Methodological and Substantive Issues in Analyses of a Dependent Nominal-Level Variable in Comparative Research. The Case of Party Choice

Transcription

1 1 Methodological and Substantive Issues in Analyses of a Dependent Nominal-Level Variable in Comparative Research The Case of Party Choice Oddbjørn Knutsen Department of Political Science University of Oslo Paper presented at the 5 th Conference of the European Survey Research Association, Ljubljana, Slovenia, July 15-19, 2013 Session on European Values Study Methodological and Substantive Applications 2

2 2 Introduction The purpose of this paper is to outline and discuss how the impact of explanatory variables for party choice or voting can be analysed given that the variable is nominal-level with more than two values. The party systems of nearly all Western democracies are multi-party systems and we assume that these comprise more than two significant parties and consequently that the party choice variable is not a dichotomous variable. This is illustrated with a research problem where five sets of value orientations are used as independent variables for explaining party choice. Two of these value orientations are labelled Old Politics values while the three others are called New Politics orientations. Since the dependent variable is a nominal level variable, ordinary least square (OLS) cannot be used. The choice given such a research problem has usually been considered to be between discriminant analysis (hereafter DA) and multinomial logistic regression (MNL). It is, however, not so easy to use these statistical techniques for analysing the specific research problems formulated in this paper. In the bivariate case, the analyses of the association between single value orientations and party choice, some measures from analysis of variance are useful. One standardised and two unstandardized measures that can be derived from analysis of variance and dummy regression are discussed. The paper discusses strengths and weaknesses of these methods for given research problems, and compares the results by using the various statistical procedures, but also reports on substantive issues regarding the topic. When the dependent variable is at a nominal level of measurement, the explained variance has no direct meaning since there is no variance in the dependent variable. However, several so-called pseudo R-squared measures have been developed to measure goodness of fit for models with a dependent nominal level variable. Here the notion of explanatory power is used for goodness of fit measured by pseudo R-squared or similar measures. Both DA and MNL contain another aspect than goodness of fit, i.e. the prediction aspect. This aspect is often referred to as predictive efficacy and refers to the ability of the models to generate accurate prediction of the status of the case on the dependent variable on the basis of the values of the independent variables. 1 The assessment of predictive efficacy will not be discussed here because this is not of direct relevance for the research problems that are formulated, and it is quite possible to have an excellent fit between the model and data without necessarily having a model with much predictive efficacy (Demaris 1992: 26). The empirical analysis in this paper is based on data from 18 West European countries with the European Values Study 2008 as the data source. The party choice variable and the five value orientations used in the empirical analyses are outlined below. The paper is organised as follows: First, five research problems are formulated. The literature on determinants of party choice is large. Various ways of analysing party choice in the literature are briefly outlined. Some basic features of DA and MNL are then outlined and 1 Sometimes a differentiation is made between prediction and classification models in this respect (Menard 2002: 27-36). In discriminant analysis Klecka (1980: 42) differentiates between an interpretation aspect and a classification aspect of discriminant analysis. In this paper the focus is on part of the interpretation aspect (see below).

3 3 the strength and weaknesses of these statistical procedures are discussed, both in general and in relation to the research problems in this paper. The data and the operationalization of the various variables are briefly outlined. In the empirical analyses each of the research problems is addressed by discussing and analysing how various measures within DA and MNL in particular can be used for analysing the particular research questions. Traditional socio-structural variables have been used as determinants of party choice in many studies. In the final section the issue of including such variables in addition to the value orientations as independent variables are briefly addressed. The research questions Given that this is comparative research, the research questions are related to cross-national differences in the strength of the associations and effects (when controlling for the other value orientations ) between value orientations and party choice (called comparative strength below). However, it is also a research question to examine the relative strength of the various value orientations within each country. Research questions should as a principle be formulated before the choice of statistical methods. One should then choose between the available statistical procedures. This should not be done the opposite way where the statistical procedures decisively influence the research problems that are formulated. The research questions are the following: 1. What is the comparative strength of the associations between party choice and value orientations? The textbooks tell us that we should use unstandardized coefficients for comparing the strength of correlations and effects, when comparisons are made between samples (countries) for the same variable. What are the unstandardized measure(s) that can be used to tap the associations between value orientations and party choice? The b-coefficient from OLS cannot be used, given that the dependent variable is a nominal-level variable. Does it make any difference whether standardised or unstandardized measures are used? The standardised coefficients are frequently used simply because the unstandardized alternatives are fairly unknown to researchers and are not part of any software statistical procedure. If we employ standardised coefficients instead, does this result in misleading conclusions? 2. What is the relative strength of the association between party choice and value orientations within the various countries? According to textbooks, standardised measures should be used for comparing the associations and effects of independent variables within the same sample (country in this context). Two standardised measures are used for analysing the relative strength of the association between each value orientation and party choice. Does it make much difference which of these measures is used? 3. What are the effects of each of the value orientations on party choice when controlling for the other value orientations? 4. What is the total explanatory power of all value orientations in a comparative setting?

