JoM-EMIT LogoButts, C. T.and Hilgeman, C.  (2003).  Inferring Potential Memetic Structure from Cross-Sectional Data: An Application to American Religious Beliefs.
Journal of Memetics - Evolutionary Models of Information Transmission, 7.
http://cfpm.org/jom-emit/2003/vol7/butts_ct&hilgeman_c.html

Inferring Potential Memetic Structure from Cross-Sectional Data: An Application to American Religious Beliefs

Carter T. Butts* and Christin Hilgeman+
*Department of Sociology and Institute for Mathematical Behavioral Sciences
University of California, Irvine [note 1]
buttsc@uci.edu
+Department of Sociology;University of California, Irvine

Abstract:

The identification and measurement of memes poses a fundamental challenge for research in memetics. Recent methodological developments regarding inference for latent algebraic structure provide a useful tool for inferring potential memetic structure from cross-sectional data. Here, we perform such an analysis on selected items from the 1988 and 1998 General Social Survey religion modules. American religious belief over the period is shown to be stable, with a complex structure which is reducible neither to a set of distinct scales nor to a model of itemwise independence. A decomposition of observed behavioral characters into latent meme-like constructs reveals underlying connections between otherwise disparate items, and demonstrates the presence of several interlocking scale-like structures. Interpretations of the resultant latent structures are provided, and some possible implications for the memetic theory of religion are discussed.

Keywords: meme theory, microbelief analysis, latent class analysis, religious beliefs

1 Introduction: The Problem of Memetic Inference

Meme theory has a number of potential strengths: it provides an intuitive way of connecting biology, culture, and the environment; it offers explanations for a wide range of social phenomena; and it is amenable to formal modeling (see, e.g., 13,11). As compelling as these factors may be, meme theory has a grave difficulty in practice: much of its long-term utility hinges upon the identification and measurement of memes (or at least small memeplexes). This - the problem of memetic inference - is a significant challenge. Unlike genes, which can be studied directly, memes are not available to inspection. (Indeed, there is considerable controversy over the precise definition of the meme itself (20,33,37).) Typically, empirical memetic accounts rely on an intuitive specification of memes, often working on the assumption that the underlying memes are synonymous with their behavioral characters (or artifactual traces thereof); see, for instance, (8,24,16). This is problematic, however, since many cultural phenomena of substantive interest involve a complex of interconnected beliefs which exhibit mutual dependence (32,26).

How, then, can we hope to proceed? As it happens, the history of genetics itself suggests an answer: well before the discovery of DNA, the genetic system was studied by indirect inference from associations among characters (31). Within sociology, indirect inference has also been used by Martin and Wiley (28) to infer the presence of "microbeliefs" from survey response data via the application of a latent lattice [note 2] model developed by Haertel and Wiley (18). (Such work is part of a long tradition of latent structure models in the social sciences, including classic work by Lazarsfeld and Henry (23); Goodman (14,15); Guttman (17), among many others.) As we shall illustrate, the latent structure model of Martin and Wiley can be employed to uncover "meme-like" structures in cross-sectional data sets. These structures consist of inferred latent elements ("candidate memes" or "quasi-memes") whose conjunction (i.e., combination by AND logic) is associated with the manifestation of behavioral characters. Thus, any given character - for instance, a professed belief in the Judeo-Christian god - is presumed to manifest within a given individual if (and only if) all of its requisite quasi-memes are present. Just as characters may depend upon multiple quasi-memes, so too may a given quasi-meme contribute to multiple characters; thus, potentially complex patterns of dependence may be inferred within the set of characters, the set of quasi-memes, and the two taken jointly.

The approach pursued here is explicitly cross-sectional, and thus we emphasize that the quasi-memes uncovered by this method should be considered only as candidates for subsequent examination by intertemporal methods. Similarly, one can posit complex expression mechanisms (e.g., incorporating XOR or NOT logic) which cannot be inferred by this technique. Where potential memetic structures can be identified using the latent lattice model, however, the model may subsequently be employed to produce estimates of the individual incidence of quasi-memes (i.e., which quasi-memes are held by which persons). These estimates may then be used longitudinally and/or in experimental settings to test for transferability of the potential memetic elements, and thereby to confirm or deny a hypothesis of genuine memetic structure. (Some suggestions for such follow-on studies are discussed in Section 3 below.) Because the initial "screening" of candidate memes under this method can be performed on large data sets, substantially greater statistical power can be brought to bear on the problem than would be possible in more traditional settings. Furthermore, the use of cross-sectional data greatly facilitates the study of representative samples from whole populations, thereby improving the prospects for successful examination of large-scale memetic ecologies. As such ecologies are of significant interest to theorists in the field (e.g., 1,13,7), it is hoped that this capability will facilitate more empirically informed development in this area.

1.1 Latent Lattices for Memetic Inference

We begin our discussion of the latent lattice model by assuming that we have identified a set of behavioral characters which span some domain of interest, and that we have obtained measures on those characters for a sample of individuals drawn at random from a given population. [note 3] (For our present purposes, we shall limit ourselves exclusively to binary characters.) Consider, for instance, a set of characters associated with responses to a standard survey instrument (e.g., agree/disagree items relating to sexual behavior). Under Martin and Wiley's model, each individual's pattern of responses to such a series of (dichotomous) items is modeled as arising from a conjunction of latent dichotomous subelements. Specifically, if $ \mathbf{z} \in \{0,1\}^n$ is a vector of "microbeliefs," $ \mathbf{x} \in \{0,1\}^m$ is a vector of responses (or "macrobeliefs"), and $ \mathbf{D}$ is an $ n \times m$ dichotomous dependency matrix, then it is assumed that

