## Religious Pluralism and Participation: Why Previous Research Is Wrong

by
David Voas, Alasdair Crockett, Daniel V. A. Olson

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Title:

Religious Pluralism and Participation: Why Previous Research Is Wrong

Author:

David Voas, Alasdair Crockett, Daniel V. A. Olson

Year:

2002

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American Sociological Review

Volume:

67

Issue:

2

Start Page:

212

End Page:

230

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Language:

English

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**Updated:**March 12th, 2013

Abstract:

## RELIGIOUSPLURALISM

AND

DAVIDVOAS DANIELV. A. OLSON

University of Sheffield Indiana University South Bend

##### ALASDAIRCROCKETT

University of Essex

Does religious pluralism undermine or promote religious involvement? Some secu- larization theories contend that diversity breeds loss of belief and lower participa- tion. The religious economies model counters that involvement is boosted by the availability of alternative religious suppliers and the competition that results, with each group working harder to gain adherents. The issue is sufficiently important that a recent review found 193 tests of this question in 26 published articles. Almost all of these findings (both positive and negative) should be abandoned. The associations

reported do not reflect the effects of pluralism but a previously overlooked math- ematical relationship between measures of religious participation and the index of pluralism. Even when pluralism has no effect on participation, the correlation be- tween these two variables is likely to be nonzero. The sign and magnitude of this expected correlation depend on the nature of the size distributions of the religious groups across the areas studied. Results from several frequently cited studies closely match what would be expected from chance alone. Various alternative methods for studying pluralism in future research are examined, but currently there is no compel-

ling evidence that religious pluralism has any effect on religious participation.

FOR MORE THAN A DECADE, soci-gious variety is limited, and especially when ologists of religion have been debating a single church is dominant? Or do people the question of whether religious pluralism become more interested and involved in re- increases or decreases religious commit- ligion when there are many different alter- ment. Are people more religious when reli- natives available? Finke and Stark (1988) sparked the debate with their assertion that religious pluralism

Direct all correspondence to David Voas, De-

increases church membership. This claim

partment of Sociological Studies, University of

ran counter to what they described as the re-

Sheffield, Sheffield S10 2TU, England (d.voas@ shef.ac.uk). The data on New York in 1865 were ceived wisdom of the day, particularly as assembled by Avery Guest and Roger Finke; the embodied in Berger's (1967) The Sacred data on U.S. counties in 1990 were collected by Canopy. Berger asserted that religious plu- the Church Growth Research Center. Both data ralism undermines the influence of "plausi- sets were made available by the American Reli-

bility structures," the social networks and

gion Data Archive. Returns from the British 185 1

institutions that reinforce the plausibility of

Religious Census were originally computerized

belief. When the state, public institutions,

by Keith Snell and Paul Ell with the collabora-

and day-to-day social contacts no longer re-

tion of Alasdair Crockett, and Deborah

inforce the truth of a given religious belief

Regulinski assisted Daniel Olson in separately entering the registration-district data. Anony- but instead expose people to a diversity of mous ASR reviewers made helpful comments. opinions, religion loses its quality as taken- The authors also thank Kirk Hadaway. for-granted truth. As a result, belief and reli-

gious involvement decline. Berger (and many classical sociological theorists) saw increased exposure to religious diversity and decreased religiosity as a natural part of the social changes accompanying moderniza- tion.

In contrast, Finke and Stark (1988, 1992) argued that such a theory fails to describe the United States, which, among the indus- trialized nations, has both the highest levels of religious pluralism and one of the highest rates of church attendance (Iannaccone 1991; Warner 1993). Instead, they proposed the notion of "religious economies" to ex- plain the American and other religious situa- tions more generally. Finke and Stark argued that religious monopolies are "lazy" and that pluralism fosters competition-with each church working harder to satisfy demand, more people are drawn into religion. More- over, increasing the number of religious op- tions raises the likelihood that any particular individual will participate. To the extent that modernization leads to religious pluralism, the implication of Finke and Stark's work is that religious involvement will also increase. Thus, this controversy is not simply between rival schools within the sociology of religion but is an important aspect of the debate over the nature of modernity.

A considerable amount of empirical work has explored this issue. The typical study looks at correlations between some measure of total religious participation (e.g., the per- centage of people who attend religious ser- vices or are church members) and an index of pluralism based on the denominational di- versity of these adherents. A negative corre- lation has been taken to corroborate secular- ization theory; a positive correlation bolsters the religious economies position. Studies have drawn data from many geographical and historical settings. Some of these appear to support one side, some the other.' The lit- erature has grown large enough to justify a major review of 193 tests of this question in 26 published articles (Chaves and Gorski 2001).

With the exception of a few studies de- scribed below, this entire body of work has relied on giving substantive interpretations

For an exhaustive list of research on this topic see Chaves and Gorski (2001).

to the observed associations between indices of pluralism and related measures of partici- pation. Methodological issues have arisen,2 but the basic premise-that a nonzero corre- lation reflects a real, probably causal, rela- tionship between religious diversity and in- volvement-has not been seriously chal- lenged. The fact that the index of pluralism is closely related to indicators used by economists and the U.S. Federal Trade Com- mission to measure competition (and more- over that it seems to be a valid measure) has contributed to that acceptance.

In what follows, we show that the ob- served relationships between diversity and involvement are predictable on the basis of wholly nonsubstantive factors. The correla- tions arise not necessarily because religious pluralism has a real effect on participation, but because of previously overlooked math- ematical associations between the pluralism index and religious participation rates. Even when pluralism has no causal influence on participation rates, the expected value of the correlation between these two variables is usually not zero. We show that this expected correlation can be positive or negative and depends on the nature of the size distribu- tions of the religious groups across geo- graphical areas in a particular data set.

