some markov models for direct observation of behavior · direct observation of behavior...

Some Markov models for direct observation of behavior

James E. Pustejovsky

Northwestern University

May 29, 2013

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

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?

Outline

• Model for behavior stream as observed

• Momentary time sampling

• Interval recording

• Some novel proposals

• Efficiency considerations

A model for the behavior stream

Session time

Interim times

Event durations

D1 D2 D3

E0 E1 E2 E3

is not occuring at time

is occu

0 behavior( )

ring at tim1 beh e avior

I Dt E D

Behavior stream MTS Interval recording Novel proposals Efficiency

Assumptions

1. Event durations:

2. Interim times:

3. Event durations and interim times are all mutually independent.

4. Process is in equilibrium.

Under this model:

• Prevalence ϕ = μ / (μ + λ)

• Incidence ζ = 1 / (μ + λ)

Alternating Poisson Process

Exp(1/ ), 1, 2, , ..~ 3 .jD j

Exp(1/ ), 1, 2, , ..~ 3 .jE j

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

( ), 0,...,k Y kL k KX

X0 = 0 X1 = 1 X2 = 0 X3 = 0 X4 = 1 X5 = 1 X6 = 1 X7 = 1 X8 = 0

Session time

Model for MTS data • Under the alternating Poisson process, X1,…,XK follow a

discrete-time Markov chain (DTMC) with two states (see e.g., Kulkarni, 2010).

X=0 X=1

p0(L) 1 - p0(L)

p1(L) 1 - p1(L)

( ) Pr( ( ) 1| (0) 0) 1 exp(1 )

( ) Pr( ( ) 1| (0) 1) (1 )exp(1 )

tt Y t Y

MTS model, continued

• Maximum likelihood estimators of ϕ and ζ have closed form expressions (Brown, Solomon, & Stephens, 1975).

• But under more general models.

• 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)

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.

U1 = 1 U2 = 1 U3 = 0 U4 = 1 U5 = 1 U6 = 1 U7 = 1 U8 = 1

Session time

0 1 , 1,..., .L

k IU Y k L t dt k K

PIR, continued • Unlike MTS, the mean of PIR data is not readily

interpretable:

(1 ) 1 exp1

Model for PIR data • Define Vk as the number of consecutive intervals where

behavior is present:

• Under the alternating Poisson process, V1,…,VK follow a DTMC on the space {0,1,2,3,…}.

max 0 : U 0 .k jk j kV

V=0 V=1

π01 1 - π01

V=2 V=3 …

1 – π12 1 – π23 1 – π34

π12 π23 π34

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.

• Equivalent to PIR for absence of event.

W1 = 0 W2 = 0 W3 = 0 W4 = 0 W5 = 1 W6 = 1 W7 = 0 W8 = 0

Session time

1 , 1,..., .L

k IW L Y k L t dt k K

Augmented interval recording (AIR)

• Divide period into K/2 intervals, each of length 2L.

• Use MTS at the beginning of each interval, to record Xk-1.

• If Xk-1 = 0, use PIR for the remainder of the interval.

• If Xk-1 = 1, use WIR for the remainder of the interval.

X0 = 0 X1 = 0 X2 = 1 X3 = 1 X4 = 0

U1 = 1 U2 = 1 W3 = 1 W4 = 0

Session time

Model for AIR data

• Define Zk = Uk + Wk + Xk .

• Under the alternating Poisson process, Z1,…,ZK/2 follow a DTMC on {0,1,2,3}, with transition probabilities πab= Pr(Zk = b | Xk-1 = a)

Z=0 U=0, W=0, X=0

Z=1 U=1, W=0, X=0

Z=2 U=1, W=0, X=1

Z=3 U=1, W=1, X=1

Intermittent transition recording (ITR)

• Divide period into K/2 intervals, each of length 2L.

• Use MTS at the beginning of each interval, to record Xk-1.

• Record time until next transition as Tk.

X0 = 0 X1 = 0 X2 = 1 X3 = 1 X4 = 0

T1 T2 T3 >2L T4

Session time

• Under the alternating Poisson process, T1,X1, …,TK/2,XK/2 have the property that

F(Tk,Xk | T1,X1,…,Tk-1,Xk-1) = F(Tk,Xk|Xk-1)

Model for ITR data

17 Behavior stream MTS Interval recording Novel proposals Efficiency

X=0 X=1

p0(L-t) 1 - p0(L-t)

p1(L-t) 1 - p1(L-t)

T | X=1

T | X=0

fμ(t)

fλ(t)

Asymptotic relative efficiency

• Procedure p, q ∈ {MTS, PIR, AIR, ITR}

• are maximum likelihood estimators based on procedure p

• are asymptotic variances based on inverse of expected information matrix.

• Asymptotic relative efficiency of p versus q

ˆVˆ ˆ,R

ˆ ˆ,p p

,ˆ ˆp pV V

PIR AIR ITR

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

asymptotic relative efficiency

Asymptotic relative efficiency: Prevalence

PIR AIR ITR

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

blue = column is more efficient red = row is more efficient

PIR AIR ITR

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

Asymptotic relative efficiency: Incidence

blue = column is more efficient red = row is more efficient

• 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?

Future work

Questions? Comments?

pusto@u.northwestern.edu

PIR model • Transition probabilities are uglier than MTS:

where and f(j) is the j-fold recursion of f.

1 1 1 (Pr 1| 0)1 L

j j k k

jV j V j fe

/ (1 )

/(1 )( )

AIR, continued

Event occurring at time (k-1)L?

Event ends before time kL?

Event begins before time kL?

Xk-1=1, Uk=1

Xk-1=0, Wk=0

AIR model

25 Extras

• Transition probabilities are given by

• Under alternating Poisson process,

ITR model

1 1 /2 /2 0

, ,..., , |

; , ,;| , |

x f x t

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