To the average person the word "model" might bring to mind someone modeling a dress, or perhaps a photographer's model, or even a model airplane or car. Let's consider these cases. The usual reason for looking at a dress on a model is to imagine what the dress would look like on one's self without having to buy it to find out. The photographer uses his camera to make a likeness of the model in the form of a picture. The model plane or car allows one to enjoy the details and perspectives of the model without the problems and expense of actual ownership of the airplane or car. Indeed, one can try experiments on the models that would be very difficult or expensive to actually perform on the real thing.

Consider the methods and purposes behind the architect's model of a building, the car designer's prototype of a new design, or the aerodynamicist's wind tunnel model of a new airplane. As one considers what is common to these ideas of a model and what purposes are served, perhaps the concept begins to take shape.

Rather than further pursue the various types of physical models, let us consider another less obvious form of model, the mental model. For example, the merchant who mentally speculates: "if I increase the price of this article from x to y, the buyers will still buy enough that I will come out ahead," and the mother who says, "if I spank my child for leaving his toys out, he will stop", are both using intuitive mental models of incredibly complicated economic, sociological, and psychological systems that even experts don't agree on. Freud, Skinner, Erikson and others have all produced models of human psychology that much of modern therapy and advertising are based on. Indeed, reflection indicates that much of human thought is involved with mental model making and the use of these models for decision making, problem solving, or merely pleasure.

The politician who tells the voters what will happen if certain policies are followed, the advertiser who tells the potential buyer the results of using his product, and the preacher who predicts the consequences of evil to his followers are all involved with the building and use of mental and verbal models.

Even the simple speculation of "if I wear these clothes I will look nice" is based on a model of how one's friends will respond to one's dress. Indeed, most processes of experience can be viewed as model building, experimentation, model modification, etc. Further reflection shows how much of one's mental activities can be viewed as involved with modeling and how many academic disciplines are based on models, even though the concept and process is poorly understood and seldom explicitly discussed.

It may seem that the idea of a model and its use is being made so general that it is useless. Our purpose in being so general is to search for what is common in these diverse examples, and to extract it for study and more efficient use.

Rather than follow further examples of mental or verbal models, we will turn to another form of model: the mathematical model. The incredible advances of the physical sciences and engineering disciplines have resulted from the development and use of mathematical models. When one describes the relation between the force applied to an object and its mass and acceleration by the familiar formula F = M ⋅ A , one is using a mathematical model of a physical phenomena. Here, mathematical functions are used to represent physical qualities, so that the interrelationship can be described by equations. If these equations are fundamental and if their solutions accurately simulate the actual phenomena, then they are given the special status of "laws". Consider the cases of Newton's laws, Kirchoff's laws, Faraday's law, Boyle's law, and in other fields, Fechner's law (psychophysics), Paneto's law (economics), etc. Indeed, model formulation and verification is the basis of the so-called scientific method.

An important feature of the various types of models we have discussed is how one can move from one to another. Consider the following hypothetical account of how a physical "law" was developed.

At some point in time it was noticed that if a heavy object was dropped, it fell down. As further experience was accumulated, it was noticed that the object always fell in a straight line toward the earth. Next, after closer observation, it was discovered that the object's speed increased as it fell. The next step was a major one. It required curiosity, mathematical ability, and a real quantum jump to move from the verbal model to a mathematical one where it was conjectured that the velocity was a linear function of time after being dropped, V = Kt . This proved to be incredibly accurate, and thus, a "law" was discovered.

The use of mathematical models has been so successful in many areas that the concept of a model was sometimes forgotten. Indeed, some models are so accurate that users can forget that they are dealing with models and not the actual phenomenon. When we work in areas where accurate models are not available, a good understanding of the modeling process becomes essential.

The first step in choosing a model is deciding what the purpose of the model will be and what questions are being asked. It is obviously an advantage to use the simplest model possible to serve a particular purpose but the danger that over-simplification will destroy the validity of the model always exists.

The second step is the actual construction of the model. Here, the various theories, laws, relations, etc. that apply must be used, and after that, the model requires that new relations be established. In other words, while building the model, one often discovers what data should be collected and what experiments must be performed, as well as what data is irrelevant or misleading. At this point, alterations are often substantial in the model.

The third step is verification or validation of the model. This usually involves some comparisons of the model with the phenomenon it models. One must be very careful at this point to test all of the characteristics the model should have, while remaining within the original goals and purposes set, not violating the assumptions that were made. A common mistake is to use models outside the area for which they were intended.

Verification often involves applying the model to data that was not used in its construction to see if it can explain the observations. If internal relations were used to derive some data, these can be compared with observations. On the other hand, if the model were built by forcing agreement with the observations, then the resulting implied internal relations can be examined for their validity.

All of these steps are done in a rather circular fashion with the attempted use of, and verification of, the model suggesting modification, restructuring and reverification, or in some cases, discarding the whole approach. Some reflection will perhaps show that these are common ideas in modeling, and we try to systematically apply them to the very interesting but very difficult problem of modeling large groups of people.