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Content created: 070719 & 120815
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Quick Essays on Theory

Variables & Models

One way to explain something is to describe how it is related to other things. The usual way to do this is to imagine that what we want to understand is dependent upon or predicted by a series of other facts about the world that we already know (or can find out). What we want to find out is conventionally called the “dependent variable.” What influences it are the “determining variables” or, more conventionally, “independent variables.”

Very pure logical example:
X = 2 + 3 (X is the dependent variable)
X = 5 - 3 (X is the dependent variable)
X = 5 - 2 (X is the dependent variable)
X = Y + Z (X is the dependent variable)

More complex real-life example:
What explains the form of the human pelvis?

pelvis form = walking constraint + birth constraint

In other words, the human pelvis must accommodate both upright walking and bearing children.

(Walking requires a joint at the hip and a way to support and balance the weight of the upper body. Giving birth requires a sufficiently large passageway for the enormous human fetus to emerge.)

So arguably the activities of walking and bearing children determine (explain) the form of the pelvis(at least partially).

Variables

Dependent variables. When, for purposes of argument, we describe one feature of the universe (such as pelvis form) as being dependent upon others (such as walking and giving birth), we call it a “dependent” variable. Our goal is to figure out what other variables it depends upon, and if possible how much it depends on each.

Determining (“independent”) variables. Seen the other way around, we want to know how well we can predict the dependent variable if we know the values of other variables. In our example the form of the pelvis is the “dependent” variable. What it depends upon (the “determining” or so-called “independent” variables) are the needs of walking and giving birth. Sometimes it is possible to be mathematically rigorous about this.

Which Variable is Which?

Since we are interested in predicting one variable from the values of other variables, we can start anywhere. For example, we can start with a different problem that involves the same features: What determines how people are able to walk? The ability to walk is dependent upon the form of the pelvis, among other things. Here the ability to walk is what we want to explain, so it is the dependent variable. The form of the pelvis is now taken as a “given,” that is, as one of the relevant determining (“independent”) variables.

Bottom Line: Nothing is inherently a dependent or a determining variable. It is dependent or determining only in the context of a particular question we are trying to answer. (Read that three times.)

An Archaeological Example: Site Distribution

In archaeology we may want to explain the location of sites in a particular part of the world. Site location becomes the dependent variable, the thing we are trying to explain. What does site location depend upon? Grossly, it depends upon resource distribution, but resources may include several different things: available water, access to hunting grounds, shelter from inclement weather, fertile soil for farming, or the location of other human groups (such as predatory raiders or helpful allies). In this particular example, each of these would be interpreted as affecting the dependent variable, i.e., the site location. Each of these, in turn, could, of course, involve other things (such as hunting grounds for different kinds of animals). All of these are determining (independent) variables.

Models

A “model” is a group of related variables and the system of relationships that we believe (or suspect) links them. Another word for a model is “theory,” but the word “model” has the advantage of being easy to envision. An architect’s model of a proposed building is not the building itself, but it is like the proposed building in some useful respects. A model (theory) of climate change is not the process of climate change itself, but it is like the process in some useful respects. Similarly, other models (theories) are created to contribute to our understanding of whatever it is that they represent.

Some models make more accurate predictions or take account of more evidence than others. For some purposes, such as a class lecture, we may want a very simple model, with only a couple of variables. For example: goats eat tree seedlings, so people who keep free-ranging goats and depend on trees tend to destroy their own environments because the goats eat the seedlings and therefore prevent the trees from reproducing themselves.

For other purposes, a far more detailed model may be necessary, one which takes all the determining variables accurately into account. For example, if you are planning an event, you want to be sure you have included all the possible expenses (determining variables) before you set the price (dependent variable). Getting the model wrong can be expensive.

An Archaeological Example: Carrying Capacity

Carrying capacity is the maximum number of people that a given piece of land can support. What are some of the determinants of carrying capacity? That is, if carrying capacity is the dependent variable, what are the determining (“independent”) variables that we need to include in our model in order to predict it? Some of them are (1) the size of the territory, (2) the climate, (3) the nature of the terrain, (4) the available plant and animal life, and so on. Other determining variables include the knowledge that the humans have of how to use the land and their preferences and beliefs. Do they know about farming or do they live by foraging? Do they think salamanders are edible? Do they trade with other regions?

Obviously each of these is potentially quite complex.

