¹ Butler, David, ‘Introduction’ in Cook, C. and Ramsden, J., eds, By-elections in British Politics (London: Macmillan, 1973).Google Scholar

² Upton, Graham, ‘The Components of Voting Change in England 1983–1987’, Electoral Studies, 8 (1989), 59–74, p. 66.CrossRefGoogle Scholar

³ Curtice, John and Steed, Michael, ‘Analysis’, in Butler, D. and Kavanagh, D., The British General Election of 1987 (London: Macmillan, 1988), p. 340.Google Scholar

⁴ See, for example, the discussion in Stray, S. J. and Upton, G. J. G., ‘Triangles and Triads’, Journal of the Operational Research Society, 40 (1989), 83–92.CrossRefGoogle Scholar

⁵ This question was changed in the early 1960s to ‘How would you vote if there were a general election tomorrow?’ and, in 06 1970Google Scholar, to ‘If there were a general election tomorrow, which party would you support?’

⁶ Stray, S. J. and Silver, M., ‘Government Popularity, By-elections and Cycles’, Parliamentary Affairs, 36 (1983), 49–55, p. 50.CrossRefGoogle Scholar

⁷ Stray, and Silver, , ‘Government Popularity, By-elections and Cycles’, p. 50.Google Scholar

⁸ For further details concerning the manner in which Gallup and other polls have been conducted, see Chapter 5 of Stray, S. J., ‘British Parliamentary By-elections, 1950–1982: An Empirical Investigation’, unpublished doctoral dissertation, University of Essex, 1986.Google Scholar

⁹ At its lowest ebb in this period, it contested just 91 of the 506 seats available in 1951.

¹⁰ For example, Butler, D. and Kavanagh, D., The British General Election of 1987 (London: Macmillan, 1988).CrossRefGoogle Scholar

¹¹ The lack of full efficiency arises because of the compositional nature of the data. For each constituency the elements in the vectors x and y sum to zero. The ‘experimental errors’ associated with each constituency vector are therefore correlated, the correlation being – 1 in the case of two-party constituencies. Within the least squares framework, this can be taken into account by introducing a variance-covariance matrix V. However, this results in a considerable increase in programming complexity that does not seem justified in the present context, where the aim is to explore a phenomenon rather than provide fully efficient estimates. The standard work on the treatment of compositional data is Aitchison, J., The Statislical Analysis of Compositional Data (London: Chapman and Hall, 1986)CrossRefGoogle Scholar. Aitchison advocates the use of so-called log-ratio models. In the context of three-party (Conservative, Labour, Liberal) data, these models would imply the use of Y variables of the form log(Conservative/Labour) and log (Labour/Liberal). However, these log-ratios are not natural quantities for use in this context, and a model such as ‘log (Conserva-tive/Labour) = constant’ does not have any relation either to reality or to the original motivation of the present model in terms of the present vote being effectively a weighted combination of past votes.

¹² The final figure of 99.1 per cent is calculated from 100(66.6818 – 0.6046)/66.6818 and simply reflects the fact that constituencies vary far more from one to another than any particular constituency varies over time. This accounts for the fact that, in their study of by-elections (Studlar, D. T. and Sigelman, L., ‘Special Elections: A Comparative Perspective’, British Journal of Political Science, 17 (1987), 247–56CrossRefGoogle Scholar). the authors found that ‘general election results can be predicted with impressive accuracy [from by-election results]’ (p. 254). It would be amazing if it were not the case that two successive elections gave broadly similar results. Studiar and Sigelman do not look at the extra explanatory power of the by-election, having taken account of the previous general election result.

¹³ With this definition for Y, the basic unexplained variation is simply V = ΣY², where the sum is over all parties and all constituencies. The variation explained by the model is M = Σ(Y – Ŷ)², where Ŷ = âX, and â is the estimated value of a. The ratio M/V is essentially equal to R ², the coefficient of determination, since it measures the percentage of variation explained by the model.

¹⁴ For models of the form Y = aX, the confidence interval takes the form â[1 ± {1 – r²)/ (n – 1)R²}^½], where t is the appropriate significance point of a t distribution having (n – 1) degrees of freedom. The interval therefore widens with increasing â and narrows with increasing R ² or n.

¹⁵ This period clearly corresponds to the ‘homing period’ described by Taylor, S. J. and Payne, C., ‘Features of Electoral Behaviour and By-elections’, in Cook, C. and Ramsden, J., eds, By-Elections in British Politics (London: Macmillan, 1973)Google Scholar. See also Stray, and Silver, , ‘Government Popularity, By-elections and Cycles’.Google Scholar

¹⁶ Curtice, and Steed, , ‘Analysis’, Table 3.Google Scholar

¹⁷ Incorporating a third a-parameter, for three-party constituencies in which the second and third places were exchanged at the by-election, results in only a marginal improvement in the overall R ², to 63 per cent.

¹⁸ Fitting three separated a-values instead of one single value improves the fit by an insignificant 0.2 per cent.

¹⁹ The problem in the earlier period may well be that some would-be Liberal supporters will be unsure how to answer the standard Gallup question, because the idea of a Liberal candidate standing in their constituency is itself a fiction.

²⁰ Mughan, A., ‘On the By-election Vote of Governments in Britain’, Legislative Studies Quarterly, 13 (1988), 29–48.CrossRefGoogle Scholar

²¹ Butler, , ‘Introduction’.Google Scholar

²² Norris, P. and Feigert, F., ‘Government and Third-party Performance in Mid-term By-elections: The Canadian, British and Australian Experience’, Electoral Studies, 8 (1989), 117–30, p. 117.CrossRefGoogle Scholar