We describe how simulation methods can be used to determine the sign and approxi- mate magnitude of the expected correlation for a particular data set under the assump- tion that pluralism has no causal influence on participation rates. Most strikingly, a re- examination of data sets analyzed in previ- ously published research shows that the ex- pected correlations (assuming no causal in- fluence of pluralism) are remarkably close to the actual correlations that have, in the past, been given substantive interpretations. All but a few studies of pluralism (described be- low) have fallen prey to the problems we de- scribe. The implication of our results is that nearly all of the evidence that has been as- sembled on both sides of the pluralism de- bate will have to be reevaluated.

Concerning, for example, the geographical scale at which rates are measured (Finke, Guest, and Stark 1996) and whether to introduce statis-

1 tical controls for the prevalence of Catholics in the population (Breault 1989; Olson 1999).

Before proceeding, it is important to clarify that we are not claiming that all re- search relating to the religious economies model is subject to the problem we de-scribe-only the work related to pluralism. Finke and Stark have used other independent variables that are not subject to these prob- lems (e.g., the religious market share of a de- nomination and measures of state regulation of religion). Nevertheless, pluralism has played a central role in their work (Finke et al. 1996; Finke and Stark 1988; Stark, Finke, and Iannaccone 1995), so much so that a positive link between religious diversity and involvement has been seen as "the core em- pirical hypothesis of the market approach to the study of religion" (Chaves and Gorski 2001:274; also see Hechter and Kanazawa 1997: 198).3

##### METHODS AND METHODOLOGICAL FINDINGS

Nearly all the research on religious diversity (or pluralism-the terms have been used in- terchangeably in this literature4) measures

More recently Stark and Iannaccone (1996) and Stark and Finke (2000) have argued that in certain circumstances pluralism may not be a good indicator of their central independent vari- able, religious competition. It might thus be sug- gested that our criticisms of research on plural- ism have no bearing on religious competition or the religious economies model. While such a dis- cussion would go beyond the scope of this ar- ticle, Olson (2002) notes that Stark and Finke (2000) fail to define religious competition and thus can argue ex post facto that sime variable or research result has no bearing on it. Olson (2002) identifies at least two, somewhat contra- dictory, meanings of religious competition used implicitly in religious economies models: the presence of close substitute religious groups and the diversity of nearby religious groups. We maintain that the pluralism index is a good mea- sure of religious competition (when defined as diversity) and that findings based on the plural- ism index are relevant to theories of religious economies when such findings are not contami- nated by the methodological errors we outline in this article.

It could be argued-and Beckford (2001) does so persuasively-that it would be preferable

the phenomenon by taking the complement of the Herfindahl index, long used in eco- nomics to measure market concentration. The denominations in a religious market each have a proportional share, pi, of all re- ligious persons in an area (i.e., pi is the size of the ithreligious group divided by the sum of all the group sizes). The Herfindahl index is Cpi2(the sum of the squared shares). The index of pluralism (essentially the same as the diversity index described by Lieberson 1969) is defined as one minus the Herfindahl index, or 1 -Cpi2.This index ranges from 0, when there is a single religious group, to a little less than 1, when there are many de- nominations of equal size."

Because the pluralism index takes even- ness of size into account, it is probably a bet- ter measure of the choices available to most potential adherents than simply the number of groups. Thus, for example, the index will remain low when one or two groups domi- nate a religious market, even though there might be scores of tiny churches in an area that most potential adherents have never come across and would be unlikely to join.

We use the term "participation" to refer to any of the standard measures of religious belonging, for example, attendance, mem- bership, formal affiliation, or declared ad- herence. Any definition of religious identity that permits intergroup comparisons is suit- able. We calculate the participation rate (or "size") of each denomination as the percent- age of the overall population belonging to that denomination; the total participation rate is defined as the sum of the participa-

to use "diversity" in this type of descriptive con- text, reserving "pluralism" as a normative term.

To offer an example, if 90 percent of the reli- gious adherents in an area belong to one group and 5 percent belong to each of two other groups, the situation is quasi-monopolistic and pluralism is low (1 -[.92+ .052+ .052]= .185). If the three denominations were of equal size, pluralism would be much higher (1 -[.332+ .332+ .332]= .67). Pluralism also increases with the number of denominations; if there were not 3 but 10 groups of equal size, the pluralism index would be .90. The index has an upper limit of 1 -lln, where n is the number of denominations in an area.

tion rates of each denomination. Here, as in most of the literature to be reexamined, reli- gious participation is therefore expressed as the percentage of the population that attends, belongs, or participates in the way specified. We emphasize that the methodological prob- lems identified below do not apply to a small number of investigations in which the de- pendent variable, participation, is measured relative to some base other than the total population. Per1 and Olson (2000), for ex- ample, use congregations as the unit of analysis and examine how the religious plu- ralism of the county in which a congregation is located affects the proportion of the congregation's members (not the county population) that typically attend.

###### SIZE VARIATION AND EXPECTED CORRELATIONS

For the reasons outlined above, the index of pluralism has much to recommend it as an indicator of diversity. Not only does it mea- sure variety or fragmentation, it can bear in- terpretations that are especially pertinent to the central debate. For example, it measures the probability that any two randomly selected persons living in an area would be of different faiths, a concept relevant to some secularization theories. A natural approach, therefore, has been to look for statistical as- sociations between this index and the level of religious participation. Depending on whether the correlation was positive or nega- tive, researchers have then claimed that di- versity either promotes or undermines reli- gious commitment.

