- Email: [email protected]
Some theoretical and methodological reasons for using Stephan-Deming adjustments in religious mobility tables
SOCIAL
SCIENCE
RESEARCH
21, 204-215 (1992)
Some Theoretical and Methodological Reasons for Using Stephan-Deming Adjustments in Religious Mobility Tables . HUGH
P. WHITT University
AND
NICHOLAS
BABCHUK
of Nebraska-Lincoln
Sherkat’s (1990) critique of the use of Stephan-Deming adjustments to control for the size of social networks in studies of religious mobility is flawed by his misinterpretation of the concepts of “opportunity structure,” “destination marginals,” and “model,” as used by Whitt, Crockett, and Babchuk (1988). These concepts are clarified and the Stephan-Deming adjustments are shown to be a theoretically and methodologically sound approach for investigating religious switching. Substantive implications for data on black Americans are explored. 0 1992 Academic
Press, Inc.
Recently, Sherkat (1990) has published a thoughtful critique of the use of Stephan-Deming (S-D) adjustments (Deming, 1964; Deming and Stephan, 1940; Stephan, 1942) in religious mobility tables, arguing that our rationale for advocating this procedure (Whitt et al., 1988) is severely flawed. While there is merit in what Sherkat has to say about the desiderata for an adequate analytic technique, his negative appraisal of the S-D adjustments cannot be supported. This paper responds to Sherkat’s evaluation, outlining theoretical and methodological reasons for using adjustments of this type in studies of religious switching. Some background is necessary. All else being equal, rates of intergroup interaction are affected by the opportunity for random contact between group members. The opportunity for random contact, in turn, is a function of the sizes of the groups in question and their distribution in physical space (Stouffer, 1940; Stewart, 1942; Zipf, 1946; Ilke, 1954; Coleman, 1964). The importance of opportunity structures has been recognized in studies of criminal victimization (Boggs, 1960; Cohen and Felson. 1979), geographical mobility (Stouffer, 1940), traffic patterns (Ilke, 1954), and religious intermarriage (Thomas, 1951). Techniques for taking opportunity structures into account and elimiCorrespondence should be sent to Hugh P. Whitt, Department of Sociology, 718 Oldfather Hall, University of Nebraska-Lincoln, Lincoln, NE 68858-0324. 204 0049-089X/92 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.
STEPHAN-DEMING
ADJUSTMENTS
205
nating their confounding effects on more theoretically interesting processes have been developed by researchers investigating occupational mobility. During the 1960s and 197Os, Goodman’s (1964, 1965, 1969a,b) model of quasi-independence (QI) was widely employed for this purpose. This analytic technique has been appropriately and fruitfully exported to criminology and studies of intermarriage. It has also been appropriated by researchers (Newport, 1979; Kluegel, 1980) investigating religious switching. Goodman’s (1964, 1965, 1969a,b) model is, however, inadequate when applied to religious mobility. In proposing it be abandoned in religious switching studies, we argued that the problems we identified could be partially corrected by using S-D adjustments for a priori marginals before fitting QI models. Sherkat (1990) disagrees, contending that our approach suffers from both theoretical and methodological shortcomings. There is, however, a serious gap between our thinking in proposing the S-D adjustments and Sherkat’s interpretation of what we say. This gap is substantial. Sherkat (1990) asserts that our procedure “goes against most theories of religious conversion and commitment,” that the S-D adjustments “significantly bias adjusted cell frequencies,” and that we are so closely tied to the dated “denominational status” model of religious switching that we ignore more recent theoretical developments. Had we been trying to do what Sherkat thinks, we would agree that the approach we used was misguided. It seems to us, however, that Sherkat and we are simply talking past each other. Sherkat’s (1990) reading of our 1988 paper differs radically from what we intended. Some of the key terms in our discussion mean something different to Sherkat than they do to us. For example, he asserts that we deem “the structures of opportunity for religious mobility . . . to be completely unrestricted” (p. 240), using this as a springboard for criticizing us for ignoring the sizes of social networks in our analysis. “Opportunity structure,” as employed in our earlier paper and herein, refers to the random opportunity for interaction under such models as QI. The term incorporates the idea that “larger denominations will have access to a larger pool of nonaffiliated friends and family members who are at increased risk for recruitment” (Sherkat, 1990, p. 242). As we use the concept, “opportunity structure” has nothing to do with the unrestricted possibilities which indeed exist for religious mobility in all but the very few sects which restrict membership. It refers instead to the size distribution of religious denominations. The pseudo-disagreement between Sherkat (1990) and ourselves can also be traced to different understandings of the word “model.” The studies we critiqued for importing the model of QI from studies of occupational mobility (Newport, 1979; Kluegel, 1980) fell into the trap of
206
WHITI
AND BABCHUK
what Carter (1971) calls “inadvertent theory.” Newport (1979) and Kluegel (1980) employed QI to remove the contaminating effects of group size so that more sociologically interesting hypotheses could be explored. In so doing, they inadvertently removed the effects of success in proselytization. Our alternative model seeks to correct this defect in previous research. It is not a confirmatory model of the type assumed by Sherkat. As in the earlier studies we critiqued, we use the term model in the sense implied by Coleman’s (1964, pp. 469-470) discussion of the “method of residues.” According to him: [I]n considering a given complex social phenomenon, certain aspects of it are explainable by matters irrelevant to the substantive matters under investigation. If we examine what part of the behavior can be explained by these factors, then the remainder stands out to be explained by less trivial factors [I]n certain structural phenomena, where the data ate in the form of numbers of people, this separating out may be done quantitatively and rigorously The approach suggested here is to make some simple and reasonable null assumptions and then to use the deviations from the predictions consequent upon these assumptions as a measure of various matters, as, for example, the “social distance” between two groups.
