# An overview of optimal scaling

### Overview of optimal scaling

Many studies involve the measurement and analysis of nominal and ordinal variables. Clearly, relative to interval variables, these levels of measurement are less amenable to analysis. Fortunately, in many situations, nominal and ordinal variables can be converted to interval variables. In particular, a technique called optimal scaling can be conducted to implement this conversion.

Optimal scaling is an umbrella term for a host of different tests. For instance, one form of optimal scaling is applicable in the context of multiple regression (see Categorical regression analysis). Another form of optimal scaling is applicable in the context of principal components analysis, and so forth.

### Shortfalls of nominal and ordinal variables

Nominal and ordinal variables are rife in psychology. To define these levels of measurement, suppose you wanted to classify individuals on the basis of religion (e.g. Christians, Muslims, etc). This measurement is regarded as nominal, because these classifications are merely names& they do not reflect order.

Suppose now we wanted to classify individuals on the basis of popularity (1 = very unpopular to 5 = very popular). This measurement is usually regarded as ordinal, because the classifications reflect order but not the degree to which individuals differ. Of course, some researchers may assume that each interval in this scale is virtually equal. As a consequence, this measurement can be regarded as interval. For the purposes of this document, however, this measurement will be designated as ordinal.

Several problems can arise from nominal and ordinal variables

• Many statistical procedures cannot be undertaken, or at least are somewhat suspect, when the variables conform to these levels of measurement (e.g., factor analysis)
• Statistical tests that can be applied to nominal and ordinal dependent variables are often less powerful or effective than statistical tests that can be applied to interval data (e.g., Mann-Whitney tests are often less powerful than t-tests)
• Statistical tests that can be applied to nominal or ordinal data are often difficult to implement and interpret definitively (e.g. multivariate permutation tests, loglinear analysis)

### Function of optimal scaling for ordinal variables.

Conceivably, the drawbacks of nominal and ordinal variables could be circumvented by converting nominal and ordinal variables to interval variables. To some degree, optimal scaling can fulfil this purpose. That is, optimal scaling derives interval measures from nominal and ordinal measures.

First, consider a researcher who wants to ascertain the relationship between wage and popularity. In this case, popularity is measured on a 5 point ordinal scale. A scatterplot of these variables is in Figure 1. This scatterplot does not conform to a straight line. Hence, linear regression would not reveal a strong relationship between wage and popularity.

Figure 1

In Figure 1, the distance between consecutive levels of popularity are deemed to be equal. Of course, these differences are entirely arbitrary, because the scale is ordinal. Conceivably, the researcher could manipulate the distance between each level of popularity to ensure the data conform more closely to a straight line.

This process is demonstrated in Figure 2. In particular, the distance between 1 and 2 is truncated considerably. Similarly, the distances between 3, 4, and 5 are also truncated. After implementing this transformation, the scatterplot conforms to a straight line.

The numbers in brackets reveal a new scale of popularity. This scale reflects the actual distances between consecutive levels of popularity that optimise the relationship between wage and popularity. Hence, this scale is deemed to be interval.

Figure 2

### Function of optimal scaling for nominal variables.

The previous section demonstrated how ordinal scales can be converted to interval scales. This section reveals how nominal scales can also be converted to interval scales. To illustrate this point, suppose we ascertain the religion, self-esteem, height, extroversion and so forth, of many individuals.

Figure 3 reveals the relationship between two of these variables, self-esteem and religion. In this case, self-esteem seems to vary across religion. Unfortunately, the reasons for this relationship are unclear. What factors are responsible for this pattern of results?

Figure 3

To resolve this issue, the researcher should derive the hypothetical variables that underlie the nominal variable, religion. To this end, the researcher can undertake a technique that investigates the degree to which the religions differ on the variables. For instance, perhaps Christians and Muslims tend to be similar on most of the variables, whereas Christians and Hindus tend to be disparate.

This process can ultimately generate the graph displayed in Figure 4. This figure essentially represents the degree to which religions differ on the variables. Religions that appear close together on this graph are similar, whereas religions that appear far apart are disparate.

Figure 4

Note that Figure 4 is two dimensional, rather than one-dimensional, and hence provides additional information. For instance, Figure 4 reveals that Hindus and Muslims are extremely different on Dimension 1, but only marginally different on Dimension 2. In other words, Hindus and Muslims are extremely different on one set of variables, but only marginally different on another set.

This plot thus suggests that each religion can be represented by two interval variables: Dimension 1 and Dimension 2. For example, Hindus can be represented as a '1' on Dimension 1 and a '4' on Dimension 2.

This ability to represent nominal categories as interval variables is futile, unless the dimensions can be interpreted. Fortunately, these dimensions can be readily interpreted. For instance, consider the first dimension, which is lowest in Hindus, moderate in Budhhists and highest in Muslims. A close scrutiny of this pattern suggests that Dimension 1 may represent age of religion, where higher scores pertain to more recent religions.

Likewise, consider the second dimension, which is highest for Hindus and Buddhists, moderate for Muslims, and lowest for the other religions. This pattern suggests that Dimension 2 may represent the primary location of each religion, where high scores reflect Eastern religions and low scores reflect Western religions. As a consequence of these interpretations, Figure 4 can be converted to Figure 5, which now labels each dimension.

Figure 5. Interpretation of the dimensions.

In short, each category of religion can now be represented by two interval variables. These variables can then be correlated with the other variables. These correlations will clarify why religion seems to correlate with self esteem. For instance, the findings may reveal that self-esteem seems to be higher in Eastern areas.

### Classes of optimal scaling

To reiterate, nominal and ordinal variables can be converted to interval scales. The precise nature of this conversion, however, is contingent upon the purpose of optimization. For instance, in the example in which wage was related to popularity, the optimization process was undertaken to maximise the relationship between the variables. In contrast, for the example in which religion was related to self-esteem, the optimization process was designed to explain or represent the similarities between each pair of religions.

In other words, you cannot simply press a button that will convert nominal and ordinal scales into interval scales. Instead, optimal scaling is usually a component of another procedure. In particular, SPSS comprises five techniques that, among other things, involve optimal scaling. For example:

• Categorical regression analysis: Undertakes multiple regression when some of the variables are ordinal or nominal.
• Nonlinear Principal Components Analysis: Undertakes conventional principal components analysis, when some of the variables are ordinal.
• Nonlinear Canonical Correlation: Undertakes canonical correlation when some of the variables are ordinal and nominal.
• Correspondence Analysis: Undertakes something like a factor analysis on two categorical variables.
• Homogeneity Analysis (also called Multiple Correspondence Analysis): Undertakes something like a factor analysis on more than two categorical variables.

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