$\displaystyle \mathbf{x} = \left(\mathbf{z}^c \odot \mathbf{D}\right)^c,$ (1)

where $ \odot$ represents the Boolean inner product and $ \cdot^c$ is the Boolean complement operator. Thus, a given individual is predicted to hold macrobelief $ \mathbf{x}_i$ if and only if they hold all microbeliefs $ \mathbf{z}_j$ such that $ \mathbf{D}_{ji}=1$. Generally, we also assume that the columns of $ \mathbf{D}$ are 1-covered, and that $ \mathbf{D}$ has no identical rows. (Obviously, $ \mathbf{D}$ is defined only up to a row permutation.) Under such a model, it can be shown that the set of observable macrobelief states forms a lattice (28); more importantly, given a lattice of observed macrobelief states, the inversion of Haertel and Wiley (18) can be employed to recover the $ \mathbf{D}$ matrix (and latent $ \mathbf{z}$ vectors). Let $ \mathbf{X}$ be a matrix whose rows contain all observable macrobelief states for some $ \mathbf{D}$, and define a row $ \mathbf{X}_{i\cdot}$ to be a meet-irreducible element (MIRE) of $ \mathbf{X}$ iff $ \mathbf{X}_{i\cdot} = \mathbf{X}_{j\cdot} \wedge \mathbf{X}_{k\cdot}$ implies $ \mathbf{X}_{i\cdot} = \mathbf{X}_{j\cdot}$ or $ \mathbf{X}_{i\cdot} = \mathbf{X}_{k\cdot}$ for all $ \mathbf{X}_{j\cdot},\mathbf{X}_{k\cdot} \in \mathbf{X}$ (where $ \wedge$ is the intersection operator [note 4]). If $ \mathbf{X}^*$ is a matrix whose rows contain the MIREs of $ \mathbf{X}$, then

$\displaystyle \mathbf{D} = \left(\mathbf{X}^*\right)^c.$ (2)

Therefore, we can readily move from observed macrobelief states to latent microbelief states, and vice versa. Of course, given real data, we do not expect things to work quite so neatly. In addition to pure measurement error, it is to be expected that the lattice model itself is only an imperfect representation of the latent structure of individual beliefs. For this reason, vectors of actual responses ($ \mathbf{y}$) are treated by Wiley and Martin as arising from the true latent state ($ \mathbf{X}$) under an error model; as a practical matter, this is treated as a problem of confirmatory latent class analysis (29).

The analogy between Martin and Wiley's microbelief analysis and meme theory is striking. Martin and Wiley's "macrobeliefs" may be likened to manifest behavioral characters, with the associated "microbeliefs" presumably being the memes themselves. The $ \mathbf{D}$ matrix (together with its associated algebraic operations) provides the expression mechanism for the underlying memes, converting memotype into behavioral phenotype. While this phenotype may be thought of as reflecting manifest beliefs, it need not be: specific behaviors, cultural practices, or the like are equally admissible. Likewise, data for the analysis can be derived from human subject experiments, direct observation, or inspection of artifacts as well as survey instruments. As mentioned earlier, this method has some significant limitations. It can only model expression by conjunctive interaction, for instance, and cannot therefore detect memes whose effects are inhibitory. Additionally, because it does not provide a stochastic model for the underlying distribution of memes among individuals, the procedure may not be robust to samples whose underlying distribution of memes is extremely biased. Despite these limitations, however, the method employed here provides an inferential framework which can discern a reasonably broad class of latent structures in the presence of limited measurement error. In applying it to the problem of memetic inference, we proceed in the spirit of Hull's argument that an effective operationalization of the notion of "meme" can "emerge only as one sets about doing memetics" (20, page 48). The formalism employed here cannot capture all conceivable memetic structures, but it may still serve as an empirically useful starting point.

Given the potential of this new tool to uncover memetic structure from empirical data, we here perform a microbelief analysis of data on American religious practice and belief drawn from the General Social Survey (GSS) for years 1988 and 1998 (6,5). While we cannot here demonstrate transmissibility - a critical requirement of the theory - we can nevertheless identify structures using the Martin and Wiley approach which could serve as "candidate memes." These structures are inferred from associations between manifest characters, reflecting a hypothetical substrate which is consistent with the observed data. Having obtained a set of hypothetical memes (and an associated expression mechanism), one may then examine subjects longitudinally to determine whether or not these elements are indeed the basic units of belief transmission. As recent work by Markovsky and Thye (25) has demonstrated, discrete beliefs regarding supernatural phenomena can be transferred under laboratory conditions. [note 5] Thus, it is in principle feasible to experimentally test the hypothesized meme structure, having initially reduced the number of possible alternatives by cross-sectional study of a large, nationally representative sample.

2 Data and Analysis

We begin our analysis with a discussion of the data set and the coding scheme employed. We then compare the 1988 and 1998 samples, and analyze general properties of the response distribution. Following this we use a confirmatory latent class model to infer a lattice structure from the religion data, and, finally, we perform a detailed structural analysis of the inferred lattice (including an examination of the potential latent memetic structure).

2.1 Data

The data for this analysis come from the religion modules of the 1988 and 1998 General Social Survey (6,5). With the exception of questions regarding friends, twelve items were asked of respondents at both time points; these comprise the first twelve rows of Table 1. Of these items, one is an "empirical" belief (GOD), four are normative judgments (GOCHURCH, BELIEVE, FOLLOW, and GOOWNWAY), five are interpretations of personal experiences (REBORN, DOUBTS1, DOUBTS2, FAITH1, and FAITH2), and two are behavioral self-reports (SAVESOUL and READWORD). Thus, the questions serve as a reasonably diverse collection of behavioral characters. Rather less diverse is the span of topics considered: three of the items deal explicitly with Christianity, seven are more broadly Judeo-Christian, and eight are at least implicitly monotheistic in nature. Thus, this sample does not allow us to study issues relating to non-monotheist worldviews (including atheism, pantheism, and polytheism), or to identify fine-grained beliefs regarding non-Judeo-Christian monotheisms (e.g., Islam). Neither does this sample focus on specific issues of denominational significance (e.g., biblical inerrancy, creationism, variations in eschatology). With this in mind, it may be most useful to consider these question items to represent a moderately general sample of beliefs and behaviors relating primarily to modern American Judeo-Christian religious practice. Latent structures identified at this level may be subsequently elaborated via more detailed questioning.

