Markovian Simulation
and the Church Growth Process
JOHN J. RASMUSSEN
Department of Operations Research
Case Western Reserve University
Cleveland, Ohio 44106

Many churches today are experiencing major increases in the number of regular attenders. Examples abound of congregations recording a 100 percent growth in just a few years. In order to plan for adequate facilities and for programs to accommodate the increased numbers of future regular attenders, those involved in managing such churches would welcome a tool to aid in membership projections. Failure to anticipate, with sufficient lead time, a congregation's growth rate will tend to overcrowd existing, and in some cases planned, facilities. In order to expand, funds must be raised, plans laid, and construction carried out. All this consumes time. Similarly, church attendance goals must be reviewed, evaluated, and perhaps revised in light of experience. Besides being costly, miscalculations also cause inconvenience, disappointment, and lost opportunity.

The model has produced three important results. First, five states have been defined to describe all persons in the population studied, according to Markovian specifications. Second, a questionnaire was developed to elicit initial information germane to a church-related forecast. Third, an example forecast was computed, with a discussion concerning its relevance to the capacity issues.

The fact that a church is a non-profit organization makes skillful planning by competent evaluators imperative.

 simulation.^1,2,3

Markovian Simulation

The following three bases of the social process problem can be bound by stochastic, or probabilistic, models. Class mobility, the relaxation of the restriction of having "kind follow after its own kind," is handled by the construction of mutually exclusive and collectively exhaustive categories of all subjects within the population of the study. A sample may then move from one category to another according to probabilistic laws. The system may be termed closed by the introduction of artificial categories from which new samples come and into which departing samples go. Finally, the measurement of social process changes in terms of generations, queue lengths, cumulations, etc., is appropriately expressed through discrete stages of time in the stochastic model. The mover-stayer model, developed by Blumen, Kogan, and McCarthy² to study the mobility of labor in American industry, is a fine illustration of these characteristics. The proportions of movers and stayers are determined for the end of a given time period for any industry category, along with the long term equilibrium distribution of the proportion of members in each category.

The choice of the proper model to describe and study a certain social process is of utmost importance. Stochastic models have properties which adequately fulfil requirements for models of social behavior. The Markov process is a stochastic model that is very adapt able to the description of social phenomena, and considering the complexity of the problem, Markovian simulation is the methodology very often employed.

The Markov process concept can now be developed. Suppose some social process is divided into a number, say n, of different categories, or states. Furthermore, each member of the population of the process belongs to one, and only one, of the states. We shall assume that this process is closed and that there is the possibility of complete mobility of the members among the n states. Then by taking a sample of these persons and observing the records of their past histories, the movement of each member of the sample among the classes can be recorded. When these movements tend to a fairly regular trend, we can say that this is the probability associated with the transition of any person from one state to another within a designated time period (month, quarter-year, year, etc.). Since only some fraction of the members of a particular state may transfer to another state, and since all the members who leave one state must be classified in one of the other previously defined states, then each transition probability must have a value between 0 (zero) percent and 100 percent, and the sum of the transition probabilities of both staying in a particular state and entering any other state must equal 100 percent. The following of these probability laws insures a stochastic process.

A Markov process is a model of a stochastic system that is characterized by a sequence of trials or periods of observance in which the results of each trial are dependent only upon the immediately preceding trial. Once a state occupied at a certain time-period is known, the histories of the states before that period are not involved in computing any subsequent probabilities of transition. The n2 transition probabilities can be concisely arranged into tabular form, called a transition probability matrix. The multiplication of this matrix with a vector containing the initial population proportions of each of the n states will yield the forecasted proportion of members in each state at the end of the time period.

Problems involving four or more states and a large number of periods to be forecasted will necessarily entail much computational complexity. The use of computer simulation becomes very attractive in this case, and establishes Markovian simulation as a powerful method of stochastic process problem solving.

The social process of church growth was modelled ⁴ and simulated in an employment of the Markovian simulation scheme as described in the previous section. A brief description of this model is presented next.

