Geographically Weighted Discriminant Analysis and the 2005 British General Election Ron Johnston, 1 Charles Pattie 2 1 School of Geographical Sciences, University of Bristol, Bristol UK. 2 Department of Geography, University of Sheffield, Sheffield UK This paper has been submitted for publication NOT TO BE CITED WITHOUT THE AUTHORS’ PERMISSION A response to a recent paper in this journal, identifying a substantial error in the empirical example used to illustrate GWDA and suggesting that it offers no improvement on standard linear discriminant analysis for that problem Recent statistical developments have enabled substantial advances in the analysis of spatial data, with novel methods being introduced – such as Geographically Weighted Regression (GWR) – which allow spatial variations in relationships to be explored. One such very recent innovatio n, building on GWR, is Brunsdon, Fotheringham and Charlton’s (2007; henceforth BFC) paper on Geographically Weighted Discriminant Analysis (GWDA). While not disputing the potential value of this approach, this brief note focuses on the empirical study deployed by BFC to illustrate GWDA’s applicability – the pattern of seats won at the 2005 UK general election in England and Wales only. Two issues are raised: 1. Does GWDA provide a better set of predictions than standard Linear Discriminant Analysis (LDA)?; and 2. Is the particular approach adopted consistent with the underlying argument regarding GWDA? Discriminant analysis and voting in England and Wales, 2005 To illustrate GWDA empirically, BFC use the results of the 2005 general election in England and Wales. Six variables (% of economically active males unemployed; % of the adult population with no qualifications; % of households in owner-occupied properties; % population pensioners; % population non-white; and % of households with a lone parent head)), reduced to two principal components, were used to predict the outcome in the 569 constituencies. According to their Table 1, the Conservative party won 196 seats, Labour won 314, and a group comprising the Liberal Democrats, Plaid Cymru and independents won 59: the actual result was – Conservative, 197; Labour, 315; Liberal Democrat, 51; Plaid Cymru, 3; Independent, 3. BFC report (p. 386) that using a ‘straightforward (nongeographically weighted) discriminant [i.e. LDA] analysis to predict the party elected in each constituency’ they predicted no seats at all for the Liberal Democrats (including Plaid Cymru and the independents). This is somewhat surprising, and we were unable to replicate their results in an LDA, using either their dataset (provided to us by Chris Brunsdon) or another widely used by electoral studies specialists (available at http://ksghome. harvard.edu/~pnorris/Data/Data.htm). Instead, we correctly predicted 21 of the 57 seats won by parties other than the Conservatives and Labour, although fewer of the seats won by those two parties (Table 1). Not only does our application of LDA produce an outcome superior to BFC’s (at least for the prediction of the geography of minor party victories) it also produces better results (again, especially for the ‘third parties’) than BFC’s two applications of GWDA to that data set – as Table 1 shows. Whereas they are only able to successfully classify 13 and 10 of the 57 Liberal Democrat and other party seats using the adaptive kernel and fixed kernel GDWA bandwiths respectively, using LDA we successfully classified 21. Given that our LDA application is ‘correct’, then the validity of GWDA – for this application at least – is open to considerable doubt. Why? One reason, we suggest, is the approach taken. BFC argue that the advantage of GWDA over LDA is that the former allows the relationships among the discriminating variables to vary over space – as is the case with GWR. So why reduce the six variables to two principal components which are, by their very nature, orthogonal and so – unless very narrow bandwidths are to be deployed – necessarily invariant in their relationship over space? We addressed this by running four separate LDAs distinguished by the number of separate groups to be identified and whether the original six variables or the two principal components were used as the discriminating variables. The former separation was undertaken because the third group in BFC’s analyses – Liberal Democrats, Plaid Cymru and Independ ents – form a chaotic conception; no strong substantive argument can be made for grouping them together and expecting them to be similar on the discriminating variables. In particular the three seats won by Independents have nothing in common: Bethnal Green & Bow was won by the Respect Party on a largely anti-Iraq War campaign against an incumbent Labour MP who had voted for the invasion; Wyre Forest was won by a local doctor standing as an independent who was first elected in 2001 on a campaign focused on retaining particular facilities at a hospital in the constituency; and the Independent Labour candidate who won in Blaenau Gwent did so against the Labour party’s candidate selected through the central party’s policy regarding all-women shortlists in certain seats. Whichever approach is taken, the results summarised in Table 2 indicate that the LDAs were superior to GWDA in predicting Liberal Democrat successes (as shown in Table 1). By incorporating the three Plaid Cymru won seats plus those won by independents, BFC are creating difficulties for any discriminant analysis, the independents because the three cases are so singular in their characteristics and Plaid Cymru because no variable (such as percentage speaking the Welsh language) is included which could discriminate the ir seats from the rest. If we exclude those six constituencie s, then we get the results shown in the second block of data in Table 2; we continue to out- perform GWDA. The