Sherkat’s attaches a different meaning to the term “model.” To him, a model includes those explanatory variables whose impact the researcher wishes to assess. In this sense the model should provide as accurate a representation as possible of a complete theory of religious switching, including all theoretically relevant predictors of the tendency to change denominations. Models of this type are becoming increasingly common. They use a confirmatory logic which differs from Coleman’s approach, which does not require the specification of theoretically relevant independent variables. Models based on the method of residues include only control variables. Because Newport (1979) and Kluegel(l980) were working within the paradigm proposed by Coleman (1964), our alternative model employed this same paradigm. Sherkat (1990) seems to think that our intent was to test a confirmatory model. It was not. As often occurs in cross-paradigm debate, miscommunication has resulted ‘in misunderstanding. Coleman’s (1964) major example of the use of the method of residues is the “distance-interaction hypothesis” or “law of social gravity” developed by Stouffer (1940), Stewart (1942), Zipf (1946)) and Ilke (1954), which states that, all else being equal, “the rate of interaction between two areas with IZ~ and n2 persons who are d,, distance apart will be proportional to nln2/d12 .” Coleman (1964, pp. 471-473) suggests that this factor represents “the expected rate of interaction which might occur if no sociological effects were operative.” He argues for its use as
STEPHAN-DEMING
ADJUSTMENTS
207
“a base line, or standardization, which can cancel out differences in interaction due to these sociologically trivial factors, and illuminate the differences due to more interesting matters . . . Without such a base line, or standardizing factor, differences which are due simply to the size of the unit or to the distance between units will be confounded with differences due to less trivial matters, and analysis of the source of the latter differences will be impossible.”
Like Newport (1979) and Kluegel (1980), we used the method of residues to construct a baseline model in Coleman’s (1964) sense. The motivation was to eliminate the confounding effects of differences in initial group size rather than to specify in advance what sociologically interesting processes might be operating.’ Under conditions of random interaction, larger groups have stronger “gravitational pulls” than smaller ones. Like Newport and Kluegel, we used the method of residues to construct a baseline model that eliminates the confounding effects of differences in group size. We did not specify ahead of time what sociologically interesting processes might be operating. As a consequence of our use of the method of residues, our approach is not wedded to any specific substantive theory of religious mobility, Sherkat’s argument to the contrary. Its only theoretical assumption is that the probability of switching between two religious groups is proportional to the product of their sizes. Sherkat (1990, p. 241) suggests that our failure to take into account two theoretical factors invalidates the use of the S-D adjustments: (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.
Although theoretically unassailable, Sherkat’s comments miss the point. If QI is used without the S-D adjustments, the theoretical processes he identifies are inadvertently controlled out of the analysis! This point requires explanation. As noted in our original paper (Whitt et al., 1988), religious mobility is logically different from occupational mobility. In most cases, jobholders do not create the positions they occupy. Their positions are established by their employers, who determine how many jobs will be available in each sector of the economy. The distribution of jobs changes over time with the expansion and contraction of particular sectors of the economy; ’ Although it would be desirable to eliminate the effects of distance as well, most surveys provide insufficient evidence to do so.
208
WHITT
AND BABCHUK
at any point in time this occupational distribution exists prior to any action by job applicants. On the other hand, religious mobility is not constrained by limitations on the number of persons switching denominations. Indeed, new memberships (the analogue of jobs) can be created at will. Growth cannot cause switching; it is a dependent variable caused by switching and, in the natural setting but not in sample surveys, by natural increase. In this regard, the proper theoretical analogy is geographical mobility rather than the dynamics of the job market. Just as population change in cities, states, and nations is completely determined by rates of in-migration, out-migration, and natural increase or decrease, changes over time in the memberships of denominations are a product of rates of conversion and defection and by gains and losses associated with birth and death. Because of this difference between occupational and religious mobility, the distinction between the distribution of religious origins and the distribution of destinations is crucially important. If, as Sherkat (1990) correctly argues, religious denominations invest resources and mobilize members to stimulate growth, it is illogical to suggest that the resources and members so invested and mobilized are those they command after they have experienced growth. As we pointed out (Whitt et al., 1988), the relevant networks are those to which the potential convert is exposed prior to switching. In survey data, there is admittedly no perfect way of measuring the sizes of denominations at the time of switching. Each person changes denominations at a different point in time. In order for the distribution of origins to be a fully accurate representation of the opportunity structure for religious switching, the S-D adjustments would need to be applied to a sample of persons all of the same age who switched denominations at the same time. Nonetheless, the distribution of religious origins comes closer to tapping the sizes of denominations at the time mobility takes place than does the distribution of destinations. Thus, the S-D adjustments with the distribution of origins specified a priori are employed to estimate the size distributions Nr and N2 in Coleman’s (1964) baseline model of the law of social gravity. Indeed, the S-D procedure assesses as a residual the effects of resource investment and mobilization of participants upon flows between denominations, while the traditional approach of Newport (1979) and Kluegel (1980) does not. When the distribution of destination denominations was rejected in favor of the distribution of religious origins as a basis for modeling the opportunity for religious switching, we intended to liken these terms to distributions of denominations at two points in time. Sherkat (1990, p. 242) apparently does not grasp this meaning when he says that “[i]t is clear that we should expect the size of destination marginals to affect the expected cell frequencies in religious switching tables.” If destination
STEPHAN-DEMING
ADJUSTMENTS
209
