# Strong Transition Model

Polls in October 2007 implied that the Liberal Democrats would get zero seats. This is obviously too low, although the model was behaving as expected. See our Lib Dem transition analysis for details of why this was.

However, the behaviour is not satisfactory. The historical record shows that the original Liberal party had its low point in 1951 with 2.5% of the national vote, but still kept six seats (though five of those had no Conservative candidate). Interestingly, the 1931 election reduced the Liberals' support from 23% to 7%, but their number of seats only decreased from 59 to 37.

We present an updated variant of the Electoral Calculus transition model which has better behaviour in the event of a steep decline in a party's support, by acknowledging the advantage of incumbency. The new model, called the Strong Transition Model (STM) continues to benefit from the existing advantages of the original Transition Model, namely that predicted votes are always positive.

## Model principles

The basic idea of the model is to divide the supporters of each party into strong and weak supporters. If the party declines in support, the model assumes that the weak supporters will defect to other parties before the strong supporters do. Seats with an incumbent from the declining party will probably have large proportion of strong supporters, so they will lose a smaller number of voters than the standard transition model expects. Seats where the declining party is already weak will continue to decline multiplicatively as usual.

As a simple model, we assume that the weak supporters of a party in any given seat are all those who voted for that party up to a threshold of 20% of the turnout. The strong supporters are those who voted for the party beyond that threshold. There will be no strong supporters in seats where the party received less than 20% of the votes cast.

## Model example

Take the case of the Liberal Democrats declining from an election vote share of 22.7% to a predicted share of 12.4%. If we count up over each seat, we discover that nationally the Lib Dem election vote of 22.7% breaks down into 17.4% weak votes and 5.2% strong votes. The model assumes that the predicted share will break down into 7.2% weak votes and 5.2% strong votes, because the weak votes defect first. So the proportion of weak votes which stay loyal is 7.2/17.4 = 41.3%.

In a particular seat where the Lib Dem election vote was 45%, that divides into 20% weak votes and 25% strong votes. As 41.3% of the weak votes stay loyal, they are predicted to now be only 0.413 x 20% = 8.3%. But all the strong votes are assumed to stay loyal, so the total Lib Dem vote is predicted to be 8.3% + 25% = 33.3%.

This compares with the standard transition model which assumes that the loyalty factor of all Lib Dem voters is 12.4/22.7 = 54.6%, so the seat with 45% support is now predicted to have 0.546 x 45% = 24.6% support.

## Model comparison

The graph shows what would happen in each constituency given that the national Lib Dem support at the last election was 22.7%, of which 5.2% were strong votes, and that the current national poll prediction is 12.4% support. The lines on the graph represent, under three possible models, the predicted Lib Dem support in any seat given its previous election vote in that seat was given by the x-axis value. The three models are:

• Additive model The predicted support in each seat is just the election support less the swing of 10.3%. Note that the predicted support goes negative if the election support level was less than 10%.

• Transition model The predicted support in each seat is just a fraction (0.546) of the election support. The predicted support is always positive, but very strong Lib Dem seats decline substantially.

• Strong Transition model (STM) The predicted support is a fraction (0.413) of the weak election voters plus all of the strong election voters. The line is piecewise linear with a change in slope at 20% (the strong voter threshold). The predicted support is always positive, but strong Lib Dem seats decline less dramatically than under the Transition Model.

## Formal model definition

Let us define the strong voter threshold α, which we have set at 20%. Let V(i,j) be the vote share for party #j in seat #i at the previous election, and let TO(i) be the turnout in that seat. Set TO to be the total number of votes cast TO = Σk TO(k). The national vote shares V(j) for party #j are

V(j) = Σk TO(k)V(k,j) / TO,

and its strong vote shares are

VS(j) = Σk TO(k) max(V(k,j) − α, 0) / TO.

Suppose that the predicted national vote shares for party #j are P(j). Then the predicted national weak and strong vote shares PW(j) and PS(j) are

PW(j) = max( P(j) − VS(j), 0 ), and

PS(j) = min( P(j), VS(j) ).

Then the predicted weak and strong votes in seat #i are given by applying the Transition model to the extended set of parties: Strong-CON, Weak-CON, Strong-LAB, Weak-LAB, etc. All parties, not just the Lib Dems, are handled using the new method. Over this extended party list, we have
1. the previous national results, VS(j) and VW(j) = V(j) − VS(j).
2. the predicted national results, PS(j) and PW(j) from above.
3. the previous seat results, VS(i,j) = max( V(i,j) − α, 0 ), and VW(i,j) = V(i,j) − VS(i,j).

The Transition Model applied to this extended party list then gives the predicted weak and strong votes for each party in each seat. We then just add up the weak and strong votes to get the total predicted vote for each party in that seat.

## Models in practice

We can run the three models for different levels of Lib Dem support. We assume that incoming or outgoing Lib Dem voters divide evenly between Labour and the Conservatives. The graph shows the number of predicted Lib Dem seats won for different levels of predicted national Lib Dem support, under each of the three models. We see that the three models are qutie similar for support levels above the general election result of 23%. This is expected because all the models behave additively for gaining parties. For declining vote shares, the additive model is most favourable to the Lib Dems. This includes the prediction that the Lib Dems will win 2 seats even if they get zero votes nationally, which is over-optimistic. On the other hand, the transition model predicts zero Lib Dem seats even if they get 10% of the votes.

The strong transition model lies between these extremes. It does predict zero seats for zero votes, but it only predicts zero seats for support levels of 3% or less. For current support levels of 12.4%, it predicts 15 Lib Dem seats, which is only a few less than the (over-optimistic) additive model.

## Summary

The strong transition model is an improvement on both the additive model and the original transition model because it takes the best features from each approach. Its features are:
• Always gives positive vote predictions, which avoids distortions when votes go negative.
• Predicts zero seats for zero votes
• Reasonable seat predictions even for significant drops in support
• Intuitive model of incumbency and local strength
• Behaves proportionally for weak seats, and additively for strong seats.
Due to these advantages, it has been adopted as the main prediction model for Electoral Calculus from 28 October 2007 onwards.