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VARIANCE COMPONENTS ANALYSIS FOR BALANCED AND UNBALANCED DATA IN RELIABILITY OF GAIT
MEASUREMENT
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Mohammadreza Mohebbi
Department of epidemiology and
preventive medicine,
Faculty of
Medicine, Nursing and Health
Sciences,
Monash University,
Melbourne .
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3-Dimensional Gait Measurement
Really expensive and fancy measurement system with lots of cameras and computers
Produces graphs of kinematics (joint angles)
Use these graphs to make important clinical & research decisions
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3-Dimensional Gait Measurement
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Kinematic measurement and the gait cycle
-40
-20
020
0 50 100
Angle
(degre
es)
% of gait cycle
Dorsi flexion
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OU
TP
UT
D
ATA
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Spatiotemporal Kinematics
KineticsMuscle length
GRFEMG
Clinical exam
Stroke 1
Pelvic Tilt60
0
Ant
Pst
deg
Hip Flexion70
-20
Flex
Ext
deg
Knee Flexion75
-15
Flx
Ext
deg
Dorsiflexion30
-30
Dor
Pla
deg
Pelvic Obliquity30
-30
Up
Dwn
deg
Hip Adduction30
-30
Add
Abd
deg
Knee Adduction30
-30
Var
Val
deg
Ankle Rotation30
-30
Int
Ext
deg
Pelvic Rotation30
-30
For
Bak
deg
Hip Rotation30
-30
Int
Ext
deg
Knee Rotation30
-30
Int
Ext
deg
Foot Progression30
-30
Int
Ext
deg
3-Dimensional Gait Measure
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3-Dimensional Gait Measure
Gait analysis is performed in motion analysis laboratories consists of physical examination, videotaping and calculation of time distance parameters.
Kinematic assessments are obtained with the use of reflective markers, multiple recording cameras, refined computer software, and force plate data.
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Conventional Biomechanical model
Limitations
Reliability
Validity
Soft tissue Artefact
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Variability in Repeated 3D Gait Measures
Major contribution to error repeated measures both within and between testers (intra-therapist, inter-therapist)
Presumes “Precise” placement of markers
Not-so-precise marker location
Not-so-consistent marker location
Skin movement
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Measurements tools
Standard clinical marker set according to Plug-in Gait model
8 camera 612 Vicon motion analysis system
2 force platforms
Subjects asked to walk at a self selected pace
Standard clinical testing protocol:
6 left clean force plate strikes
6 right clean force plate strikes
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Sources of variability
OUTPUT DATA
2 measurement sessions
1 therapist
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Sources of variability
OUTPUT DATA
2 measurement sessions
2 therapists
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6 therapists, 2 sessions, 6 trials, one subject!
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One single point in gait cycle
05
10
15
A B C D E F A B C D E F
Session 1 Session 2
An
gle
(d
eg
ree
s)
Assessor
1 person, Assessors A-F, 6 trials:One point in gait cycle
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Study Patients
Stroke Population referred to CGAS for assessment:
Subject Age Height
(cm)
Weight (kg) Side Yrs
Post
1 53 185.8 97.3 L 0.5
2 59 174.3 85.1 R 3.5
3 58 175.6 74.6 R 2.5
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A hierarchical structure
Patient
Session
Therapist 1
Therapist 2
Therapist 3
Level 1
Level 2
Level 3
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Data structure
Patient
Session
Therapist
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Therapist
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Therapist
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Session
Therapist
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Therapist
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Therapist
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
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Interaclass correlation coefficient: ICC
ICC: the ratio of the between-cluster variance to the total variance.
The reliability of a measurement is formally defined as the variance of the true values between individuals to the variance of the observed values, which is a combination of the variation between individuals and measurement error.
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ANOVABetween group variability
Variability of group means around the OVERALL MEAN (of all observations)
Within group variability
Variability of a group's observations around the group's mean (i.e. the group’s SD)
Within group variability
Variability of a group's observations around the group's mean (i.e. the group’s SD)
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Hierarchy of the model
level 1 has N people to be measured
for the n’th person, there are In assessors, the level-2 variable
for the n’th person’s i’th level-2 repeat, there are Jni sessions (days on which measurement is repeated), the level-3 variable
for the n’th person’s i’th level-2 repeat, and j’th level-3 repeat, there are Knij trials, the level 4 variable
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The variance components model
At time m of gait cycle
is measurement of some aspect of gait from person n, assessor i, session j, trial k
is an average measurement of the gait parameter and
is the residual error term.