refers to the size distribution of all denominations other than one’s own, then we completely agree. However, in our usage, “destination marginals” refers instead to the distribution of denominations at the time of the survey, while origin marginals are those based on the distribution of denominations prior to switching. We should perhaps have called them the distributions at Time 1 and Time 2. The marginals at Time 1 establish the opportunity structure for switching, which in turn generates the marginal distribution at Time 2. As we showed by reanalyzing Newport’s (1979) data, the two methods yield different results. Sherkat (1990) claims this is because the S-D adjustments introduce bias in adjusted cell frequencies which is proportional to the denomination’s rate of growth. This bias, he maintains, “is substantial, systematic, and unacceptable.” True, the differences in frequencies are substantial and systematic, but they are far from unacceptable and do not constitute bias. The altered frequencies represent a necessary correction for differential denominational growth. As Sherkat (1990) points out and amply demonstrates, the adjusted cell frequencies in the main diagonal differ systematically from the raw frequencies on the basis of their rate of growth or decline. This is not accidental. Those denominations which have grown from the origin to the destination distribution have their destination marginals reduced by the S-D adjustments to reflect the smaller size of their networks when switching took place. Conversely, the greater influence in the past of declining denominations is captured by increasing their adjusted marginal frequencies. These modifications in the marginal distribution of destinations affect all cell frequencies, including those for cells in the main diagonal. As a consequence, the index of status persistence, an indicator of retention rates, is systematically affected by denominational growth or decline. As Sherkat shows, the indices of status persistence for the adjusted distributions are systematically different from those calculated over the raw data. While Sherkat (1990) is correct in assessing the impact of the S-D adjustments on marginal and cell frequencies, his critique fails to recognize the crucial distinction between description and causal inference. “Clearly,” he says, “the biases described above are artificially creating patterns of denominational switching which do not exist.” When compared with the raw data, the hypothetical and essentially fictional distributions of cell frequencies resulting from the S-D adjustments facilitate inferences about the operation of causal processes. They are not intended to describe real rates of denominational growth and decline, which are best assessed from the original raw frequency distribution. The hypothetical distributions generated by the S-D procedure form the basis for the application of the method of residues. In applications
210
WHIT-I- AND BABCHUK
of Coleman’s method, “observed” frequencies incorporating theoretically based adjustments are compared with equally fictional expected frequencies generated by relatively simple and straightforward causal assumptions. The differences between observed and expected frequencies are the residues. One family of models of this type is chi-square analysis. In these models, including QI and the S-D adjustments, artificially created patterns of denominational switching which do not exist are generated as part of the analysis. No one claims that these hypothetical distributions correspond to reality-they do not. Instead, they represent the cell distributions which would exist if the theoretical processes specified by the researcher were operating. Traditional chi-square generates the expected frequencies by assuming (sometimes contrary to the fact) that the distribution of cases to the cells is random given the observed marginal distributions. The method of quasi-independence adds the assumption that this random process applies only to the off-diagonal cells. S-D adjustments add to the QI model the assumption that the marginal frequencies are known a priori. These marginal frequencies are estimated from the distribution of religions of origin in the sample. Contrary to Sherkat (1990), there is nothing inherent in the method to prevent the use of uny set of a priori marginals as an assumption. The distribution of origins is used because it mirrors the theoretical processes both we and Sherkat are concerned with modeling, i.e., the effects of the size of social network on religious switching. In his analysis of supposed bias due to S-D adjustments, Sherkat (1990) confines his attention to the adjusted frequencies, failing to note that the procedure also affects the chi-square expected frequencies and the mobility ratios derived from the adjusted data. As measures of residues, the mobility ratios are the only measures having theoretical importance; neither the adjusted frequencies (which play the role of observed frequencies in calculating the QI model) nor the expected frequencies are in themselves of any intrinsic interest. Their sole purpose is to serve as intermediate steps in the calculation of mobility ratios. Tables 1 through 6 provide material to illustrate the above discussion. Table 1 (Sherkat’s Table 1) presents a crosstabulation of current religion by religion of origin drawn from the 1979-1980 National Survey of Black Americans (Jackson and Gurin, 1987). Tables 2 and 3 show, respectively, the expected frequencies and mobility ratios based on this tabulation under the QI model. Tables 4-6 give the comparable data under the S-D adjustments. As Sherkat (1990) suggests, the cell frequencies in Table 3 indeed differ systematically from the raw data in Table 1. The differences between cell values in the two tables correlate modestly and negatively with the growth or decline of one’s religion of origin (r = - .488), but are uncorrelated with gains and losses of the denomination chosen (r =
STEPHAN-DEMING
211
ADJUSTMENTS
TABLE 1 Current Religion by Religion of Origin, Survey of Black Americans, 1979-1980 Religion of origin Current religion
Lib
Meth
Bapt
Cons
Other
Lib Meth Bapt Cons Other Cath None Total
41 0 4 2 2 5 9 63
8 212 58 20 5 7 28 338
14 23 980 103 42 20 95 1277
1 5 15 120 4 0 17 162
3 9 6 19 1 8 46
0
Cath
None
Total
1 2 7 5 8 97 14 134
1 2 10 6 5 2 61 87
66 247 1083 262 85 132 232 2107
.OOO). This pattern is expected given that the S-D adjustments maintain the initial patterns in the data subject only to constraints based on a priori marginals. The differences between Tables 2 and 5 and those between Tables 3 and 6 are less systematically based upon denominational growth and decline. The expected values in Tables 2 and 5 are completely dependent upon the marginal frequencies in these tables, which are identical within limits of rounding to those in their respective parent tables. These two tables serve only as a basis for calculating the mobility ratios in Tables 3 and 6, which show the ratio of observed to expected frequencies. The mobility ratios in Tables 3 and 6 represent the extent to which particular cells are over- or underrepresented given the assumptions of their respective underlying models. Changes in the sizes of both origin and destination denominations are positively associated with changes between tables in mobility ratios; the mobility ratios in Table 6 particularly exceed those in Table 3 when both denominations are growing. Thus, Table 6 yields high mobility ratios for growing denominations, while Table TABLE 2 Expected Frequencies under Quasi-Independence
Lib Meth Bapt Cons Other Cath None Total
Lib
Meth
Bapt
Cons
Other
Cath
None
Total
41.00 1.28 7.53 4.60 2.07 1.11 5.40 62.99
4.56 212.00 44.13 26.98 12.15 6.52 31.66 338.00
15.14 24.97 980.00 89.66 40.36 21.68 105.19 1277.00
1.80 2.96 17.40 120.00 4.79 2.57 12.48 162.00
1.01 1.67 9.82 6.00 19.00 1.45 7.04 45.99
1.33 2.19 12.86 7.86 3.54 97.00 9.22 134.00
1.16 1.92 11.27 6.89 3.10 1.67 61.00 87.01
66.00 246.99 1083.01 261.99 85.01 132.00 231.99 2106.99
212