For each of the twelve initial questions, a dichotomous coding was constructed which differentiated positive assertions (coded 1) from a lack of the same (coded 0). (Where "don't know" responses could be identified with the lack of an assertion, these were coded 0. Otherwise, such responses were treated as missing data.) A brief description of the coded items is provided in Table 1.

As noted above, the set of questions asked of respondents at both time points involved a series of items regarding the religious beliefs and practices of friends nominated by the respondent. Although the data does not permit the type of diffusion analysis which would be ideal for meme-theoretic purposes, it nevertheless provides us with additional information regarding the respondent. With this in mind, we constructed a thirteenth item, FRNDCON, which was coded as 1 if the respondent indicated having at least one friend who was a member of his or her congregation, and 0 otherwise. (Respondents not belonging to a congregation were also coded 0.)

For the 1988 and 1998 samples, 1481 and 1445 respondents (respectively) were asked the twelve questions (plus friendship) employed here. Removing all cases containing missing data reduced the respective totals to 1384 and 1150. With thirteen dichotomous variables, the number of potential response patterns was 8192  ($ 2^{13}$), of which 718, 646, and 1056 were observed for the 1988, 1998, and combined samples (respectively). Our analysis, then, centers on the frequencies of these behavioral phenotypes, and on their associated characters.

Table 1: Description of coded GSS items
Item Description
GOD Believes in a personal god, at least some of the time
GOCHURCH Believes that regular religious service attendance is important to being a good Christian or Jew
BELIEVE Believes that believing in god without doubt is important to being a good Christian or Jew
FOLLOW Believes that faithfully following church/synagogue teachings is important to being a good Christian or Jew
GOOWNWAY Believes that following one's conscience (even against church/synagogue) is important to being a good Christian or Jew
REBORN Believes self to have had a born again experience
SAVESOUL Reports attempting to convert others to Christianity
READWORD Reports reading the Bible at home within the last year
DOUBTS1 Reports evil in the world has never caused religious doubts
DOUBTS2 Reports personal suffering has never caused religious doubts
FAITH1 Reports religious faith strengthened by death in family on some occasion
FAITH2 Reports religious faith strengthened by birth of a child on some occasion
FRNDCON Reports belonging to and having at least one friend in own congregation

2.2 Distributional Comparison

The first question which presents itself regarding behavioral phenotypes for American religious belief over the 1988-1998 interval is that of structural stability: is there evidence of substantial change in the distribution of phenotypes, or is the frequency distribution constant over the period (for these characters, at least)? With this in mind, we begin our analysis by comparing the distribution of observed phenotypes in the 1988 sample with that observed in 1998.

Taking the 8192 possible response patterns, we begin by accumulating the frequency distribution of observed response states for each sample. A simple  $ \chi^2$ test of homogeneity indicated no significant difference between the distributions (1070.039 on 8191 degrees of freedom, $ p \approx 1$). Due to the sparseness of the sample, additional tests were performed as well. A Monte Carlo $ \chi^2$ conditioning on row/column marginals with 5000 replicate draws [note 6] did not indicate significant differences between the samples (p>0.998), nor did similar tests of the L1 (the absolute difference - sometimes called "manhattan" metric in reference to city block distances), L2 (Euclidean distance), maximum norm, or rank correlation (all $ p \ge 0.9756$). Taken together, these tests suggest that any observed deviations between the 1988 and 1998 phenotype distributions are typical of what would be expected by chance variation, and that this conclusion is quite robust to choice of similarity measure.

Another indication of the strong correspondence between distributions is provided by the similarity of rank structure among the most commonly observed patterns. Specifically, let us take $ F_{88}(S)$ to be the fraction of cases from the 1988 sample accounted for by pattern set $ S$ (with $ F_{98}$ defined analogously). Next, we define $ r_{88}(i)$ and $ r_{98}(i)$ to be the set of all patterns for the 1988 and 1998 samples (respectively) having (ascending) sample rank less than or equal to $ i$. Given this, the "coverage ratios" $ \tfrac{F_{88}\bigl(r_{88}\left(i\right)\bigr)}{F_{98}\bigl(r_{88}\left(i\right)\bigr)}$ and $ \tfrac{F_{88}\bigl(r_{98}\left(i\right)\bigr)}{F_{98}\bigl(r_{98}\left(i\right)\bigr)}$ can be used to assess the detailed agreement of the two distributions regarding the most frequent cases. Where the two distributions are precisely equal, both coverage ratios are equal to 1; as the distributions deviate from equality, the ratios will shift correspondingly. Plots of coverage ratios for the GSS religion data are shown in Figure 1 for ranks one through one hundred. For reference, 95% and 5% quantiles have also been shown for the upper and lower ratios (respectively), based on 5000 Monte Carlo replications of the data set fixing marginal totals. As can be seen, the coverage ratios remain reasonably close to 1 throughout the interval, indicating 1) that the same patterns are common in both samples, and 2) that these patterns account for a fairly similar proportion of their respective samples. The observed coverage ratios are also well within the natural range of variability which might be anticipated by the margins alone, further suggesting that the observed deviation is not extreme. This further reinforces the result of the more general test procedures, indicating that belief patterns for the two samples do not differ greatly overall, nor in regions of high probability.

Figure 1: Coverage ratios for 1988 and 1998 phenotype distributions, by threshold rank (PDF Version)
\begin{figure}\begin{center} \resizebox{4in}{6in}{\includegraphics{cov.ratio.ps}}\end{center}\end{figure}

The finding that - for these characters - American religious belief was fairly stable over the period from 1988-1998 would seem to suggest that the present questions tap into relatively "deep" belief structures whose distributional properties change (at best) slowly with time. In order to uncover these general latent structures, then, we combine the two samples for the analyses which follow.

2.3 Distributional Properties

Combining the religious belief data for the 1988 and 1998 samples, we first investigate the global population distribution across the set of possible behavioral phenotypes. To what extent, for instance, is the distribution heavily concentrated on a small number of realized states, versus being widely dispersed? How does the state frequency decay as one moves from the most frequent to the least frequent states? While they do not deal with the properties of particular characters, the answers to such questions nevertheless provide information regarding what sort of process may have given rise to the observations at hand.

