Modeling Physical Activity Data
Dave Osthus and Bryan Stanfill
Center for Survey Statistics and Methodology, Iowa State University
March 28, 2011
Dave & Bryan
PAMS
Outline
Modeling Usual Energy Expenditure
Motivation
Potential groupings and influential factors
A look at the data
Analysis outline
Results
Modeling Physical Activity Behavior
Set up of problem
Binomial regression idea
Logistic regression idea
Dave & Bryan
PAMS
Motivation
Due to the current obesity epidemic there has been heightened
interest in energy consumption exceeding expenditure, especially in
certain groups of the population. Nutritionists have developed
models to better understand energy consumption, but a similar
treatment of energy expenditure (EE) is less developed. Dr. Beyler
and others developed a model to account for the various biases in
self-report questionnaires and result in an estimated distribution of
the usual energy expenditure for groups of people.
Dave & Bryan
PAMS
Potential Groupings and Influential Factors
Gender
Race/Ethnicity
Age
BMI/Weight
Education
Number of adults in a household
Smoking status
Dave & Bryan
PAMS
Survey Objectives
The Physical Activity Measurement Study (PAMS) is a survey
designed to obtain information on physical activity patterns of
Adult women and men (21-70)
Hispanic and African American populations (limited sample
size)
Rural and non-rural adults
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PAMS
Survey Objectives, Continued
More specifically,
Individuals are sampled from four counties in Iowa: Marshall,
Black Hawk, Dallas and Polk.
Goal is 1200 participants at the end of the study, spanning
two years.
Approximately equal number of males and females.
Approximately 10% African American and 10% Hispanic.
Minorities oversampled.
Dave & Bryan
PAMS
Data Collection Process
Data collection was intended to sample individuals uniformly over
two years, partitioned into eight quarters.
At the individual level:
Data were collected on two non-consecutive installments.
For each installment:
Individual wore SenseWear Monitor for 24 hours.
24-hour activity recall administered via phone the following
day.
Dave & Bryan
PAMS
A Look at the Data
Below are the daily energy expenditure distributions (in kilocalories
per day) by gender and measurment type. The top row
corresponds to measured EE from SenseWear Monitor and the
bottom row reported EE from the 24-hour recall. The red vertical
lines indicate the mean EE for each cell.
Female
Male
50
40
Measured
30
20
count
10
0
50
40
Reported
30
20
10
0
2000
4000
6000
8000
10000
12000
2000
4000
Energy Expendature
Dave & Bryan
PAMS
6000
8000
10000
12000
Scatter plots
Scatter plots of reported EE vs. measured EE by gender.
Female
Male
12000
Reported EE
10000
8000
6000
4000
2000
2000
4000
6000
8000
2000
Measured EE
Dave & Bryan
PAMS
4000
6000
8000
A Look at the Data, Continued
Box-plots of the reported EE and measured EE for women in age
groupings with ranges of 21-40, 41-50, 51-60 and 61-70 and
sample sizes of 69, 76, 103 and 78, respectively. A clear pattern is
present in the monitor EE, which indicates that grouping data of
this kind is justified.
Monitor Energy Expenditure by Age Group
61−70
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Reported Energy Expenditure by Age Group
61−70
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51−60
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Age (years)
Age (years)
51−60
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41−50
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21−40
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1500
2000
2500
3000
3500
●
4000
41−50
●
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●
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21−40
4500
3000
4000
5000
Reported EE
Dave & Bryan
PAMS
●
● ●
2000
Monitor EE
●
6000
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7000
●●
8000
Procedure for Estimating Usual Daily EE Parameters
1
Transform measured and reported EE to approximate
normality
2
Test for nuisance effects (e.g. day of week, season, replicate)
3
Fit a model to the adjusted transformed measurements for
each group to account for sources of variation and bias
4
Fit a population model based on group-level estimates to
reduce the number of parameters (where appropriate)
5
Estimate a distribution of usual daily EE for each group on
the original scale
For what follows, we only consider females with two complete
replicates, i.e. two sets of monitor and 24-hour recall data, with at
least 99% of their monitor data accounted for. Also, all females
with either a meaured or reported daily energy expenditure above
8000 kcals/day were considered outliers and removed.