WHIT-T AND BABCHUK TABLE 3 Mobility Ratios under Quasi-Independence
Lib Meth Bapt Cons Other Cath None
Lib
Meth
Bapt
Cons
Other
Cath
None
-
1.76
0.53 0.43 0.97 4.49 1.67
1.31 0.74 0.41 1.07 0.88
0.92 0.92 1.15 1.04 0.92 0.90
0.56 1.69 0.86
1.79 0.92 1.00
0.75 0.91 0.54 0.64 2.26
0.86 1.04 0.89 0.87 1.61 1.20 -
0.84 0.69 1.14
1.36
1.52
3, based on QI, controls out the effect of growth over and above that expected given the denomination’s initial size. A number of cells radically differ between the two tables. Differences in magnitude of .20 or greater occur in 15 of the 38 non-null cells. Most such changes indicate only variations in the strength of flows without any change in substantive interpretation. For example, either traditional QI techniques or the S-D approach would conclude that black American Baptists are relatively unlikely to become apostates. The S-D analysis, however, more strongly underscores this point through a very low mobility ratio (.69 vs .90 under QI). Both techniques point to a strong flow to no religion from conservative denominations, but this pattern is much clearer (a mobility ratio of 2.44 vs 1.36) under S-D assumptions. More importantly, choice of method influences substantive conclusions about which cells indicate above-chance frequencies of switching. In Tables 3 and 6, mobility ratios above 1.00 indicate that the flow between the two denominations represented by the cell is greater than expected given the tables’ respective assumptions. Seven cells shift from above to below 1.00 or vice versa between Table 3 and Table 6. In summary, sometimes choice between methods makes a large difference in mobility TABLE 4 Observed Frequencies under Stephan-Deming
Lib Meth Bapt Cons Other Cath None Total
Adjustments
Lib
Meth
Bapt
Cons
Other
Cath
None
Total
44.2 0.0 5.9 1.3 1.3 5.6 4.1 62.4
4.2 277.9 54.0 2.3 0.6 4.0 -2.5 340.5
10.8 35.7 1148.7 36.5 15.3 16.3 14.5 1277.8
1.3 10.5 25.8 108.2 3.7 0.0 11.9 161.4
0.0 5.9 14.4 4.7 15.0 1.2 4.6 45.8
1.0 3.6 10.0 3.1 5.0 103.9 5.7 132.3
1.4 4.4 18.2 6.0 5.1 2.9 48.6 86.6
62.9 338.0 1277.0 162.1 46.0 133.9 86.9 2106.8
STEPHAN-DEMING
TABLE 5 Expected Frequencies under Stephan-Deming
Lib Meth Bapt Cons Other Cath None Total
213
ADJUSTMENTS
Adjustments
Lib
Meth
Bapt
Cons
Other
Cath
None
Total
2.57 9.47 2.24 1.22 1.17 1.53 18.20
2.13 36.27 8.59 4.66 4.49 5.86 62.60
9.75 35.13 30.66 16.63 16.01 20.92 129.10
2.28 8.20 30.22
1.24 4.47 16.48 3.90 2.04 2.66 30.79
1.14 4.11 15.16 3.59 1.95
1.56 5.62 20.70 4.90 2.66 2.56 38.00
18.70 60.10 128.30 53.88 31.00 30.01 38.30 360.29
3.88 3.74 4.88 53.20
2.45 28.40
which may or may not alter substantive conclusions. At other times, even small changes can modify interpretation of the data. Sherkat (1990) maintains that religious switching can be more fruitfully analyzed through individual-level analytic techniques such as event history analysis. There is merit in this position, but typical event history models, limited as they are to individual-level variables, fail to control for structural factors such as the sizes of social networks. Care should be taken to incorporate structural as well as individual attributes. This may be done by including either mobility ratios drawn from a S-D analysis or products of initial group sizes as characteristics of individuals in a proportional hazard model. With this caveat, event history models should probably be used when the data permit, i.e., when the precise timing of switching is available for each respondent. Although some data of this sort are available, the exact timing of religious mobility is usually unknown. In such situations the S-D adjustments allow modeling at least an approximation of the opportunity structure. Even with samples of mixed ages, the distribution of origins is theoretically preferable to the distribution of destinations because it provides a partial control for size rather than inadvertently controlling out the effects of denominational growth. ratios,
TABLE 6 Mobility Ratios under Stephan-Deming
Lib Meth Bapt Cons Other Cath None
Lib
Meth
Bapt
Cons
-
1.54
1.11 1.02
0.57 1.28 0.85
0.62 0.58 1.06 4.78 2.67
1.49 0.27 0.13 0.89 -
1.19 0.91 1.02 0.69
Adjustments Other
Cath
None
1.31 0.87 1.20
0.88 0.88 0.66 0.86 2.57
0.90 0.78 0.88 1.22 1.91 1.13 -
0.95 2.44
0.59 1.73
2.33
214
WHI’IT
AND BABCHUK
The foregoing underscores the theoretical reasoning underlying the use of S-D adjustments. Rather than “following in the footsteps of the denominational status paradigm, failing to recognize the theoretical insights of other sociological theories of conversion and participation in voluntary organizations,” the model is tied to no theory other than the law of social gravity. It permits the assessment of any structural theory the researcher chooses through the examination of residues as embodied in mobility ratios. Rather than a “flawed model” “ without theoretical justification” which makes “no contribution to the scientific understanding of social processes,” (Sherkat, 1990, p. 248), the Stephan-Deming adjustments are well-suited, given imperfect data, to map the theoretical notion that social networks make a difference. REFERENCES Boggs, S. L. (1960). “Urban crime patterns,” American Sociological Review 30, 899-908. Carter, L. F. (1971). “Inadvertent sociological theory,” Social Forces 50, 12-25. Cohen, L. E., and Felson, M. (1979). “Social change and crime rate trends: A routine activity approach,” American Sociological Review 46, 588-608. Coleman, J. S. (1964). Introduction to Mathematical Sociology, Free Press of Glencoe, London. 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 tables are known.” Annals of Mathematical Statistics 11, 427-444. Goodman, L. A. (1964). “A short computer program for the analysis of transaction flows,” Behavioral Science 9, 176-178. Goodman, L. A. (1965). “On the statistical analysis of mobility tables,” American Journal of Sociology 70, 564-585. Goodman, L. A. (1969a). “How to ransack social mobility tables and other kinds of crossclassification tables,” American Journal of Sociology 75, l-40. Goodman, L. A. (1969b). “On the measurement of social mobility: An index of status persistence,” American Sociological Review 34, 831-850. Ilke, F. C. (1954). “Sociological relationship of traffic to population and distance,” Traffic Quarterly 8, 123-136. Jackson, J. S., and Gurin, G. (1987). National Survey of Black Americans, 197991980, Interuniversity Consortium for Political and Social Research, Ann Arbor, MI. Kluegel, J. R. (1980). “Denominational mobility: Current patterns and recent trends.” 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. Sherkat, D. E. (1990). “Some theoretical and methodological objections to Stephan-Deming adjustments to religious mobility tables.” Social Science Research 19, 239-249. Stephan, F. F. (1942). “An iterative method of adjusting frequency tables when expected marginal totals are known,” Annals of Mathematical Statistics 13, 166-178. Stewart, J. Q. (1942). “A measure of the influence of a population at a distance.” Sociometry 5, 63-71. Stouffer, S. A. (1940). “Intervening opportunities: A theory relating mobility to distance.” American Sociological Review 5, 845-867.
STEPHAN-DEMING
ADJUSTMENTS
215
Thomas, J. L. (1951). “The factor of religion in the selection of marriage mates,” American Sociological Review 16, 487-491. Whitt, H. P., Crockett, H. J., Jr., and Babchuk, N. (1988). “Religious switching: An alternative model,” Social Science Research 17, 206-218. Zipf, G. K. (1946). “The P,P*/D hypothesis: On the intercity movement of persons,” American Sociological Review 11, 677-686.
