Religious pluralism and religious participation: a median-based approach to the non-substantive problem

Abstract

Sociologists of religion have long been interested in the interaction between religious pluralism and religious vitality. Previous empirical studies approach this theme by drawing on data of denominational participation rates across geographical units, investigating the property of association between the quantity of one minus the Herfindahl–Hirschman Index (religious pluralism), and the total religious participation rate (religious vitality). However, this association could be theoretically spurious. Taking advantage of the median’s statistical property of being less sensitive to the variations of extreme values, this study proposes to apply the median instead of the arithmetic summation of religious participation rates to measure geographical-unit-level religious vitality. This method is illustrated by analyzing the New York State census of religion 1865 and the U.S. county survey 1990.

Appendix

In this appendix, we will be showing the process of computing the threshold denominational size. Denote $x_{p }$ to be the threshold, we compute the first-order derivative of $1 - \sum\nolimits_{i = 1}^{n} {\left( {\frac{{x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{n} x_{i} }}} \right)^{2} }$ to find the value of $x_{p }$. Specifically, we have

$$f = 1 - \mathop \sum \limits_{i = 1}^{n} \left( {\frac{{x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{n} x_{i} }}} \right)^{2} = 1 - \frac{{\mathop \sum \nolimits_{i = 1}^{n} x_{i}^{2} }}{{\left( {\mathop \sum \nolimits_{i = 1}^{n} x_{i} } \right)^{2} }}.$$

$${\text{For}}\,{\text{any}}\, 1 \le p \le n,\,f_{{x_{p} }}^{'} = - \frac{{2x_{p} \left( {\mathop \sum \nolimits_{i = 1}^{n} x_{i} } \right)^{2} - 2\mathop \sum \nolimits_{i = 1}^{n} x_{i} \mathop \sum \nolimits_{i = 1}^{n} x_{i}^{2} }}{{\left( {\mathop \sum \nolimits_{i = 1}^{n} x_{i} } \right)^{4} }}.$$

$${\text{Since}}\,\left( {\mathop \sum \limits_{i = 1}^{n} x_{i} } \right)^{4} > 0,$$

$$f_{{x_{p} }}^{'} = 0$$

$$\Leftrightarrow 2x_{p} \left( {\sum\limits_{i = 1}^{n} {x_{i} } } \right)^{2} - 2\left( {\sum\limits_{i = 1}^{n} {x_{i} } } \right)\left( {\sum\limits_{i = 1}^{n} {x_{i}^{2} } } \right) = 0$$

$$\Leftrightarrow x_{p} \mathop \sum \limits_{i = 1}^{n} x_{i} = \mathop \sum \limits_{i = 1}^{n} x_{i}^{2}$$

$$\Leftrightarrow x_{p}^{2} + x_{p} \sum\limits_{i = 1,i \ne p}^{n} {x_{i} } = x_{p}^{2} + \sum\limits_{i = 1,i \ne p}^{n} {x_{i}^{2} }$$

$$\Leftrightarrow x_{p} = \frac{{\mathop \sum \nolimits_{i = 1,i \ne p}^{n} x_{i}^{2} }}{{\mathop \sum \nolimits_{i = 1,i \ne p}^{n} x_{i} }} = \frac{{x_{1}^{2} + x_{2}^{2} + \cdots + x_{p - 1}^{2} + x_{p + 1}^{2} + x_{p + 2}^{2} + \cdots + x_{n}^{2} }}{{x_{1} + x_{2} + \cdots + x_{p - 1} + x_{p + 1} + x_{p + 2} + \cdots + x_{n} }}.$$

To show that religious pluralism reaches its maximum at $x_{p} ,$ we need to compute the second-order derivative of $f$. To facilitate computation, denote ${\text{H}} = \sum\nolimits_{i = 1}^{n} {x_{i} }$, ${\text{S}} = \sum\nolimits_{i = 1}^{n} {x_{i}^{2} }$, ${\text{h}} = H - x_{p} = \mathop \sum \limits_{i = 1,i \ne p}^{n} x_{i}$, and ${\text{s}} = S - x_{p}^{2} = \mathop \sum \limits_{i = 1,i \ne p}^{n} x_{i}^{2}$. By definition, both h and s are positive. Then, we have

$$f_{{x_{p} }}^{'} = - \frac{{2x_{p} H^{2} - 2HS}}{{H^{4} }} = - \frac{{2x_{p} H - 2S}}{{H^{3} }} = - \frac{{2x_{p}^{2} + 2x_{p} h - 2x_{p}^{2} - 2s}}{{H^{3} }} = - \frac{{2x_{p} h - 2s}}{{H^{3} }}.$$

$$f_{{x_{p} }}^{''} = - \frac{{2hH^{3} - 3\left( {2x_{p} h - 2s} \right)H^{2} }}{{H^{6} }} = - \frac{{2hH - 3\left( {2x_{p} h - 2s} \right)}}{{H^{4} }} = - \frac{{2h\left( {x_{p} + h} \right) - 6\left( {x_{p} h - s} \right)}}{{H^{4} }} = - \frac{{6s + 2h^{2} - 4x_{p} h}}{{H^{4} }}.$$

Since $x_{p} = \frac{s}{h}, f_{{x_{p} }}^{''} = - \frac{{2s + 2h^{2} }}{{H^{4} }} < 0$, so $x_{p}$ corresponds to the value of denominational participation rate threshold that maximizes religious pluralism of a given geographical unit.