Some theoretical and methodological objections to Stephan-Deming adjustments to religious mobility tables

SOCIAL

SCIENCE

RESEARCH

19, 239-249 (1990)

Some Theoretical and Methodological Objections to StephanDeming Adjustments to Religious Mobility Tables DARRENE. SHERKAT Duke

University

Whitt, Crockett, and Babchuk’s(1988) model of religious switching calls for an adjustment of the marginal values of denomination of destination to the values of the denomination of origin using the Stephan-Deming algorithm. The theoretical and methodological shortcomings of their alternative model are examined in this paper. Theoretically, the alternative model fails to account for differences in investments and social networks which should make us expect an effect of denominational size on expected cell values. Methodologically, Whitt et al. (1988) misuse the algorithm to adjust marginal values to theoretical values, causing systematic bias in the adjusted tables. Q ITBO Academic PKSS. IUIC.

Until the late 197Os, researchers interested in the intergenerational transmission of occupations frequently used Goodman’s (1965, 1969) mode1 of quasi-independence. This technique allows researchers to compare observed mobility with what was expected if mobility among mobile individuals was independent from father’s occupation. Later, Newport (1979) and Kluegel (1980) applied the mode1 of quasi-independence to religious mobility tables. Recently, Whitt, Crockett, and Babchuk (1988) proposed that religious switching can be more reasonably modeled by using quasi-independence techniques on frequency tables in which the marginal totals for religion of destination have been adjusted to the marginal totals of the religion of origin. Whitt et al.‘s (1988) alternative mode1 is inadequate for both theoretical and methodological reasons. First, the theoretical justification for the restriction of marginal values for the denomination of destination to the marginal values for the denomination of origin goes against most theories of religious conversion and recruitment. Second, the Stephan-Deming adjustments to origin marginals systematically bias adjusted cell frequencies. This article sugAddress correspondence to Darren E. Sherkat, Department of Sociology, Duke University, Durham, NC 27706. Comments from Anthony Oberschall and Thomas DiPrete on previous work encouraged this research. Data from the National Survey of Black Americans were provided by the Interuniversity Consortium for Political and Social Research. 239 0049-089X/90 $3 .OO Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.

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gests that more care should be taken when using quasi-independence models to describe patterns of religious switching, and that more complex log-linear and time-series models must be used to go beyond these descriptions. THE ALTERNATIVE

MODEL

The Whitt et al. (1988) model for religious mobility is an attempt to control for size of denominations of destination, a problem which cannot, as Kluegel(l980) claimed, be solved by the model of quasi-independence on unadjusted frequency tables. The problem for religious mobility was seen to be the elimination of the effects of size of denomination of destination completely, because the structures of opportunity for religious mobility were deemed to be completely unrestricted. Following the logical methods outlined in Zipf (1946), Whitt et al. (1988:208) claim that the religion of origin marginal accurately represents the population at risk (N,) in a two-way table with blocked diagonals. They note that in the blocked table the marginal distribution of destinations represents not the size of religious categories, but the relative attractiveness of the denominations. Whitt et al. (1988:209) then search for an adequate model for N2, the size of the destination or choice group. The objection which Whitt et al. (1988:209) have to the calculation of expected cell values using destination marginals is that the destination marginals are a function of religious mobility. Switching inflates (or deflates) the expected cell frequencies, depending on whether a denomination has gained (or lost) adherents. Hence, we no longer obtain expected frequencies for switching from one category to another, but rather for switching from one category to another given that some specific amount of switching has already occurred. They settle on the origin marginals for the unblocked table because they are: (1) “predetermined by forces outside the model; (2) . . . logically prior to denominational switching; and (3) represent(s) a size distribution.” Basically, this means that the origin marginals are only a function of demographic factors (presumably you were born into the religion with which you are affiliated at the age of 16). Origin marginals are, generally, not a function of denominational switching since most switching occurs in adulthood. And denomination of origin is a nonarbitrary size distribution relevant to some similar life event across respondents. The alternative model uses an iterative method to adjust destination marginal totals to the unblocked marginal values for the denominations of origin. The iterative method which Whitt et al. select is the StephanDeming procedure outlined in Stephan (1942) and in Deming (1964). This method was originally formulated as an algorithm to adjust sample cell frequencies to known population marginals, but it has also been suggested as a method for adjusting cell frequencies to theoretical marginals (Dem-