SCIENCE
RESEARCH
21, 204-215 (1992)
Some Theoretical and Methodological Reasons for Using Stephan-Deming Adjustments in Religious Mobility Tables . HUGH
P. WHITT University
AND
NICHOLAS
BABCHUK
of Nebraska-Lincoln
Sherkat’s (1990) critique of the use of Stephan-Deming adjustments to control for the size of social networks in studies of religious mobility is flawed by his misinterpretation of the concepts of “opportunity structure,” “destination marginals,” and “model,” as used by Whitt, Crockett, and Babchuk (1988). These concepts are clarified and the Stephan-Deming adjustments are shown to be a theoretically and methodologically sound approach for investigating religious switching. Substantive implications for data on black Americans are explored. 0 1992 Academic
Press, Inc.
Recently, Sherkat (1990) has published a thoughtful critique of the use of Stephan-Deming (S-D) adjustments (Deming, 1964; Deming and Stephan, 1940; Stephan, 1942) in religious mobility tables, arguing that our rationale for advocating this procedure (Whitt et al., 1988) is severely flawed. While there is merit in what Sherkat has to say about the desiderata for an adequate analytic technique, his negative appraisal of the S-D adjustments cannot be supported. This paper responds to Sherkat’s evaluation, outlining theoretical and methodological reasons for using adjustments of this type in studies of religious switching. Some background is necessary. All else being equal, rates of intergroup interaction are affected by the opportunity for random contact between group members. The opportunity for random contact, in turn, is a function of the sizes of the groups in question and their distribution in physical space (Stouffer, 1940; Stewart, 1942; Zipf, 1946; Ilke, 1954; Coleman, 1964). The importance of opportunity structures has been recognized in studies of criminal victimization (Boggs, 1960; Cohen and Felson. 1979), geographical mobility (Stouffer, 1940), traffic patterns (Ilke, 1954), and religious intermarriage (Thomas, 1951). Techniques for taking opportunity structures into account and elimiCorrespondence should be sent to Hugh P. Whitt, Department of Sociology, 718 Oldfather Hall, University of Nebraska-Lincoln, Lincoln, NE 68858-0324. 204 0049-089X/92 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.
STEPHAN-DEMING
ADJUSTMENTS
205
nating their confounding effects on more theoretically interesting processes have been developed by researchers investigating occupational mobility. During the 1960s and 197Os, Goodman’s (1964, 1965, 1969a,b) model of quasi-independence (QI) was widely employed for this purpose. This analytic technique has been appropriately and fruitfully exported to criminology and studies of intermarriage. It has also been appropriated by researchers (Newport, 1979; Kluegel, 1980) investigating religious switching. Goodman’s (1964, 1965, 1969a,b) model is, however, inadequate when applied to religious mobility. In proposing it be abandoned in religious switching studies, we argued that the problems we identified could be partially corrected by using S-D adjustments for a priori marginals before fitting QI models. Sherkat (1990) disagrees, contending that our approach suffers from both theoretical and methodological shortcomings. There is, however, a serious gap between our thinking in proposing the S-D adjustments and Sherkat’s interpretation of what we say. This gap is substantial. Sherkat (1990) asserts that our procedure “goes against most theories of religious conversion and commitment,” that the S-D adjustments “significantly bias adjusted cell frequencies,” and that we are so closely tied to the dated “denominational status” model of religious switching that we ignore more recent theoretical developments. Had we been trying to do what Sherkat thinks, we would agree that the approach we used was misguided. It seems to us, however, that Sherkat and we are simply talking past each other. Sherkat’s (1990) reading of our 1988 paper differs radically from what we intended. Some of the key terms in our discussion mean something different to Sherkat than they do to us. For example, he asserts that we deem “the structures of opportunity for religious mobility . . . to be completely unrestricted” (p. 240), using this as a springboard for criticizing us for ignoring the sizes of social networks in our analysis. “Opportunity structure,” as employed in our earlier paper and herein, refers to the random opportunity for interaction under such models as QI. The term incorporates the idea that “larger denominations will have access to a larger pool of nonaffiliated friends and family members who are at increased risk for recruitment” (Sherkat, 1990, p. 242). As we use the concept, “opportunity structure” has nothing to do with the unrestricted possibilities which indeed exist for religious mobility in all but the very few sects which restrict membership. It refers instead to the size distribution of religious denominations. The pseudo-disagreement between Sherkat (1990) and ourselves can also be traced to different understandings of the word “model.” The studies we critiqued for importing the model of QI from studies of occupational mobility (Newport, 1979; Kluegel, 1980) fell into the trap of
206
WHITI
AND BABCHUK
what Carter (1971) calls “inadvertent theory.” Newport (1979) and Kluegel (1980) employed QI to remove the contaminating effects of group size so that more sociologically interesting hypotheses could be explored. In so doing, they inadvertently removed the effects of success in proselytization. Our alternative model seeks to correct this defect in previous research. It is not a confirmatory model of the type assumed by Sherkat. As in the earlier studies we critiqued, we use the term model in the sense implied by Coleman’s (1964, pp. 469-470) discussion of the “method of residues.” According to him: [I]n considering a given complex social phenomenon, certain aspects of it are explainable by matters irrelevant to the substantive matters under investigation. If we examine what part of the behavior can be explained by these factors, then the remainder stands out to be explained by less trivial factors [I]n certain structural phenomena, where the data ate in the form of numbers of people, this separating out may be done quantitatively and rigorously The approach suggested here is to make some simple and reasonable null assumptions and then to use the deviations from the predictions consequent upon these assumptions as a measure of various matters, as, for example, the “social distance” between two groups.
Sherkat’s attaches a different meaning to the term “model.” To him, a model includes those explanatory variables whose impact the researcher wishes to assess. In this sense the model should provide as accurate a representation as possible of a complete theory of religious switching, including all theoretically relevant predictors of the tendency to change denominations. Models of this type are becoming increasingly common. They use a confirmatory logic which differs from Coleman’s approach, which does not require the specification of theoretically relevant independent variables. Models based on the method of residues include only control variables. Because Newport (1979) and Kluegel(l980) were working within the paradigm proposed by Coleman (1964), our alternative model employed this same paradigm. Sherkat (1990) seems to think that our intent was to test a confirmatory model. It was not. As often occurs in cross-paradigm debate, miscommunication has resulted ‘in misunderstanding. Coleman’s (1964) major example of the use of the method of residues is the “distance-interaction hypothesis” or “law of social gravity” developed by Stouffer (1940), Stewart (1942), Zipf (1946)) and Ilke (1954), which states that, all else being equal, “the rate of interaction between two areas with IZ~ and n2 persons who are d,, distance apart will be proportional to nln2/d12 .” Coleman (1964, pp. 471-473) suggests that this factor represents “the expected rate of interaction which might occur if no sociological effects were operative.” He argues for its use as
STEPHAN-DEMING
ADJUSTMENTS
207
“a base line, or standardization, which can cancel out differences in interaction due to these sociologically trivial factors, and illuminate the differences due to more interesting matters . . . Without such a base line, or standardizing factor, differences which are due simply to the size of the unit or to the distance between units will be confounded with differences due to less trivial matters, and analysis of the source of the latter differences will be impossible.”
