I was recently having a conversation with another political scientist who showed me a graph that suggests the effective number of vote-earning parties in established democracies has been increasing over time. I was skeptical that it was, *relative to baseline*. Of course, if we do not have a baseline, we do not really know what is causing any such possible increase. The baseline should be the Seat Product Model, which tells us what we should expect the effective number of parties to be, *given the electoral system*. When we do the baseline, the increase over time remains, but is not significant.

Here is a graph with no baseline. It is just the the effective number of vote-earning parties (N_{V}) in Western Europe (most countries–see below for notes on coverage). The scatterplot marks elections by a three-letter abbreviation for each country. The *x*-axis is years since 1945, the earliest election year in the dataset. The graph’s *y*-axis is unlogged, but the plotted regression curve and 95% confidence intervals are based on a logged N_{V}.

(The regression is a GLS with random effects by country. It would not be much different if OLS were used.)

There does seem to be an increase over time. The regression estimates N_{V} averaging around 3.42 in 1945 and around 4.60 in 2011. The 95% confidence intervals on those estimates are 2.96 – 3.96 and 3.98 – 5.32, respectively. So, yes, the vote is getting more fragmented over time in Western Europe!

But hold on a moment. We should look at the fragmentation relative to baseline. As shown in *Votes from Seats*, the seat product (mean district magnitude times assembly size; in a two-tier system, also taking into account the size of the compensatory tier) explains around 60% of the variance in key party-system outcomes, including the effective number of parties. It would be useful to know if the Seat Product Model (SPM) is on its way to being unable to account for party-system fragmentation if current trends continue. It would be useful to know if recent fragmentation is part of that other 40% (i.e., the amount of variance in N_{V} that the SPM can’t account for). That is, are we witnessing some inexorable fragmentation of party systems that is resulting from the breakdown of existing party alignments in the electorate, and which electoral systems have begun to lose their ability to constrain? Should Western European countries go so far as to reduce their proportionality, in order to contain fragmenting trends?

So the next data visualization asks the question from a different perspective. Is the ratio of observed fragmentation to the SPM prediction increasing over time? We can take any given election’s actual N_{V}, divided by the SPM-predicted N_{V} to arrive at a ratio, which is equal to 1.00 for any election in which the result exactly matches the predicted value. (In other words, if R^{2}=100%, all elections would have a ratio of 1.00.)

Here it is, for N_{V}, again with the estimates from a GLS regression and the 95% intervals. In the regression, the ratio is entered as its decimal log, but the graph uses the underlying values for ease of interpretation.

What we see is indeed an increase (note the slope of the dashed line). However, the reference line at 1.00 (the log of which is, of course, zero) is easily within the 95% confidence interval of the regression throughout the six and a half decades of the data series. The regression estimates a ratio of actual to SPM of 0.911 in 1945 and 1.054 in 2011. The 95% confidence intervals are 0.782 – 1.062 and 0.905 – 1.228, respectively.

In other words, the increase is not statistically significant. There may in fact be an increase, which is to say that something in that other 40% is driving, over time, the SPM to be less successful at predicting the fragmentation of the vote. However, it could just be “noise”; we really can’t say, statistically, because of 1.00 remaining well within the confidence interval.

If it continues on current pace, then 1.00 (or rather its log) will be outside the confidence interval on N_{V} as soon as the year **2065**. I will put it on my calendar to check how we are doing at that time.

Independent of the statistical significance, there could be something of interest going on. Note that the regression trend does not cross the 1.00 line till about 42 years into the time series (i.e., 1987). This suggests that, prior to that time, the average election in Western Europe saw the vote be less fragmented than it “should have been”, according to its electoral system. That could suggest that major party organizations were partially overriding the electoral-system effect (producing party systems on average around 90% as fragmented as expected) in the early post-war years. In more recent times, the weakening of party alignments could be making the electoral system expectation finally be realized, with some tendency to exceed in recent times. But we really can’t say, given that the main conclusion is the SPM is all right, and should be for a little while yet, even if the current trend continues (which, of course, it might not).