Our model of the carrying capacity of a given region must take account of all of the most important factors that might influence it. Once we have confidence that we have included most of what matters, we can then use the output of the model (the carrying capacity we have found) as a determining (“independent”) variable in some other model. We can ask, for example, whether human populations tend to grow till they approximate the carrying capacity of the land, or we can ask whether, as population size approaches the carrying capacity, the rate of innovation increases, thereby further increasing the carrying capacity. Or we can ask whether, as population approaches the carrying capacity the number of judicial proceedings rises, or malnutrition results in lower birth weights, or other imaginable effects.

Here are a few dependent variables of interest in many World Civ courses:

Other Models: An Ethnographic Example: Matchmakers

Are there other kinds of models? Of course. Probably none is as useful or broadly used as the model of dependent and determining variables. But here is an example of a different kind of model, one that focuses on a metaphor:

I have done extensive studies of arranged marriage. Over most of the world, most marriages involve a “window of opportunity” beginning when a young person is considered marriageable (or engageable) and ending when he or she is regarded as “over the hill” or when the available potential mates are used up. At the beginning of this period, the young person (or the parents or the matchmaker) can afford to be picky, looking over many possible spouses and seeking the most desirable one. Towards the end, as the number of potential spouses drops off, a certain desperation sets in, and the criteria for what constitutes a desirable match become less stringent.

I hypothesize that when a region has more matchmakers, they seek to finalize marriages before their competitors do while there are still potential spousal candidates to be had. Therefore when there are more matchmakers (perhaps because custom awards them a high remuneration), the minimum age at marriage tends to drop, until in some cases children are “spoken for” even before they are born.

It would be possible to recast this model in terms of independent and determining variables, and it would prove in fact to be a series of chained propositions. But doing so is not necessarily the easiest way to envision what is happening. A metaphorical phrase like “window of opportunity” turns out to be a much more efficient way of describing the situation and making sense of the motives of the individuals involved.

Models & Predictions

By definition, a model is a description of a system of relationships that is abstracted from actual cases to describe a “general” or “typical” situation. The marriage example shows one more important feature of models: it allows one to hypothesize yet more relationships. The proposition that having more matchmakers decreases the average age of marriage (or engagement) is not a fact about the world; it is a prediction based on the “window-of-opportunity” metaphorical model. If that model is “robust” then predictions derived from it should turn out to be true. When the logically derived predictions turn out to be wrong, it suggests that there is something the matter with the model.

But without the model, we wouldn’t have any derivative predictions, and we might very well not collect the data that would allow some of our most interesting discoveries. In this example, nobody has ever collected a data set that can be used to correlate the number of competing matchmakers in a region with mean marriage age. Such a finding would perhaps help to explain customs of pre-natal or childhood engagement in many parts of the world. But since nobody has thought of it, the research has not (yet) been done. (Here’s your chance!)

  1. In other words, until there is a model, there is no logical way to decide what constitutes data.Imagining a model is the creative act that establishes what bits of the world constitute relevant data.
  2. Making predictions from the model that turn out to be correct gradually moves the model from random guess to informed hunch to scientific “fact.”
  3. Making predictions from the model that are logical but turn out to be wrong shows that the model is deficient and inspires investigators to modify it and try again.

Range of Convenience of Definitions and Models

Footnote
George A. Kelly 1963 A Theory of Personality: The Psychology of Personal Constructs. New York: W.W. Norton.

The influential mid-XXth-century psychologist George A. Kelly coined the terms “focus of convenience” and “range of convenience” to refer to properties of an individual’s mental model of some part of the world, a model that Kelly called a “personal construct.”Footnote Confronted with a new phenomenon, a person seeks to understand it using the personal constructs already part of his or her repertoire. (For example, people who smile are usually friendly.) However some phenomena are not well accommodated by a person’s pre-existing models. (That smiling person is pointing a gun-like object at me and making a lot of noise. Smiling people I have met before don’t do that.) A prime act of mental creativity and maturation comes in making or borrowing a new mental model better suited to the new situation.

Since Kelly coined it, the phrase “range of convenience” has proven to be a useful idea, applicable far beyond the realm of cognitive psychology, since all models (and definitions), not just personal ones, obviously have a focus and range of convenience, and all prove inapplicable as they move outside their range of convenience, where new concepts become necessary.

For example, is it useful to see the political system of the traditional Nyoro kingdom of Uganda or of the Bronze Age Zhōu dynasty of China as a “feudal system”? Do those cases fall outside the range of convenience of our model (or definition) of feudalism, created, as it was, to understand the dynamics of feudalism in medieval Europe? Or do they merely fall further from the model’s focus of convenience than the European case does?

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