4 4 5. What is the impact of New and Old Politics orientations in a comparative setting? For the research problems, measures from multinomial logistic regression and discriminant analysis will be compared. One way of indicating the similarities and differences between correlation coefficients, controlled effect and explanatory power is to correlate the various measures with each other using the countries as units for the analyses. A macro file has been generated with the various measures tapping these aspects of the relationship between value orientations and party choice. These correlations between the various measures will be reported below. Statistical methods with a nominal dependent variable: discriminant analysis and multinomial logistic regression Introduction The analyses of party choice has for a long time been made by dichotomising the variable into leftist and non-leftist parties. Perhaps most famous within this approach is the Alford index for tapping class voting (Alford 1964). The index is based on a dichotomisation of party choice into leftist and non-leftist parties, and also a two-class schema. This is simply the percentage difference between the workers and the other classes in voting for the leftist parties. However, this tradition goes far beyond the studies of class voting. Large parts of important works and even whole books have been written on the basis of this dichotomisation of party choice (Rose 1974; Franklin, Mackie, Valen 1992). More recent and advanced contributions are also based on dichotomous party choice variables (Elff 2007). Another approach has been to use the left right location of the parties as a dependent variable and then use conventional OLS on a party choice variable where the parties are located on a left right scale. Another more sophisticated approach is to change the dependent variable from actual party choice to the propensity to vote for each of the parties in the system. This is measured by a battery of questions which ask for each party how likely it is (on a scale from for example 1 to 10) that a respondent will ever vote for a given party. The approach is based on the assumption that parties, as objects of electoral choice, represent utilities to voters and the responses to the questions are supposed to tap electoral utilities. Electoral utilities are then supposed to determine which party a voter will vote for (van der Eijk 2002: ; van der Eijk et al. 2006). In order to analyse party choice between different countries the data are reordered in the form of a stacked data matrix so that each voter appears as many times as there are parties for which their vote propensity has been measured (van der Brug et al. 2009: ). This approach it is argued is well-suited for cross-national analyses and OLS-regression. Similar approaches can be used as statistical methods since the dependent variable has a high level of measurement (van der Eijk et al. 2006: ). However, electoral utilities are important and a valuable supplement to the dichotomous or discrete choice which choosing a given political party represent. In the electoral utility approach, the individuals considerations of electoral utilities are considered as prior variables in a causal model that influences party choice (van der Eijk et al. 2006: ),

5 5 and it is still highly legitimate to focus on the dichotomous or discrete party choice as the crucial choice and variable in electoral research. The two main statistical analyses that can handle dependent variable at a nominal level are discriminant analysis and multinomial logistic regression. Discriminant analysis is a statistical method that can be considered as older than multinomial logistic regression, but has not been frequently used in connection with analyses of party choice. Multinomial logistic regressions have been quite popular in more recent analyses of party choice, but it is an ongoing debate how useful this statistical procedure and various variants are for analysing party choice. Discriminant analysis Description of the statistical method Discriminant analysis has long been used as a statistical procedure for analysing determinants of party choice since this method only requires that the dependent variable is at nominal level. The nominal-level variable is called the group variable and the variables that are supposed to discriminate between the groups are called discriminating variables. Central in the analysis are the so-called discriminant functions. A canonical discriminant function is a linear combination of the discriminating variables, derived such that differences between the groups (according to the group means) are maximized for the first function; the second function is also derived to maximize the differences between the groups, but under the added condition that values of the second function are not correlated with values of the first function. The third function is derived in similar fashion, under the condition of being uncorrelated with the previous functions, and so forth. Each discriminant function can be considered to correspond to one dimension in a spatial perspective. The basic statistics are associated with each of these functions; the strength of the functions are measured by the canonical correlations, the location of the groups on each function indicated by the group centroids and the effects of the discriminating variables that contribute (or discriminate most) on the given functions the discriminant coefficients. On the basis of the two latter statistics the various functions are interpreted (Klecka 1980: 23-42). In the context of analyses of determinants of party choice, the party choice variable is the group variable and various determinants of party choice the discriminating variable. Discriminant analysis is first and foremost a powerful tool for examining conflict dimensions in party systems. Each discriminant function can be considered as a conflict dimension. 2 The research problems in this paper do not address conflict dimensions but the overall strength of different determinants of party choice (value orientations). The statistics associated with the functions are not directly relevant. For example, the central standardised statistical measure, the canonical correlation, in discriminant analysis is associated with each of the discriminant functions, not with the overall discriminating power of all discriminating variables. In a discriminant analysis with only one discriminating variable, this coefficient is identical to the eta-coefficient from analysis of variance (Klecka 1980: 37) (see further discussion of this below). 2 For analyses of conflict dimensions by means of discriminant analysis, see Barnea and Schwartz (1998), Evans and Whitefield (1998), Knutsen (1989, 2012), Moreno (1997) and Rosas (2010).