In contrast to the usual interpretation, we find that nonzero correlations will occur for mathematical reasons that depend only on the size distributions of the denominations in a data set across geographical units. The general principle is that when the larger de- nominations have the greatest size variation, correlations tend to be negative, but when the smaller denominations are more vari- able, correlations tend to be positive. For the sake of illustration consider three hypotheti- cal areas, beginning with town A, where An- glicans make up 50 percent and Methodists 20 percent of the population. (For present purposes, it makes no difference whether those values are levels of adherence, mem- bership, or even attendance, as a percentage of the total population.) We calculate overall participation and the index of pluralism and locate these two values as a point on a scatterplot (see Figure la). Imagine that in a neighboring town, B, the Methodists are still at 20 percent, but here the Anglicans reach 55 percent. Clearly, total participation is higher, but pluralism is lower, because the size imbalance between the two groups is greater than in A. Conversely, if in nearby town C the Methodists are again 20 percent of the population while the Anglicans have only 45 percent, overall participation will be lower but pluralism would go up because the size difference between the groups is now smaller than before. Thus, the correlation between pluralism and participation is nega- tive for these three towns.

Consider now two more hypothetical towns, D and E, in which it is the smaller denomination (the Methodists) that varies (Figure lb). Anglicans comprise 50 percent of the population in both D and E, as they do in town A above. In town D, the Method- ists have more participants than in A, with 25 percent. As a result, not only is total par- ticipation higher than in A, the index of plu- ralism is also higher, because the denomina- tional shares are more evenly balanced. In town E, the Methodists are at only 15 per- cent, hence overall participation is lower, as is pluralism.

Thus, where the large denominations are more variable than the others, participation and pluralism will tend to be negatively cor- related.6 When the small denominations show more variability, then a positive corre- lation tends to arise. (See Appendix A for technical details.) These statements merely express a mathematical truth; nothing is as- serted about how the religious groups came to be larger or smaller, or more or less vari- able in size. It is evident, however, that non- zero correlations can no longer be inter- preted as resulting only from an effect of pluralism on participation.

The sign and magnitude of the correlation will depend on the interplay of the means, vari- ances, and shapes of the distributions for each denomination. Analysis becomes especially com- plicated when, as in many real data sets, there are numerous denominations of varying sizes.

(a) Large Religious Denominations Are Most Variable

TOWN A | Anglicans | 50% |

Methodists | 20% | |

TOWN B | Anglicans | 55% |

Methodists | 20% | |

TOWN C | Anglicans | 45% |

Methodists | 20% |

0 .40 .45 .50

Total participation | 70% |

Index of pluralism | .41 |

Total participation | 75% |

Index of pluralism | .39 |

Total participation | 65% |

Index of pluralism | .43 |

lndex of Pluralism (b)Small Religious Denominations Are Most Variable

TOWN A Anglicans 50% Methodists 20%

TOWN D Anglicans 50% Methodists 25%

TOWN E Anglicans 50% Methodists 15%

0 .40

Total participation 70% Index of pluralism .41

Total participation 75% Index of pluralism .44

Total participation 650h Index of pluralism .36

.45 .50

###### lndex of Pluralism

Figure 1. Hypothetical Towns with Two Denominations: The Association between Pluralism and Participation Depends on Whether the Large or Small Denomination Is Most Variable

To anticipate our conclusions, we suggest that the usual interpretation of earlier find- ings is wrong. The conventional approach assumes that, as a result of social and politi- cal history, religious diversity would be high or low (see the left side of Figure 2a), and in consequence individual denominations (and hence all groups collectively) would have higher or lower participation rates (see the right side of Figure 2a). It now seems easier to defend a simpler explanation, one in which the only substantive causal link is be- tween a host of historical, social, and essen- tially random factors and the participation

(a) Usual Interpretation: Causal

Causal Causal

Historical & Denominational ,Denominational random forces -pluralism participation rates

Calculation Causal

###### Total participation rate

(b) Revised Interpretation: Noncausal

Causal Calculatlon

Historical & ,Denominational ,Denominational random forces participation rates pluralism

ICalculation ,.' Artlfact

###### Total participation rate

Key:

Causal= Sociologically significant causal connection Ca/cu/ation= Result of calculation using formula An'ifad= Correlation resulting from mathematically necessary association

Figure 2. Interpretations of the Relationship between Denominational Pluralism and Total Participation Rate

rates of each religious group at any particu- Stark et al. 1995) expressed concern that this lar time and place (see the left side of Figure feature might bias correlations in a negative 2b). The total participation rates are then direction. In related investigations, we find simply the sum of the sizes of the groups the situation to be problematic only when present in a geographical area (see the measurement error is severe. This separate middle part of Figure 2b). Likewise, de- issue is, in any case, rendered moot by the nominational pluralism simply describes the more serious problems identified above. collection of participation rates for the Our investigations also render moot an groups present in an area, being related to otherwise important difficulty we discov- them by nothing more than the formula for ered. The correlations between pluralism and the index (see the right side of Figure 2b). participation rates can differ dramatically, Finally, when examined across many geo- even shifting sign, depending on which mea- graphical areas, denominational pluralism sure of religious involvement (e.g., atten- and the total participation rate will usually dance or membership) is used to calculate have a nonzero correlation for reasons that pluralism and participation rates. Although are purely mathematical rather than substan- the rank order of the mean sizes of the de- tive and causal. nominations in a data set may change little

between different measures of participation, the relative amount of variation in each denomination's size may change substan- Note that the problem outlined above does tially depending on the choice of measure. not emerge simply because the total number As we argue, it is the relative amounts of of religious participants is used to calculate variation by denomination that most affect both the participation rate (in the numerator) the value of the correlation to be expected. and the pluralism index (in the denomina- Because all such correlations are also in- tor). Some analysts (e.g., Finke et al. 1996, fected by the mathematical, noncausal influ-

ences highlighted above, we will not dwell further on this issue.

In describing these methodological prob- lems, we do not wish to imply that the index of pluralism is intrinsically unsatisfactory. The index is not, by itself, the source of the problems we have found. The pluralism in- dex appears to be as valid a measure of reli- gious diversity as the total attendance rate is of religious involvement. The problems de- scribed above arise only when the pluralism index is used together with a dependent vari- able that expresses religious participation relative to the total population.

