Ratio Scaling

Ratio scaling refers to a family of direct scaling methods in which subjects directly estimate the magnitude of stimuli or adjust the magnitude of stimuli. The methods we have looked at previously resulted in interval scales. Ratio scaling produces ratio scales. If you remember, ratio scales are only invariant to multiplicative transforms.

Here are 5 ratio scaling methods:

Magnitude Estimation
Subjects make direct numerical estimations of the sensory magnitude of the stimuli.
Magnitude Production
The subjects adjust the magnitude of a stimulus to match a numerical value given by the experimenter.
Direct Ratio Estimation
The subjects respond to a pair of stimuli with a number representing the apparent sensory magnitude.
Direct Ratio Scaling
Subjects adjust the magnitude of one stimulus relative to another to match a ratio specified by the experimenter.
Cross-Modality Matching
Subjects match the magnitude of a stimulus across a stimulus modality (loudness —> brightness).

We will focus on Magnitude Estimation.

Magnitude Estimation

Here is an example of typical instructions for subjects taking part in a magnitude estimation experiment. This will give you an idea of how this type of experiment is implemented.

 "You will be presented with each, in turn, of a series of circular lights. Tell me how bright each light is by assigning a number to its brightness. Call the 1st light any number that seems appropriate to you. Then assign numbers to successive lights in such a way that they reflect your impression of how bright they are. If the second light is twice as bright as the first, the number you assign should be twice as large; if it is only one-fourth as bright, the number should be one-fourth as large; and so on. There is no limit to the range of numbers that you may use. You may use whole numbers, fractions, or decimals. But try to make each number match the brightness that you see."

For each stimulus you compute the geometric mean of the subjects' responses:

(If values of 0 are used, use the mode rather than the geometric mean; better yet, avoid zero as a response.)

So in this technique the subjects directly assign numbers to the magnitude of sensation. As in other psychophysical procedures described previously, there are a number of issues to take into consideration.

Beware of adaptation effects. Since you may be assessing a wide range of stimuli, there may be changes in adaptation due to the changes in magnitude of the stimuli. There may also be hysteresis effects in which the order of the stimuli affect the responses. Subjects may readjust their responses due to the subsequent stimuli as they settle on a range of scale values. It is sometimes useful to present the subjects with the range of stimuli so they can get an idea of what numbers they may use in their responses. Different random orders of stimulus presentation can also be used for different subjects to control for hysteresis.

Often the derived scale in a magnitude estimation experiment is not linearly related to an interval scale derived with another method. For example, the square-root of the magnitude estimation scale may be linearly related to the interval scale.

The results from ratio scaling are typically power functions. We will explore this in the next section.

Correction for the perception of numerosity

The subject's task is to assign a number to match the subjective magnitude of the stimulus intensity. Sometimes the relationship between the sensation magnitude and the numbers assigned to them may not be linear. This may because the perception of numerosity, or the size of numbers, may not be linear with numbers (this is especially true with large numbers, for example, thinking about the distances of celestial objects or Bill Gates' wealth).

One way to correct for this is to have subjects scale length and use this data to correct scaling in the experimental data. It has been shown that our perception of line length is directly proportional to actual line length (exponent = 1.0).

An example is shown in this figure:

In (a) you see that the subjects estimations are overestimating the sensation magnitude. Reading values (or fitting a function) from the ordinate to the abscissa is used to correct the data in (b).

Magnitude Production

Magnitude production is like the inverse of magnitude estimation. In magnitude production, the experimenter tells the subject the numerical value of some sensory magnitude and then requires him to adjust the stimulus to produce it. Preliminary experimentation is usually needed to determine the appropriate range of numbers for the subject.

Both magnitude estimation and magnitude production are susceptible to the regression effect in which the subjects' judgments tend to regress towards the mean. Here is an example:

In a magnitude estimation experiment, subject will tend to give numbers that are less extreme which would compress the scale along the y-axis. In a magnitude production experiment, subjects will tend to adjust the stimulus intensity towards the middle of the range compressing the data along the x-axis. Averaging the results from both methods is often taken as the correct scale if you are trying to determine the true scale relationship.

Direct Ratio Estimation & Production

Ratio estimation is a form of magnitude estimation in which a subject assigns a ratio to the magnitudes of two presented stimuli. For example, if flash A is perceived as 5 times brighter than flash B, the subject should respond with the number 5.

The geometric means of the ratios are computed to create the scale values.

In ratio production, the subject adjusts the ratio of one stimulus relative to another to a value given by the experimenter.

In addition to the readings, a book by George A. Gescheider, "Psychophysics: The Fundamentals," 1997, Lawrence Erlbaum Assoc., ISBN: 0-8058-2281-X, gives a good accounting of ratio scaling methods.

Let's move on...

 Continue on to Stevens' Power Law