6 6 There are few statistical measures that tap the overall explanatory or discriminating power of single or groups of variables in discriminant analysis. There is however one measure that can be used for this purpose. This measure is labelled Wilks lambda transformed below. Wilks lambda is a multivariate measure of group differences covering several variables (the discriminating variables). Wilks lambda is the ratio of within-groups variance to total variance (sum of squares), and is therefore the percentage of variance in discriminant scores not explained by group membership. Wilks lambda measures the residual discrimination in the system (Klecka 1980: 38-39). In the SPSS output this statistics is associated with the remaining cumulative discriminating power of all the discriminating functions and then with the remaining discriminating power when the most important functions is dropped and so on. Since there are five discriminating variables in the analysis of the value orientations, there is a maximum of five functions, and the Wilks lambda that is used here is based on the residual discriminating power when all these functions (significant and insignificant) are included. This measure simply taps the residual discrimination when the discriminating power of all five value orientations is taken into consideration. The measure taps the residual discrimination and is then an inverse measure. As lambda increases to its maximum value of 1.0, it is reports progressively less discrimination power, and when it decreases towards zero, it denotes high discriminating power (Klecka 1980: 38-39). Here we simply use the inverse value of Wilks lambda so that the measure of discriminating power is calculated as 1-W. In a discriminant analysis with only one discriminating variable, Wilks lambda transformed is identical to the square of the canonical correlation coefficient. Thus, when additional discriminating variables are added to the analysis, the canonical correlations are associated with the various functions, while Wilks lambda registers how much discriminating power that is left when the various functions are included in the analysis. Wilks lambda transformed then taps the discriminating power or explained variance of the variables that are included in the analysis. This measure has been used previously in Knutsen (1988), Knutsen and Scarborough (1995) and Knutsen 1995b: 45-48). Toka (1998: ) has, however, argued that the measure can in no way interpreted as analogous to R-squared statistics. Other scholars have, however, used this measure (transformed as indicated above) for reporting on the explanatory power of a model in a discriminant analysis (Dattalo1994: ) and considered Wilks lambda as the percentage of variance in discriminant scores not explained by group membership (Betz 1987: 397). Strength and weaknesses The strength of discriminant analysis for research problems is first and foremost associated with discriminant functions and for analyses of party choice conflict dimension. Wilks lambda transformed enables the researcher to estimate the total impact of the discriminating variables across the various discriminant functions. The assumptions for discriminant analyses can be considered difficult to obtain for all variables and groups in concrete data sets. It is required that the discriminating variables are at interval or ratio level of measurement so that means and variances can be calculated and

7 7 legitimately employed in mathematical equations. Nominal level variables cannot be used (Klecka 1980: 9). There is no coefficient which indicates the overall controlled effect of a given discriminant variable. The effects of the various discriminant variables are associated with the various discriminant functions. If there is only one significant discriminant function, the effect of the variables on that function approaches the total effect of that variable. A discussion follows below of how Wilks lambda can be used for the purpose of examining the effect of one discriminating variable by running a model with all discriminating variables included and another model with the given variable omitted from the analysis. 3 Furthermore there are some assumptions regarding the group variable. It is assumed 1) that the population covariance matrices for the discriminating variables are equal for each group, and 2) that each group is drawn from a population which has a multivariate normal distribution (Klecka 1980: 9-10). There is some disagreement about the performance of the statistical procedure when these latter assumptions are violated. Some argue that the results can be misleading when these assumptions are violated, while Klecka (1980: 60-62) argues that discriminant analysis is a fairly robust method and that the violation of the assumptions may have consequences for specific aspects of the procedure. 4 Multinomial logistic regression Description of the statistical method The main preoccupation of binary logistic regression in the case of party choice is to examine the probability for voting for a given party in comparison to a reference category when other independent variables are controlled for. Binomial logistic regression can be used for a dichotomous dependent variable that is a two-party system. The essence of the analysis is to calculate the probability of voting for a party for different social groups such as, for example, workers, employers, higher level non-manuals and so on for a class variable. It is important to understand that the probability, the odds and the logit are three different ways of expressing exactly the same thing in logistic regression (Menard 2002: 13). Multinomial logistic regression allows for more than two categorical values on the dependent variable and is then suitable for analysing predictors of party choice in a multiparty setting. MNL is a straightforward extension of binomial logistic regression. One value on the dependent variable is designated as the reference category and the probability of membership in other categories is compared to the probability of membership in the reference category (Menard 2002: 91-92). This is indeed the essence of the MNL model: the various measures in MNL are generally calculated for each logistic function (pair of groups). It has been argued that this is similar to calculating the separate canonical discriminant function coefficients for each linear discriminant function in discriminant analysis (Menard 2002: 94), but it is nevertheless quite different functions. In discriminant analysis each group is located in the various groups on a dimension in relation to each other, while the functions in MNL are the calculated probabilities for different independent variables and 3 The F-to-remove statistics that are associated with each variable can also be used for this purpose (Dattalo 1994: 132; Knutsen 1989), but this is not a standardized coefficient. 4 For a review of this debate, see Dattalo (1994: 22-23).