In answer to the first question, we note that the estimated entropy for the phenotype distribution [note 7] is approximately 9.28 bits, substantially below the 13 bits associated with the uniform density. Put another way, we can immediately discern that the information content of the observed phenotype distribution is slightly in excess of the maximum for 9 binary characters, signifying just under a 4 character "redundancy" (or an entropy reduction of 29%). This suggests a fairly high degree of concentration in the distribution, which we display visually in Figure 2. Figure 2 provides a log-log plot of the frequency of each observed phenotype, sorted in descending order of probability density; a least-squares line has been provided for reference. As the linear fit suggests, the observed frequency distribution decays approximately as a power law over much of its range ($ y=0.044 x^{-0.709}$, $ R^2=0.94$). Although some minor deviations are present (partially due to granularity effects), the relationship is clearly not exponential in form ($ R^2=0.69$). Thus, the overall frequency distribution displays an extremely high degree of concentration on the most common values, combined with a "long tail" of numerous patterns observed in only a few instances. This should be contrasted with an equivalent exponential model, in which less concentration would be observed on the most common patterns, coupled with a lighter tail.

Figure 2: Frequency distribution of phenotypes (combined GSS data) and 95% Monte Carlo confidence intervals, independence model (PDF Version)
\begin{figure}\begin{center} \resizebox{4in}{6in}{\includegraphics{rel.distrib.ps}}\end{center}\end{figure}

This observed pattern of extreme concentration strongly suggests underlying configural structure in the data; in particular, it is interesting to note that models based on itemwise independence tend to produce exponential frequency distributions. To compare against one such model, Figure 2 shows the mean and 95% Monte Carlo confidence interval for the replicate frequency distributions under an independence model with character frequencies matching the observed data (50,000 replications). The null frequency distribution is clearly different in form from the observed frequency distribution, decaying roughly exponentially rather than as a power law. Furthermore, the observed distribution is far beyond the 95% confidence interval for the independence model over the vast majority of its range, suggesting that the latter is not a credible model for the former. Thus, we see from examining the overall distribution of behavioral phenotypes that 1) the population distribution is highly structured, and 2) this structure cannot be accounted for by raw character frequencies alone. We now proceed to an investigation of the underlying configural properties of the combined belief data.

2.4 Lattice Model Fit

To uncover latent configural structure in the combined GSS data, we employ the microbelief analysis strategy of Martin and Wiley (27,28). In this context, this technique centers on the identification of a lattice of phenotypes - a set of phenotypic states which is closed under intersection of characters - such that the observed distribution of states can be be expressed in terms of the lattice together with a simple error process. To accomplish this, a "candidate lattice" is initially formed from the original data set, and is subsequently fit to the observed phenotype distribution via confirmatory latent class analysis (29). Multiple candidate lattices may be evaluated in this manner, with the final choice being made by optimization of some prespecified model fit criterion e.g., the Akaike's Information Criterion (AIC) or Bayesian Information Criterion (BIC) [note 8]. The selected model may then be compared against other, non-lattice alternatives, and/or analyzed in other respects. Here, we shall focus on the latent class analysis of the GSS belief data; a detailed structural analysis of the inferred lattice structure will be performed in the succeeding section.

The general procedure employed was as follows. For $ n=1,\ldots,50$, the $ n$ most frequent patterns were extracted from the combined data set and closed under intersection to form a lattice ( $ \mathcal{L}_n$). A confirmatory latent class model with itemwise false positive/false negative error parameters (29) was fit for each such lattice, and the associated model selection statistics retained. (Note that all estimates shown are maximum likelihood estimators.) Finally, $ n^*$ was selected such that the BIC of the model on $ \mathcal{L}_{n^*}$ was minimized. For the combined GSS religion data, $ n^*$ was found to equal 25, resulting in a lattice of size 55. The elements of this lattice (represented as binary strings) along with estimated class probabilities are shown in Table 2. Note that although only 25 of the lattice elements were drawn from the data (the rest being constructed via closure), 40 of the elements are estimated to have non-negligible rates of incidence in the population. This demonstrates that the closure operation is able to uncover viable phenotypic states [note 9], but the zero-weight patterns are of substantive interest as well. Such patterns reflect phenotypes which are admissible under the assumed memetic model, but which are essentially unobserved within the population. These "empty" states may reflect the action of a social process such as clustering (22), which renders certain admissible combinations of beliefs more probable than others, or they may result from some as yet unidentified process. By providing a baseline against which to compare our observations, the lattice model may be able to aid in the discovery of new social phenomena.

Table 2: Estimated class probabilities, combined GSS data
Phenotype $ p($state$ )$ Phenotype $ p($state$ )$
0000000000000 0.0159 1010001100110 0.0230
1000000000000 0.0250 1111001100110 0.0036
0000100000000 0.0305 1010011100110 0.0192
1000100000000 0.0001 1111011100110 0.1160
0000000010000 0.0000$ *$ 1010111100110 0.0382
1000000010000 0.0000$ *$ 1111111100110 0.0029
0000100010000 0.0000$ *$ 1111000110110 0.0000$ *$
1000100010000 0.0077 1111100110110 0.0000$ *$
0000000011000 0.0304 1111001110110 0.0000$ *$
1000000011000 0.0381 1111011110110 0.0000$ *$
0000100011000 0.0182 1111111110110 0.0000$ *$
1000100011000 0.0159 1111000111110 0.0387
0000000000010 0.0000$ *$ 1111100111110 0.0098
1000000000010 0.0000$ *$ 1111001111110 0.0000$ *$
1010000000010 0.0000$ *$ 1111011111110 0.0703
0000100000010 0.0040 1111111111110 0.0037
1000100000010 0.0130 1111000100111 0.0196
1010100000010 0.0002 1111100100111 0.0039
0000000000110 0.0334 1111001100111 0.0153
1000000000110 0.0707 1111011100111 0.0664
1010000000110 0.0553 1111111100111 0.0048
0000100000110 0.0227 1111001110111 0.0000$ *$
1000100000110 0.0386 1111011110111 0.0177
1010100000110 0.0009 1111111110111 0.0000$ *$
1010000100110 0.0000$ *$ 1111001111111 0.0095
1111000100110 0.0358 1111011111111 0.0548
1010100100110 0.0000$ *$ 1111111111111 0.0109
1111100100110 0.0153    
$ *$  probability $ <1 \times 10^{-4}$

As noted above, the model employed was a confirmatory latent class analysis with itemwise asymmetric error probabilities. Estimates for these probabilities are provided in Table 3. Note that with very few exceptions, the estimated error rates are quite low, indicating that most observations are quite close to the lattice. False positive probabilities were found to be somewhat higher than false negative probabilities in general, suggesting that deviations from the lattice were more often in the direction of reporting too many beliefs rather than too few. Such asymmetries may reflect differential availability of error opportunities, as well as a general demand effect for reporting high levels of religiosity (given the cultural significance placed on religion within the United States).

