1 Models
For our purposes, “quantitative thinking” refers to habits of mind that have been found helpful and illuminating when thinking about things that can be measured and the relationships between them. Central to quantitative thinking is the habit of building, exercising, improving, and drawing conclusions about ☞ models ☜.
The everyday word model is used in several distinct ways. A human model might be displaying clothing or jewelry, giving potential customers an idea of how the items look. Medical research uses animal models to study the physiology of disease progression or to test potential treatments ultimately intended for use in humans. A baby doll is a model of an infant human used by children to develop their understanding of how the various parts of a person come together or to mimic or practice interactions with babies. There are model trains and planes that represent large real-world objects and bring them down to a child’s scale. A blueprint is a model of a building; model houses or apartments let people see a building at full scale without intruding on neighbors’ privacy.
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1.1 What is a model?
All of these examples of models have in common two key properties. First, a model is a ☞ representation ☜ that is more convenient in some ways than the original object. For instance, model planes and trains comfortably inhabit a small playroom. A table and a pencil suffice to explore and annotate blueprints.
Second, a model is made with a ☞ purpose ☜ in mind. The purpose is manifest in the ways that the model is more convenient than the original. For instance, with a blueprint model, a designer can quickly check whether nearby doors will interfere with one another and fix the problem, speculate whether there is space for an additional closet in the hallway, and measure the areas of rooms to calculate the amount of building material needed. To implement a design change, use a pencil or an eraser to add it to the model. Biologists build models of ecological systems to explore their robustness and vulnerabilities, engineers build models of new designs or existing structures to see how they will react to usual and unusual loads and stresses, and financiers build models of investments and the future cash flows those investments are expected to yield. Weathermen rely on models of atmospheric processes to make predictions and measure uncertainty.
A blueprint is convenient for planning and communicating a building’s structure, but it is useless for providing shelter. A model airplane conveys the shape and configuration of wings, fuselage, and tail, but will not carry passengers.
We employ quantitative models for many purposes, for instance, testing theories about how real-world systems work and identifying missing elements in those theories, translating partial knowledge of a system into information needed for decision-making, or checking how interventions in a real-world system might change outcomes.
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1.2 Example: Modeling an epidemic
The features of an epidemic, such as COVID-19, are familiar. A new pathogen, such as SARS-CoV-2, emerges from an unknown origin and infects a handful of people. The pathogen is contagious: infective people transmit it, somewhat at random, to susceptible people, some of whom become infective and can transmit the virus to others. At the same time, infective people recover, losing their ability to convert susceptibles into new infectives.
The verbal description in the previous paragraph is a model of an epidemic. The mechanism is called “Susceptible, Infective, Recovered,” or ☞ SIR ☜ for short. Using techniques introduced in the following chapters, we have translated the verbal description into a quantitative model that allows us to simulate the outbreak. We will not go into the details here: The model describes a bookkeeping process that, based on the current numbers of susceptible and infective people, calculates the next day’s tally of new cases and recoveries.
To explore the model, we have connected the graph below to a computer program that implements the bookkeeping and plots, day by day, the numbers of susceptible and infective people over the time course of the epidemic.
The graph shows an epidemic in a population of 100,000 people. It starts on day 0 with the arrival of the initial cases and progresses day by day for 50 days. The number of infectives (red) grows at first, then declines. The susceptible population (black) decreases in proportion to the number of infectives.
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1.3 Quantity, value, variable
We will use the word “system” a lot. A system is a collection of factors—each representing a characteristic of the system—that interact with one another. Drawing from the world of statistics and data, we will use the word “☞ variable ☜” to denote the kind of characteristic. For example, temperature, humidity, and comfort are all variables.
A ☞ quantity ☜ is an ☞ operationalized ☜ form of a variable. Sometimes the operationalization is obvious and familiar. The measurement of temperature is a case in point. Before 1600, “temperature” was sensed by skin or the glowing color of hot metal, but not measured. Around 1610, the “thermoscope” enabled visualization of temperature using the level of water in a tube. Not until more than a decade later was a quantitative scale—a ruler—added to the thermoscope, turning it into a thermometer. From a modeling perspective, a thermometer is a device that operationalizes the quantitative measurement of temperature.
Age and height are familiar characteristics. We operationalize height, for instance, by marking the top of a person’s head on a wall or door frame, then using a ruler to measure (that is, “quantify”) the distance from the floor to the mark. We operationalize age—this will be obvious—by calculating the time difference between birth and the moment of measurement. (More commonly, we ask the person!)