8 8 values on independent nominal-level variables for voting for a party versus the voting for the parties that comprise the reference group. The impact of the whole independent variable (in particular for nominal level variables with more than two categories), is frequently lost in MNL analyses. While there is a b-coefficient for the binary logistic regression analysis that indicates the effect of the independent variable, there is no such coefficient in multinomial logistic regression. The focus is then more on a comparison of the probability of voting for a given party as indicated above, and there are numerous examples whereby the research problem is adopted to exactly this in the literature given the use of MNL. 5 Pseudo R-squared as measures of goodness of fit However, there are pseudo R-squared measures which are supposed to tap the goodnessof-fit or the explanatory power for the full model of explanatory variables. These measures have received relatively little attention in the literature concerning binary logistic regression and MNL, but some textbooks deal significantly with this aspect (see Menard 2002: 20-27). These measures are calculated based on the predicted probabilities and observed classification for all categories on the dependent variable (Menard 2002: 94). There are several such measures and these produce quite different results concerning explanatory power (DeMaris 2002; Hagle and Mitchell II 1992, Menard 2000, see also Goeman and Cessie (2006) and Fagerland, Hosmer and Bofin (2008). Three pseudo R-squared measures are reported in the SPSS NOMREG program, McFadden, Cox and Snell and Nagelkerke. The McFadden measure is recommended by Menard (2002: 27), but it is well-known that it produces quite low explanatory power, much lower than R-squared when these measures are compared (Demaris 1992: 54). Cox and Snell s and Nagelkerke s measures are based on the same approach. The R-squared is based on the improvement of the likelihood from a null model to a fitted model. While Cox and Snell s measure can never be 1.00 even if the full model predicts the outcome perfectly and has the likelihood of 1, Nagelkerke s R-squared adjusts for this and can achieve the level of 1.00 if the full model perfectly predicts the outcome. Nagelkerke s measure is therefore chosen to tap the explanatory power based on multinomial logistic regression. It will however be compared with the McFadden measure, but this is not done in any detail in this paper. In the case of only one independent variable, the pseudo R-squared measure can be considered as a standardized measure of the explanatory power of a given independent variable, and the square root can be considered as a coefficient similar to r from regression analysis and eta from analysis of variance. In the bivariate analyses between the value orientations and party choice below, the Nagelkerke pseudo-measure will be used in this way. Strengths and weaknesses Multinomial logistic regression is generally well-suited to analyse independent variables on different levels of measurement. Both nominal (factor-variables) and interval level variables (covariates) can be used as independent variables. 5 Analyses of these aspects of determinants of party choice by means of MLR are found in Alwarez and Nagel (1998), van den Berg and Coffé (2012) Dow and Endersby (2004), Quinn, Martin and Whitford (1999), Whitten and Palmer (1996).

9 9 MNL does not make any assumptions of normality, linearity, and homogeneity of variance for the independent variables. Because it does not impose these requirements, it is argued that it is preferable to discriminant analysis when the data does not satisfy these assumptions (Demaris 1992: 61). The main preoccupation in MNL is that the statistics that are based on the comparisons of probabilities for one and one category with the reference category. The lack of a coefficient that taps the impact of the independent variable on the whole dependent variable may be considered as a major problem for many research problems in MNL. Empty cells are a significant or even major problem in logistic and multinomial logistic regression in particular. If a cell is empty in the contingency table that the analysis is based upon, the odds and logit for that particular category will be +- indefinitely ( ) and the results will be a very high estimated standard error for the coefficient associated with the category. Generally, this results in instability of the estimates of coefficients and their standard errors. The problem applies especially to categorical variables, and appears in particular when such variables have many categories since this generally increases the likelihood for empty cells. This applies both to the dependent variable and categorical independent variables. This problem occurs frequently in analyses of party choice as the dependent variable, in particular in systems with many parties. The solution is often to collapse categories on the dependent party choice variable or independent variables. This can, however, result in a cruder measurement of the independent variable and may bias the strength of the relationship between the predictor and the dependent variable towards zero (Menard 2002: 78-79, 93). Menard s textbook provides an illustrative example: In an analysis of a variable with four values on the dependent variable and six values on an independent ethnicity variable, four of the groups on the ethnicity variable had to be collapsed into an other category. Failure to do this would have resulted in problems with zero cells, and instability in estimates of coefficients and their standard deviations (Menard 2002: 93). One might argue that important aspects of the original analysis have to be dropped in order to satisfy the assumptions for the statistical procedure. This problem emerges frequently when dealing with many parties. The estimates become problematic and the method cannot be used for the purpose. It should be emphasized that this problem does not arise seriously in the analyses here with five interval level variables. In one of the standard articles about MNL, the method (and also probit modelling) is illustrated with an analysis based on the election survey from the 1994 Dutch election survey (Alvarez and Nagel 1998). The analyses are based on the five largest parties. This is considered good because the data were rich enough to allow us to explore voting for give of the partiers (italics added here). The other parties are simply dropped from the analyses and this is not considered as an important problem at all in the paper. Van der Eijk et al. (2006: 438) have pointed out that this analysis had excluded 4 smaller parties, and 10% of the sample that had indicated a party choice was consequently omitted. This will bias the estimated coefficients since the probabilities for voting for these parties are not included in the calculations. When these authors use their electoral utility approach and exactly the same independent variables that are used by Alvarez and Nagel (1998), R- squared was 0.48 for an analysis based on the five parties and which then omits 10% of those who had indicated a party choice in the survey. R-squared increased to 0.59 when all party voters and voting for the four smaller parties was included in the analysis. The reason