Table 3: Estimated error probabilities by character, combined GSS data
Character $ p($false neg$ )$ $ p($false pos$ )$
GOD 0.0192 0.1373
GOCHURCH 0.2088 0.0817
BELIEVE 0.0554 0.3490
FOLLOW 0.1422 0.2241
GOOWNWAY 0.0001 0.4887
REBORN 0.1636 0.0658
SAVESOUL 0.1163 0.1042
READWORD 0.1722 0.3140
DOUBTS1 0.0000$ *$ 0.2709
DOUBTS2 0.0000$ *$ 0.2392
FAITH1 0.2305 0.1271
FAITH2 0.1309 0.1109
FRNDCON 0.0022 0.1987
$ *$  probability $ <1 \times 10^{-4}$

Model fit statistics for the optimal lattice model, along with several reference models, are shown in Table 4. In addition to the saturated model, the fit statistics are shown for the null model of independence and a Guttman scale (17). The Guttman scale constitutes an obvious null model of unidimensional structure, and was here implemented as a confirmatory latent class model in the same manner as the general lattice model above [note 10] (35). (The Guttman scale itself was constructed using item frequencies.) As Table 4 indicates, the unrestricted lattice model is comfortably favored over all three reference models by both the AIC and BIC. Likelihood ratio tests against the saturated model fail to reject both the unrestricted lattice and Guttman scale models ( $ p \approx 1$ in both cases), with the former showing a significant improvement over the latter ( $ p \approx 0$). Taken together, these results indicate that: the lattice model is an acceptable representation of the data; the unrestricted lattice is substantially superior to a simple, unidimensional structure; and the unrestricted lattice is also substantially superior to a null model of itemwise independence.

Table 4: Model fit statistics, combined GSS Data
    Residual Likelihood    
Model $ \ln L$ Df Ratio $ \chi^2$ AIC BIC
Saturated Model -16300.12 0 0 48982.24 96797.65
Lattice Model -18281.96 8110 3963.685 ($ p=1$) 36725.93 37198.77
Guttman Scale -19041.04 8151 5481.835 ($ p=1$) 38162.07 38395.58
Null Model -21143.36 8178 9686.488 ($ p=0$) 42312.72 42388.62

2.5 Structural Analysis

Having inferred a lattice for the combined GSS belief data, we now perform a detailed structural analysis of the lattice itself. Strictly speaking, the lattice is merely a small, idealized set of response patterns which are assumed to lie behind the complete set of observed response patterns. By closely examining this set, however, we may uncover various relationships between behavioral characters. Furthermore, because this set is (by construction) a lattice, it follows that we can identify a dual set of items such that the former can be formed out of the latter by an operation of conjunction as described in Equation 2 (28,18). It is the elements of this dual set which serve as our "candidate" memes, for the simple reason that each observed character can be seen as arising from the interaction of one or more such elements. By analyzing the relationships between behavioral characters and the inferred memes, [note 11] then, we hope to uncover latent structural properties of the religious belief system. These properties may, in turn, suggest secondary procedures to test the hypothesis that the candidate memes are, in fact, genuine units of social replication.

Note that, in keeping with the fundamental duality of the memotype/phenotype set, the inferred memes are non-trivially structured as well. Formally, the memotype set is a distributive lattice, which in the present context indicates that it is closed under union and intersection (28). Less formally, we may be interested in relations among memes such as precedence, in which one meme is held [note 12] only if another is held first. By studying such relationships among inferred memes, we may gain insight into their empirical interpretation.

We begin our analysis by employing the Haertel-Wiley inversion (28,18) to extract the mapping from memotypes to behavioral phenotypes (the $ \mathbf{D}$ matrix) from the lattice of phenotypic states. The $ \mathbf{D}$ matrix for the combined GSS religion data is shown in Table 5. Interpretation of the $ \mathbf{D}$ matrix is as follows: $ \mathbf{D}_{ij}=1$ if and only if character $ j$ depends on meme $ i$, and an individual will exhibit a given character if and only if he or she holds all memes on which said character depends. The $ \mathbf{D}$ matrix thus encodes the putative expression mechanism for the underlying meme system.

Table 5: Inferred $ \mathbf{D}$ matrix, combined GSS data
  GOD GOCHURCH BELIEVE FOLLOW GOOWNWAY REBORN SAVESOUL
a
 1 1 1 1 0 1 1
b 0 1 1 1 0 1 1
c 0 1 0 1 0 1 1
d 1 1 1 1 0 1 1
e 0 1 1 1 0 1 1
f 0 1 0 1 0 1 1
g 0 1 0 1 0 0 0
h 0 0 0 0 0 1 1
i 0 0 0 0 0 0 0
j 0 0 0 0 0 1 1
k 0 0 0 0 0 0 0
l 0 0 0 0 0 0 0
m 0 0 0 0 1 1 0
n 0 0 0 0 1 0 0
  READWORD DOUBTS1 DOUBTS2 FAITH1 FAITH2 FRNDCON  
a 1 0 0 1 1 1  
b 1 0 0 1 1 1  
c 1 1 1 1 0 1  
d 1 1 1 0 0 1  
e 1 1 1 0 0 1  
f 1 1 1 0 0 1  
g 0 1 1 0 0 1  
h 0 0 0 0 0 1  
i 0 0 0 0 0 1  
j 0 1 1 0 0 0  
k 0 1 1 0 0 0  
l 0 0 1 0 0 0  
m 0 0 0 0 0 0  
m 0 0 0 0 0 0  

