Uni-dimensional and Multidimensional Scales
“Unidimensionality” is used to describe a specific type of measurement scale. A unidimensional measurement scale has only one (“uni”) dimension. In other words, it can be represented by a single number line. Some examples of simple, unidimensional scales:
- Height of people.
- Weight of cars.
- Volume of liquid.
Unidimensionality can also refer to measuring a single ability, attribute, construct, or skill. For example, a unidimensional mathematical test would be designed to measure only mathematical ability (and not, say, grasp of English grammar, knowledge of sports, or other non-mathematical subjects or concepts).
Some concepts (like height or weight) are obviously unidimensional. Others can be forced into a unidimensional status by narrowing the idea into a single, measurable construct. For example, self-worth is a psychological concept that has many layers of complexity and can be different for different situations (at home, at a party, at work, at your wedding). However, you can narrow the concept by making a simple line that has “low self worth” on the left and “high self worth” on the right.
Multidimensional scaling is a visual representation of distances or dissimilarities between sets of objects. “Objects” can be colors, faces, map coordinates, political persuasion, or any kind of real or conceptual stimuli (Kruskal and Wish, 1978). Objects that are more similar (or have shorter distances) are closer together on the graph than objects that are less similar (or have longer distances). As well as interpreting dissimilarities as distances on a graph, MDS can also serve as a dimension reduction technique for high-dimensional data (Buja et. al, 2007).
The term scaling comes from psychometrics, where abstract concepts (“objects”) are assigned numbers according to a rule (Trochim, 2006). For example, you may want to quantify a person’s attitude to global warming. You could assign a “1” to “doesn’t believe in global warming”, a 10 to “firmly believes in global warming” and a scale of 2 to 9 for attitudes in between. You can also think of “scaling” as the fact that you’re essentially scaling down the data (i.e. making it simpler by creating lower-dimensional data). Data that is scaled down in dimension keeps similar properties. For example, two data points that are close together in high-dimensional space will also be close together in low-dimensional space (Martinez, 2005). The “multidimensional” part is due to the fact that you aren’t limited to two dimensional graphs or data. Three-dimensional, four-dimensional and higher plots are possible.