The word church is used as a general descriptor. Since the sanctuary is usually the first facility to be affected by overcrowding, and since the sanctuary is almost entirely used for church services in local church bodies, the model will count or project church attenders. Other facilities could be substituted.

There is some limit to the population size and to the geographical location of parishioners of interest to a given church. For example, a church in Chicago should definitely not expend resources in an attempt to obtain regular attenders from the Greater Chicago area, and especially not from, say, Cleveland. The drawing area, then, consists of the population tract(s) from which a church desires to obtain its attenders. A smaller, non-suburban town would most likely comprise the entire drawing area for every church in that town. Once the drawing area is established, its total growth must be forecasted with a high degree of care. Usually these forecasts are available from a city's public offices and/or from census figures. If a constant rate is known, it is incorporated into GOCHURCH. Otherwise a regression or an exponential smoothing extrapolation of previous population measurements must be made.

Five classes, or states, of residents within the drawing area are defined for the model. Each person under consideration must belong to one, and only one, of the states.

State I includes persons who attend the church regularly.
State 2 includes persons who attend irregularly and/or infrequently.
State 3 includes persons who do not visit the church at all and do not attend another church, but may visit with encouragement from the church or its attenders.
State 4 includes those who attend another church and do not visit your church at all.
State 5 includes those who are entirely disinterested in attending any church.

A major feature of the model is a questionnaire which has been formulated to elicit the initial information necessary to run GOCHURCH. Included in the completed form are the data that are needed to construct the probability transition matrix. The information requested by the questionnaire must be as accurate as possible, as every iteration in GOCHURCH is based upon the initial transition probabilities. It is recognized that data concerning the movements of persons among States 3, 4, and 5 may be difficult to obtain, since church records are generally not that exhaustive. Hence, some evaluative procedure may be required to quantify ambiguous or previously unrealized occurrences.

In the absence of any theoretically justifiable and relevant method for generating seemingly intangible values, expert judgments using the Delphi technique may be considered the best available method, The Delphi technique' is used as a method for eliciting, refining, and integrating the subjective opinions of a panel of experts without sacrificing or comprising any individual's suggestions. This approach is helpful in reaching a consensus via a series of voting rounds using anonymous feedback. It relies on experts who make rational analyses rather than merely guessing. The experts take into account new or discrepant information and construct logically sound deductions about the future based on a thorough and disciplined understanding of a particular phenomenon. The successive rounds of Delphi facilitate resolution of any controversies, ambiguities, or redundancies inherent in the nature of future events. This consensus of experts definitely enhances one's ability to make decisions upon which forecasts may be based,

The model assumes that such a technique has been applied in determining the initial transition probability data. At the initial period, each of the States 1 through 5 will contain a certain percentage of the drawing area. To determine the distribution of the members of the drawing area after their movements during the first time period, one must multiply the transition probability matrix with the vector containing the initial population proportions. In order to find the population proportions at the end of the second period, the same procedure is followed. The most recently calculated proportion vector is multiplied by the original transition probability matrix (since the probabilities are intended to represent regular trends of movement among the states) to yield the new proportion vector.

This iterative technique is continued for the desired number of forecasted periods, with the new population proportion values replacing the ones previously used. In other words, only the current distribution of the population is necessary and sufficient to predict the distribution for the next period.

The author has performed the simulation for a hypothetical church named Monte Vista Chapel 4. In this example the Chapel was anticipating expanding its facilities three years hence, but wanted to investigate the possibility of overcrowding within just two years.

Using the time unit of one simulated period to represent three months, the GOCHURCH program calculated the forecasts for 20 periods, or five years. The number of persons classified by each of the five states was determined by multiplying each component of the proportion vector by the forecasted drawing area population. The total attendance (States I and 2) after two years just exceeded the existing auditorium seating constraint, while the three year forecast was well in excess. The results of continuing with the effects of the current transition probabilities were now evident, and provided the church planners the forecasting information they needed. Such insight provides a more certain incentive for preparing adequate facilities for future church attenders, hence maintaining a high likelihood of their continuing to attend regularly.