I also wanted to checked the parliamentary party systems, that is, the effective number of seat-winning parties (N_{S}).

Here it is even more clear that the SPM is doing all right! It is only about now that the regression estimate has finally reached 1.00, but the rate of increase is more minor than with N_{V}, and clearly of minimal significance.

The regression estimates a ratio of actual N_{S} to SPM prediction of around 0.909 in 1945 and 0.991 in 2011. Confidence intervals are 0.773 – 1.068 and 0.843 – 1.164, respectively.

It is somewhat interesting that the trend in the ratio for N_{V} is rising above 1.00 before the ratio for N_{S}. Perhaps there’s an explanation of interest in there. The electoral system more directly constrains N_{S}, after all, and voters perhaps are more willing to “waste” votes as party alignments decrease. But it could just be noise.

(If I do a graph like the first one, with N_{S} with the baseline, there is an increase, but less significant than for N_{V}.)

The conclusion is that there is indeed some truth to the notion that West European party systems are fragmenting. However, relative to the Seat Product Model, they are fragmenting at a slow and hardly significant pace. How can that be? Well, perhaps it is obvious, or perhaps it is not. But a country’s seat product tends to increase over time. Most countries included here have expanded their assemblies over time, and some have also increased district magnitudes and/or adopted upper (compensatory) tiers. So, the effective number of parties should increase to some degree over time, even if voters were just as moored to their party organizations and identities as they ever were!

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Appendix: some details.

On the last point above: Specifically, a GLS regression on *expected N _{S}* says we should have seen on average N

_{S}=3.58 in 1945 but 3.67 in 2011. That is not much, but it means some increase is “baked in” even before we look at how actual voters behave. Some part of the increase is in the 60% rather than the 40%.

I dropped Belgium and Italy from the regressions, although they are included in the scatterplots for recent years. The reasons for dropping are that we could not obtain data for the share of seats allocated in upper (compensatory) tiers for the years when these countries used multi-tier PR systems; without that, we can’t calculate the extended version of the SPM (for 2-tier PR). In the later years in the Italy series, when we have such data, these are actually even more complex rules (involving a majoritarian component and alliance vote-pooling), and so the SPM really can’t predict them. In Belgium, the electoral system has been “simple” since 2003, but I think we can agree that there is no semblance of a national party system in that country.

France is also not included, partly due to the importance of the elected presidency (after 1965) and partly due to the two-round system for assembly (after 1958). We do show in

*Votes from Seats*that the SPM works pretty well for France nonetheless. So I doubt its inclusion would have altered the results much. But I wanted to stick to the PR systems and FPTP, which the SPM is designed to handle.

“The regression trend does not cross the 1.00 line till 1987.” And the Greens entered the Bundestag in 1983. They were not the first Green Party in Europe to have members elected nationally but the impression was created that they had been, because they attracted the most media attention. The Finnish Green League elected four MPs in 1987. The Italian Green Lists made their debut at the 1987 general election. The Swedish Greens first won seats in the Swedish Riksdag in the 1988 general election. The Irish Green Party first entered the Dáil in 1989. By 1989, 30 Green MEPs were elected to the European Parliament, enough to form their own Group: West Germany 8, Belgium 3, France 8, Italy 7, Netherlands 2, Spain 1, Portugal 1. I am not convinced that the Seat Product Model could predict the rise of the Greens.

You need not to be convinced of that, because the purpose of the SPM is not to predict the presence or absence any specific parties!

The Seat Product Model is not independent of the data you have used to test it on. Since afaik you’ve already tested it on that data and found it convincingly able to predict the values. So the fact that they’re significant is more by design than by chance.

Surely under such a circumstance, you would expect better performance, and therefore reduce the threshold for rejecting the null hypothesis, when analysing its performance according to some additional factor.

I have two assumptions here (conveniently marked with “afaik” and “surely”) and I’m not a modeller or a statistician, but your confidence seems unreasonably high in this circumstance.

The SPM is derived independent of data.