##### INVESTIGATIVE PROCEDURE

###### DENOMINATIONAL

DISTRIBUTIONS

Mathematically, all that is needed to produce a noncausal nonzero correlation between pluralism and total participation rates is some variation in total participation from one geographical area to the next. This ex- ceedingly weak assumption is satisfied in essentially every data set that does not forc- ibly assign everyone to some religious group (as is done in some surveys measuring reli- gious affiliation). The sign and magnitude of the correlation will depend primarily on the relative variation in size of each denomina- tion across different geographical units.

A huge range of possible causes and com- binations of causes could produce this variation, from historical patterns of immi- gration, reformation, or persecution, to tran- sitory features of choirs, ministers, or youth groups. It is even possible that religious pluralism influences the size distributions of different religious groups. If so, how- ever, it is only one of many such factors, and there appears to be no way, using cross- sectional data, of separating out the past ef- fects of pluralism from the effects of other variables influencing denominational size distributions.

Our claim that variability in denomina- tional sizes is in itself sufficient to produce correlations between the measures of plural- ism and participation raises an important question. How similar are the correlations observed in actual data sets to those correla- tions one would expect if pluralism had no effect on participation rates? If the values are very different, then it may still be pos- sible to discern a real influence of pluralism. If, however, the observed correlations are similar to what would be expected given only the size distributions of the denomina- tions involved, then on the face of it plural- ism has little effect. Such a result would im- ply not only that past interpretations of re- search in this area should be abandoned, but also that theories about pluralism's effects on religious involvement are suspect.

EXPERIMENTAL DESIGN

To interpret the observed pluralism-partici- pation relationships, we need to determine the expected value of the correlation under the assumption that pluralism has no sub- stantive effect on participation rates and only the denominational size distributions affect the correlation. The problem is far too com- plex to solve analytically (see Appendix A for a basic theoretical treatment). Instead, we use computer simulations to approximate the expected value of the correlation for par- ticular data sets.7

The starting point is the data matrix D(i j) containing the actual participation rates. Each D(i j) represents the proportion of the population belonging to the jth denomina- tion in the ith geographical area. A new ma- trix of simulated data, S(i j), is created for a set of artificial "areas" using a technique similar to the "bootstrapping" or "re-sampling" methods used elsewhere for other purposes. Each S(ij) is chosen randomly from among all the actual participation rates in the jth column (for the same denomina- tion) of the data matrix D. Importantly, the selection of the participation rates S(i j)for each denomination in a geographical area is totally independent of the participation rates selected for the other denominations in the same area. Thus, the size of any one denomi- nation in an area is unaffected by any char- acteristic of the simulated religious environ- ment, including the religious pluralism of the simulated area. Pluralism can have no substantive effect on the choice of any S(i j) or on the main dependent variable, the total participation rate of a simulated area (i.e.,

'Details and examples are available from the authors.

Table 1. Statistics for Actual and Simulated Data

Mean Value Value for for Actual Simulated Country and Area Data Data

TOWNS IN NEW YORK STATE, 1865 (N = 847) Mean pluralism ,564 ,578

Mean (attendancelpopulation) .275 .276

Correlation ,494 ,496

Slope ,309 ,388

U.S. COUNTIES, 1990 (N = 3,104)

Mean pluralism | .702 | ,727 |
---|---|---|

Mean (adherents/population) | ,588 | .588 |

Correlation | -.329 | -.297 |

Slope | -.435 | -.791 |

ENGLAND, 185 1

All Areas (N = 576)

Mean pluralism ,605 ,605 Mean (attendance/population) .63 1 .63 1 Correlation ,027 ,054 Slope .037 ,086

###### Areas Where Pluralism 1.6 (N = 236)

Mean pluralism ,474 .491 Mean (attendancelpopulation) .624 ,621 Correlation ,259 .I78 Slope .383 ,318

###### Areas Where Pluralism > .6 (N= 340)

Mean pluralism .697 .690

Mean (attendance/population) ,635 .639

Correlation -.260 -.I05

Slope -.911 -.405

Y-intercept 1.270 ,919

Note: All simulated values are the means of 500 repetitions for New York State and U.S. Counties, and of 2,000 repetitions for England. Each simula- tion run generated the same number of cases as in the actual data. The number of cases above or be- low the pluralism value of .6 varied across the simu- lated data sets for England in 185 1.

market, for which reason Finke et al. (1996:208) commend the data set as offer- ing "an ideal unit of analysis."

Figure 3a shows the proportion of the population who are church attenders (the de- pendent variable) plotted against the index of pluralism (the independent variable) cal- culated from the attendance figures for all 847 towns with valid data. Because people sometimes crossed town boundaries to attend church, a few towns have attendance propor- tions greater than 1. The most striking fea- ture of the scatterplot is the strong positively sloping linear pattern. There is no observable "ceiling" effect of pluralism, a level beyond which additional pluralism has no additional impact on attendance as maintained by Finke and Stark concerning these data (Finke et al. 1996; Finke and Stark 1998; Stark and Finke 2000:225-26). The simple correlation be- tween pluralism and the proportion of the population attending church is .494, a result that remains highly statistically significant even when controls for other possible expla- nations are introduced using multiple regres- sion (Finke et al. 1996). On the surface, the results appear consistent with Finke et al.'s claim that greater religious pluralism leads to greater religious involvement.

