# No, the parties:largest-size relationship is not different for two-tier PR

Might as well graph it.

No reason here to doubt that the logical model, NS = s1–4/3, applies equally well to two-tier systems as it does to simple, single-tier systems. This was a question I raised in the earlier planting on the revision of the extended Seat Product Model (incorporating two-tier systems without an empirical constant).

Thus any deviations of regression output from the precise predictions of the models–as reported in that earlier post–are not caused by some systematic difference in this relationship for two-tier systems. Such deviations are just noise. For instance, the regression intercept on these 472 elections is significantly greater than zero. Yet a nonzero intercept is impossible. It can’t be that the effective number of parties is any different from 1.0 (the log of which is 0) if the largest party has 100% of the seats.* More to the point for the question I had, the regression shows no significant difference in slope (or intercept for that matter) between single-tier and two-tier systems. They behave the same in this sense, meaning that when the compensation tier increases the effective number of parties and reduces the seat share of the largest, it does so while preserving NS = s1–4/3, on average. And, by the way, for those who care about such things, the R2=0.899.

Bottom line: there is no statistically significant difference between single-tier and two-tier electoral systems in how the effective number of parties is related to the size of the largest.

* If I suppress the constant (while also eliminating the binary for “simple”) the coefficient is –1.341, or almost precisely the logically required –4/3. When run with the constant, it is –1.235, but the 95% confidence interval includes –1.333.

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