10 10 for omission of these smaller parties is not explained in the Alvarez and Nagel article, but this is surely caused by problems with empty cells. 6 The use of log odds ratios as a basis for calculating cleavage strengths The Alford index for class voting has been criticized for being sensitive to the distribution of the two variables (dichotomous class and party choice), and it has been suggested that logodds ratios should be used instead to measure so-called relative class voting in contrast to the absolute class voting tapped by the Alford index (Hout, Brooks and Manza 1993: ; Weakliem 1995; Nieuwbeerta 1995: 39-42). When more than two classes or social groups are used to tap the social class or social variables, the analyses become more complicated. Hout, Brooks and Manza (1993: ; 1995) suggest using the kappa-index. The kappa index calculates several log-odds ratios between a reference category on the class variable and each of the other classes, and uses the standard deviation of these log-odds ratios as a measure of class voting. The higher the value of the kappa-index, the higher is the level of class voting. The kappa-index has several desirable statistical properties. The most desirable property is that the index is based on logodds ratios and is therefore not dependent on the marginal distributions of the independent or dependent variables. This way of measuring class voting is to some extent also found in analyses of other social cleavages. For example, in more recent research on the impact of various structural variables on party choice the kappa index is frequently used (Brooks, Nieuwbeerta and Manza 2006; Jansen 2011). The calculation of kappa is, however, based on a nominal level independent variable (such as a class variable based, for example, on the EGP class scheme) or religious denomination, not for variables that are at the interval or ratio level of measurement. It is unclear how interval level cleavage variables can be included in the analyses. It is the researcher who calculates the kappa-indices but it is based on the logic of logistic regression and multinomial logistic regression. Kappa values can be calculated for each political party. It has also been proposed that the average kappa coefficient for each significant party in a party system can be used as a measure of the overall cleavage strength simply by calculating the average kappa across the various parties. These overall cleavage strengths are supposed to have properties that allow comparison of strength across social conflict variables (at a nominal-level of measurement) (Brooks, Nieuwbeerta and Manza 2006; Jansen 2011). There are, however, two important critical questions regarding these measures when they are aggregated from a given party to the whole party system. It is logical that all social groups shall count equally when the focus is on the analysis of a single party, but when the kappa-measure is aggregated to the whole party system, the question arises whether all classes and also the kappas for all parties should count equally even though the classes are different sizes and the parties have different levels of support in the surveys. This is not discussed in the literature on these measures of social cleavages. 6 For comparisons between DA and MLR, see Bull and Donner (1987), Dallatto (1984) and Press and Wilson (1977).