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Sometimes an operationalization is technical, obscure, and to some extent arbitrary. For instance, the economic variables “cost of living” and “poverty level” have official operationalizations in the US. An officially defined, standardized “market basket” serves as the basis for calculating the cost of living. The government defines poverty level by a formula that involves the cost of living, family size, and composition, and is based on pre-tax income and does not depend on location, even though housing costs, food, energy, and such vary markedly from one place to another.
In a quantitative model of a system, each characteristic of the system is operationalized into a quantity that measures that characteristic. Being attentive to the details of the operationalization can be an important component of critical thinking.
The ☞ value of a quantity ☜ is a specific numerical level.1 Typically, the value will differ across contexts, say, from hour to hour or place to place. For example, as I write this, the outdoor temperature is 52 degrees F, as operationalized by the US Weather Service from a thermometer in a ventilated, shielded box at a specific location in my city. Values can change; the temperature tomorrow will likely be different. In reading “the temperature tomorrow will be different,” the underlying idea is not that the variable or operationalization will change—the Weather Service will keep doing what it has done for many years. Instead, the idea of “different” is that the value of the quantity will be different.
1.4 Example: Quantifying comfort?
Imagine a hot, humid, sultry summer afternoon in the US. The relative humidity is 90%, the temperature 95◦F (equivalent to 35◦C). Sultry weather makes us uncomfortable.
It is helpful to provide people with a quantitative scale of comfort so they can make decisions about outdoor activities. The context is a system composed of three quantities: humidity, temperature, and comfort. (There may be other quantities that we will need eventually to tell a complete story—wind speed, for instance—but for simplicity, we work with just the three.)
It may be evident that humidity and temperature are inherently quantities, but what about comfort? In modeling, we often deal with concepts that are not inherently quantitative and may be vague. There are many ways to operationalize comfort to a quantitative scale, each imperfect in its own way. A favored operationalization of weather-related comfort is the “feels like” scale, known as the “heat index.” The heat index derives from a technical model of convective heat transfer from the skin. The heat index is not a comprehensive operationalization of the general concept of comfort. Still, the heat index captures an important aspect of comfort quantitatively.
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Another example of an imperfect, but practical operationalization is the pain-rating scale shown in Fig 1. 2.
A common criticism of quantitative analysis is that any particular quantitative operationalization is imperfect and incomplete. This criticism can be valid and serve as a prompt to refine the operationalization and the analysis method, but it by no means invalidates quantitative analysis. The test of operationalization is not whether it is perfect, but whether it is good enough to contribute meaningfully to the purposes for which we are conducting the quantitative analysis.
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Note 1: Varying meanings of variable
A source of potential confusion for students is the different meanings of words across different fields. Statistics and algebra assign different meanings to the word “variable.” In this book, “variable” will be used in the statistical sense. However, our distinction between “variable” and “quantity,” that is, the distinction between a characteristic and the operationalization of that characteristic, is not made systematically in statistics.
- Statistics
- A variable is any characteristic that, in principle, could be recorded as data. The value of a variable is what is recorded, typically one of a set of categories (e.g. “red,” “blue,” “brown) or a number. A variable varies in the sense its recorded value may differ from one”unit of observation” to another. The word “unit” refers to a specimen, such as a single book in a library catalog or a single patient in a clinical study. Although the value may vary from specimen to specimen, the variable itself is defined in a meaningfully consistent way across specimens.
- Algebra
- A variable is a symbolic placeholder such as a letter or name for a quantity that might not yet be known numerically. A variable changes depending on the circumstances. For instance, the area of a circle is \(\pi\, r^2\), where \(r\) is the radius. Different circles can have different radii, so \(r\) is a variable. In contrast, \(\pi\) is a ☞ constant ☜, that is, a quantity that is the same in all circumstances. A constant has a value; a variable is a placeholder for a value.
Sometimes algebraic variables are called “unknowns.” The goal of many algebra tasks is to find a value for a variable, that is, to “solve” for an unknown. In contrast, in statistics, one measures a variable on each member of a group of specimens.
- Quantitative reasoning
- As in statistics, a variable is a characteristic. A quantity is a numerically measurable representation of that characteristic as defined by an operationalization. A value is a specific level of the quantity. “Unit” in quantitative reasoning has an entirely different meaning than in statistics. A unit is a standard for measurement, such as a centimeter or a liter. The unit is an essential component of a quantity, as we will see in Chapter 2.