ADJUSTMENTS

TO RELIGIOUS

MOBILITY

241

TABLES

ing and Stephan, 1940; Stephan, 1942). After such adjustments to the frequency table the method of quasi-independence is used to obtain expected frequencies for off diagonal cells and mobility ratios are calculated from these expected frequencies. THEORETICAL

OBJECTIONS

TO THE ALTERNATIVE

MODEL

If it is the case that destination marginals for religious mobility tables should be adjusted to the values for the origin marginals then there should be no theoretical reason for us to suspect that the size of an organization is conducive to the attraction of new members. Whitt et al. (1988) take the position that in occupational markets size distributions of destination marginals are the result of structural shifts in the economy, but that religious denominations cannot be affected by structural factors. This is incorrect for two reasons: (1) just as industries make investments toward expansion of certain sectors of the economy, religious organizations invest in future growth through proselytization and the development of an infrastructure of churches to handle current needs and expected growth; and, (2) the importance of social networks for the recruitment of new members makes it reasonable that we should expect denominational categories with higher destination marginal totals to receive more converts than denominations with lower destination marginals. Finke and Stark (1989) have found that the investments of denominations in the religious marketplace drastically changed the structure of the American religious marketplace from the 18th to the 19th century. Methodists and Baptists transformed themselves from small sects into the core of the American religious marketplace through their active efforts and flexible adaptation to the changing religious economy of the American frontier (Finke and Stark, 1989). Since religious organizations make investments which create gains in membership from other denominations (or from nonaffiliation) the marginal differences between origin and destination for a given denomination reflect, at least in part, a structural shift in religious affiliation. If a set of firms employ large numbers of individuals in a population for job type A at time T, but job A is on the decline while job type B is rapidly expanding, then we should expect to find more individuals switching from Job A (or any other job, for that matter) to Job B at time T + 1. Similarly, if the Methodists make infrastructural improvements and recruitment drives toward a segment of the population at time T, and then fail to maintain their drives for retention and recruitment, while the Baptists increase investments toward recruitment and retention in the same population, then we should expect to find more people in the population who have switched to Baptism at time T + 1. Just as occupational mobility is not merely the result of individual volition, religious mobility is similarly constrained by structural factors. Individuals cannot attend churches which do not exist or

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do not make themselves known to potential or actual adherents. And attractive and outgoing churches will pull members away from churches which have not made similar investments as surely as higher salaries in emergent sectors or the lack of jobs in declining sectors will force workers into particular sectors of the economy. Sociological theory also tells us that social networks play an important role in religious conversion, and participation in social movements (Rochford, 1985; Snow, Zurcher, and Eckland-Olson, 1980; Stark and Bainbridge, 1980). In this view, ties of friendship and kinship pull people away from former lifestyles and beliefs and push them toward new ideas and behaviors. Denominations that have more members will have more networks of families and friends which would give them a recruitment advantage over denominations which have fewer members, and therefore fewer sets of networks, on which to build a recruitment base. Unless larger denominations are exclusionary and have sparser networks of friends and family who are not members of the denomination, we should expect that larger denominations will have access to a larger pool of nonaffiliated friends and family members who are at increased risk for recruitment. It is clear that we should expect the size of the destination marginals to affect the expected cell frequencies in religious switching tables. It is also clear that not doing so will result in the undercalculation of expected cell values for denominations which have gained members, and the overcalculation expected values for denominations which have lost members. METHODOLOGICAL

OBJECTIONS

TO THE ALTERNATIVE

MODEL

The Stephan-Deming algorithm was originally formulated as a method of adjusting frequency tables to reflect known population marginal totals (Deming and Stephan, 1940:427).’ Since the algorithm was designed to readjust sample cell frequencies to reflect their distribution in a population, cell frequencies were always adjusted to larger marginal totals, since sample sizes are, by definition, smaller than the population itself. Obviously, Whitt et al.‘s (1988) proposition, to adjust cell frequencies to reflect marginal totals which are a product of the sample itself, goes beyond the primary intended use of the adjustment procedure. A note of caution issued by Deming and Stephan (1940:444) on the use of adjusted tables is wise advice, “It must not be supposed that any or all ’ While Deming and Stephan (1940) and Stephan (1942) mention the possibility of using this procedure to ensure consistency between crosstabular tables, and for adjusting tables to theoretically deduced marginal totals, they do not discuss the statistical implications of this. (Deming’s (1964) later work drops all mention of theoretically driven marginal adjustments.) By adjusting cell frequencies to theoretical marginal totals, you compound the sampling error of the original table with the error present in the theory driving the adjustment.

