Transition Model 2004
This page first posted 8 July 2004
The model has been modified on 28 October 2007 to become the Strong
Transition Model, which itself has been superseded by modern
regression methods. This article is now of historical
interest only.
Since 8 July 2004, predictions have been based on a new model
of voter movement. This new model is quite similar in effect to
the simple additive Uniform National Swing model, but has two
major advantages:
- Predicted share of the vote cannot go unreasonably low or negative
- Predicted share of the vote cannot go unreasonably high or above 100%
Under the additive model, both of these features could theoretically happen
and the first often happened in practice.
The new model avoids these problems, whilst retaining the good properties
of the old model, and making very similar predictions for which party will
actually win each seat. Backtesting each model against the last four elections
showed quite good performance from both, with the Transition Model being
slightly, but not significantly, more accurate.
Below is a technical description of both models, plus the results of Backtesting,
for those who may be interested.
Sections
- Introduction and Notation
- Additive Model
- Transition Model
- Comparison and Backtesting
1. Introduction and Notation
Suppose we have parties $i=1,\ldots,n$, and seats $k=1,\ldots,N$.
We measure all party votes as a percentage of turnout, so support values are real numbers
between zero and one.
Let us define our observables as
- $E(i)$ national (percentage) support for party $i$ at the last election,
- $C(i,k)$ (percentage) support for party $i$ in seat $k$ at the last election, and
- $P(i)$ current (percentage) support for party $i$ at the moment, as measured by the opinion polls.
Our aim to estimate the unknown variable of current support for each party in each seat:
- $A(i,k)$ current support for party $i$ in seat $k$.
There is no way we can be certain of this figure (without holding an election now), but
we can estimate it in a variety of ways.
2. Additive Model
The simplest model is that of additive Uniform National Swing. It says that if a party
is up 3% nationally, that it will be up 3% in each seat. In symbols,
$$
A(i,k) = C(i,k) + P(i) - E(i).
$$
This model, although simple, is quite robust, easy to implement, and seems to work
well in practice. One problem with the model is that the prediction $A(i,k)$ can be
higher than 100% or (more commonly) negative. Whilst this may not alter who wins the seat,
it is an unattractive feature.
For example, suppose Labour's support has dropped by 6% nationally, but in a particular
seat their support at the last election was only 4%. Then their predicted vote in that
seat will be -2% which is obvious nonsense.
3. Transition Model
The Transition Model aims to get around this problem, without moving too far from
the overall behaviour of the Additive Model.
The general idea is to categorise parties into those whose support has increased and those
whose support has decreased. The two groups of parties are handled separately.
- Parties who decline will be assumed to decline multiplicatively. That is, if their national
supports drops by a tenth (say from 40% to 36%), then their vote in each seat
will drop by a tenth.
- Parties who increase will gain those voters from the declining parties in proportion
to the amount of that party's increase compared with the other increasing parties.
Note that this is not a simple multiplicative model for the increasing parties. They only
increase votes in any one seat to the extent that the declining parties have lost votes.
To express the model in symbols we need to define two quantities. They are
- Share(i) is the relative national share of gains made by party i
relative to all the gaining parties:
$$
\hbox{Share}(i) = {\max\left( P(i)-E(i), 0\right) \over \sum_{j=1}^n \max\left( P(j)-E(j), 0\right)}.
$$
- SwingVoters(k) is the fraction of voters in seat $k$ who have changed parties
$$
\hbox{SwingVoters}(k) = \sum_{j=1}^n C(j,k) \max\left(1 - {P(j)\over E(j)}, 0\right).
$$
The predicted support levels are now
$$
A(i,k) = {C(i,k) P(i)\over E(i)},\qquad\hbox{if party $i$ declines},
$$
$$
A(i,k) = C(i,k) + \hbox{Share}(i)\times\hbox{SwingVoters}(k),\qquad\hbox{if party $i$ increases}.
$$
It is called a transition model, because we are modelling the transition of
each voter from their vote at the election to their vote now. Probabilistically, we
are saying that a voter from party $i$
- will definitely stay with party $i$, if party $i$ increases; otherwise
- will randomly stay with party $i$, with chance $P(i)/E(i)$, or else
- will change to party $j$, with chance $\hbox{Share}(j)(1-P(i)/E(i))$.
This is not fully perfect, as defectors from a particular party may be more likely to vote
for some parties rather than others, but it does eliminate the problems of the Additive Model.
4. Comparison and Backtesting
To test the two models, they were each run over the last four elections. In each case
the model was told the full results of the previous election, plus the total
votes received for each party at the new election, and asked to predict the detailed results
of the new election. The results were:
| Seats Wrong | Ave Vote Error |
Election | Additive | Transition | Additive | Transition |
2001 | 31 | 30 | 2.9% | 2.8% |
1997 | 50 | 42 | 3.1% | 3.1% |
1992 | 49 | 51 | 2.9% | 3.1% |
1987 | 40 | 40 | 3.1% | 3.2% |
The table shows the number of seats in each election which were wrongly predicted by each
model, and the average major party vote share estimation error of each model.
Apart from the better performance of the Transition Model over the Additive Model in 1997, both
models behave very similarly. There is no significant difference between their predictions.