Dave & Bryan
PAMS
Transform EE Measurements to Approximate Normality
150
150
1500
2500
3500
Monitor EE
4500
7.2
7.6
8.0
8.4
100
0
0
0
20
50
40
50
Frequency
100
Frequency
80
60
Frequency
60
40
20
0
Frequency
80
100
100
120
120
The distributions of measured EE and reported EE on the left and
right, respectively are plotted on both the original and log scale.
The log transformation produces a distribution that is passably
normal.
1000
Monitor EE on Log Scale
3000
5000
Reported EE
Measured EE
7000
7.5
Reported EE
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PAMS
8.0
8.5
9.0
Reported EE on Log Scale
Adjust for Nuisance Effects
A weighted linear regression taking into account both survey
weights and the stratified sampling design was fit to the log
transformed EE data. In particular we’re interested to see if the
nuisance effects: Rep, Quarter 1, Quarter 2 or Weekend (day of
week) are significant.
Reported EE
Measured EE
Covariate
Intercept
Rep
Age
Age2
Quarter1
Quarter2
Hispanic
Black
College
Smoke
Weekend
Estimate
7.8559767
0.0041662
0.0027932
-0.0000836
0.0029372
0.0157222
-0.0257205
0.0399515
-0.0120230
-0.0342825
0.0263776
p-value
<.0001
0.7127
0.6471
0.1707
0.9101
0.5753
0.7194
0.2422
0.5864
0.1428
0.2246
Covariate
Intercept
Rep
Age
Age2
Quarter1
Quarter2
Hispanic
Black
College
Smoke
Weekend
No nuisance effects appear to be significant.
Dave & Bryan
PAMS
Estimate
7.7928713
0.0183775
0.0091680
-0.0001002
0.0412050
-0.0052034
-0.1240973
0.0925985
-0.0274299
0.0024525
-0.0080371
p-value
<.0001
0.1954
0.3965
0.3446
0.2965
0.8956
0.2308
0.0274
0.3968
0.9421
0.7660
Measured EE Model
xgij = µg + tgi + dgij + ugij
where
xgij is a measurement of daily EE in normal scale for individual i on day j in group g from
an unbiased reference instrument
µg is the mean daily EE in the normal scale for group g
tgi is individual i’s mean deviation from the mean daily EE of group g, in the normal scale
2
tgi ∼ N(0, σtg
)
dgij is individual i’s deviation from his of her mean daily EE on day j in the normal scale
2
dgij ∼ N(0, σdg
)
ugij is random measurement error for individual i on day j in group g, in the normal scale
2
ugij ∼ N(0, σug
)
and
2
2
2
Var(xgij ) = Var(µg + tgi + dgij + ugij ) = σtg
+ σdg
+ σug
under the assumptions that
Cov(tgi , dgij ) = Cov(tgi , ugij ) = Cov(dgij , ugij ) = 0
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Reported EE Model
ygij = µyg + β1g (tgi + dgij ) + rgi + egij
where
ygij is the measurement of daily EE in normal scale for individual i on day j in group g
from the self-report instrument
µyg is the group mean of self-reported daily EE in the normal scale for group g
β1g is the slope that accounts for the systematic error in the relationship between
self-report and actual daily EE in group g
rgi is individual i’s deviation from the self-reported group-level mean in the normal scale
2
rgi ∼ N(0, σrg
)
egij is random measurement error in the self-report on the normal scale for individual i
on day j in group g
2
egij ∼ N(0, σeg
)
and
2
2
2
2
2
2
Var(ygij ) = Var(µyg + β1g (tgi + dgij ) + rgij + egij ) = β1g
σtg
+ β1g
σdg
+ σrg
+ σeg
under the assumptions that all random effects are uncorrelated.