What, though, is the expected value of the correlation given the denominational distri- butions here? We did 500 separate runs of our simulation using the New York data. The mean of the 500 correlations recorded from each simulation is .496 (standard error = .023),9 a result that is almost identical to the correlation found in the real data (.494). These results strongly suggest that the pro- nounced positive association in the actual New York data is purely a by-product of the denominational size distributions that hap- pen to be found in that data set and does not reflect any additional substantive effects of pluralism on religious participation rates. To demonstrate visually how closely the simu- lations mimic the real data, Figure 3b shows a scatterplot from one "typical" run of the simulation.lo

We refer to the standard deviations of the sta- tistics recorded across all runs of our simulation as the "standard error" of these statistics.

lo We chose as "typical" a run that produced a correlation and a regression slope that were

nothing was done to prevent total adherence in a simulated county from exceeding 100 percent of the p~pulation.'~

The scatterplot in Figure 4b is again from a typical run of our simulation for the 1990

U.S. county data. The differences between the simulation results and the actual data (see Table 1) appear to be in part a conse- quence of our decision not to place an up- per limit on the total adherence rate, but may also follow from other relationships found in the real but not the artificial data because of the limited assumptions built into our simulation.14 Nevertheless, we find that nearly all of the association between pluralism and participation can be explained as the expected result of the de- nominational size distributions found in the actual U.S. county data.

The 185 1 Religious Census is the only fully comprehensive religious census in the mod- ern history of Great Britain.15 It was taken on Sunday, March 30, 1851, and documents attendance at 34,467 places of worship in England and Wales. These data were re- corded for 38 religious groups (with an ad- ditional category for minor or unidentifiable religious groups), which allows us to assess religious pluralism for that day quite pre- cisely. We examine attendance for the 576 English census registration districts (the

l3 Although we have constructed models fea- turing such a restriction, we chose not to use these simulations because they introduce addi- tional assumptions about the interrelationships among the sizes of denominations in a given simulated area. In particular, restricting the total participation rate to no more than 100 percent of the population tends to make the intercorrelations among the sizes of different denominations nega- tive, thus violating the assumptions of our null model in which the sizes of denominations are independent of one another.

l4 For example, the simulated data (shown in Figure 4b) differ from the real U.S. data in that there are no counties that are religious near-mo- nopolies (on the left side of the scatterplot).

l5 The 200 1 census of England and Wales con- tained a question on religion, but the categories are extremely broad: none, Christian, Buddhist, etc.

smallest unit for which the data have been published).

The 1851 Religious Census has already loomed large in the pluralism debate (see Bruce 1992; Finke and Stark 1998; Stark et al. 1995), and this single data source has yielded apparently contradictory results. As will become apparent, there is a curvilinear relationship between pluralism and partici- pation rates in the English data that, it had previously been assumed, could be explained by an analysis similar to that given in Crockett (2000). We shall see, however, that the pattern is once again explicable in terms of noncausal mathematical associations.

Figure 5a shows a scatterplot of the real data for the 576 English registration dis- tricts, with church attendance plotted against pluralism. Participation can be greater than 100 percent because it reflects total atten- dances at all services on the given Sunday, and in 185 1 attending multiple services was common. Although the correlation across all English districts is close to zero (.027), the results do not follow a linear pattern. The best-fit line shown in Figure 5a is locally weighted (using the SYSTAT LOWESS pro- cedure) to highlight the nonlinear relation- ship between pluralism and attendance rate.

I The pronounced curvilinearity of the trend is clear and can be substantiated using more formal statistical procedures. The vertical

I line in Figure 5a-at a pluralism value of .6-was fitted by eye at the approximate point of peak participation rates, with 236 districts scoring below and 340 above that level.

We used the same simulation method to estimate the expected correlation as before, 1 this time using 2,000 repetitions. Not only is the mean correlation of .054 very close to the real value of .027 (the difference being less than the standard error of .037), it fur- ther emerges that, just as in the real data, the pluralism-participation relationship appears to be different depending on whether the in- dex of pluralism is high or low. The correla- tion among simulated areas with a pluralism value of less than or equal to .6 is positive, while that recorded among observations with a pluralism index greater than .6 is negative. The actual and expected values of the corre- lation are .259 and .I78 respectively for the areas with pluralism values below .6, and

(a) Actual Data (b) Simulated Data

###### 1.5 1.5

I I I I Mean r= ,054

iI

BOO

, e

0 | 0 | I | I | f | I | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | .2 | .4 | .6 | .8 | 1.O | 0 | .2 | .4 | .6 | .8 | 1.O | |

Attendance Pluralism | Attendance Pluralism |

##### Figure 5. Relationship between Pluralism and Attendance: Registration Districts in England, 1851

Note: Figure 5a shows results using actual data from 576 registration districts in England in 1851. Figure 5b shows a typical run from the total of 2,000 simulations. The curvilinearity of the actual data is also present, in somewhat weaker form, in the simulated data (see Table 1 for a detailed comparison).

-.260 and -. 105 for the areas with pluralism values above .6.

The general correspondence between the real and simulated data is made clear in the scatterplot of a typical simulation run shown in Figure 5b. While the curvilinear- ity in the simulated data is slightly weaker than in the real data, it is clearly present, with a pluralism level of .6 again appearing as the critical breakpoint figure. Our simu- lation, based entirely on random sampling from the attendance rates for each denomi- nation, has produced essentially the same nonlinear relationship found in the actual data. The result is impressive corroboration for the argument that denominational size distributions alone can generate associa- tions that seem on the surface to be socio- logically meaningful.16

##### ALTERNATIVE METHODS

As we have shown, the usual empirical ap- proach adopted in the debate over religious

l6 The curvilinearity in the English data is ex- plained by mathematical relationships that are similar to, if somewhat more complex than, those already described. Details are available from the authors.

pluralism and participation is deeply flawed. However, because pluralism is central to a variety of competing theories of religion and modernization, the identification of more appropriate methods will be a priority for re- search in this area. Here we review several possibilities, rejecting some and noting dif- ficulties with others, but suggesting various potential ways forward.

