Every mathematical model is an attempt to replicate aspects of reality. Mathematics is the language used to express that model and capture the complexity of the real world. The type of replication required will determine the nature of the model produced – the model’s intended purpose. Likewise, the ability of the mathematical framework to handle real-world complexity will affect the model – the theoretical and practical limitations of mathematics. Complexity and purpose help determine the type of model produced.
Models can be classified according to their:
- Fidelity, the extent to which models are intended to replicate reality;
- Theoretical Support, a logical framework of consistent laws which enable models to explain a wide range of behaviour;
- Methodology, a logical collection of methods and principles, mathematical and computational in this case, that allows a model, and any associated theory, to be expressed and analysed.
In summary: “One or more methodologies are used to express a theory of a certain strength which in turn supports a model of a certain fidelity.”
A model can be described as “successful” by the degree to which it replicates reality and usefully fulfils its purpose. A model does not need to use the richest methodology, have the strongest theoretical support or the highest fidelity to be useful. For example, a simple back-of-the-envelope calculation that improves one process, in one industry, at a given time is highly useful and thus successful. Even if a model is not “rocket science”, it can still instil confidence.
Model fidelity describes the extent to which a model can replicate reality. Fidelity depends on the purpose of the model. Some models, such as an aircraft flight simulator need to strongly replicate reality to fulfil their purpose of training people. Thus, they have very high fidelity, sometimes called analogue. However, other models may only need to illustrate a principle, such as the motion of a train. It could be made analogue for a specific train, but people often learn more from a simpler illustration.
Model fidelity also depends on the ability of the mathematics used to handle complexity. A more illustrative model may only be possible, such as the interaction of many species, as the governing laws may not be fully known, or data is not detailed enough to calibrate completely.
Further details in Morecroft J. (2007). Types of fidelity:
This is the highest fidelity. The model has a high degree of detail and scaling. Parameter values can be determined numerically with realistic units. Algebraic and functional relationships are closely constrained. Model variables can match measured data—almost a point by point comparison with the real-world situation.
This is the mid-range fidelity. Parameter values and variables can be given plausible values. Although numerical accuracy is less critical than the analogue case, the theoretical mechanisms to explain growth, such as feedback loops, transition laws and rate changes, are realistic.
This is the lowest fidelity. Here a general principle is taken from one domain and applied to another, without too much regard for numerical values or scale. The important issue is the type of structure, such as feedback loops, and the corresponding behaviour, such as the shape of a graph over time.
The Limited Enthusiasm model of church growth is towards the high fidelity end of the spectrum as all its variables and parameters have well-defined meaning and measurable values, more analogue than illustrative.
With demographics added the model becomes a little less analogue as some of the parameters are harder to estimate. Nevertheless, they are, in principle, measurable. The absence of cohorts makes the birth/death process a little more illustrative than analogue. The model provides principles and can be calibrated for congregations and denominations and fits within the standard genre of population modelling.
The Spiritual Life model of church growth is illustrative. Realistic values can be given for the population numbers and compared with data, but Spiritual Life, which is a soft variable, is in principle hard to quantify. However, the model can tell plausible stories with the level of Spiritual Life being compared with proxy variables. The model provides principles, but would not be easy to calibrate for a particular church grouping.
The Limits to Church Growth models are metaphorical. Although they could be made illustrative, and perhaps even analogue, that is not their purpose. They are constructed to show how growth limits from other sciences, such as the environment and business, are also significant for understanding barriers to church growth.
Not all models can be elevated to the lofty position of a “theory”. However, all models have a degree of theoretical support. For some situations, strong theoretical support is essential to build confidence. This is particularly true for models which are hard to calibrate or have soft variables, ones that are hard to measure. However, for models of specific situations, where a particular problem needs to be solved, and/or where there is a wealth of data to calibrate the model, then strong theoretical support is not always necessary. Confidence is built from the model’s ability to solve the problem.
What follows is my own rough outline of the degree of theoretical support for a model, no doubt incomplete!
The theory has been tested in a wide variety of situations, it is basic enough that it pervades many domains and is the basis for many other theories and models. The theory is “atomic” describing simple principles from which more complex situations are constructed. The theory is universally accepted such that there are no realistic alternatives and has been accepted for a long time.
Newtonian mechanics is an obvious example. A universal theory would be unlikely in a subject as complex as social modelling, which depends on the highly variable and sometimes independent behaviour of actors.