reason why the LDAs outperform the GWDAs reported by BFC can readily be appreciated by a comparison of the main maps in BFC’s paper – showing the actual election result (Figure 3), the predicted result using LDA (Figure 4), the predicted result using fixed bandwidth GWDA (Figure 8), and the predicted result using adaptive bandwidth GWDA (Figure 9). Although the country can readily be divided into blocks according to which of the two main parties is likely to win there, LDA (according to BFC but not our results reported here) cannot identify the block of seats won by Liberal Democrats and others in far southwest England and in west Wales. The GWDAs does partially identify the latter (although only three of the eight constituencies won by neither the Conservatives nor Labour in Wales), but not most of the isolated seats/small groups of constituencies won by the Liberal Democrats and others elsewhere (e.g. southwest London and central southern England). In addition, the GWDAs divide the country into more cohesive blocks of Conservative and Labour territory than is actually the case. A further reason for the relative failure of GWDA with regard to the Liberal Democrats is that although it explores whether the relationships among the variables vary across different parts of the country it does not also investigate whether the differences between Liberal Democrat and Conservative and Labour seats vary: the relationships are the same, but the allocation is not. Many of the seats won by the Liberal Democrats are in areas where the Conservatives dominate (mainly in the south and west of England, but also in some suburban areas), and it is very likely that the Conservatives would win them if there were not a strong Liberal Democrat performance. But increasingly over recent elections the Liberal Democrats have also been challenging very strongly in some Labour strongholds – in 2005, for example, against government policies on the Iraq War and charging top-up fees to University students. Those constituencies are very different in their population characteristics from the first group. It is very unlikely indeed that either an LDA or a GWDA could separately identify both blocks; instead, they are most likely to identify the largest of the two only. Because of this, no LDA or GWDA can predict most of the Liberal Democrat seats; as a consequence, because of the local weighting incorporated, the latter predicts many more of the Conservative and Labour seats than the former – but also mis-allocates most of the Liberal Democrat seats to those categories as well. GWDA gets the big picture right, but cannot identify the residuals within it. BFC claim, in the abstract to their paper, to have shown that ‘similar social conditions can lead to different voting outcomes in different parts of England and Wales’: this discussion shows that in fact they have not, that GWDA has not captured that (well- known) situation, notably with regard to the very different sources of support for the Liberal Democrats. Discussion Is GWDA a necessary sophistication in order to improve the predictive capability of a discriminant analysis in this example? If support for political parties is spatially clustered, over and above what one would anticipate from knowledge of the constituencies’ characteristics, then perhaps reflecting that clustering by inclusion of regional variables would do the job as well. But the results of the analyses reported here suggest that is unnecessary; because BFC’s LDAs appear to have been wrongly conducted, we have found no evidence at all that GWDA is superior to LDA for their chosen empirical task – estimating the outcome of the 2005 British general election in England and Wales, in particular for the smallest of the three main parties. This conclusion, of course, does not negate the potential use of GWDA in other contexts, where there is a prima facie case for arguing that the relationships among the discriminating variables vary over space. But this is not the case with British elections – and in any case would not be if the potential for such variation were denied (or at least very substantially reduced), as BFC did, by reducing the number of variables through orthogonalisation. The new generation of local spatial statistical approaches has very considerable potential to advance our appreciation of a wide range of geographical patterns – but only if they are applied to problems where the key argument (spatially- varying relationships) is valid. Sadly this was not the case with BFC’s example for GWDA, hence its potential awaits further elucidation. References Brunsdon, C., S. Fotheringham, and M. Charlton (2007) “Geographically Weighted Discriminant Analysis”. Geographical Analysis 39, 376-396. Table 1. Predicted outcome (number of seats won) of the 2005 general election in England and Wales, using DA and GWDA with two principal components Seats won by C L LD+ Using DA BFC 150 290 0 JP 131 236 21 Using GWDA BFC (Adaptive kernel) 160 297 13 BFC (Fixed kernel) 161 291 10 ACTUAL 197 315 57 Key: C – Conservative; L – Labour; LD+ Liberal Democrats and others. BFC – Brunsdon, Fotheringham and Charlton; JP – Johnston and Pattie; LDA – Linear Discriminant Analysis; GWDA – Geographically Weighted Discriminant Analysis. The BFC results are taken from their Tables 4-6, p. 393. Table 2. Results of various LDA analyses Seats won by C L LD+ LD PC I All constituencies 6 variables – 3 groups 146 240 28 5 groups 140 178 21 3 1 2 components – 3 groups 131 236 21 5 groups 124 77 11 3 2 Constituencies won by Conservative, Labour and Liberal Democrat only 6 variables 147 245 30 2 components 143 227 14 ACTUAL 197 315 57 51 3 3 Key: C – Conservative; L – Labour; LD+ Liberal Democrats and others; LD – Liberal Democrats; PC – Plaid Cymru; I – Independents.
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