11 11 Another limitation of these measures is that they are developed for nominal level independent variables, not variables at a higher level of measurement. When, for example, frequency of church attendance is included as a determinant of party choice, it is dichotomized. The data, countries and variables The data used in this paper is from the 2008 European Values Study (EVS 2008). The eighteen West European countries included in the analysis in this paper are outlined in Table 1. They are grouped into four regions for descriptive purposes and for being able to indicate some broad comparative patterns. < Table 1 about here > The party choice variable In the EVS 2008 surveys, respondents were first asked the traditional question about voting intention: If there was a general election tomorrow, can you tell me if you would vote. If the respondents answered Yes, I would vote, they were then asked which party they would vote for. There was then a follow-up question for those who replied that they would not vote. These respondents were asked Which party appeals to you most. Those who indicated a specific party on this question are added to those who indicated a party on the voting intention question to obtain the party choice variable that is used here. The party choice variable aims at including as large a portion of the samples as possible. On the country-level average, 69% of the respondents indicate a party choice based on the two questions. The regional means vary from 79% in the Nordic countries to 61% in Southern Europe. The variation between the countries varies from 45% in Portugal to 89% in Norway. See the Appendix Table. Parties receiving less than 2% support are grouped into other parties. All respondents who indicated a party choice are included in all analyses; none are left out. The national weight variables are used for the various countries to adjust for sampling biases. 7 The weighted number of cases sometimes deviates from the unweighted N, Old Politics and New Politics value orientations Values are prescriptive beliefs signifying that certain end-states or modes of conduct are personally or socially preferable to other end-states or modes of conduct (Rokeach 1973: 5-8). Value orientations may be the most central feature of culture because they express shared conceptions of what is good and desirable. Some important political value orientations can be derived from the structural cleavages incorporated in the well-known model of Lipset and Rokkan (1967). The religious cleavage is related to religious as opposed to secular values. The most important political value orientations that emerged from the Industrial Revolution were economic left right values. These value orientations are economic in nature, and refer in particular to the role of government in creating more economic equality in society versus the need for economic incentives and efficiency (Knutsen 1995a). 7 For Belgium and Germany the samples were drawn as disproportional stratified samples and the weighting variable to take this into account was also used.

12 12 Religious secular values and economic left right values are often referred to as Old Politics since they capture the essence of the traditional lines of conflict in industrial society. In contrast, New Politics refers to value conflicts emerging from post-industrial society. Here, New Politics values are conceptualised along three different dimensions. The value conflict between environmental protection versus economic growth values is firmly rooted in the public mind, and in many West European countries conflicts over environmental values seem to be the most manifest expression of the New Politics conflict (Dalton 2009). In a series of articles, Scott Flanagan emphasised that a libertarian/authoritarian dimension is the central New Politics dimension (Flanagan 1987; Flanagan & Lee 1988, 2003). The libertarian/authoritarian value orientations are also the central components in Herbert Kitschelt s (1994, 1995) important works on changes in the party systems of Western democracies. The third set of New Politics orientations is related to immigration and immigrants. This has become a major policy area in Europe with different views among the mass publics. One might argue that these orientations are attitudes rather than values, but here they are considered as basic orientations which are closer to values. Comparative research has shown that these orientations are closely related to, and reflect basic values and beliefs concerning different conceptions of national identity, ethnicity and multiculturalism (Hainmueller & Hiscox 2007: ). On the basis of the above discussion, I use the following central value dimensions for analysing the relationship between value orientations and party choice: Old Politics orientations 1) Religious versus secular values (Relsec.) 2) Economic left right values (Eclr) New Politics orientations 3) Libertarian/authoritarian values (Libaut) 4) Environmental protection versus economic growth, higher taxation etc. (Environ) 5) Immigration and immigrant orientations (immigration orientations, Immigr) All value orientations are tapped by multiple items from the survey. All indices are constructed as equal-weighted additive indices with values from 0 to 10. All respondents are assigned a score on the indices. 8 8 Respondents who have not answered a question are assigned the mean value for their country on that particular question. Details regarding the question wording for the items and the index constructions can be obtained from the author.

13 13 The comparison of the associations between party choice and value orientations in a cross-national setting (research question 1) Introduction In this section two standardised and two unstandardized coefficients are compared for tapping the comparative strength of the associations. The measures that can be used from discriminant analysis and multinomial logistic regression will be discussed, but first and foremost measures that are associated with analysis of variance and partly also OLS (dummy-regression) will be used and explained below. Discriminant analysis: In a discriminant analysis with only one discriminant variable, the canonical correlation is identical to the eta-coefficient in analysis of variance (Klecka 1980: 36-37) so it is not worth reporting this coefficient since the eta-coefficient is used (see below). Wilks lambda transformed is the square of the canonical correlation with only one discriminating variable. This measure then only taps additional information when more than one discriminating variable and more than one discriminant function is derived from the analysis. Multinomial logistic regression: As mentioned above, there is no standardised or unstandardized coefficient that measures the strength of the relationship between the nominal level dependent variable and the independent variable. However, the pseudo R- squared measures can be useful in this context. Nagerkerke s R-squared is equivalent to explained variance, and the square-root of this measure will be used as a bivariate correlation between party choice and each of the value orientations. Eta from analysis of variance: The eta-coefficient, also called the correlation ratio, is closely associated with analysis of variance, but eta can be a useful coefficient outside the context of ANOVA. 9 The eta-coefficient requires that the dependent variable is at interval or ratio level, while the independent variable is at a nominal level. Eta squared is the explained variance in a one-way analysis of variance (with one independent variable) and is equivalent to R squared in OLS (Iversen and Norpoth 1980: 30-37). In practice, the ratio-level variable (which in the analyses here is the value orientation) has to be treated as the dependent variable and the nominal level variable (party choice in this case) as the independent variable when eta-coefficient is calculated. This has the important consequence that there is no multivariate coefficient which the eta can be compared with when other independent variables are included in the analysis. The eta-coefficient is then very useful for examining the bivariate correlation between party choice and independent variables at a ratio-level. 10 Granberg and Holmberg (1988: 50) examine for example the impact of single issues on voting and use the eta-coefficient, which they label the issue voting coefficient for their purposes. However, there is no direct equivalent coefficient that can be used in multivariate analyses with a dependent variable that is at a nominal level of measurement. 9 The discussion of the eta-coefficient and the two unstandardized coefficients are based on Knutsen (1998: 9-15). 10 The strength of the relationship between various independent variables and party choice which uses the eta-coefficient is found in Granberg and Holmberg (1988), van der Eijk, Schmitt and Binder (2005) and Knutsen (1995a,b).