COMPARING AND ACTUAL

###### EXPECTED CORRELATIONS

Perhaps the most straightforward approach to avoiding the problems described above would be to measure the effects of religious diversity by comparing the actual correla- tions between pluralism and participation with the expected correlations based on de- nominational size distributions. Unfortu- nately, several technical problems limit the usefulness of this method. First, when the standard error of the expected value pro- duced by our simulations is large (as it is likely to be when the number of geographi- cal areas in the data is small), such tests may lack the statistical power to detect any real effects of pluralism. Second, it is not clear exactly which "expected value should be used as the standard of comparison. For ex-

ample, one could argue that the benchmark should include real-world constraints (not included in our simulations), such as the re- quirement that total participation not exceed the population. However, introducing this or other plausible constraints makes the de- nominational participation rates dependent on each other, thus violating our stipulation for the "expected" model that no denomina- tion is affected by any aspect of the religious environment. Third, even if one could decide which yardstick is best, problems arise if it is necessary to control for other variables (e.g., through regression) to arrive at a good estimate of any direct, nonspurious, effect of pluralism.

Finally, the "expected" value of the corre- lation may, to some unknown degree, actu- ally reflect the past influences of pluralism. Other simulations and analyses we have done suggest that if pluralism has a real ef- fect on participation, it may also alter the denominational size distributions in ways that tend, over time, to shift the expected correlation in the direction of the actual cor- relation.'' However, we also find that any genuine effect of pluralism will almost al- ways have a greater impact on the observed correlations than on the expected correla- tions. Hence, the real effects of pluralism should be evident in a difference between the actual and expected values,18 but the strength of these effects would be uncertain.

Note that the potential influence of plural- ism on the denominational size distributions, and hence on the expected association with participation, does not undermine our cri- tique of previous research. We have not de- nied that pluralism could influence partici- pation rates; we deny only that the associa- tions reported in most previous research count as evidence that it does.

I' Descriptions of these simulation experi- ments are available from the authors. If plural- ism has a negative influence on participation, for example, it would tend to depress group sizes in high-pluralism areas and boost group sizes in low-pluralism areas, which is likely to translate into greater relative variability of the largest de- nominations, and hence a negative expected plu- ralism-participation correlation.

l8 The excess should be most readily apparent if greater pluralism acts to increase (rather than reduce) participation.

###### COMPARING TYPES

DIFFERENT

OF INVOLVEMENT

Another approach that might circumvent the difficulties is to use one measure of partici- pation (e.g., affiliation) in calculating the pluralism index and a different measure (e.g., church attendance) in calculating total participation. Recall that in our empirical examples (as in most published work) both pluralism and participation were based on the same measure of religious involvement. One might hope that when the two statistics are based on different types of involvement, nonzero correlations will be meaningful. With one important exception, such tactics unfortunately do not avoid the mathematical problems we have exposed.

The difficulty is that measures such as at- tendance, membership, affiliation, number of churches per county, and so on are gener- ally correlated with each other. Higher atten- dance figures tend to be associated with higher membership figures, and vice versa; it is hard to see that both could be valid in- dicators of religious involvement if they were not systematically related. The exact nature of the relationship may change from one denomination to the next, but the key

I point remains: Different indicators of in- volvement are not independent. We already know, however, that pluralism and participation calculated from the same data (let us call the variables Plur, and

1 Part,) tend to be correlated given a pattern of denominational sizes. If the two measures of participation (Part, and Part,) are also correlated, then it follows (because correla- tion is transitive) that the expected value of

I the correlation between Plur, and Part2 will also tend to be nonzero. The one exception comes when religious involvement (by some definition) is univer- sal (as when a religious census defines ev- eryone as having some religious affiliation). In this case, the resulting measure of in- volvement (Partl) can have no association with the corresponding index of pluralism (Plur,). In fact, the correlation would sim- ply be undefined because when participation is a constant 100 percent it can have neither variance nor covariance with pluralism. While it would therefore be futile to look for a relationship in the conventional way be-

tween Plurl and Partl, it might be worth ex- amining the index of pluralism based on this type of involvement (Plurl) in conjunction with a separate kind of participation (Part2).19 Thus, for example, if data from a census forces everyone to have a religious affiliation, one might compare religious plu- ralism based on affiliation with the rates of church attendance in the same areas. The re- sulting correlation would not be subject to the problems we have outlined above. It must be stressed, however, that such a pro- cedure relies on the independence (in total levels) of the two types of involvement used, and in this special case that independence is assured by the ubiquity of belonging as mea- sured in one of those ways.

###### ADDITIONAL APPROACHES

Although there appears to be no easy way of sidestepping the methodological difficulties we have pointed out, we can offer some pos- sible avenues for future research. One ap- proach is to avoid cross-sectional data alto- gether. Where available, "panel" studies of the same set of geographical areas at differ- ent periods might enable researchers to in- vestigate whether changes in participation rates over time are related to the pluralism of an area (or conversely, whether participa- tion rates influence subsequent levels of plu- ralism). Unfortunately, there are few data sets that provide comparable data for well- separated periods, and ideally such studies should be based on data from at least three points in time (Finkel 1995). Studies do ex- ist using data from two separate points to in- vestigate changes in participation: For ex- ample, Christian0 (1987) found no statisti-

l9 Note that it does not help to achieve 100percent coverage by including a "no religion" cat- egory when calculating the pluralism index. If pluralism is thus made to depend in part on the size of the nonreligious group, then there will be a necessary mathematical relationship between the index and the overall participation rate be- cause the number participating is merely the to- tal population minus the size of the nonreligious group. In terms of the theory underlying the gen- eral debate, though, it seems useful to consider the possible influence of the unchurched (who may function either as a source of secularization or as a ready market for religion).

cally significant effects of pluralism in con- sidering U.S. cities circa 1900. Pettersson and Hamberg (1997), Blau, Land, and Redding (1992), and Blau, Redding, and Land (1993) reported on longitudinal work that apparently showed significant effects, but once again our analysis points to the role of noncausal influences. Our tests suggest that artifactual relationships can be elimi- nated only when the independent variable is pluralism at the beginning of the period over which change in participation is measured. In contrast, Blau et al. (1992) and Blau et al. (1993) measured pluralism at the end of the period, while Pettersson and Hamberg (1997) used change in pluralism over time to predict change in attendance rates. Both of these latter methods fail to avoid incorpo- rating noncausal mathematical relationships into the estimates of pluralism's effects.