ADJUSTMENTS

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243

of the adjusted m,j (adjusted cell frequencies) are necessarily “closer to the truth” than the corresponding sampling frequencies rZ;,j, even under ideal conditions.” I suspect that by “ideal conditions” Deming and Stephan (1940) refer to adjustments to relatively error free population data. When cell values in frequency tables are adjusted to reflect sample values, such as those for the denomination of origin, rather than to reflect population values (which, of course, are unknown), the matrix of adjusted cell frequencies takes on some peculiar properties. Contingent on the direction of adjustments to the marginal totals (downward for denominations which gained adherents, and upward for denominations which lost adherents) adjusted cell frequencies are systematically biased. Following Whitt et al. (1988), let mi and mj represent the a priori row and column marginals to which cell frequencies are to be adjusted (where i and j represent the number of rows and columns, respectively). In the alternative model, mi and mj are set to equal Izi, the sample row marginal representing denomination of origin. The adjusted cell frequencies, mij, are arrived at iteratively through calculation of a set of factors, p(l), = m,/2n, for each row. Original cell frequencies njj are multiplied by the corresponding row factors, p(l),. Cell values are then summed for each column and column totals are then subtracted from the a priori column marginal mj. The difference, between the sum of the cell values for columns and the a priori mj, is divided by the original column marginal rzj to give the first stage column factor 4(1)j. In the second cycle, the cell frequencies are multiplied by their corresponding S(l)j and these are summed for the rows. Row sums are then subtracted from the mi and divided by the nj to give the second stage row factors p(2)i. The cycle is then repeated until row and column marginals equal the a priori totals, leaving a mixi matrix of adjusted cell frequencies. It is easily seen that, since the a priori column totals (mj) are smaller than the corresponding sample totals (nj) for denominations which gained members and larger than the corresponding sample totals (nj) for denominations which lost members, the column factors in the first stage will be smaller for growing denominations. This is because the difference, between the sum of the cell frequencies from the row adjustment in the first stage from the a priori mj, is divided by a larger denominator (nj) for denominations which have grown. Similarly, in later cycles of the iterative procedure the column factors will be systematically biased by the influence of large denominators for the growing denominations and small denominators for the shrinking denominations. The resulting bias, as it will later be shown, is substantial, systematic, and unacceptable. RESULTS

Table 1 and Table 2 present data from the National Survey of Black Americans (NSBA). This survey was conducted in 1979-1980 by the

41 0 4 2 2 5 9

63 3 ,651

Religion Liber Meth. Bapt. Cons. Other Cath. None

Total Gain 9% Stay

Liberal

338 -91 ,621

8 212 58 20 5 7 28

Methodist

1277 - 194 ,767

14 23 980 103 42 20 95

Baptist

162 100 .741

1 5 15 120 4 0 17 46 39 ,413

Other

2107

232 87 145 .701

61 134 -2 .724

262 85 132

6 5 2

5 8 91 14

66 241 1083

Total 1 2 10

None 1 2 7

Catholic

of Religion by Religion of Origin

Religion of Origin Conservative

TABLE 1 National Survey of Black Americans-Crosstabulation

if2 E

P 5 E ?’

