Scaling in Psychological Measurement PDF

Okay, I will convert the provided text into a structured Markdown format, keeping the important information, summarising where needed, and converting formulas to LaTeX format. ## 2 SCALING If something exists, it must exist in some amount (Thorndike, 1918). Psychologists generally believe that peo...

Okay, I will convert the provided text into a structured Markdown format, keeping the important information, summarising where needed, and converting formulas to LaTeX format. ## 2 SCALING If something exists, it must exist in some amount (Thorndike, 1918). Psychologists generally believe that people have psychological attributes, such as thoughts, feelings, emotions, personality characteristics, intelligence, learning styles, and so on. If we believe this, then we must assume that each psychological attribute exists in some quantity. Psychological measurement can be seen as a process through which numbers are assigned to represent the quantities of psychological attributes. The measurement process succeeds if the numbers assigned to an attribute reflect the actual amounts of that attribute. The standard definition of measurement (borrowed from Stevens, 1946) found in most introductory test and measurement texts goes something like this: "Measurement is the assignment of numerals to objects or events according to rules." In the case of psychology, education, and other behavioral sciences, the "events" of interest are generally samples of individuals' behaviors. The "rules" mentioned in this definition usually refer to the scales of measurement proposed by Stevens (1946). This chapter is about **scaling**, which concerns the way numerical values are assigned to psychological attributes. Scaling is a fundamental issue in measurement, and it involves a variety of considerations. This chapter discusses the meaning of numerals, the way in which numerals can be used to represent psychological attributes, and the problems associated with trying to connect psychological attributes with numerals. Psychological tests are intended to measure unobservable psychological characteristics such as attitudes, personality traits, and intelligence. Such characteristics present special problems for measurement, and this chapter discusses several possible solutions for these problems. These issues are fundamental to psychological measurement, to measurement in general, and to the pursuit and application of science. More specifically, they are important because they help define scales of measurement. That is, they help differentiate the ways in which psychologists apply numerical values in psychological measurement. In turn, these differences have important implications for the use and interpretation of scores from psychological tests. the way scientists and practitioners use and make sense out of tests depends heavily on the scales of measurement being used and even into the nature of numbers. ## FUNDAMENTAL ISSUES WITH NUMBERS In psychological measurement, numerals are used to represent an individual's level of a psychological attribute. For example, your numerical score on an IQ test is used to represent your level of intelligence, your numerical score on the Rosenberg Self-Esteem Inventory is used to represent your level of self-esteem, and a numerical value can even be used to represent your biological sex (e.g., males might be referred to as "Group 0" and females as "Group 1"). Thus, psychological measurement is heavily oriented toward numbers and quantification. Numerals can represent psychological attributes in different ways, depending on the nature of the numeral that is used to represent an attribute. Properties of numerals influence the ways in which numerals represent psychological attributes. As shown in Figure 2.1, this section outlines three important numerical properties, and it discusses the meaning of zero. In essence, the numerical properties of identity, order, and quantity reflect the ways in which numerals represent potential differences in psychological attributes. Furthermore, zero is an interestingly complex number. this complexity has implications for the meaning of different kinds of test scores. A "score" of zero can have extremely different meanings in different measurement contexts. Here's a description of Figure 2.1: The image is a diagram outlining the properties of numbers, showing a progression from "Least information" to "Most information." Each level includes a property: * **Least information**: Identity - Same vs. different * **More information**: Order - Relative amount of attribute * **Most information**: Quantity - Exact amount of attribute ### The Property of Identity The most fundamental form of measurement is the ability to reflect "sameness versus differentness." Indeed, the simplest psychological measurements are those that differentiate between categories or groups of people. For example, you might ask first-grade teachers to identify those children in their classrooms who have behavior problems.The children who are classified as having behavior problems should be similar to each other with respect to their behavior. In addition, those children should be different from the children who are classified as not having behavioral problems. the individuals within a category should be the same as each other in terms of sharing a psychological feature, but they should be different from the individuals in another category. In psychology, this requires that we sort people into at least two categories. The idea is that objects, events, or people can be sorted into categories that are based on similarity of features. In many cases, these features are behavioral characteristics reflecting psychological attributes, such as happy or sad, introverted or extroverted, and so on. Certain rules must be followed when sorting people into categories. The first and most straightforward rule is that, to establish a category, the people within a category must satisfy the property of identity. That is, all people within a particular category must be "identical' with respect to the feature reflected by the category. For example, everyone in the "behavioral problem" group must, in fact, have behavioral problems, and everyone in the “no behavioral problem" group must not have behavioral problems. Second, the categories must be mutually exclusive. If a person is classified as having a behavioral problem, then they cannot simultaneously be classified as not having a behavioral problem. Third, the categories must be exhaustive. If