The decision had been made to build in three years, or after 12 time periods. The primary additional cost of building one year earlier is the cost of securing the down-payment capital one year earlier. Suppose, however, the church planners choose to wait the three year interval before building, even though the attendance would exceed the seating capacity after two years. Since only a certain maximum number of attenders can be accommodated, gross growth will be zero for the one year interval. Besides losing possible revenue from potential attenders, the transitition probabilities may be drastically altered, so that the growth rate after the construction of the building may be less than before. Church planners, then, must weigh the effects of this interrupted attendance growth in determining the church growth characteristics after the building is completed.

But what of church shrinkage? Any change in size of a church hinges upon the function of the number of people becoming attenders versus the number becoming non-attenders. GOCHURCH will effectively evaluate the transition probability matrix describing a certain church, indicating decrements in attendance as well. This prediction of decline should stimulate church planners to augment current attendance procedures, if they so desire, to increase the probability of persons becoming regular attenders. In effect, they must alter their attendance developing procedures to match the transition probability matrix which expresses their growth goals for the future.

Occurrences which may accelerate or decelerate the rate of change of either the transition probabilities or the size of the drawing area should also be considered. Two methods may be followed to adapt GOCHURCH to handle dynamic probabilities of transition among the five states.

First, it would not be wise to allow the incorporation of a continuous change per period of each of the transition probabilities, because of the unreliability of such estimates by church planners. Such rates of change cannot be realistically calculated, especially concerning States 3, 4, and 5, since churches generally do not collect this type of data. Hence, revising one or more rows of the transition probability matrix after an appropriate number of periods is the suggested method of effecting dynamic probabilities of transition. The proportion values calculated at the period of change would become those for the new period zero, and GOCHURCH would be run again.

A second possibility concerns a drastic influx into the church's drawing area of any certain ethnic or religious group of persons, coupled with a possible outflow of other categories of people. Since GOCHURCH is intended to be used as a middle-range forecasting instrument, these population movement trends may not affect the transition probability matrix for the 1~ngth of the forecast. However, in the event one wishes to study the effects of such trends, the program can be suitably adapted. Instead of altering the 5 x 5 = 25 values of the matrix, only the 5 terms in the population proportion vector Deed be changed, according to a specification of the trend values.

Similarly, a change in the rate of growth of the drawing area may be handled by running the simulation in sections. For the period during which the population rate change parameter must be altered, a new rate is substituted, the most recent population proportion vector is inputted, and the problem is simulated from the new time zero.

Several interesting results have thus been developed. First, it was possible to limit the size of the population with which a church is involved, and to categorize each person of this population according to one, and only one, of five distinct states. Second, a questionnaire was formulated to obtain the data required to describe the cbureb growth problem as a Markov process. Although this process is valid as long as the transition probabilities remain static, certain refinement of the initial information, perhaps through a Delphi technique⁵, may be necessary for a more accurate forecast. Third, the forecasts produced by the model may aid analysis and church planners in assessing capacity excesses of various facilities. Consequences of altering or of not altering future plans should then be considered, with much reliance given to the predictions by the model GOCHURCH.

The fact that a church is a non-profit organization makes skillful planning by competent evaluators imperative. This planning should incorporate all relevant tools which are available. GOCHURCH is but one such tool.

REFERENCES

¹Bartholomew, D. J., Stochastic Models for Social Processes, John Wiley and Sons, London, 1967.
²BIumen, T., M.Kogan, and P. McCarthy, The Industrial Mobility of Labor as a Probability Process, Cornell University, Ithaca, New York, 1955.
³Pool, L, et al., Candidates, Issues and Strategies, MIT Press, Cambridge, 1965.
⁴Rasmussen, J., "GOCHURCH-A Simulation Model for Forecasting Church Growth," Technical Memorandum No. 305, Department of Operations Research, Case Western Reserve University, Cleveland, 1973.
⁵Reisman, A., Managerial and Engineering Economics, Allyn and Bacon, Boston, 1971.