Like Newport (1979) and Kluegel (1980), we used the method of residues to construct a baseline model in Coleman’s (1964) sense. The motivation was to eliminate the confounding effects of differences in initial group size rather than to specify in advance what sociologically interesting processes might be operating.’ Under conditions of random interaction, larger groups have stronger “gravitational pulls” than smaller ones. Like Newport and Kluegel, we used the method of residues to construct a baseline model that eliminates the confounding effects of differences in group size. We did not specify ahead of time what sociologically interesting processes might be operating. As a consequence of our use of the method of residues, our approach is not wedded to any specific substantive theory of religious mobility, Sherkat’s argument to the contrary. Its only theoretical assumption is that the probability of switching between two religious groups is proportional to the product of their sizes. Sherkat (1990, p. 241) suggests that our failure to take into account two theoretical factors invalidates the use of the S-D adjustments: (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.
Although theoretically unassailable, Sherkat’s comments miss the point. If QI is used without the S-D adjustments, the theoretical processes he identifies are inadvertently controlled out of the analysis! This point requires explanation. As noted in our original paper (Whitt et al., 1988), religious mobility is logically different from occupational mobility. In most cases, jobholders do not create the positions they occupy. Their positions are established by their employers, who determine how many jobs will be available in each sector of the economy. The distribution of jobs changes over time with the expansion and contraction of particular sectors of the economy; ’ Although it would be desirable to eliminate the effects of distance as well, most surveys provide insufficient evidence to do so.
208
WHITT
AND BABCHUK
at any point in time this occupational distribution exists prior to any action by job applicants. On the other hand, religious mobility is not constrained by limitations on the number of persons switching denominations. Indeed, new memberships (the analogue of jobs) can be created at will. Growth cannot cause switching; it is a dependent variable caused by switching and, in the natural setting but not in sample surveys, by natural increase. In this regard, the proper theoretical analogy is geographical mobility rather than the dynamics of the job market. Just as population change in cities, states, and nations is completely determined by rates of in-migration, out-migration, and natural increase or decrease, changes over time in the memberships of denominations are a product of rates of conversion and defection and by gains and losses associated with birth and death. Because of this difference between occupational and religious mobility, the distinction between the distribution of religious origins and the distribution of destinations is crucially important. If, as Sherkat (1990) correctly argues, religious denominations invest resources and mobilize members to stimulate growth, it is illogical to suggest that the resources and members so invested and mobilized are those they command after they have experienced growth. As we pointed out (Whitt et al., 1988), the relevant networks are those to which the potential convert is exposed prior to switching. In survey data, there is admittedly no perfect way of measuring the sizes of denominations at the time of switching. Each person changes denominations at a different point in time. In order for the distribution of origins to be a fully accurate representation of the opportunity structure for religious switching, the S-D adjustments would need to be applied to a sample of persons all of the same age who switched denominations at the same time. Nonetheless, the distribution of religious origins comes closer to tapping the sizes of denominations at the time mobility takes place than does the distribution of destinations. Thus, the S-D adjustments with the distribution of origins specified a priori are employed to estimate the size distributions Nr and N2 in Coleman’s (1964) baseline model of the law of social gravity. Indeed, the S-D procedure assesses as a residual the effects of resource investment and mobilization of participants upon flows between denominations, while the traditional approach of Newport (1979) and Kluegel (1980) does not. When the distribution of destination denominations was rejected in favor of the distribution of religious origins as a basis for modeling the opportunity for religious switching, we intended to liken these terms to distributions of denominations at two points in time. Sherkat (1990, p. 242) apparently does not grasp this meaning when he says that “[i]t is clear that we should expect the size of destination marginals to affect the expected cell frequencies in religious switching tables.” If destination
STEPHAN-DEMING
ADJUSTMENTS
209
refers to the size distribution of all denominations other than one’s own, then we completely agree. However, in our usage, “destination marginals” refers instead to the distribution of denominations at the time of the survey, while origin marginals are those based on the distribution of denominations prior to switching. We should perhaps have called them the distributions at Time 1 and Time 2. The marginals at Time 1 establish the opportunity structure for switching, which in turn generates the marginal distribution at Time 2. As we showed by reanalyzing Newport’s (1979) data, the two methods yield different results. Sherkat (1990) claims this is because the S-D adjustments introduce bias in adjusted cell frequencies which is proportional to the denomination’s rate of growth. This bias, he maintains, “is substantial, systematic, and unacceptable.” True, the differences in frequencies are substantial and systematic, but they are far from unacceptable and do not constitute bias. The altered frequencies represent a necessary correction for differential denominational growth. As Sherkat (1990) points out and amply demonstrates, the adjusted cell frequencies in the main diagonal differ systematically from the raw frequencies on the basis of their rate of growth or decline. This is not accidental. Those denominations which have grown from the origin to the destination distribution have their destination marginals reduced by the S-D adjustments to reflect the smaller size of their networks when switching took place. Conversely, the greater influence in the past of declining denominations is captured by increasing their adjusted marginal frequencies. These modifications in the marginal distribution of destinations affect all cell frequencies, including those for cells in the main diagonal. As a consequence, the index of status