Dave & Bryan
PAMS
Parameter Estimates for Females in Log Scale
The following are age group fixed effect parameter estimates
obtained via method of moments.
Interpretation
(Parameter)
Usual PA (µg )
Reported PA (µyg )
Pop Bias (β1g )
Group 1 (<41)
n=69
7.878 (0.019)
8.010 (0.035)
0.47 (0.24)
Group 2 (41-50)
n=76
7.791 (0.018)
7.987 (0.027)
0.89 (0.22)
Group 3 (51-60)
n=103
7.753 (0.015)
8.016 (0.024)
1.13 (0.14)
Group 4 (61-70)
n=78
7.700 (0.018)
7.985 (0.026)
0.72 (0.14)
The following are age group random effect parameter estimates
obtained via method of moments.
Interpretation
(Parameter)
2
Usual PA (σtg
)
2
Day to day (σdg
)
2
Monitor ME(σug
)
2
Recall ME (σeg
)
2
Person-Bias(σrg
)
Group 1 (<41)
n=69
0.0271 (0.0056)
0.028 (0.016)
-0.012 (0.015)
0.0160 (0.0045)
0.0579 (0.0104)
Group 2 (41-50)
n=76
0.0147 (0.0045)
0.0082 (0.0025)
0.0019 (0.0024)
0.0060 (0.0021)
0.0275 (0.0045)
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PAMS
Group 3 (51-60)
n=103
0.0208 (0.0037)
0.0041 (0.0009)
0.0053 (0.0020)
0.0060 (0.0016)
0.0222 (0.0056)
Group 4 (61-70)
n=78
0.0242 (0.0052)
0.0040 (0.0017)
0.0001 (0.0015)
0.0053 (0.0015)
0.0198 (0.0057)
Age Plots
Estimated lines relating mean daily EE and reported daily EE by
age groups.
7.5
8.0
8.5
9.0
8.5
8.0
24PAR EE (log scale)
7.5
7.5
8.0
8.5
Age Group 4
7.5
8.0
9.0
8.0
8.5
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24PAR EE (log scale)
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Monitor EE (log scale)
Dave & Bryan
9.0
9.0
Age Group 3
9.0
8.5
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Monitor EE (log scale)
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24PAR EE (log scale)
Age Group 2
7.0
24PAR EE (log scale)
Age Group 1
7.5
8.0
8.5
Monitor EE (log scale)
PAMS
9.0
Estimated Distributions of Usual Energy Expenditure
0.0004
0.0008
Age Group 1
Age Group 2
Age Group 3
Age Group 4
0.0000
probability density
0.0012
Estimated distributions of usual daily EE for all four age groups
1500
2000
2500
3000
3500
usual daily EE (kcal/d)
Dave & Bryan
PAMS
4000
4500
Estimated Distributions of Usual Energy Expenditure
Estimated distributions of usual daily EE, individual means of daily
monitor EE, and individual means of reported daily EE for age
group 1 (age < 41)
1e−03
Age Group 1 (Age 21 − 40)
6e−04
4e−04
2e−04
0e+00
probability density
8e−04
usual daily EE
monitor means
24PAR means
2000
4000
6000
daily EE (kcal/d)
Dave & Bryan
PAMS
8000
Modeling Physical Activity Behavior
The Center for Disease Control (CDC) and World Health
Organization (WHO) make physical activity recommendations
for age groups based on intensity (METs) and type (aerobic
or muscle-training) of physical activity. This brings two
questions to mind:
1
Are people complying with these recommendations?
2
What covariates are related to compliance?