One of the first questions we might ask about both the memotype and phenotype structures is that of whether we can identify patterns of similarity and dissimilarity among the various elements. Consider, for instance, the set of behavioral characters. A natural measure of dissimilarity between observed characters is the Hamming distance (19) between the respective columns of $ \mathbf{D}$; that is, the number of differing dependencies between the columns. Under such a measure, a distance of zero would reflect absolutely identical patterns of dependencies - both characters would depend upon precisely the same memes [note 13] By turns, a large Hamming distance would indicate that the two characters "load" on very different memes, thus suggesting substantive dissimilarity. Interestingly, the duality of the $ \mathbf{D}$ matrix implies that we can apply a similar logic to the memes. In the latter case, we take Hamming distances between the rows of $ \mathbf{D}$, thereby measuring the extent to which various memes share similar patterns of connection to the character set.

Given these matrices of Hamming distances, we can attempt to represent the pattern of similarities/dissimilarities via multidimensional scaling (MDS) (34). Two-dimensional MDS solutions for the memes and behavioral characters are shown in Figure 3; note that the Euclidean distances between elements in the MDS solution approximate the underlying Hamming distances, and therefore elements which are close together are generally more similar than those which are far apart. Thus, we see that the the characters DOUBTS1 and DOUBTS2, while closely related to each other, differ substantially from the other characters. Furthermore, a "chain" of elements leads from FAITH2 to SAVESOUL, suggesting an underlying linear similarity pattern. Substantively, this pattern seems loosely associated with increasing Christian religiosity, moving from a reported strengthening of faith from the birth of a child at the one extreme to reported conversion attempts at the other. Two associated characters - FOLLOW and GOCHURCH - are juxtaposed, owing to the fact that they have identical memetic dependencies. In contrast, GOOWNWAY is isolated from all of the other memes in the set, suggesting that it has relatively little relation to the rest of the items.

Figure 3: MDS of latent meme-like structures (right panel) and behavioral characters (left panel) by Hamming distance between association patterns, combined GSS data; note that FOLLOW and GOCHURCH are superimposed (see text) (PDF Version)
\begin{figure}\begin{center} \resizebox{4in}{6in}{\includegraphics{rel.mds.ps}}\end{center}\end{figure}

Turning to the putative memes, we observe four loose clusters and one intermediate point (l). c and e are the most closely related memes, both being similar to d and f. a and b are closely related to each other, but relatively distant from the other clusters which are present. Interpretation of these patterns is also aided by the dotted arrows connecting various memes within the set. These arrows indicate precedence, such that x precedes y if and only if any character which depends on y also depends on x. [note 14] Interestingly, we can see here that precedence is only loosely related to similarity: many memes share a precedence relation with other memes which have very different patterns of dependency. This is in strong contrast to precedence among the characters, which follows similarity quite closely.

A better view of the precedence relations may be had by turning to Figure 4, which shows the set of memes, partially ordered by minimal precedence. (Such sets are referred to as "posets.") As the figure makes clear, the structure of dependency among memes is quite intricate. Given the complexity of the precedence graph, it is useful to consider the interpretation of various common structural features; one list of such features (and their interpretations) is given in Table 6. With this in mind, we can immediately see that the meme poset has two components, indicating that the "miniature" Guttman scale formed by m and n has no strict precedence relation with any other memes in the set. Turning to the larger component, a large number of intersecting scale-like elements can be seen, having unique origins in memes a, c, and d. The chief enabling element within the poset is clearly f, which serves as the critical linchpin for three memes and numerous chains. f also acts to join the chain de with c, a linking role which is similar to that played by meme h (which joins f with the ab chain). The terminal nodes for the chains of the main component are i and l, which are thus the memes lying at the most extreme ends of their associated scales. As we shall see, this is reflected in their phenotypic associations.

Figure 4: Inferred meme poset, combined GSS data (PDF Version)
\begin{figure}\begin{center}

Table 6: Behavioral interpretations of common precedence graph features
Graph Feature Interpretation
Vertex Belief
Edge Minimal precedence
Path Embedded "Guttman-like" scale
Component Maximal set of related beliefs
In-isolate "Elementary" belief with no dependencies
Out-isolate "Terminal" belief on which nothing depends
In-star Belief with multiple distinct dependencies
Out-star Belief which "enables" multiple beliefs

Returning to the behavioral characters, we can use the same logic to identify and interpret precedence based on memetic dependencies. A plot of the relevant poset is provided in Figure 5. The graph structure shown in Figure 5 displays a striking pattern of what appear to be interlocking but partially distinct religiosity scales. The critical nexus for the behavioral characters seems to be READWORD, which depends on a number of more basic religion/spirituality characters and which in turn appears to act as an enabler for more high-commitment Christian beliefs. These last characters, in turn, seem to be split into two chains, one stressing individual experiences (SAVESOUL and REBORN) and the other emphasizing attention to collective activities (GOCHURCH/FOLLOW and FRNDCON). REBORN and FRNDCON, then, would seem to reflect the extremes of the individual experience and collective involvement Christian belief scales (respectively). Interestingly, reporting that one has never had religious doubts due to personal suffering or "evil in the world" does not have a strict precedence interaction with other characters, and GOOWNWAY seems to be entirely distinct. It seems particularly noteworthy that the line of demarcation here is not associated with the type of character in question: the main component contains a tightly integrated mix of personal experience, normative judgment, and pure belief items. Rather, it would seem that the distinction between components is based either on the direction of the belief (positive/negative) or on some other latent structural factor. As we shall see, there is evidence for the latter interpretation.

