# Scale Factor

A number that multiplies some quantity. For example, if one respondent is estimated to have appeal scores for three alternatives of 1, 2 and 3 and another respondent has appeal scores for the same alternatives of 2, 4 and 6 respectively, the scale factor for the first respondent may be considered to be 1 and for the second respondent to be 2 (or, more precisely, the relative scale factor of the first respondent to the second respondent is 2).

Sometimes a scale factor is defined as a linear transformation (i.e., adding a constant and multiplying another).

## Scale factors in models of preference

When a model is used to measure preferences (e.g., MaxDiff and Choice Modeling), a practical challenge is that it can be difficult to interpret scale factors. For example, if one respondent's data is used to estimate coefficients of 2, 4 and 6 for three alternatives and another's estimates coefficients of 20, 40 and 60, there is typically ambiguity regarding how to interpret the ratio of the scale factors. In the context of an experiment the scale factor's meaning is typically straightforward - the respondent with a relatively large scale factor has exhibited more consistent preferences. However, there is typically no reason to believe that such differences in behavior in a survey will reflect differences in real-world behavior.

The main approaches to addressing scale factors in applied research are to:

• Ignore them.
• To make no explicit attempt to address the scale factor in the analysis but to keep it in mind when interpreting the data (e.g., focusing on relativities rather than absolute values of parameters).
• Use statistical models which explicitly assume that respondents differ in terms of scale factors and that these scale factor differences are artifacts of the research process. Although on face value this can initially seem more appealing than the first two approaches, a practical problem with this approach is that it is only possible to interpret any resulting models by making explicit assumptions regarding the correct scale factor (for example, the probabilities computed from logit models using the Inverse Logit Transformation are related to the assumed scale factor).
• Apply a transformation to any estimated parameters. For example, scaling utilities so that they have a minimum of 0 and add up to 100. As with the use of statistical models that explicitly assume scale factors, this involves replacing data provided by respondents with assumptions about respondents' true underlying preferences.
• Calibrating scale parameters using revealed preference data .