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The Perception of Correlation
in Scatterplots
Ronald A. Rensink
Departments of Computer Science and Psychology
University of British Columbia
Vancouver, Canada
(Work done with Gideon Baldridge)
2CPSC 444, 15 Apr 10
What’s the best way to visualize information?
Look at scatterplots - perception of correlation- accuracy & precision
Need a way to objectively evaluate performance
3CPSC 444, 15 Apr 10
Approach: Use techniques from vision science - decades of experience
- basis of modern theories of perception
Vision Science: Threshold Techniques
E.g., Relation between intensity and brightness
Adjust intensity of test until its brightness just begins to differ from the standard…
One light at fixed intensity (standard)Second light (test) at a different intensity
Light 1 Light 2
Intensity = 10 W Intensity = 10 W
Light 1 Light 2
Intensity = 10 W Intensity = 11 W
Light 1 Light 2
Intensity = 10 W Intensity = 12 W
Light 1 Light 2
Intensity = 10 W Intensity = 13 W
Standard Test
Just noticeable difference (jnd) - 75% correct1 W above standard
A bit more rigorously…
Forced-choice Technique
Q: Which one has the greater intensity?
Look at how accuracy of response varies with difference between test and standard
Intensity = 10 W Intensity = 15 WIntensity = 13 W Intensity = 10 WIntensity = 11 W Intensity = 10 W
Psychometric function
-describes accuracy (probability of correct)as a function of intensity of test light
.
50%
100%
Intensit y of test light
Standard = 10W
10 11 12 13
.
50%
100%
Intensit y of test light
Standard = 10W
10 11 12 13
75%
Threshold (75%) = 10.8 W
.
50%
100%
Intensit y of test light
Standard = 10W
10 11 12 13
75%
Threshold (75%) = 10.8 W
jnd = Δ = 0.8 I W
For I = 10 W, jnd ΔI = 0.8W
For a large range of intensities,
ΔI / I = k = constant
Weber’s Law
For I = 30 W, jnd ΔI = 2.5W
For I = 50 W, jnd ΔI = 4.0W
ΔI / I = .08
ΔI / I = .08
ΔI / I = .08
Weber Assumption:
Each ΔI / I corresponds to a unit of sensation (ΔS)
ΔI / I = k = ΔS
Integrating…
S = log(I) Fechner’s Law
S
I
S = Psychological quantityI = Physical quantity
Each unit of S corresponds to an increase in I by factor of 10
9CPSC 444, 15 Apr 10
Approach: Apply this technique to correlation
Forced-choice: Which one has the higher correlation ?
10CPSC 444, 15 Apr 10
1. Fix a particular base correlation (e.g., r = 0.6)
2. Find the jnd Δr (75% correct level) for that correlation- keep showing pairs of scatterplots to observer- adjust differences until jnd is reached- criterion: level in three subwindows is the same
Experimental Procedure - Precision
11CPSC 444, 15 Apr 10
Results.
1.00.50
base correlation (rA)
0.2
0.1
0
from above
from below
Δr = k(1/b - r)
k: variability (= 0.24)
b: offset (= 0.907)
Δu = ku
Δu u
= k
Weber’s Law(with u = 1/b - r)
Only 2 parameters (k,b)to specify precision over all correlations
12CPSC 444, 15 Apr 10
Weber Assumption:
Each ΔI / I corresponds to a unit of sensation (ΔS)
ΔI / I = k = ΔS
Integrating…
S = log(I) Fechner’s Law
Relating physical quantity I to psychological quantity S
13CPSC 444, 15 Apr 10
Assumption:
Each Δu / u corresponds to a unit of sensation (Δg)
Δu / u = k = Δg
Integrating…
g = log(u)
Relating physical quantity r to psychological quantity g
g(r) = log(1 - br) log(1 - b)
14CPSC 444, 15 Apr 10
Adjust test plot to be midway between reference plots
Experimental Procedure - Accuracy
Start with 0.0 & 1.0. Then 0.0 & 0.5 and 0.5 & 1.0. Then 0.0 & 0.25 and 0.25 & 0.5, etc, etc.
16CPSC 444, 15 Apr 10
Results
.
1.00.50
1.0
0.5
0
objective correlation (r)
g(r) = log(1 - br) log(1 - b)(b = 0.9)
Fechner’s law(with u = 1-br)
17CPSC 444, 15 Apr 10
Conclusions
Precision of correlation estimate given by linear function
Δr = k(1/b - r) 0 < k, b < 1
Accuracy of correlation estimate given by log function
g(r) = log(1 - br) / log(1 - b) 0 < b < 1
These performance curves are completely described by k, b
Thus, to evaluate a given scatterplot design:1. Measure jnds at two base correlations2. Calculate k and b3. Plug into performance curves