persistence, an indicator of retention rates, is systematically affected by denominational growth or decline. As Sherkat shows, the indices of status persistence for the adjusted distributions are systematically different from those calculated over the raw data. While Sherkat (1990) is correct in assessing the impact of the S-D adjustments on marginal and cell frequencies, his critique fails to recognize the crucial distinction between description and causal inference. “Clearly,” he says, “the biases described above are artificially creating patterns of denominational switching which do not exist.” When compared with the raw data, the hypothetical and essentially fictional distributions of cell frequencies resulting from the S-D adjustments facilitate inferences about the operation of causal processes. They are not intended to describe real rates of denominational growth and decline, which are best assessed from the original raw frequency distribution. The hypothetical distributions generated by the S-D procedure form the basis for the application of the method of residues. In applications
210
WHIT-I- AND BABCHUK
of Coleman’s method, “observed” frequencies incorporating theoretically based adjustments are compared with equally fictional expected frequencies generated by relatively simple and straightforward causal assumptions. The differences between observed and expected frequencies are the residues. One family of models of this type is chi-square analysis. In these models, including QI and the S-D adjustments, artificially created patterns of denominational switching which do not exist are generated as part of the analysis. No one claims that these hypothetical distributions correspond to reality-they do not. Instead, they represent the cell distributions which would exist if the theoretical processes specified by the researcher were operating. Traditional chi-square generates the expected frequencies by assuming (sometimes contrary to the fact) that the distribution of cases to the cells is random given the observed marginal distributions. The method of quasi-independence adds the assumption that this random process applies only to the off-diagonal cells. S-D adjustments add to the QI model the assumption that the marginal frequencies are known a priori. These marginal frequencies are estimated from the distribution of religions of origin in the sample. Contrary to Sherkat (1990), there is nothing inherent in the method to prevent the use of uny set of a priori marginals as an assumption. The distribution of origins is used because it mirrors the theoretical processes both we and Sherkat are concerned with modeling, i.e., the effects of the size of social network on religious switching. In his analysis of supposed bias due to S-D adjustments, Sherkat (1990) confines his attention to the adjusted frequencies, failing to note that the procedure also affects the chi-square expected frequencies and the mobility ratios derived from the adjusted data. As measures of residues, the mobility ratios are the only measures having theoretical importance; neither the adjusted frequencies (which play the role of observed frequencies in calculating the QI model) nor the expected frequencies are in themselves of any intrinsic interest. Their sole purpose is to serve as intermediate steps in the calculation of mobility ratios. Tables 1 through 6 provide material to illustrate the above discussion. Table 1 (Sherkat’s Table 1) presents a crosstabulation of current religion by religion of origin drawn from the 1979-1980 National Survey of Black Americans (Jackson and Gurin, 1987). Tables 2 and 3 show, respectively, the expected frequencies and mobility ratios based on this tabulation under the QI model. Tables 4-6 give the comparable data under the S-D adjustments. As Sherkat (1990) suggests, the cell frequencies in Table 3 indeed differ systematically from the raw data in Table 1. The differences between cell values in the two tables correlate modestly and negatively with the growth or decline of one’s religion of origin (r = - .488), but are uncorrelated with gains and losses of the denomination chosen (r =
STEPHAN-DEMING
211
ADJUSTMENTS
TABLE 1 Current Religion by Religion of Origin, Survey of Black Americans, 1979-1980 Religion of origin Current religion
Lib
Meth
Bapt
Cons
Other
Lib Meth Bapt Cons Other Cath None Total
41 0 4 2 2 5 9 63
8 212 58 20 5 7 28 338
14 23 980 103 42 20 95 1277
1 5 15 120 4 0 17 162
3 9 6 19 1 8 46
0
Cath
None
Total
1 2 7 5 8 97 14 134
1 2 10 6 5 2 61 87
66 247 1083 262 85 132 232 2107
.OOO). This pattern is expected given that the S-D adjustments maintain the initial patterns in the data subject only to constraints based on a priori marginals. The differences between Tables 2 and 5 and those between Tables 3 and 6 are less systematically based upon denominational growth and decline. The expected values in Tables 2 and 5 are completely dependent upon the marginal frequencies in these tables, which are identical within limits of rounding to those in their respective parent tables. These two tables serve only as a basis for calculating the mobility ratios in Tables 3 and 6, which show the ratio of observed to expected frequencies. The mobility ratios in Tables 3 and 6 represent the extent to which particular cells are over- or underrepresented given the assumptions of their respective underlying models. Changes in the sizes of both origin and destination denominations are positively associated with changes between tables in mobility ratios; the mobility ratios in Table 6 particularly exceed those in Table 3 when both denominations are growing. Thus, Table 6 yields high mobility ratios for growing denominations, while Table TABLE 2 Expected Frequencies under Quasi-Independence
Lib Meth Bapt Cons Other Cath None Total
Lib
Meth
Bapt
Cons
Other
Cath
None
Total
41.00 1.28 7.53 4.60 2.07 1.11 5.40 62.99
4.56 212.00 44.13 26.98 12.15 6.52 31.66 338.00
15.14 24.97 980.00 89.66 40.36 21.68 105.19 1277.00
1.80 2.96 17.40 120.00 4.79 2.57 12.48 162.00
1.01 1.67 9.82 6.00 19.00 1.45 7.04 45.99
1.33 2.19 12.86 7.86 3.54 97.00 9.22 134.00
1.16 1.92 11.27 6.89 3.10 1.67 61.00 87.01
66.00 246.99 1083.01 261.99 85.01 132.00 231.99 2106.99
212
WHIT-T AND BABCHUK TABLE 3 Mobility Ratios under Quasi-Independence
Lib Meth Bapt Cons Other Cath None
Lib
Meth
Bapt
Cons
Other
Cath
None
-
1.76
0.53 0.43 0.97 4.49 1.67
1.31 0.74 0.41 1.07 0.88
0.92 0.92 1.15 1.04 0.92 0.90
0.56 1.69 0.86
1.79 0.92 1.00
0.75 0.91 0.54 0.64 2.26
0.86 1.04 0.89 0.87 1.61 1.20 -
0.84 0.69 1.14
1.36
1.52