Another general approach involves ex- ploiting the associations between different measures of participation to work around the problems. Given the overall ratios of atten- dance to membership for each denomination, for example, we could calculate the "ex- pected" attendance for each denomination by area based on the appropriate member- ship figures. Hence, if across all the areas studied, Baptist attendance averages 80 per- cent of membership, one could determine whether Baptists systematically "over at- tend" or "under attend in more pluralistic areas. Using the same procedure for all the denominations, one might be in a position to see if the total attendance figures in more pluralistic areas were generally higher or lower than the expected attendance figures based on membership. Once again, there are practical constraints, as few data sets include such complete information on both member- ship and attendance by denomination for a large number of areas.

Another potential approach uses the reli- gious participation (e.g., frequency of church attendance) of randomly sampled in- dividuals as the dependent variable and the contextual pluralism of the county (or coun- try) of residence as the independent variable (e.g., Verweij, Ester, and Nauta 1997). If no additional steps are taken, this method also succumbs to the problems we have identi- fied. The participation rate of a randomly sampled individual is probabilistically re

lated to the overall participation rate for the entire geographical unit (e.g., county or na- tion), which, as we have already shown, is noncausally related to the index of pluralism for the same area. However, using a method that is essentially an extension of the "over" and "under" attendance method described in the previous paragraph, one could examine the attendance rate of individuals controlling for the particular denomination to which they belong. The aim would be to determine whether, for example, Baptists attend less frequently or more frequently when they live in more religiously pluralistic areas. We therefore recommend using dummy vari- ables for each denomination in individual- level regression models such as those used by Verweij et al. (1997); these controls are most effective when the index of pluralism derives from the same data as the respon- dents' denominational identities.

Finally, a related alternative is to study the effects of pluralism on subsets of the popu- lation (e.g., particular congregations or reli- gious groups). In this instance, participation is expressed relative to the number of per- sons in the group rather than the total popu- lation, and no mathematical relationship arises with the pluralism of the surrounding area. As examples, Zaleski and Zech (1995) and Per1 and Olson (2000) examined whether the pluralism of the county in which a church is located affects the per capita giv- ing (controlling for income) and attendance rates of members of those congregations. They found no statistically significant influ- ences. Phillips (1998) used the percentage of male Mormons who had been ordained into the priesthood of Melchizedik (an indicator of commitment) as the dependent variable, but the strong negative correlation with plu- ralism among Utah counties is not simple to inter~ret.~'Findings from work on particu- lar denominations or groups will not settle the question of whether the main impact of

20 Phillips (1998) also reported a very strong positive relationship between the Melchizedik ordination rate and the proportion of the popula- tion that is Mormon, a variable that is strongly negatively correlated with religious pluralism among Utah counties. We suspect that it is the extent of Mormon dominance, rather than the de- gree of non-Mormon diversity, that accounts for the variation in involvement in this data set.

diversity on religious involvement is favor- able or unfavorable (Chaves and Gorski 2001:264), but a collection of such studies could help to shed light on the (possibly var- ied) effects of pluralism.

##### CONCLUSIONS

For more than a decade, research into the re- lationship between religious diversity and religious involvement has relied on an im- plicit assumption that the statistical associa- tions discovered reflect causal processes. That belief cannot be sustained: Random variation in the sizes of the various religious groups across the areas studied will lead to correlations between the pluralism index and total participation rates that may be substan- tially different from zero, in either a posi- tive or a negative direction. By implication, all evidence of the usual kind that has been offered on both sides of the debate is now suspect.

This particular discovery might simply initiate a ground-clearing exercise, with pro- ponents of the rival theories going back to the empirical drawing board. We believe, however, that our additional findings may also represent a substantive advance. The fact that the pluralism-participation relationship is heavily contaminated by artifactual effects gave us no reason to predict that the observed and expected correlations would resemble each other so closely; to identify a bias does not generally imply that the bias is going to account for the entire phenomenon. The natural conjecture resulting from the re- markable congruence of the actual and simu- lated associations between pluralism and participation-that the correlations may be wholly explained by the distributional prop- erties of the data-is a significant empirical statement, not a corollary of the method- ological result.

As we have noted above, a degree of simi- larity between observed and expected corre- lations might occur if religious diversity does in fact have an influence on participation, as such effects could change the denominational size distributions and hence the expected cor- relations. However, our simulations of this potential problem suggest that any real ef- fects of pluralism should still be evident in a difference between the actual and the ex-

pected correlations. The absence of such dif- ferences in data drawn from two different countries and two different centuries is po- tentially informative. Proponents of secular- ization and religious economies theories could both be wrong: Pluralism may have no effect at all on religious participation.

Such suspicions receive some corrobora- tion from the limited amount of evidence available that has not been undermined by our methodological critique. Although we advise including additional controls in the individual-level analysis of Verweij et al. (1997), pluralism is unlikely to reemerge as a significant effect in that work. The longi- tudinal study of Christian0 (1987) found no effect. Likewise, Zaleski and Zech (1995) and Per1 and Olson (2000) reported no ef- fect of pluralism on an alternative measure of commitment (congregational giving). We find no compelling methodologically unproblematic research that shows a genuine relationship between pluralism and partici- pation.

Although the evidence is highly circum- stantial, a case can therefore be made that pluralism actually has little or no effect on participation. Much more work using alter- native methods is needed to test this conjec- ture, especially since supporters of both secularization and the religious economies model have articulated plausible reasons why diversity should influence participation. There is little doubt that the search will con- tinue: The question of whether religious di- versity promotes or undermines commit- ment-and by implication how moderniza- tion affects traditional belief and practice- remains one of the most interesting problems in the field. Nevertheless, the onus is now on proponents of the rival theories to dem- onstrate that pluralism actually has an effect on religious involvement.

