Scaling

Steven's Power Law

Scaling: Assigning numbers to the magnitude of psychological events. It's a measurement procedure. That is, the numbers are assigned to allow particular arithmetic operations to stand in for manipulations of the events.

As we add energy to a stimulus (a light for example) it changes in psychological magnitude (becomes brighter). What is the correspondance between changes in energy and changes in the pyschological quality? If we can express the changes in psychological magnitude in terms of changes on a physical dimension (as changes in energy, area, wavelength, etc) we are performing psychophysical scaling.

Other kinds of stimuli cannot be meaningfully measured in terms of any single physical dimension. But in a collection of stimuli we can easily see they differ along a psychological dimension--colors can differ in pleasantness; cars in attractiveness; instructors in appropriateness; pizza toppings in appeal; etc. If we want to quantitatively describe the psychological difference of this collection of stimuli, we can perform psychometric scaling.

With Psychometic scaling the set of stimuli are assigned numbers which reflect the psychological intensity. With psychophysical stimuli there is an equation which relates changes in the stimulus value to changes in the psychological value.

Fechner's Law:

Equal stimulus ratios produce equal sensed differences.
Stimuli which look equally different are produced by stimulus energies which differ by a particular ratio.
S = k log R.
Sensation increases as the logrithm of physical intensity
- the "k" simply makes the units of measurement work out. "k" is known as the scaling constant.
  - X calories of heat is converted to Y foot-pounds of work by the appropriate "k"
  - Why doe we multiply inches by 2.54 to get centimeters? "2.54" is a scaling constant and makdes the units of measure work out

Stevens' Power Law:

Equal stimulus ratios produce equal senses ratios.
Stimuli which appear to stand in a particular ratio (twice as large; twice as bright; one-fourth as loud; etc), are produced by stimulus energies which are in a another particular ratio (triple the energy to make the stimulus twice as bright, for instance).
S = k R ⁿ
That is, psychological intensity ("Sensation") increases as the nth power of stimulus intensity; again, "k" is simply a scaling constant.
When the logarithms of both the stimulus and response values are taken, the judgments should fall on a straight line.

S = k Rⁿ [1]

Now, take the logarithms of both sides. (You learned to do this in highschool).

log S = log (k R ⁿ ) [2]

log S = log K + log (R ⁿ ) [3]

log S = log K + n Log R [4]

log S = n Log R + log k [5]

Equation [5] is simply the equation for a straight line.

Let Log S = Y
Let Log K = b
Let n = m
Let Log R = X

log S = n Log R + log k [5]

Y = m X + b

Y = b + m X

The most general form of the equation for a straight line is actually:

To get the value of Y, multiply the value of X by a constant then add another constant.

Scaling Methods

Fechner developed "Summated jnd scales" and "Fechner Scales". Scales derived using these methods tell us about how different stimuli are from one another in terms of how easily confused they are.

Stevens developed Magnitude Estimation, Ratio Estimation, Magnitude Production, and Ratio Production (and additional scaling methods). Scales developed using these methods describe the apparent differences in magnitude.

There are many kinds of psychometric scaling procedures (paired comparisons, rating, ranking to name a few). Each procedure generates a scale providing particular information about the psychological differences between the stimuli.

Click here to view a table of representative exponents of power-functions of sensory dimensions.

Calculation of the Exponent for Steven's Power Law

Express the sensation and stimulus in ratios--not abolute amounts. If the stimulus goes from 1 tsp of salt dissolved in a quart of water to 3 tsps, then the stimulus has increased by a ratio of 3 times If the less concentrated liquid is judged as "12" on a saltyness scale and the more concentrated as "24" then the sensed ratio is 2 times.
"Y" is the sensed ratio	"X" is the ratio of stimulus intensities	"n" is the exponent of the power function.

Given the exponent of the power law specify the sensed ratio for any stimulus intensity ratio:

Example:If I increase the sound pressure level of a sound so it is 6 times the energy, how many times louder will the new stimulus be?

X = 6 (this stimulus is this many times the energy of the original stimulus)

n = 0.67

Y = 6^0.67 = 3.32 (times louder)

Given the stimulus ratio and the sensed ratio, determine the power law exponent.

Example: I presented the subject with a range of small weights (the smallest weight was 30 g; the largest was 4750 g). The heaviness of each weight was judged 10 times and the median judgment for each weight determined. The median judgment for the smallest weight was 3; the median judgment for the largest weight was 5000. What is the power law exponent?

Ratio of Sensations = (Ratio of Stimuli)ⁿ

Y = Xⁿ

X = (4750 ÷ 30) = 158.33 [which is approximately] 160 (that is, the largest stimulus was 160 times the smallest)

Y = (5000 ÷ 3) = 1667 (the largest stimulus was judged 1667 times larger than the smallest.

1667 = (160)ⁿ	Set up the relationship of ratios	Sensed ratio = (stimulus ratio)ⁿY = Xⁿ
log 1667 = n log 160	Take the logs of both sides	log y = n log X
n log 160 = log 1667	Rearrange terms	n log X = log Y
n = log 1667 ÷ Log 160	Divide both sides by log X	n = log Y ÷ log X
	log 1667 = 3.2219 log 160 = 2.2041
n = 3.2219 ÷ 2.2041
n = 1.4618	For small lifted weights, sensed intentisty increases as the 1.4618 power of the mass of the stimulus.