3, based on QI, controls out the effect of growth over and above that expected given the denomination’s initial size. A number of cells radically differ between the two tables. Differences in magnitude of .20 or greater occur in 15 of the 38 non-null cells. Most such changes indicate only variations in the strength of flows without any change in substantive interpretation. For example, either traditional QI techniques or the S-D approach would conclude that black American Baptists are relatively unlikely to become apostates. The S-D analysis, however, more strongly underscores this point through a very low mobility ratio (.69 vs .90 under QI). Both techniques point to a strong flow to no religion from conservative denominations, but this pattern is much clearer (a mobility ratio of 2.44 vs 1.36) under S-D assumptions. More importantly, choice of method influences substantive conclusions about which cells indicate above-chance frequencies of switching. In Tables 3 and 6, mobility ratios above 1.00 indicate that the flow between the two denominations represented by the cell is greater than expected given the tables’ respective assumptions. Seven cells shift from above to below 1.00 or vice versa between Table 3 and Table 6. In summary, sometimes choice between methods makes a large difference in mobility TABLE 4 Observed Frequencies under Stephan-Deming
Lib Meth Bapt Cons Other Cath None Total
Adjustments
Lib
Meth
Bapt
Cons
Other
Cath
None
Total
44.2 0.0 5.9 1.3 1.3 5.6 4.1 62.4
4.2 277.9 54.0 2.3 0.6 4.0 -2.5 340.5
10.8 35.7 1148.7 36.5 15.3 16.3 14.5 1277.8
1.3 10.5 25.8 108.2 3.7 0.0 11.9 161.4
0.0 5.9 14.4 4.7 15.0 1.2 4.6 45.8
1.0 3.6 10.0 3.1 5.0 103.9 5.7 132.3
1.4 4.4 18.2 6.0 5.1 2.9 48.6 86.6
62.9 338.0 1277.0 162.1 46.0 133.9 86.9 2106.8
STEPHAN-DEMING
TABLE 5 Expected Frequencies under Stephan-Deming
Lib Meth Bapt Cons Other Cath None Total
213
ADJUSTMENTS
Adjustments
Lib
Meth
Bapt
Cons
Other
Cath
None
Total
2.57 9.47 2.24 1.22 1.17 1.53 18.20
2.13 36.27 8.59 4.66 4.49 5.86 62.60
9.75 35.13 30.66 16.63 16.01 20.92 129.10
2.28 8.20 30.22
1.24 4.47 16.48 3.90 2.04 2.66 30.79
1.14 4.11 15.16 3.59 1.95
1.56 5.62 20.70 4.90 2.66 2.56 38.00
18.70 60.10 128.30 53.88 31.00 30.01 38.30 360.29
3.88 3.74 4.88 53.20
2.45 28.40
which may or may not alter substantive conclusions. At other times, even small changes can modify interpretation of the data. Sherkat (1990) maintains that religious switching can be more fruitfully analyzed through individual-level analytic techniques such as event history analysis. There is merit in this position, but typical event history models, limited as they are to individual-level variables, fail to control for structural factors such as the sizes of social networks. Care should be taken to incorporate structural as well as individual attributes. This may be done by including either mobility ratios drawn from a S-D analysis or products of initial group sizes as characteristics of individuals in a proportional hazard model. With this caveat, event history models should probably be used when the data permit, i.e., when the precise timing of switching is available for each respondent. Although some data of this sort are available, the exact timing of religious mobility is usually unknown. In such situations the S-D adjustments allow modeling at least an approximation of the opportunity structure. Even with samples of mixed ages, the distribution of origins is theoretically preferable to the distribution of destinations because it provides a partial control for size rather than inadvertently controlling out the effects of denominational growth. ratios,
TABLE 6 Mobility Ratios under Stephan-Deming
Lib Meth Bapt Cons Other Cath None
Lib
Meth
Bapt
Cons
-
1.54
1.11 1.02
0.57 1.28 0.85
0.62 0.58 1.06 4.78 2.67
1.49 0.27 0.13 0.89 -
1.19 0.91 1.02 0.69
Adjustments Other
Cath
None
1.31 0.87 1.20
0.88 0.88 0.66 0.86 2.57
0.90 0.78 0.88 1.22 1.91 1.13 -
0.95 2.44
0.59 1.73
2.33
214
WHI’IT
AND BABCHUK
The foregoing underscores the theoretical reasoning underlying the use of S-D adjustments. Rather than “following in the footsteps of the denominational status paradigm, failing to recognize the theoretical insights of other sociological theories of conversion and participation in voluntary organizations,” the model is tied to no theory other than the law of social gravity. It permits the assessment of any structural theory the researcher chooses through the examination of residues as embodied in mobility ratios. Rather than a “flawed model” “ without theoretical justification” which makes “no contribution to the scientific understanding of social processes,” (Sherkat, 1990, p. 248), the Stephan-Deming adjustments are well-suited, given imperfect data, to map the theoretical notion that social networks make a difference. REFERENCES Boggs, S. L. (1960). “Urban crime patterns,” American Sociological Review 30, 899-908. Carter, L. F. (1971). “Inadvertent sociological theory,” Social Forces 50, 12-25. Cohen, L. E., and Felson, M. (1979). “Social change and crime rate trends: A routine activity approach,” American Sociological Review 46, 588-608. Coleman, J. S. (1964). Introduction to Mathematical Sociology, Free Press of Glencoe, London. 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 tables are known.” Annals of Mathematical Statistics 11, 427-444. Goodman, L. A. (1964). “A short computer program for the analysis of transaction flows,” Behavioral Science 9, 176-178. Goodman, L. A. (1965). “On the statistical analysis of mobility tables,” American Journal of Sociology 70, 564-585. Goodman, L. A. (1969a). “How to ransack social mobility tables and other kinds of crossclassification tables,” American Journal of Sociology 75, l-40. Goodman, L. A. (1969b). “On the measurement of social mobility: An index of status persistence,” American Sociological Review 34, 831-850. Ilke, F. C. (1954). “Sociological relationship of traffic to population and distance,” Traffic Quarterly 8, 123-136. Jackson, J. S., and Gurin, G. (1987). National Survey of Black Americans, 197991980, Interuniversity Consortium for Political and Social Research, Ann Arbor, MI. Kluegel, J. R. (1980). “Denominational mobility: Current patterns and recent trends.” 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. Sherkat, D. E. (1990). “Some theoretical and methodological objections to Stephan-Deming adjustments to religious mobility tables.” Social Science Research 19, 239-249. Stephan, F. F. (1942). “An iterative method of adjusting frequency tables when expected marginal totals are known,” Annals of Mathematical Statistics 13, 166-178. Stewart, J. Q. (1942). “A measure of the influence of a population at a distance.” Sociometry 5, 63-71. Stouffer, S. A. (1940). “Intervening opportunities: A theory relating mobility to distance.” American Sociological Review 5, 845-867.
STEPHAN-DEMING
ADJUSTMENTS
215
Thomas, J. L. (1951). “The factor of religion in the selection of marriage mates,” American Sociological Review 16, 487-491. Whitt, H. P., Crockett, H. J., Jr., and Babchuk, N. (1988). “Religious switching: An alternative model,” Social Science Research 17, 206-218. Zipf, G. K. (1946). “The P,P*/D hypothesis: On the intercity movement of persons,” American Sociological Review 11, 677-686.