Data Analysis for an Absolute Identification Experiment. Randomization with Replacement. Randomization without Replacement

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1 Data Analysis for an Absolute Identification Experiment 1 Randomization with Replacement Imagine that you have k containers for the k stimulus alternatives The i th container has a fixed number of copies (n i, proportional to P(S i ) ) of the i th stimulus On each trial, one of the S n i (i=1,, k) stimuli is selected to be presented to the subject That stimulus is immediately replaced in its corresponding container Then, the a priori probability for S i (i=1,, k) remains the same for all trials The stimulus uncertainty remains the same on all trials k IS= P(S i )log P(S i ) i=1 Randomization without Replacement Imagine that you have k containers for the k stimulus alternatives The i th container has a fixed number of copies (n i, proportional to P(S i ) ) of the i th stimulus On each trial, one of the S n i (i=1,, k) stimuli was selected to be presented to the subject That stimulus is NOT replaced in its corresponding container Then, the a priori probability for S i may change from trial to trial The stimulus uncertainty IS may change from trial to trial On the last trial, the subject knows exactly what stimulus to expect (whichever stimulus is the last one left in a container) 3

2 More on Randomization We prefer the method of randomization with replacement because It ensures constant IS for each trial It makes data analysis easier With the method of randomization with replacement, equal a priori probability no longer guarantees equal number of occurrences for all stimulus alternatives. Note that frequency of occurrence probability The advantage of randomization without replacement is that the experimenter controls the exact number of times each stimulus alternatives is presented. 4 R 1 R R 3 R 4 R 5 S 1 S S 3 S 4 S Estimation of IT IT est Average information transfer: k k P(S IT= P(S i,r j )log i R j ) P(S i ) j=1 i=1 Its maximum-likelihood estimate: k k IT est = ( n ij n )log ( n ij n n j=1i=1 i n ) j where n ij k k n i = n ij n j = n ij j=1 i=1 k k k n = n ij = n = k i n j j=1i=1 i=1 j =1 Interpretation of IT or ITest (compare with k= U ) 6

3 Percent-correct scores and IT est (A) k k P(S IT= P(S i,r j )log i R j ) j=1 i=1 P(S i ) (B) (C) (D) % 0 bits 5% 0 bits 100% bits 0% bits 7 Channel Capacity 8 Maximum Information Transmission Mathematically, IT IS. Intuitively, if the input and output are perfectly correlated, then IT = IS (= IR). Assume that there exists a maximum information transmission For small values of IS, IT = IS. As IS increases, IT = constant regardless of the value of IS. This maximum IT is accepted as the channel capacity. 9

4 4 Maximum Achievable Information Transmission Information Transmission IT (bits) 3 Channel Capacity:.5 bits Stimulus Uncertainty IS (bits) 10 The Magic Number 7 11 What does the Magic Number Mean? The magic number is derived from an IT range of.3 3. bits The magic number summarizes the typical channel capacityfor uni-dimensional stimuli Uni-dimensional stimuli Only one physical variables (target) is manipulated to form the stimulus set Other physical variables (background) are either held constant or randomized 1

5 How Magic is the Magic Number? The Magic Number does NOT apply to Absolute pitch Over-learnt stimuli Human face recognition Multi-dimensional stimuli 13 Dimensionality 14 How to Achieve High IT IT for uni-dimensional stimuli is limited IT(multi-D) is not limited by 7 In general, try Lots of dimensions A few values ( to 3) per dimension Examples? Speech perception Face recognition 15

6 How do you define dimensionality? From literature never explicitly defined Read between lines number of independently manipulated physical variables But physical and perceptual dimensionality may not be the same!! 16 Dimensionality a Visual Example Orientation of lines: 1D or D? IT for direction, or angle of inclination is 3.3 bits for a 5-sec exposure time (ref. p. 86, Miller s 7 paper) This is clearly at the high end of 7 ( 3.3 =9.8) 17 Dimensionality an Auditory Example Lateralization Rough Two Clicks t (msec) Interaural Time Delay 18

7 Dimensionality a Haptic Example Motion Flutter Vibration Frequency (Hz) 19 IT and Channel Capacity For Different Sensory Modalities AL and DL are in modality-specific physical units IT and channel capacity are in bits: We can compare apples with oranges! 0 Readings W. R. Garner, Uncertainty and Structure as Psychological Concepts. New York: Wiley, 196. G. A. Miller, The magical number seven, plus or minus two: Some limits on our capacity for processing information, The Psychological Review, vol. 63, pp ,