you think that all first-graders can be classified as either having behavioral problems or not having behavioral problems, then these categories would be exhaustive. To summarize the second and third rules, each person should fall into one and only one category. When numerals have only the property of identity, they represent sameness vs. differentness, and they serve simply as labels of categories. The categories could be labeled with letters, names, or numerals. You could label the category of children with behavior problems as "Behavior Problem Children,” you could refer to the category as "Category B," or you could assign a numeral to the category. For example, you could label the group as "0," "1," or "100." When having only the property of identity, numerals are generally not thought of as having true mathematical value. For example, if "1" is used to reflect the category of children with behavioral problems and "2" is used to represent the category of children without behavioral problems, then we would not interpret the apparent 1-point difference between the numerical labels as having any form of quantitative significance. The latter point deserves some additional comment. When making categorical differentiations between people, the distinctions between categories represent differences in kind or quality rather than differences in amount. Again returning to the teachers’ classifications of children, the difference between the two groups is a difference between types of children-those children who have behavioral problems and those who do not. In this example, the classification is not intended to represent the amount of problems (e.g., a lot vs. a little) but rather the presence or absence of problems. In this way, the classification is intended to represent two qualitatively distinct groups of children. Of course, you might object that this is a rather crude and imprecise way of measuring or representing behavioral problems. You might suggest that such an attribute is more accurately reflected in some degree, level, or amount than in a simple presence/absence categorization. This leads to additional properties of numerals. ### The Property of Order Although identity is the most fundamental property of a numeral, the property of order conveys more information. if numerals have only the property of identity, they convey information about whether two individuals are similar or different but nothing more. In contrast, when numerals have the property of order, they convey information about the relative amount of an attribute that people possess. When numerals have the property of order, they indicate the rank order of people relative to each other along some dimension. In this case, the numeral 1 might be assigned to a person because they possess more of an attribute than anyone else in the group. The numeral 2 might be assigned to the person with the next greatest amount of the attribute, and so on. For example, teachers might be asked to rank children in their classrooms according to the children's interest in learning. Teachers might be instructed to assign the numeral 1 to the child who shows the most interest in learning and 2 to the child whose interest in learning is greater than all the other children except the first child, continuing in this way until all the children have been ranked according to their interest in learning. When numerals are used to indicate order, they again serve essentially as labels. For example, the numeral 1 indicated a person who had more of an attribute than anyone else in the group. The child with the greatest interest in learning was assigned the numeral 1 as a label indicating the child's rank. Each person in a group of people receives a numeral (or letter) indicating that person's relative standing within the group with respect to some attribute. Although the property of order conveys more information than the property of identity, it is still quite limited. While it tells us the relative amount of differences between people, it does not tell us about the actual degree of differences in that attribute. They are still a rather imprecise way of representing psychological differences. ### The Property of Quantity Although the property of order conveys more information than the property of identity, the property of quantity conveys even greater information. when numerals have the property of order convey information about which of two individuals has a higher level of a psychological attribute, but they convey no information about- the exact amounts of that attribute. In contrast, when numerals have the property of quantity, they provide information about the magnitude of differences between people. At this level, numerals reflect real numbers or, for our purposes, numbers. The number 1 is used to define the size of the basic unit on any particular scale. All other values on the scale are multiples of 1 or fractions of 1. Each numeral (e.g., the numeral 4) represents a count of basic units. Think about a thermometer that you might use to measure temperature. To describe how warm the weather is, your thermometer reflects temperature in terms of "number of degrees" (above or below 0). The degree is the unit of measurement, and temperature is represented in terms of this unit. Units of measurement are standardized quantities; the size of a unit will be determined by some convention. For example, 1 degree Celsius ($1^\circ C$) is defined (originally) in terms of 1/100th of the difference between the temperature at which ice melts and the temperature at which water boils. Real numbers are also said to be continuous. In principle, any real number can be divided into infinitely small parts. In the context of measurement, real numbers are often referred to as scalar, metric, or cardinal, or sometimes simply as quantitative values. The power of real numbers derives from the fact that they can be used to measure the amount or quantity of an attribute of a thing, person, or event. ### The Number 0 The number 0 is a strange number (see Seife, 2000), with at least two potential meanings. To properly interpret a score of 0 in any particular situation, you must understand which meaning is relevant in that situation. **Absolute Zero:** Zero reflects a state in which an attribute of an object or event has no existence. Zero in this context is referred to as absolute zero. In psychology, the best example of a behavioral measure with an absolute 0 point might be reaction time. **Relative or Arbitrary Zero:** A zero of this type is