Figure 5: Inferred character poset, combined GSS data (PDF Version)
\begin{figure}\begin{center}

So far, we have focused exclusively on structure within meme and character sets; much of our interest, however, lies on the relationship between the two. Here, as before, our guide is the $ \mathbf{D}$ matrix, which provides an explicit accounting of the macro/micro dependencies within the data set. While a good deal can be learned from direct perusal of $ \mathbf{D}$, it is also helpful to examine the graph formed by one or more posets together with the cross-type dependencies. Such a graph is shown in Figure 6, which combines the meme poset with the associated characters. Following the minimalist logic employed elsewhere, cross-type dependencies are shown minimally: if character $ A$ depends upon memes $ a$ and $ b$, and if $ a \to b$, then only $ b \to A$ is displayed. This visualization allows one to use a "path tracing" approach to interpreting the dependency structure; by starting with one or more elementary memes and following the outgoing paths, one can observe at what points one "picks up" various behavioral characters. The figure also makes evident certain non-obvious relationships between meme and character structures which are of clear substantive import. Consider, for instance, REBORN and GOOWNWAY. Considering only the MDS solution or within-set precedence orderings, these two characters could not seem more dissimilar: while REBORN is at the extreme end of what appears to be a Christian religiosity scale, GOOWNWAY seems eponymously distant from all of these characters. Yet, as Figure 6 reveals, REBORN and GOOWNWAY are linked by their joint dependence on the $ m \to n$ scale! Similarly, DOUBTS1 and DOUBTS2 share a number of dependencies with FOLLOW/GOCHURCH, SAVESOUL, REBORN, FAITH1, GOD, and BELIEVE, a fact which is not obvious from the within-type posets.

Figure 6: Character dependencies with inferred meme poset, combined GSS data (PDF Version)
\begin{figure}\begin{center}

Putting all of this together, we can discern much regarding the latent structure of American religious belief in the period from 1988-1998. First, there is a clear structure of increasing commitment to Christian religiosity which begins with the acceptance of a personal god and an experience of having one's religious faith strengthened by events such as the birth of a child, leads through periodic Bible reading in the home, and then divides into two high-commitment scales. One of these involves reports of individual experiences and activities usually identified with evangelical protestantism, including personal attempts at conversion of others and the experience of being "born again." The other involves normative judgments and experiences which are more collective in nature, including a belief in the importance of church attendance and obedience to church authority, and reporting having a friend who is a member of one's congregation. A deep structural linkage exists between belief in the importance of following one's conscience even when it conflicts with religious authorities and reporting a born again experience, despite the many differences in dependencies between the two. The inferred memes associated with this linkage may be interpreted as reflecting a Guttman scale for belief in the importance of personal religious experiences, with m constituting the belief that such experiences constitute valid religious knowledge, and n constituting the more extreme notion that one's personal experience or gnosis is more reliable than the teachings of religious authorities. (See Table 7 for one set of possible interpretations of the meme system.) Similarly, reporting that one has never had religious doubts due to personal suffering or "evil in the world" appears to depend on a mixture of Guttman scales involving religious conviction, with precursors g (apparently reflecting confidence in the authority of the church) and j (reflecting an immunity to doubt based on the opinions of others) culminating first in an immunity of conviction to others' actions (k) and finally in an immunity of conviction to one's own suffering (l). Belief in the importance of church attendance and the obedience to religious authority appears to follow from a combination of a belief in the potential of religion to bring positive benefits, combined with a conviction in the reliability of religious institutions. A similar crossover between the immunity of doubt to others' opinions and the efficacy of the Bible would seem to lie behind self-reported conversion activity. Other crossovers between general religious conviction and belief in the personal, positive benefits of religion appear to be underlie articulated belief in a personal god, the notion that such belief is an important part of being a good Christian or Jew, and the report that one's religious faith has been strengthened by various personal events. Thus, by examining the latent algebraic structure of American religious belief, we are able not only to discern direct relationships among manifest behavioral characters, but also more subtle connections based on presumptive memetic dependencies.

Table 7: Possible interpretations of inferred meme-like structures, combined GSS data
Meme Possible Interpretation
a Religion/spirituality has personal significance
b Religion/spirituality brings positive benefits
c Religion/spirituality reduces suffering in times of trouble
d There is a god
e One must be confident that there is a god
f Belief in god is an important bulwark against adversity
g One must have confidence in religious institutions
h The Bible should be consulted regularly
i One must associate with co-religionists
j One should counter doubt regarding one's religious beliefs
k One's religious conviction should not be affected by others' actions
l One's religious conviction should not be affected by one's experiences
m Personal religious experiences are a valid source of knowledge (gnosis)
n Personal religious experiences outweigh religious authority

As a final note, we reiterate the observation that our interpretation of the potential memetic structure uncovered is limited to its association with the observed behavioral phenotypes. These, in turn, are limited by the available character set. Thus, as noted earlier, we cannot discern fine-grained distinctions among those not carrying any of the four "root" memes (a, c, d, and m), because (by construction) the absence of these memes corresponds to a phenotype which evidences none of the characters under study. [note 15] Broadening the character set (e.g., to encompass subtle distinctions among varieties of atheism or agnosticism) would presumably result in the discovery of additional candidate memes associated with the new characters; some of these would likely be root memes, while others may have varying degrees of dependency on novel or extant candidate memes. It is thus important to bear in mind that the candidate set identified here is not exhaustive of the religious domain.

3 From Cross-Sectional Analysis to Diffusion

The basic strategy followed here might be described (with apologies to Firth (10)) by the notion that "you shall know a meme by the company it keeps." As opposed to inspecting longitudinal data for patterns of persistence which might imply an underlying diffusion process, we instead seek indirect evidence of latent discrete structures in the distribution of behavioral characters. This latter strategy has the tremendous advantage of leveraging existing data resources (which are overwhelmingly cross-sectional), but at some point it is critical to determine which (if any) of the latent meme-like structures inferred by the method are truly transmissible. Although a detailed treatment of this topic is beyond the scope of this paper, we here make some suggestions which may be of use for subsequent work in this area.