David Voas is Lecturer in Sociology at the Uni- versity of Sheffield, England. He has research in- terests in demography, ethnicity, and the sociol- ogy of religion, and is working to connect these fields. He recently started a project funded by the Arts and Humanities Research Board on com- puter simulation and the study of religious change.

Daniel V. A. Olson is Associate Professor of So- ciology at Indiana University South Bend. His re- cent articles in the Journal for the Scientific Study of Religion and Sociology of Religion focus on religious characteristics of the geographic environment (e.g., pluralism, denominational market share) and their influences on individual and congregational levels of religious involve- ment. A chapter in Sacred Canopies, Sacred Mar- kets: Essays on Religious Markets and Religious Pluralism (Ted Jelen, ed., Rowman and Little-

field, 2002) reviews the theoretical and empiri- cal status of religious economies models. His fu- ture research will explore how characteristics of the religious environment shape the religious composition of close social ties and the effect of such ties on religious behavior.

Alasdair Crockett is Depositor Services Manager and Research Fellow at the UK Data Archive, University of Essex, England. His research inter- ests are in the sociology of religion, particularly from a historical perspective. He is currently completing a book on religion in nineteenth-cen- tury Britain and co-editing a book on religion

and modernization.

APPENDIX A

A Technical Outline

While it should be clear from Figure 1 (p. 216) that the correlation between total participation and the index of pluralism will tend to be nonzero, the more technical account that follows may help to explain how and why the phenomenon occurs. (Proofs or more detailed analyses are available from the first author.) Consider once again a town in which there are only two denominations. We can construct a di- agram (Figure A-1) in which each axis represents the level of participation in one of the denomina- tions as a percentage of the population. The pair of denominational sizes in the town will appear as a point within the triangular region bounded by the two axes and the diagonal line representing 100percent participation.

We can construct two lines running through this point: the set of points that would produce an equal level of total participation (a line with slope -I), and the set of points that would produce the same index of pluralism (a line with positive slope ylx, where x and y are the participation rates in the two denomi- nations). The first thing to notice is that these two lines are not, in general, perpendicular.

Given a set of values (e.g., for different towns or areas), the direction of the correlation can be deter- mined as a function of the dispersion of those val- ues. Recall that in an ordinary two-dimensional scatterplot, one can gain a quick impression of the correlation by drawing a vertical line showing the mean of the x variable and a horizontal line show- ing the mean of the y variable. If most points are

0 Participation in Denomination A 100

Figure A-1. Associations between Pluralism and Participation Produced by Size Variation in Two Denominations

above average on both variables or below average on both, then the correlation is likely to be positive. If most points are higher than average on x but low- er than average on y or vice versa, then the correla- tion is likely to be negative. In Figure A-1, the "equal participation" and "equal pluralism" lines perform this same function. Where most points fall either above both lines or below both lines, then the correlation is likely to be positive. If more points fall to one side or the other, a negative correlation usually results. (We let x represent the larger de- nomination here and in what follows.)

What will determine whether the points fall one way or the other and thus whether the correlation will be positive or negative? While the relative size of the denominations (and hence the degree of ob- liqueness of the angle made by the two lines) is rel- evant, the principal determinant is the relative vari- ability of the denominations. Note that if the region in which the points are scattered (shown by an oval in Figure A-1) is elongated vertically as pictured, so that in this instance the small denomination is more variable in size, then most points fall above or below the two lines, producing a positive correla- tion. When the region is elongated horizontally, as would occur when the larger denomination is more variable in size, the correlation will be negative. In- dependent random variation of participation in each denomination is not sufficient to produce a zero cor- relation between total participation and the index of pluralism.

If both denominations vary in size by the same absolute amount, then the obliqueness of the angle made by the two dotted lines would ensure that most points fell above or below both, resulting in a posi- tive correlation. If the magnitude of variation is pro- portional to the size of the denominations, the scat- ter of points is elongated horizontally in such a way that a negative correlation is likely. The fact that the magnitude of variation is typically associated with size may explain why most previous studies have reported a negative correlation.

When is the expected correlation positive, and when negative? The exact calculation will depend on the nature of the distributions (i.e., their shape, skew, etc. as well as their dispersion), but it is pos- sible to produce analytical solutions for simple cas- es. Assume, for example, the size of the large de- nomination is a & u, and of the small denomination, b _+ v, where a and b are the mean sizes and u and v are uniformly distributed random variables that vary between 0 and m and 0 and n respectively. The plu- ralism-participation correlation is then 0 when m2/a= n2/b.In the case of inequality, the correla- tion will be positive if n2/b(the ratio for the small denomination) is greater, and negative otherwise. For example, if the two denominations are uniform- ly distributed in the ranges [30, 501 and [lo, 201,

then a = 40, b = 15, m = 10, and n = 5. Because m21 a (= 2.5) is larger than n21b(= 1.67), the correlation between total participation and the index of plural- ism will be negative.

This observation may be generalized to a situa- tion with more denominations and different distri- butions using the following rough rule of thumb: The relative variability of each denomination for these purposes can be measured by the ratio of its variance to its mean. These ratios can then be com- pared for the set of "large" and "small" denomina- tions. Whether a denomination is large or small can be decided by calculating the sum of squared mean sizes for all the other denominations and dividing by the sum of those sizes (again excluding the one in question). If the participation rate of the denomi- nation is above that threshold, it can be classified as "large," and vice versa. The variance-to-mean ratios can be averaged for each group and used as a guide to the expected sign of the pluralism-participation correlation. We stress that there are too many com- plicating conditions (not least the fact that variation of denominations close to the size threshold can have unpredictable effects) to produce a simple al- gorithm for determining the direction, let alone the magnitude, of the correlation in every situation, but these principles will give an indication of what to expect in most situations.

One final difficulty must be mentioned. Even in the special case when the denominations are of the

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