14 14 Since this measure and two associated unstandardized measures are discussed in some detail below, the formula for the eta-coefficient is outlined here: f i is the percentage of the vote for each party (party no. i), x ij is the observed placement on the value scale of individual j voting for party i; is the mean score on the value scale for those individuals voting for a given party (i) and is the overal The eta-coefficient is a standardized coefficient which ranges from 0.00 to It corresponds precisely to the r-coefficient when the categories on the party choice variable are given the mean score on ratio-level variable and this variable is correlated with the ratio-level variable. One of the two unstandardized measures is fairly closely associated with the eta-coefficient, while the other is coupled to regression analysis with dummy variables. Both of these measures are based on the same logic as the eta-coefficient and the assumption that the ratio-variable is the dependent variable; they cannot be used in multivariate analyses with party choice as the dependent variable, The squared measure (Taylor and Herman): A measure used to tap polarisation in the party system was proposed by Taylor and Herman (1971), and used by Sigelman and Yough (1978) and Lane and Ersson (1994: ). This is an unstandardized measure which is simply the numerator in the formula for the eta-coefficient, i e.: f i ( - ) 2 where: f i is the percentage of the vote for each party (party no. i), is the (mean) score on a value dimension for a given party (i) and is the overall mean of the scale for all parties calculated on the basis of the percentage of the vote. The measure is unstandardized because it is based only the (squared) deviations from the mean, not the variance in the ratio or interval level variable (the denominator for the etacoefficient) The absolute magnitude measure (The Huber measure): The other alternative measure is derived from a work of John Huber on the left right scale (Huber 1989: 615). Huber presents a measure which is based on the logic of dummy regression, 11 but which can easily be transformed to an equivalent measure to the logic of variance statistics. This measure can be written as follows: f i ( - ) 11. Huber s measure was developed to tap the total effect of partisanship on left right self-placement scale from a regression analysis where the support for the different parties was the dummies. His measure of what he called the weighted mean party component (WMPC) was calculated as follows: WPMC= b i * VP i b i is the dummy coefficient for party i VP i is the percentage of the vote received by party i.

15 15 ( - ) is the absolute value of the difference between the mean score on the left right scale of voters for party i and the "grand mean", i.e. the mean (of voters) for all parties. On this measure, unlike the measure advanced by Taylor and Herman, deviations from the overall mean are not squared. The differences between these two measures might appear trivial, but in fact they are substantial. The left right polarization measure of Taylor and Herman implies that large deviations even for a small party will count substantially, since the deviations from the overall mean is squared. This is not the case for the Huber measure. Here the deviation from the mean and the size of the party count equally, and significant deviations from the grand mean for larger parties count relatively more than large deviations for smaller parties. Empirical analysis Table 2A-E show the strength of the correlations between party choice and each of the five value orientations for the 18 countries based on the four measures, the two unstandardized measures and the standardised eta- and square root of Nagelkerke s R-squared coefficient. < Table 2 about here > Religious secular values: There are large comparative variations in the strength of the correlations. For all measure the correlations are strongest for The Netherlands and Germany, and the correlations are outstanding in The Netherlands, much higher than in the other countries. The smallest correlations are found in Denmark and Iceland, but somewhat surprisingly the correlations are comparatively strong in the other Nordic countries, in particular in Finland and Norway, but also in Sweden. We note that the ranking of the three mentioned Nordic countries are higher for the square measure than for the absolute magnitude measure. A closer examination of this which is not discussed in detail here, is that the Christian parties in the Nordic countries are extremist parties on this dimension with much higher scores than religions parties in the other countries and, of course, relative to the other parties in the Nordic countries. These parties are small in terms of electoral support and therefore their extremist position at the voter level contributes to a comparatively higher coefficient for the squared measure than for the absolute magnitude measure. The ranking of the regions is almost precisely the same for the four measures: the southern region has, however, a higher average score than the Nordic countries on the absolute magnitude measure. The correlations between these correlation coefficients for the 18 units are very high, between the absolute magnitude measure and the other three measures, and between the three other coefficients. Economic left right values: Table 2B shows the strength of the correlations between the economic left right index and party choice based on the four measures. According to all four measures the strength of the correlations is strongest in the five Nordic countries, followed by France, Switzerland and The Netherlands 12 and weakest in Ireland and Portugal and then Luxembourg and Spain. 12 The correlation in Italy is however larger than in The Netherlands for the absolute magnitude measure and the correlation in Austria is exactly the same as in The Netherlands for the same measure.