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Recommendations in More Detail
Recommendations for adults (18-64 years) are of the following
form:
Aerobic exercise should be in bouts of at least 10 minute
durations of the following intensity/duration levels:
1
2
3
At least 150 minutes of moderate-intensity (3-6 METs) aerobic
exercise OR
At least 75 minutes of vigorous-intensity (over 6 METs)
aerobic exercise OR
An equivalent combination of moderate and vigorous aerobic
exercise
Muscle-strengthening activities should be performed two or
more days a week.
How could compliance with these recommendations be modeled?
Dave & Bryan
PAMS
First Idea: Model Relationship between P(Bout) and Age
Binomial Regression
exp(β0 + β1 ∗ xi )
1 + exp(β0 + β1 ∗ xi )
pi is the probability age i adults will participate in a
continuous bout of some defined period of time on any given
day
xi is age i
log (pi /(1 − pi )) = β0 + β1 ∗ xi ⇒ pi =
y
2
0
7
10
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P(Hour Bout)
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0.2
n
2
1
11
14
8
6
0.0
Age(x)
20
21
22
23
24
25
1.0
P(Hour Bout) vs. Age
●
●
20
30
40
50
Age
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PAMS
60
70
Fitted First Idea
Fitting the previous model we get the following parameter
estimates
β0
β1
Estimate
1.60326
-0.02416
Std. Error
0.34769
0.00672
z-value
4.611
-3.595
Pr (> |z|)
4.00e-06
0.000325
We can interpret the β1 parameter as follows: All else being
equal, the odds of an individual with age x + 1 engaging in 60
minutes of physical activity on a given day are a factor of 0.98
times that of an individual with age x.
1.0
P(Hour Bout) vs. Age
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P(Hour Bout)
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30
40
50
Age
Dave & Bryan
PAMS
60
70
A Bayesian Approach to First Idea
A Bayesian approach to this model considers β0 and β1 to be
random variables which makes pi a random quantity. A
general procedure is as follows:
Define pi as before with likelihood
L(β0 , β1 ) ∝
N
Y
piyi (1 − pi )ni −yi .
i=1
Put a prior on β0 and β1 . Here we put Beta priors on two
specific ages which effectively adds two observations to our
dataset and is given by
a1
a2
g (β0 , β1 ) ∝ page25
(1 − page25 )b1 page65
(1 − page65 )b2 .
Simulate from the joint posterior which is given by
f (β0 , β1 |y ) ∝
N+2
Y
i=1
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piyi (1 − pi )ni −yi .
Bayesian Results
Below is the joint posterior distribution of β0 and β1 along with
the fitted line to the data with a 95% credible bands. The
Bayesian approach allows us to make statements such as: we are
95% sure the true probability that a 24 year old will engage in one
hour of physical activity is between 65% and 80%.
Joint Posterior for β0 and β1
●
0.01
1.0
P(Hour Bout) vs. Age Curve
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0.8
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P(Hour Bout)
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0.2
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−6.9
0.5
1.0
1.5
2.0
2.5
0.0
β1
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3.0
●
20
β0
30
40
50
age
Dave & Bryan
PAMS
60
70
Second Idea: Model Number of Bouts per Person
Let Yij ∼ Poisson(λi ) where Yij is the number of bouts
recorded for person i on day j
Poisson Regression
log(λi ) = β0 + β1 Agei + β2 Genderi + β3 BMIi + · · ·
Distribution of Number of Bouts
500
400
count
300
200
100
0
0
1
2
3
4
5
Number of Bouts
Dave & Bryan
PAMS
6
7
Poisson Regression continued
We would expect a lot more 0’s than this model allows so
we’d make this a zero-inflated Poisson model as follows:
(
Yij ∼ Poisson(λi ) with probability θ
Yij = 0
with probability 1 − θ
In this case we’re interested in the value of θ and identifying
the sub-populations (according to the value of λi ) that deflate
θ by not getting the recommended level of activity.
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Join In
Ideas?
Thoughts?
Recommendations?
Dave & Bryan
PAMS