A strong theory is one that has been tested in a wide variety of situations and is the basis for other theories and models. It is atomic, describing simple principles. It has a long track record as the accepted theory, although alternatives are possible.
Einstein’s General Relativity is a good example. It is the accepted theory of gravitation, although there are alternatives. It is well tested in a variety of situations, but there are some observed anomalies in astrophysics for which a modification of the theory might be needed as an alternative to the dark matter and dark energy hypotheses.
Although the underlying theory for the model may be only one of many, the principles it expresses could be applied across a wide variety of domains and scenarios. Many metaphorical models would be in this class. However, the theory is often only partial, describing universal phenomena but in a restricted environment.
The negative exponential law is one such example, used for cooling, radioactive decay and deaths in populations. The limits to growth metaphor is another, describing, for example, species, human populations and market penetration of companies.
Similar to the above, but without the cross-domain generality. Capable of being repeated for other examples of the same phenomena in the same domain. The People’s Express model (Morecroft, 2007) is an example. Although a model of a specific company (People’s Express Airlines) was in mind, the model was used to illustrate the consequences of under-investment and could thus be applied to a range of companies. However, the model structure has less relevance outside the business world.
Similar to above but the model is only for one example in a given domain. An example would be a model of a specific company without concern as to whether this may represent a class of similar companies.
A set of individual hypotheses that make sense, but do not constitute a cohesive set of rules.
The Limited Enthusiasm model is part of the general theory of word-of-mouth models, which are in turn based on models of the spread of disease. These are based on the mass action principle that expresses how two differing subpopulations interact. Other examples of models based on this theory are in the social diffusion references and the epidemic modelling references. The model with demographics added has the same theory.
The Discipleship model is phenomenological. It is a model of a generic church congregation. It could be applied to most congregations (so not “specific”), but has no applicability outside a church context (thus not “general”). It uses various familiar model constructs.
A methodology is a collection of rules and principles that allow a model to be constructed, expressed and analysed. The rules must make logical sense and thus be expressed in terms of mathematical and computational constructs. Mathematics contains a range of constructs from calculus to statistics, probability to symbolic logic. A modelling methodology must have rules that relate its mathematics to the real world.
The following are three of many dynamical modelling methodologies. They are dynamical as the type of modelling concerns behaviour over time.
Models have well-defined state variables, each of whose rates of change is influenced by combinations of one or more of the other state variables and exogenous effects. There is a strong emphasis on the analysis of equilibrium states and oscillations, and how they change as parameters are adjusted. Analysis involves mathematics and, where necessary, computer simulation. The state variables and time are modelled continuously, with state variables at later times determined exactly by variable values at an earlier time, that is they are deterministic. If time were modelled discretely, the equations would be replaced by difference/recurrence relations. State variables generally count numbers of agents, such as people, animals, plants, etc., but may also include physical quantities such as energy, surface area and money.
Models have well-defined state variables which are thought of as accumulations. Rules of accumulation are determined by causal connections from other state variables, including intermediate dependent, variables. As such, the model is a network of causal relationships. Often the rules of accumulation involve feedback between variables, that is causal loops, leading to endogenously generated behaviour. The network structure generates system behaviour. System dynamics models can be reduced to differential equations; however, the information about the model structure is mostly lost. Analysis is usually through computer simulation and feedback loop methods. The state variables and time may either be modelled continuously or discretely. Models are deterministic. State variables may count numbers of agents, represent physical quantities, or more intangible “soft” variables, such as stress, enthusiasm, attractiveness etc. That is variables that are well defined but may not have any clear or unique measure, such as those that occur in social sciences.
These model the changes in each individual agent, rather than in the numbers of agents. These models are thus disaggregated. Changes in the state of an agent may come through values in its own state, the aggregated sum of one or more sets of neighbouring agents, and the whole environment. If numbers of agents are computed then the agent-based model can be approximated to an aggregate, or mean-field, model, such as those in differential equation modelling and system dynamics. Models are usually stochastic where state changes are determined by probability rules, although deterministic models are also possible. State variables and time may be continuous or discrete. Analysis is through computer simulation and a variety of probabilistic and statistical physics methods.
To date, all the church growth models presented on this site are built using either differential equation or system dynamics. However, there are also agent-based versions of some of these models.