44.2 0 5.9 1.3 1.3 5.6 4.1

62.5 ,707

Religion Liber. Meth. Bapt. Cons. Other Cath. None

Total % Stay

Liberal

National

Survey

340.7 ,816

4.2 277.9 54.0 2.3 .6 4.0 -2.5

Methodist

of Black

1277.1 ,899

10.8 35.7 1148.7 36.5 15.3 16.3 14.5

Baptist

Americans-Stephan-Deming

161.4 .670

1.3 10.5 25.8 108.2 3.7 0 11.9

Religion Conservative

TABLE 2 Adjusted at age 16

Crosstabulation

45.9 .327

0 5.9 14.4 4.7 15.0 1.2 4.6

Other

of Religion

132.4 ,785

1.0 3.6 10.0 3.1 5.0 103.9 5.7

Catholic

by Religion

86.5 .562

1.4 4.4 18.2 6.0 5.1 2.9 48.6

None

of Origin

2107

63 338 1277 162 46 134 87

Total

246

DARREN

E. SHERKAT

Survey Research Center at the University of Michigan (Jackson and Gurin, 1987). In these tables respondents are grouped by religious affiliation when they were growing up and current religious affiliation. (The General Social Survey. used by Newport (1979), specifically asks for religion at the age of 16.) Table 1 gives the crosstabulation of religion by religion of origin, and Table 2 presents the Stephan-Deming adjusted table.’ In Table 1 and 2, I use rows to indicate current religion and columns to indicate original religion (the analysis above still holds with the row factor now being the biasing agent). The differences between the categories presented here and those used by Newport (1979) are in the classification of medium status denominations (restricting medium status protestants to Methodist affiliation). and the pooling of nondenominational Protestants with nontraditional sects, Moslems, and Jews into the “Other” category. A more detailed explanation for the categorization of denominations for Black Americans can be found in Ellison and Sherkat (1990). The patterns outlined above are readily apparent when Tables 1 and 2 are examined together. Denominations which lost members in Table 1 appear to lose fewer of them in Table 2. For instance the Methodists, who lost 91 members, have lost only 62.6 members in Table 2. The retention rate (reported under “% stay”) of the Methodists is .627 in Table 1, but it jumps to .816 in the adjusted table. Conversely, categories which experience gains from switching appear to lose in the adjusted table. The nonreligious category has a remarkably high rate of retention, compared to the general population analyzed in Newport (1979), in Table 1, .701, but in Table 2 the rate of retention falls to .562. We can see that whereas in Table 1 only 10 respondents who originated in the nonreligious category actually switched to Baptism, the adjusted Table 2 reports that 18.2 nonreligious respondents became Baptists. Clearly the biases described above are artificially creating patterns of denominational switching which do not exist. Of particular interest is cell rn,,* in the adjusted table, indicating Methodists who switched out of religion. In Table 1 we see that 28 Methodists switched out of religion, representing 8.3% of all original Methodists in the sample, and 22.2% of Methodists who switched to another category. Yet, in Table 2 the adjustment procedure arrives at a value of -2.5 for cell rn,.* .3 Needless to say this is an unacceptable value. It is clear from this example that the misuse of the algorithm creates not only substantial bias, but also produces meaningless results. * I am grateful to Hugh P. Whitt for supplying me with a listing in BASIC of a program for performing Stephan-Deming adjustments to two-way frequency tables. 3 Similar results were obtained when Black Americans in the General Social Survey 1972-1988 were examined, including the negative value for m7.2 in the adjusted table.

ADJUSTMENTS

TO RELIGIOUS

Index of Status Persistence-From

High SES Medium SES Baptist Low SES No denomination Catholic None

MOBILITY

247

TABLES

TABLE 3 Newport (1979) and from the Adjusted Table Newport (1979) G

Adjusted table G

.610 .567 ,718 ,498 ,523 .805 .299

,668 ,738 .826 ,520 ,441 .899 ,276

Table 3 presents Goodman’s (1969) index of status persistence from Newport’s (1979) Table 5 and from a quasi-independence model using the data from Table 3 in Whitt et al. (1988). The index shows the degree to which people stay in their denomination of origin over what is expected given the quasi-independence assumption. While it is less apparent here, the pattern of bias created by the method is reflected in the differences between Newport’s (1979) findings and the findings based on Whitt et al.‘s adjustment of Newport’s data. The Baptists and Medium status denominations which experienced losses have augmented status persistence in the adjusted table. The categories which experienced large gains, the none and nondenominational Protestant categories, lose status persistence because of the adjustments to the matrix. The empirical findings show that the patterns of bias created by the adjustments advocated by Whitt et al. (1988) create the systematic biases suggested in the methodological objections. Further, the empirical findings show that the biases are substantial, and that they alter the interpretations of the data which they analyzed in predicted ways. SUMMARY

Previous examinations of religious switching have focused primarily on theoretical models which ordered religious denominations along status lines (cf. Newport, 1979; Stark and Glock, 1968). Unfortunately, these studies have failed to take into account structural factors which have an impact on the growth and decline of denominations. The result of the continued emphasis on denominational status in predicting patterns of affiliation has been a lack of interface between theories of religious conversion, social movement participation, and religious switching. Whitt et al.‘s (1988) alternative model followed in the footsteps of the denominational status paradigm of religious switching, failing to recognize the theoretical insights of other sociological theories of conversion and participation in voluntary organizations. The failure to recognize theoretical

