator and denominator between the two timepoints are categorized exclusively into either type A or type B so that:

\[ P_{ \Delta } = \frac{ n_1 + n_{ \Delta A} + n_{ \Delta B } }{ d_1 + d_{ \Delta A } + d_{ \Delta B } } - \frac{ n_1 }{ d_1 } = \frac{n_1 + n_{\Delta A} + n_{\Delta B} }{ d_2} - \frac{n_1}{d_1} \]

Perhaps A is the entries into a population and B is the exits (and the existing population members do not change). Or A might just be the net effect of entries and exits from the population and B might be change within existing population members (in which case B would not affect the denominator).

We want to be able to partition the overall change \(P_{\Delta}\) into the change attributable to A and B.

\[ P_{\Delta} = P_{\Delta A} + P_{\Delta B} \]

We start by defining \(n_{\Delta A 0}\) as the A numerator required to not affect the numerator-denominator ratio from \(P_1\) given the observed A denominator changes \(d_{\Delta A}\):

\[ \frac{n_{\Delta A 0}}{d_{\Delta A}} = \frac{n_1}{d_1} \]

Rearranging gives us:

\[ n_{\Delta A 0} = n_1 \cdot \frac{ d_{\Delta A}}{d_1} \]

We then define \(n_{\Delta A \epsilon}\) as the difference between the numerator change to keep the proportion constant(\(n_{\Delta A 0}\)) and the numerator change observed (\(n_{\Delta A }\)):

\[ n_{\Delta A \epsilon} = n_{\Delta A } - n_{\Delta A 0} \]

Using the equivalent identities for B, we can rewrite \(P_{\Delta}\) as:

\[ P_{\Delta} = \frac{n_1 }{ d_2} + \frac{n_{\Delta A 0} + n_{\Delta A \epsilon} }{ d_2} + \frac{n_{\Delta B 0} + n_{\Delta B \epsilon} }{ d_2} - \frac{n_1}{d_1} \\ \]

and then:

\[ P_{\Delta} = \frac{n_1 + n_{\Delta A 0} + n_{\Delta B 0}}{ d_1+ d_{\Delta A} + d_{\Delta B}} + \frac{n_{\Delta A \epsilon} }{ d_2} + \frac{n_{\Delta B \epsilon} }{ d_2} - \frac{n_1}{d_1} \\ \]

Since we defined \(n_{\Delta A 0}\) to maintain \(\frac{n_{\Delta A 0}}{d_{\Delta A}}=\frac{n_1}{d_1}\), it follows that:

\[ \frac{n_1 + n_{\Delta A 0} + n_{\Delta B 0}}{ d_1+ d_{\Delta A} + d_{\Delta B}} = \frac{n_1}{d_1} \]

which allows the following simplification:

\[ P_{\Delta} = \frac{n_{\Delta A \epsilon} }{ d_2} + \frac{n_{\Delta B \epsilon} }{ d_2} \\ \]

where we have now partitioned \(P_{ \Delta}\) into parts attributable to A and B. Substituting in observed variables gives us:

\[\begin{equation} \label{finaldecomp} P_{\Delta} = \frac{n_{\Delta A } - \big ( n_1 \cdot \frac{ d_{\Delta A}}{d_1} \big ) }{ d_2} + \frac{n_{\Delta B } - \big ( n_1 \cdot \frac{ d_{\Delta B}}{d_1} \big )}{ d_2} \\ \end{equation}\]

So we can now define the two additive components of change:

\[ P_{\Delta A} = \frac{n_{\Delta A } - (n_1 \cdot \frac{ d_{\Delta A}}{d_1}) }{ d_2} \\ \]

\[ P_{\Delta B} = \frac{n_{\Delta B } - (n_1 \cdot \frac{ d_{\Delta B}}{d_1}) }{ d_2} \\ \]

This decomposition can be applied again to the change components themselves so that we might for instance decompose the A changes further to:

\[P_{\Delta A} = P_{\Delta C} + P_{\Delta D}=\frac{n_{\Delta C } - (n_1 \cdot \frac{ d_{\Delta C}}{d_1}) }{ d_2} + \frac{n_{\Delta D } - (n_1 \cdot \frac{ d_{\Delta D}}{d_1}) }{ d_2}\]

To show a quick example, let’s imagine we have a group of 80 people, 40 of whom are women. By the next time point, 10 people leave, 6 of whom are women. In addition, 110 new people join the group but only 11 of these are women. We go from a 50% female group to 25% by the second timepoint a change in the proportion of -0.25.

When we substitute in these numbers into the equation we get:

\[ P_{\Delta} = \frac{-6 - \big ( 40 \cdot \frac{-10}{80} \big ) }{ 180} + \frac{11 - \big ( 40 \cdot \frac{110}{80} \big )}{ 180} \\ \]

\[ P_{\Delta} = \frac{-1}{ 180} + \frac{-44}{ 180} = -0.0055 + -0.2444 \\ \] In other words, the overall -0.25 in the proportion of women is able to be decomposed into a -0.0055 effect from differential exits and a much larger -0.244 from differential entries.

Decomposing changes in parties’ vote shares

Now we can consider how to decompose a change in a party’s vote share \(V_{\Delta}\) into the contributions from voters switching parties (\(V_{\Delta vote}\)), switching to or from non-voting (\(V_{\Delta DNV}\)), and population replacement (\(V_{\Delta pop}\)).

\[\begin{equation} \label{votedecomp} V_{\Delta} = \frac{n_{\Delta vote } - \big ( n_1 \cdot \frac{ d_{\Delta vote}}{d_1} \big ) }{ d_2} + \frac{n_{\Delta DNV } - \big ( n_1 \cdot \frac{ d_{\Delta DNV}}{d_1} \big )}{ d_2} + \frac{n_{\Delta pop } - \big ( n_1 \cdot \frac{ d_{\Delta pop}}{d_1} \big )}{ d_2} \\ \end{equation}\]

Let’s imagine we want to decompose a change in party A’s vote share between two elections. In election 1, party A gets 40% of the vote and the other parties get a combined 60%. Turnout is 50%. In election 2, party A gets 50% of the vote, a 10 percentage point increase in A’s share. Turnout increases to 80% (a 30 percentage point increase).

Table 1 shows the transition table for this example. I show the table as counts here, but it would work equally well with proportions of people in each cell.

Table 1: Transition table for hypothetical example showing counts of people in the population. Rows show vote choices of individuals at the first election and columns show vote choices at the second election.

	A	Other	DNV	Dead
A	1,318,916	358,207	90,604	132,273
Other	721,284	1,763,057	148,648	217,011
DNV	1,602,996	1,550,975	1,445,313	150,716
Ineligible	156,804	127,760	215,436	0

Applying this to real data

So what happens when I apply this to the 664 party-election observations in the data? Party switching is a larger contributor to changes in party vote shares 78% of the time, meaning that party switching is more important for most parties as well as most elections.

But maybe certain types of parties rely more on turnout? The following graph shows that’s not the case. Higher numbers on this graph indicate that party switching contributed more to change than turnout switching. On average, parties in every family (e.g. conservative, social democrat, christian democrat, radical right etc) in all 18 countries is more affected by party switching than turnout switching.

So my results continue to show that elections are overwhelmingly won and lost by party switching rather than turnout switching. This is true for parties everywhere on the political spectrum and all around the world.