)()3,()2,()1,()()( mnijk
mnij
mni
mn
mmnijkY
)(mnijkY
)(m
)(mnijk
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AssumptionsThe model assumes that the level-4 random component
(residual error) follow a Normal distribution with mean zero and standard deviation i.e .
Similarly at level-1, level-2, and level-3 with , and being the standard deviation of random effects at level 1, 2 and 3 respectively.
)(m),0( )()( mm
nijk trialN )1,(m
patient )2,(mtherapist )3,(m
session
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Assumptions
are mutually independent, and are independently distributed from the residual errors
Sets of repeats can each be viewed as random selections of repeats over their respective levels of measurement
)3,()2,()1,( , mmm and
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Assumptions
Can pool patients to estimate
” patients & therapists to estimate
” patients & therapists & sessions to estimate
)2,(mtherapist
)3,(m
session
)(mnijk
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An example
We used the 80th gait cycle point of Hip Rotation measurements for the unaffected side of three patients.
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4 level random effect model fitted to the 80th percentage point of gait cycle for hip rotation
Estimate 95% Conf. Interval]Fixed partIntercept -0.6 -7.11 5.9Random partPatient 5.6 2.0 15.8Therapist 1.7 0.5 6.0Session 2.2 1.2 3.7Residual 2.1 1.8 2.5
)80(
)1,80(
)3,80(
)2,80(
)80(
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ICC
The ICC between measurements for the same patient, but different therapists is
whereas for the same therapist and patient we get
86.0)()80()3,80()2,80()1,80(
)1,80(
patient
96.0),()80()3,80()2,80()1,80(
)2,80()1,80(
patienttherapist
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Average across the gait cycle
If m=1 to M where M is a fixed number of sampling points, e.g. 50 or 100, for every gait cycle, then the following model can be used
a “fixed” effect, is an average value of the gait parameter for the m’th point
)()3()2()1( mnijknijninmnijkY
m
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Assumptions
The random effects , and and their standard deviations are “averaged” across the gait cycle
Can be thought of loosely as each being an average of the respective sets of variance components , and or m=1 to M.
),0( )1()1( Nn ),0( )2()2( Nni ),0( )3()3( Nnij
)1,(m )2,(m )3,(m
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Another example
Foot rotation measurements for the unaffected side of patients
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4 level random effect model for all percentage point of the gait cycle: foot rotation
Estimate 95% Conf. Interval
Fixed part
-10.8 -18.3 -3.3
-10.1 -17.6 -2.6
...
-10.3 -17.7 -2.8
...
-12.5 -20.0 -5.0
Random part
Patient 6.2 2.2 18.4Therapist 3.0 1.5 6.2Session 2.1 1.3 3.4Residual 3.7 3.6 3.8
0
)1()2(
)3(
2
50
100
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The alternatives to hierarchical models
Ignore group membership and focus exclusively on inter-individual variation and on individual-level attributes.
ignoring the potential importance of group-level attributes
the assumption of independence of observations is violated
focus exclusively on inter-group variation and on data aggregated to the group level
eliminates the non-independence problem
ignoring the role of individual-level variables
Both approaches essentially collapse all variables to the same level and ignore the multilevel structure
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The alternatives to hierarchical models, continued
Define separate regressions for each group
Allows regression coefficients to differ from group to group
does not examine how specific group-level properties may affect / interact individual-level outcomes
not practical when dealing with large numbers of groups or small numbers of observations per group
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The alternatives to hierarchical models, continued
include group membership in individual-level equations in the form of dummy variables
analogous to fitting separate regressions for each group
treats the groups as unrelated
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Advantages of Hierarchical models
simultaneous examination of the effects of group-level and individual level
the non-independence of observations within groups is accounted for
groups or contexts are not treated as unrelated
both inter-individual and inter-group variation can be examined
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Limitation
Sample size
Missing values
Study Design
Functional data analysis
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Questions?