3.1 Phenotypic Consequences of Memetic Transmission

An important implication of meme theory which sets it apart from many other diffusion models (e.g. 12,21,3,4) is that uniform diffusion on the level of the memotype can give rise to non-uniform change at the level of the phenotype. By way of illustration, consider the expression mechanism of Table 5. If an individual (A) exhibiting the character SAVESOULinteracts with an individual (B) exhibiting the character GOOWNWAY, standard diffusion models would generally predict some chance that each would adopt the others' behavioral state, i.e., that the A would gain GOOWNWAY or drop SAVESOUL, or that B would gain SAVESOUL or drop GOOWNWAY. Assuming that diffusion occurs at the memetic level, however, Table 5 would suggest the possibility that one or both individuals might exhibit the character REBORN; this follows from the fact that A must hold memes a-f, h, and j, B must hold memes m and n, and the union of these sets covers REBORN. Such a prediction would not arise from any of the classical diffusion models cited above (but see the evolutionary learning models of 9,30). Given this, one strategy for testing hypothesized memetic structures would be to identify novel phenotypic consequences from memetic crossovers (using the $ \mathbf{D}$ matrix), place individuals with the appropriate preconditions in contact with one another, and compare the individuals' post-encounter phenotypes with the predictions of the theory. Since these tests would focus on effects related to specific characters, reasonable power could be obtained with fairly small samples - this ability to screen populations for easily evaluated phenotypic effects is another benefit of the cross-sectional modeling strategy.

3.2 Lattice Walk Models

Beyond testing novel phenotypic predictions for specific characters, the model of Equation 1 could also be used as the starting point for a more general diffusion model, the predictions of which could be compared with intertemporal data. In particular, it is noteworthy that one trivial consequence of the conjunctive memetic model employed here is that behavioral phenotypic change - if motivated by change at the memetic level - should take the form of a walk on the phenotype lattice. Placing a probability distribution on the rate of meme transfer would yield a matrix of transition probabilities between lattice states. This, in turn, could easily be used to predict such things as first passage times for various phenotypic states, predictions which could be compared with data arising from experimental or other sources. Here, as before, it is important to emphasize the fact that the initial cross-sectional study dramatically reduces the number of states which must be considered for the transition matrix, and provides reasonable initial clues as to the possible transition probabilities. This "rough cut" of the meme system can thus serve to greatly improve the efficiency of subsequent inference based on intertemporal data.

4 Conclusion

Examination of data from the 1988 and 1998 GSS religion modules using Martin and Wiley's latent algebraic approach (27,28) reveals a complex structure of discrete latent constructs beneath the surface of the question responses. Meme theory suggests the possibility that these inferred constructs are actually memes, in which case the posets shown here provide a plausible model of the paths by which particular religious memes may come to "infect" a given host (7). While longitudinal data will be required to determine whether these inferred constructs meet the transmissibility criterion, the general approach shown here should be a useful preliminary step in future memetic studies of religion (particularly where more diverse sets of characters may be considered). Combined with studies on the evolution of religious characters across time (e.g., 24,16) inference of latent structure from large, cross-sectional studies such as the GSS may greatly improve the prospects for a memetic theory of religious belief.

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Acknowledgements

The authors would like to thank John Levi Martin, Philip Cohen, and Kim Romney for their helpful comments regarding this work.

Notes

1.  Full address: Department of Sociology and Institute for Mathematical Behavioral Sciences; University of California, Irvine; Irvine, CA 92697; Tel: +1 (949) 824-8591

2. A lattice is a partially ordered set for which all finite subsets have a least upper bound and a greatest lower bound. In the limited context considered here, a lattice can be thought of as a collection of binary vectors which are closed under elementwise intersection (e.g., indicators for a set of features which may or may not be present). See the appendix of Martin and Wiley (28) for an application-centered elaboration.

3. It is possible to generalize the sampling assumptions, but we shall restrict ourselves to the simple case for the present.

4. I.e.,$ \mathbf{a} \wedge \mathbf{b}$ is the binary vector whose ith position equals 1 iff $ \mathbf{a}_i=\mathbf{b}_i=1$.

5. Although the differing framework of Markovsky and Thye (25) does not allow us to assess whether this transference involved microbeliefs in the sense used here, it is interesting to note that the indirect measures used in their experiments (subjective assessments of an object which had purportedly been exposed to "pyramid power") would seem a priori likely to have tapped latent belief change of some sort.

6. Draws were subsampled from a Markov Chain Monte Carlo simulation of length 2,500,000, every 500th step being accepted.

7.I.e.,$ -\sum_x f_x \log_2 f_x$, where $ f_x$ is the empirical frequency of phenotype $ x$.

8. Akaike's Information Criterion (AIC) and the Bayesian Information Criterion (BIC) are generalized fit indices which can be used to compare non-nested models. Both criteria are based on tradeoffs between fit to the data at hand (i.e., likelihood of the data under the proposed model) and parsimony (e.g., the expected decrease in predictive accuracy due to overfitting); the two differ in the precise assumptions involved. An excellent overview of the AIC (and related information-theoretic criteria) can be found in Bozdogan (2), with a similar introduction to Bayesian model selection techniques in Wasserman (36).

9. Approximately 29 elements would be expected to have non-negligible class probabilities if closure had no relationship to the distribution of phenotypes.

10. Note that Guttman scales are lattices, and hence are special cases of the general lattice model.

11. These are more properly thought of as "latent meme-like constructs," though for reasons of brevity we refer to them here simply as memes. It is hoped that the reader will not forget their provisional status, however!

12. In the present context, we refer to memes as being "present" or "held" if they take allele 1, and "not held" or "absent" if they take allele 0. Such usage should not imply that a meme's state necessarily implies a physical presence or absence.

13. Insofar as can be inferred from the data at hand, at least.

14. Note that the precedence relations shown in the included figures are minimal, in the sense that $ x \to y$ only if $ \nexists z\neq y :$ $ x \le z, z \le y$ (where precedence is signified by $ \le$ and where $ z=y$ iff $ z \le y$ and $ y \le z$).

15. Deviations from this prediction are taken to arise at random with probabilities given in Table 3. Thus, some persons carrying none of the candidate memes would be expected to report having read the Bible at home in the past year (apx 30%, in this case), but this is still much lower than the rate which would be expected for persons carrying all of the identified memes (apx 83%).

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