Some Markov models for direct observation of behavior James E. - PowerPoint PPT Presentation

Some Markov models for direct observation of behavior James E. Pustejovsky Northwestern University May 29, 2013

2 Direct observation of behavior • Quantities of interest • Prevalence : proportion of time that a behavior occurs • Incidence : rate at which new behavioral events begin • Intensity, contingency, others • Applications in psychology and education research • Measurement of teaching practice • Measurement of student behavior • Evaluating interventions for individuals with disabilities • Other examples in animal behavior, organizational psychology, social work, exercise physiology

3 Observation recording methods • How to turn direct observation of a “behavior stream” into data? • Continuous recording methods • Produce rich data, amenable to sophisticated modeling • Effort-intensive • Discontinuous recording methods • Less demanding methods needed in field settings • Momentary time sampling • Interval recording • Other possibilities?

4 Outline • Model for behavior stream as observed • Momentary time sampling • Interval recording • Some novel proposals • Efficiency considerations

Behavior stream MTS Interval recording Novel proposals Efficiency 5 A model for the behavior stream Interim times E 1 E 2 E 0 E 3 Session time D 2 D 1 D 3 Event durations  0 behavior is not occuring at time t   ( ) Y t  1beh avior is occu ring at tim e t     1 j          0   I t D E D i i j     1 0 j i

Behavior stream MTS Interval recording Novel proposals Efficiency 6 Alternating Poisson Process Assumptions   Event durations: ~ Exp(1/ ), 1,2, , .. 3 . 1. D j j   Interim times: 2. ~ Exp(1/ ), 1,2, , .. 3 . E j j Event durations and interim times are all mutually 3. independent. Process is in equilibrium. 4. Under this model: • Prevalence ϕ = μ / ( μ + λ ) • Incidence ζ = 1 / ( μ + λ )

Behavior stream MTS Interval recording Novel proposals Efficiency 7 Momentary time sampling (MTS) ( K + 1) moments, equally spaced at intervals of length L . • Observer records the presence or absence of a behavior at • each moment Recorded data are • X   ( ), 0,..., Y kL k K k X 0 = 0 X 1 = 1 X 2 = 0 X 3 = 0 X 4 = 1 X 5 = 1 X 7 = 1 X 6 = 1 X 8 = 0 Session time

Behavior stream MTS Interval recording Novel proposals Efficiency 8 Model for MTS data • Under the alternating Poisson process, X 1 ,…, X K follow a discrete-time Markov chain (DTMC) with two states (see e.g., Kulkarni, 2010) . 1 - p 1 ( L ) p 1 ( L ) X =0 X =1 1 - p 0 ( L ) p 0 ( L )      t        ( ) Pr( ( ) 1| (0) 0)  1 exp    p t Y t Y    0  (1 )       t          ( ) Pr( ( ) 1| (0) 1) (1 )exp   p t Y t Y    1   (1 )

Behavior stream MTS Interval recording Novel proposals Efficiency 9 MTS model, continued • Maximum likelihood estimators of ϕ and ζ have closed form expressions (Brown, Solomon, & Stephens, 1975) .     • But under more general models. E X • Extensive literature, lots of generalizations • stopping rules for observation time (Brown, Solomon, & Stephens, 1977, 1979; Griffin & Adams, 1983) • Irregular observation times (e.g., Cook, 1999) • Random effects to describe variation across subjects (e.g., Cook et al., 1999)

Behavior stream MTS Interval recording Novel proposals Efficiency 10 Partial interval recording (PIR) Divide period into K intervals, each of length L . • For each interval, observer records whether behavior • occurred at any point during the interval. Recorded data are •             0 1 , 1,..., . U I Y k L t dt k K     k [0, ) L U 4 = 1 U 1 = 1 U 2 = 1 U 3 = 0 U 5 = 1 U 6 = 1 U 7 = 1 U 8 = 1 Session time

Behavior stream MTS Interval recording Novel proposals Efficiency 11 PIR, continued • Unlike MTS, the mean of PIR data is not readily interpretable:         L       (1 ) 1 exp 1     E U      

Behavior stream MTS Interval recording Novel proposals Efficiency 12 Model for PIR data • Define V k as the number of consecutive intervals where behavior is present:   V      max 0 : U 0 . k j k k j • Under the alternating Poisson process, V 1 ,…, V K follow a DTMC on the space {0,1,2,3,…}. 1 – π 34 1 – π 23 1 – π 12 V =0 V =1 V =2 V =3 … 1 - π 01 π 23 π 34 π 01 π 12

