## Scaling

Stevens (1951) proposed an influential classification of measurement scales. Beginning with the lowest level in the hierarchy, nominal scales assign each participant to an unordered category (e.g., marital status: single, married, divorced, widowed). Ordinal scales assign each participant to one of several ordered categories (e.g., clothing size: 1 = small, 2 = moderate, 3 = large). Interval scales assign participants a number such that a one-unit difference at any point on the scale represents an identical amount of change (e.g., a change from 3 to 4 degrees or from 30 to 31 degrees represents the same change in temperature on the Celsius scale). Finally, ratio scales share the same equal interval property as the interval scale, but in addition have a true 0 point where 0 represents absence of the measured quantity (e.g., height in centimeters).

Stevens originally argued that the level of measurement limits the type of statistical analysis that may be performed. This position is potentially disturbing because many measures in psychology may not greatly exceed an ordinal level of measurement. Indeed, Krosnick and Fabrigar (in press) have shown that labels used to represent points on Lik-ert-type items often do not come close to approximating equal spacing on an underlying dimension. On the other hand, several authors (e.g., Cliff, 1993; McDonald, 1999) have noted that for t-tests and analysis of variance, whether the measurement scale is ordinal, interval, or ratio, makes only a modest difference in the conclusions about the existence of differences between groups, so long as the assumptions of the analysis (e.g., normality and equal variance of residuals) are met. Similarly, for linear regression analysis or structural equation modeling, the level of measurement also does not have a profound effect on tests of the significance of coefficients. These results occur because monotonic (order preserving) transformations typically maintain a high correlation between scores on the original and transformed scales. Often, ordinal measurement will be "good enough" to provide an adequate test of the existence of a relationship or group difference even with statistical tests originally designed for interval level data.

However, if we have hypotheses about the form of the relationship between one or more independent variables and the dependent variable, ordinal measurement is no longer "good enough." Longitudinal analyses testing trend over time require interval level measurement. The origin and units of the scale must be constant over time; otherwise, the test of the form of the relationship will be confounded with possible effects of the measuring instrument. When standard statistical procedures designed for interval-level data are used with ordinal-level data, estimates of parameters of the growth model will be seriously biased. Special methods designed explicitly for ordinal-level data and large sample sizes are required (Mehta, Neale, & Flay, 2004).

Changes in the origin or units of the scale can happen because raters explicitly or implicitly make normative judgments relative to the participant's age and gender.1 Consider the trait physically active. Informants may rate the second author as being very physically active—a rating of 8 on a 9-point scale ranging from "not at all" to "extremely" active at age 25 and then again at age 50. Yet, physical measures of activity (e.g., a pedometer) may show twice as much physical activity at age 25 as at 50. In effect, such ratings may be "rubber rulers" that correctly describe the standing of the individual relative to a same age comparison group. However, when changes occur in either the origin or the units of the scale, clear interpretation of the results of longitudinal analyses focused on the form of change is precluded. These problems do not characterize all longitudinal studies. Physical measures (e.g., height, blood pressure) and many cognitive measures provide invariant measurement at the interval level. Some rating scale measures may approximate interval-level measurement and be suitable for short-term longitudinal studies. But, few investigators consider this fundamental issue— the origin and units of the measure must be constant over time. Such invariance is fundamental in interpreting the results of longitudinal studies of change. We revisit this issue later in the chapter.