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developments led to the creation of a flawed model. Methodological innovations without theoretical justification make no contribution to the scientific understanding of social processes. This is not the only problem with methodological innovation. Even if the theoretical reasoning behind an innovation is plausible, the methods themselves may be flawed beyond repair. Iterative proportional fitting has previously been used in studies of occupational mobility to attempt to sort out circulation mobility from structural mobility (Hazelrigg, 1974; Hazelrigg and Garnier, 1976). In these cases the fitting method was used to adjust origin and destination marginals to arbitrary equivalent values. These methods have ultimately failed to accomplish their task, despite the theoretical attractiveness of the concepts of circulation and structural mobility, they cannot be disentangled by this or any other methodology (Hauser and Grusky, 1988; Sobel, 1983). In the case of religious mobility, the method suggested by Whitt ef al. (1988) clearly yields untenable results. There is still some value in the method of quasi-independence for describing general patterns of religious switching but researchers must be aware of the limitations of the model being used. The model of quasiindependence should remain as a tool for suggesting patterns of affiliation. and for comparing these patterns over time, but to disentangle the relationships between life events and religious switching more complex log-linear and event-history models will be needed. Many of the theories mentioned above could more appropriately be tested using logistic or probit regression on individual-level data. An even better method would be to analyze the risk of switching over time using continuous-time stochastic models such as Cox’s proportional hazards model and parametric survival models (Sherkat, 1990). Unfortunately, little data is available which allows an analysis of the hazard of religious switching over time. REFERENCES Deming, W. E. (1964). Statistical Adjustment of Data, Dover, New York. Deming, W. E., and Stephan, F. F. (1940). “On a least squares adjustment of a sample frequency table when expected marginal totals are known,” Annals of Mathematical Statistics 11, 427-444. Ellison. C. G., and Sherkat. D. E. (1990). “Patterns of religious mobility among black Americans.” The Sociological Quarterly, forthcoming. Finke, R., and Stark. R. (1989). “How the upstart sects won America: 1776-1850.” Journal for the Scientific Study of Religion 28, 27-44. Goodman, L. A. (1965). “On the statistical analysis of mobility tables,” American Journal of Sociology 70, 564-585. Goodman, L. A. (1969). “On the measurement of social mobility: An index of status persistence,” American Sociological Review 34, 831-850. Hauser, R. M., and Grusky. D. B. (1988). “Cross-national variation in occupational dis-

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tributions, relative mobility chances and intergenerational shifts in occupational distibutions,” American Sociological Review 53, 723-741. Hazelrigg, L. E. (1974). “Partitioning structural effects and endogenous mobility processes in the measurement of vertical occupational change,” Acta Sociologica 17, 115-139. Hazehigg, L. E., and Gamier, M. A. (1976). “Occupational mobility in industrial societies: A comparative analysis of differential access to occupational ranks in seventeen countries.” American Sociological Review 41, 498-511. Jackson, J. S., and Gurin, G. (1987). National Survey of Black Americans, 1979-1980, Interuniversity Consortium for Political and Social Research, Ann Arbor, MI. mobility: current patterns and recent trends,” Kluegel, J. R. (1980). “Denominational Journal for the Scientific Study of Religion 19, 26-39. Newport, F. (1979). “The religious switcher in the United States,” American Sociological Review 44, 528-552. Rochford, E. B. (1985). Hare Krishna in America, Rutgers University Press, New Brunswick, NJ. Sherkat, D. E. (1990). “Parametric and proportional hazards models of religious switching,” Working paper, Department of Sociology, Duke University, Durham, NC. Snow. D.. Zurcher, L., and Eckland-Olson, S. (1980). “Social networks and social movements: A microstructural approach to differential recruitment,” American Sociological Review 45, 787-801. Sobel, M. E. (1983). “Structural mobility, circulation mobility and the analysis of occupational mobility: A conceptual mismatch,” American Sociological Review 48, 721727. Stark, R., and Glock. C. Y. (1968). American Piety: The Nature of Religious Commitment, University of California Press, Berkeley, CA. Stark, R.. and Bainbridge, W. S. (1980). “Networks of faith: Interpersonal bonds and recruitment to cults and sects,” American Journal of Sociology 85, 1376-1395. Stephan. F. F. (1942). “An iterative method of adjusting frequency tables when expected marginal totals are known,” Annals of Mathematical Statistics 13, 166-178. Whitt, H. P., Crockett, H. J., and Babchuk, N. (1988). “Religious switching: An alternative model.” Social Science Research 17, 206-218. Zipf, G. K. (1946). “The PIPZ/D hypothesis: On the intercity movement of persons,” American Sociological Review 11, 677-686.