Behavior stream MTS Interval recording Novel proposals Efficiency 13 Whole interval recording (WIR) Divide period into K intervals, each of length L . • For each interval, observer records whether behavior • occurred for the duration of the interval. Recorded data are •             1 , 1,..., . W I L Y k L t dt k K     k [0, ) L W 1 = 0 W 2 = 0 W 3 = 0 W 4 = 0 W 5 = 1 W 6 = 1 W 7 = 0 W 8 = 0 Session time Equivalent to PIR for absence of event. •

Behavior stream MTS Interval recording Novel proposals Efficiency 14 Augmented interval recording (AIR) Divide period into K /2 intervals, each of length 2 L . • Use MTS at the beginning of each interval, to record X k -1 . • If X k -1 = 0, use PIR for the remainder of the interval. • • If X k -1 = 1, use WIR for the remainder of the interval. X 2 = 1 X 0 = 0 X 1 = 0 X 3 = 1 X 4 = 0 U 1 = 1 U 2 = 1 W 3 = 1 W 4 = 0 Session time

Behavior stream MTS Interval recording Novel proposals Efficiency 15 Model for AIR data Define Z k = U k + W k + X k . • Under the alternating Poisson process, Z 1 ,…, Z K /2 follow a DTMC on • {0,1,2,3}, with transition probabilities π ab = Pr( Z k = b | X k -1 = a ) π 00 π 12 π 02 Z =0 Z =2 U =0, W =0, X =0 U =1, W =0, X =1 π 01 π 13 π 02 π 11 π 00 π 12 Z =1 Z =3 π 11 π 13 U =1, W =0, X =0 U =1, W =1, X =1 π 01

Behavior stream MTS Interval recording Novel proposals Efficiency 16 Intermittent transition recording (ITR) Divide period into K/2 intervals, each of length 2 L . • Use MTS at the beginning of each interval, to record X k -1 . • Record time until next transition as T k . • X 2 = 1 X 0 = 0 X 1 = 0 X 3 = 1 X 4 = 0 T 2 T 4 T 3 >2 L T 1 Session time

Behavior stream MTS Interval recording Novel proposals Efficiency 17 Model for ITR data • Under the alternating Poisson process, T 1 , X 1 , …, T K /2 , X K /2 have the property that F ( T k , X k | T 1 , X 1 ,…, T k -1 , X k -1 ) = F ( T k , X k | X k -1 ) 1 - p 0 ( L-t ) p 0 ( L-t ) T | X =1 f μ ( t ) X =0 X =1 f λ ( t ) T | X =0 1 - p 1 ( L-t ) p 1 ( L-t )

Behavior stream MTS Interval recording Novel proposals Efficiency 18 Asymptotic relative efficiency • Procedure p , q ∈ {MTS, PIR, AIR, ITR} ˆ ˆ •   are maximum likelihood estimators based on , p p procedure p     ˆ ˆ • are asymptotic variances based on inverse of   , V V p p expected information matrix. • Asymptotic relative efficiency of p versus q     ˆ ˆ   V V     ˆ ˆ ˆ ˆ q q       A R E , A R E ,     ˆ ˆ p q p q   V V p p

Behavior stream MTS Interval recording Novel proposals Efficiency 19 Asymptotic relative efficiency: Prevalence PIR PIR AIR AIR ITR ITR 0.8 0.8 0.6 0.6 MTS MTS 0.4 0.4 True incidence (frequency per interval) True incidence (frequency per interval) 0.2 0.2 0.0 0.0 asymptotic asymptotic 0.8 0.8 relative relative efficiency efficiency 0.6 0.6 2.0 PIR PIR 1.05 0.4 0.4 1.0 1.00 0.2 0.2 0.95 0.5 0.0 0.0 0.8 0.8 blue = column is more efficient 0.6 0.6 red = row is more efficient AIR AIR 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 True prevalence True prevalence

Behavior stream MTS Interval recording Novel proposals Efficiency 20 Asymptotic relative efficiency: Incidence PIR AIR ITR 0.8 0.6 MTS 0.4 True incidence (frequency per interval) 0.2 0.0 asymptotic 0.8 relative efficiency 0.6 2.0 PIR 0.4 1.0 0.2 0.5 0.0 0.8 blue = column is more efficient 0.6 red = row is more efficient AIR 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 True prevalence

21 Future work Evaluating these models & methods • Field testing • When is it okay to treat ML estimates from individual sessions as • “pre -processing ”? Lots still to do • Build data-collection software • Extensions to between-period regression models • Random period/subject effects • PIR, AIR, ITR under other distributional assumptions? •

Questions? Comments? pusto@u.northwestern.edu