16 16 According to the two unstandardized measures three groups of countries can be identified. The correlations are strongest in the Nordic countries and France and weakest in Ireland, Portugal, Luxembourg and Spain, while the remaining countries comprise an intermediate group. This division can to some degree also be found for the two standardized measures, but is not so clear. The ranking of the four regions is also for economic left right values identical across the four measures although the distances between the groups are somewhat different. The Nordic counties are most outstanding for the squared measure and the average correlation is relatively smallest for the standardised measures. Again, the correlations between the four measures based on countries as the units are close to perfect: Environmental values: Concerning environmental values, the correlations for the 18 countries are shown in Table 2C. The strongest correlations are found in Belgium, Switzerland and the Nordic countries, while the weakest correlations are found in the south European countries, in Ireland and somewhat surprisingly, Germany. The correlations are particularly weak in Greece, Germany, Spain and Portugal. As to the average correlations for the four regions we find similar rankings for all four measures and the relative distance for these average correlations are impressively similar.. There is some variation in the ranking of the countries among those with both the highest and lowest correlations, but the correlations between the four measures are again very strong, Libertarian authoritarian values: The correlations between party choice and libertarian values are shown in Table 2D. The correlations are largest in Austria, The Netherlands and Switzerland, followed by Greece and Denmark. Among the countries where the correlations are smallest, we find Portugal, Ireland, Britain and Luxembourg and for the squared measure also Spain. There is a consistent ranking of the regions across the four measures. The correlations are largest in the central western regions followed by the Nordic region and smallest on the islands. The largest average correlations are found in the Central western region and are more pronounced for the unstandardized measure, in particular for the squared measure. The correlations between the strength of the coefficients for the 18 units are again very strong, Immigration orientations: Regarding immigration orientations, Table 2E shows considerable variation in how these orientations and important for individuals party choice. The correlations are strongest in Austria, Italy and Switzerland followed by several of the Nordic countries and France and weakest in Ireland and the Southern European countries, Portugal, Greece and Spain. The correlations in Norway are comparatively large for the standardised measures compared to the unstandardized. We find again a consistent ranking of the regions: the average correlation is largest in the Nordic countries and then the central western region then is the south and finally in the Island countries.

17 17 The correlations between the strength of these measures are somewhat less strong than for the other correlations between value orientations and party choice, with one exception. It is, however the correlation between the two unstandardized measures that is smaller (0.89) while the correlations between the standardised measure and the two other measures are somewhat larger ( ). The correlation between the two standardised measures is perfect (1.00). A comparison of the standardised measures for examining the relative importance of value orientations within countries (research question 2) In this section the two standardised coefficients eta and the square root of Nagelkerke s R- squared are compared with respect to their correlations between party choice and the five value orientations within each of the 18 countries. According to the textbook, such standardised coefficients should be used for examining the relative importance of various independent variables on a dependent variable. It should be emphasized that the eta coefficient is the same as in the previous section and is calculated with the value indices as the dependent variable. It is sometimes argued that unstandardized measures can be used to examine the relative impact of variables within a sample when the variables have the same number of values. From the analyses of the 18 countries, it is evident that the two unstandardized coefficients cannot be used even though they have the same number of categories (0-10). This has been examined but is not shown here in any table. These coefficients show very different strength compared with the two standardised measures. The highest correlation between the four measures tapping the comparative strength of the value orientations between the 18 countries in the previous section is found for the two standardised measures which are 1.00 for all four measures ( when three decimals are used instead of two) in the previous cross-national analysis of the 18 countries. This implies that these have identical patterns with regard to the strength for each of the value orientation across countries. However, these coefficients might nevertheless be of different sizes but very similar in the sense that they rank the countries in the same way and have the same relative strength. This issue is also interesting regarding which of the pseudo R-squared measures that are most similar to R-squared. The comparison between the measure from analysis of variance (eta) and the various pseudo R-squared is relevant here since they are equivalent. Only a few comments on this topic will be formulated at the end of the section. In Tables 3 A-E the strength of the two standardised measures are ranked according to their strength within each of the 18 countries. The average correlations for the various regions and also in total for all 18 countries (in Table 3E) are also calculated and ranked. Table 3 about here > Below are some substantive comments to the findings on the basis of the table. The Nordic countries: In all the Nordic countries economic left right orientations are decisively correlated most strongly with party choice (Table 3A). The strength of these