called a relative or arbitrary zero. In the physical world, attributes such as time (e.g., calendar, clock) and temperature measured by standard thermometers are examples. Celsius scale represents the melting point of ice, but it does not represent the "absence" of anything The psychological world is filled, at least potentially, with attributes having a relative 0 point. For example, it is difficult to think that conscious people could truly have no (zero) intelligence, self-esteem, introversion, social skills, attitudes, and so on. Despite the fact that most psychological attributes do not have an absolute 0 point, psychological tests of such attributes could produce a score of 0. In such cases, the zero would be considered arbitrary, not truly reflecting an absence of the attribute. A $z$ score of 0 indicates an average score within the set of score. In this case, zero represents an arbitrary or relative zero. In psychology, there can be a problem in determining whether a test score of zero should be thought of as relative or absolute. The problem concerns the distinction between the test being used to measure a psychological attribute and that psychological attribute itself. Consider an example that Thorndike (2005) used to illustrate this problem. Thorndike describes a scenario in which a sixth-grade child takes a spelling test and fails to spell any of the words correctly. The child thus receives a score of 0 on the test. The test itself has an absolute 0 point, indicating that the child failed to spell any words correctly. The question then becomes how we are going to treat the child's test score. Should we consider it an absolute zero or a relative zero? This is important because the type of zero associated with a test affects how we interpret and use the test scores. For example, we might plan to conduct statistical analyses on test scores for a research study and if we assume zero scores then we can feel performing operations such as multiplication and division on the test scores. In sum, the three properties of numerals and the meaning of zero are fundamental issues that shape our understanding of psychological test scores. If two people share a psychological feature, then we have established the property of identity. ## UNITS OF MEASUREMENT The property of quantity requires that units of measurement be clearly defined. Quantitative measurement depends on our ability to count these units. Before discussing the process and implications of counting the units of measurement, we must clarify what is meant by a unit of measurement. In many everyday cases of physical measurement, the units of measurement are familiar. In contrast, in many cases of psychological measurement, units of measurement are often less obvious. presumed to related to the psychological attributes themselves? Imagine that you are building a bookshelf and you need to measure the length of pieces of wood. Unfortunately, you cannot find a tape measure, yardstick, or a ruler of any kind-how can you precisely quantify the lengths of your various pieces of wood? One solution is to create your own unique measurement system. First, imagine that you happen to find a long wooden curtain rod left over from a previous project. You cut a small piece of the curtain rod; let us call this piece an "xrod" (see Figure 2.2). Let call this piece an "xrod". Because your pieces of bookshelf wood are longer than the xrod, you will need several xrods. Therefore, you can use this original xrod as a template to produce collection of identical xrods. That is, you can cut additional xrods from the curtain rod, making sure that each xrod is the same exact length as your original xrod. You can now use your xrods to measure the length of all your pieces of wood. For example, to measure the length of one of your shelves, place one of the xrods at one end of the piece of wood that you will use as a shelf. the length of the object. Now count the number of xrods, and you might find that the shelf is "8 xrods long." Here's a description of Figure 2. 2: The image shows a diagram of measuring a shelf using "xrods." It illustrates how multiple xrods are placed end-to-end to measure the length of a shelf. The shelf is labeled as being 8 xrods long. You have just measured length in "units of xrods “and the measure is good as any measure except the is the only one who representation! Arbitrariness is an important concept in understanding units of measurement, and it distinguishes between different kinds of measurement units ,the unit size can be arbitrary-The size of your unit of measurement, the xrod, was completely arbitrary A second form of arbitrariness is that some units of measurement are not tied to any one type of object. Our xrods can be used to measure the spatial extent of anything that has spatial extent. A third form of arbitrariness is that, when they take a physical form, some units of measurement can be used to measure different features of objects. ## ADDITIVITY AND COUNTING The need for counting is central to all measurement. Whether we are measuring a feature of the physical world or of the psychological world, all measurement involves counting. Similarly, when you use a behavioral sampling procedure (i.e., a test) to measure a person's self-esteem, you count responses of some kind. For example, you might might interpret the number of "true" marks as indicating the level of the respondent's measurement. counting as a facet of measurement involves that might the unit same Recall again the xrod example and not a good measure of the amount of actual In addition, one's mind the is not as easy with quantities where quantities cant the amount of actual Counts: and measurement 1997 for a of mathematics measurement only when one is counting to reflect of that amount student student ## FOUR SCALES OF MEASUREMENT As discussed earlier, measurement involves the assignment of numbers to observations to in particular ways. scales of measurement (Bartholomew, 1996). Framework measurement rule Stevens these Nominal: fundamental level of measurement The and The we of social what you Ordinal ranking one amount you require to amount has the measure of the Interval one not only one amount has the measure of the ## Scales interval (Ghiselli to we the intervals on one will the ratio indicate distances 80 This require to is is measuring two underlying that for one the on for it that for and is the in the ## to for that the of and that for and you that The hair: $(1+1+1+1+ 1+1+1+1+1+1+1+1+1+2+2+2+ 2+3+3+3)/20$ $= 30/20 = 1.5$