Client Report
Follow-Up to November 3, 20I5 Consultation Meeting
Mindfulness and Eating Behavior
Tammy Turner, Ph.D.
Department of Nutritional Sciences
University of Arizona
Consultants: Jessika Ava, Kyle Carter, Yifei Yuan
Advisor: Dr. Dean Billheimer
Client Report 2
1 Executive Summary
Dr. Tammy Turner, University of Arizona Department of Nutritional Sciences, visited the
Statistical Consulting Lab free student session to receive advice on the most effective analytical methods
for an upcoming intervention based pilot study. Suggested methods, with sample analyses using
simulated data, are presented in this report.
2 Detailed Summary
2.1 Overview of Study
A pilot study will consist of 30 adolescents participating in a single-arm non-randomized
mindfulness video intervention. Each video, varying in lengths up to 20 minutes will be viewed once a
day for six weeks. Video content will focus on mindfulness skills related to eating, physical activity, sleep,
stress, and general techniques (e.g. meditation). The impact of the intervention on change in dietary
behavior, intrinsic motivation, perceived stress, and mindfulness will be assessed. Secondary outcomes
included acceptability, feasibility, adherence, usefulness, likelihood of adoption, and technical feasibility.
2.2 Research Question
To determine if a significant difference occurs in mindfulness, stress, motivation, and eating
behaviors, as measured by questionnaires, between pre- and post- intervention, in adolescents.
2.3 Client’s Needs
This report will answer Dr. Turner’s following questions:
1) How to analyze repeated measures over 6 time points?
2a) Is there a more robust way than paired t-test to measure the changes in a single treatment group
with pre-post questionnaire data?
2b) and/or is there a better way to do subgroup analysis on this data besides stratifying by covariates
and re-running paired t-tests if I use this as the primary analysis
2.4 Study Design
Pre Treatment
Baseline Scores: Mindfulness, Stress, Eating Behavior, Motivation, (each as measured by separate
questionaires), Prior Mindfulness Knowledge, age, BMI, race, other demographics
Intervention
Healthy Eating Videos: 1 Video/Day for 6 weeks
“Dose” can be measured by number of videos watched
Client Report 3
Table 1: Measurement of Mindfulness Scale
Week
1
2
3
4
5
6
Video Length
3 min 6 min +
+
+
15 min
# times NOT mindful
X1
X2
X3 X4 X5 X6
Mindful scale= length/times not mindful 3/X1
Post Treatment
Outcome: Mindfulness, Stress, Eating Behavior, Motivation, each as measured by separate
questionnaires
3 Suggested Methods
3.1 Random Mixed Effects Model to Assess Mean Mindfulness Score Values over Time
To assess mean mindfulness score values over time, we define mindfulness score as
𝑁𝑜. 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑛𝑜𝑡 𝑚𝑖𝑛𝑑𝑓𝑢𝑙
𝑀𝑖𝑛𝑑𝑓𝑢𝑙𝑛𝑒𝑠𝑠 𝑆𝑐𝑜𝑟𝑒 =
𝑡𝑜𝑡𝑎𝑙 𝑣𝑖𝑑𝑒𝑜 𝑙𝑒𝑛𝑔𝑡ℎ 𝑖𝑛 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
(Please note this figure is the inverse of the original analysis plan, explanation below). A linear
mixed effects model with random intercept and unstructured covariance matrix will provide an
adequate model. Time is modelled linear and the random effects allow for the subject specific
heterogeneity.
Data was simulated using a random number generator within Excel software, taking random
values 0-10 for number of times not mindful divided by video length measured 3, 6, 9, 12, and 15
minutes. SAS Statistical Software is used to assess the simulated data.
The random mixed effects model is:
Mindfulness_Scoreij=B1 +B 2(timeij) + b1i + eij , where:
i=subject 1,2,….,30; j=time 1, 2, …., 6
B1 = fixed intercept
B 2 = fixed time effect
b1i = subject-specific random effect of intercept for subject i
The simulated data provides this model:
Mindfulness_Scoreij=1.34 – 0.20(timeij) + b1i + eij
Using simulated data, the results show a negative association with time; this is what would be
expected if intervention is successful and number of non-mindful moments decrease with time.
Client Report 4
The assumptions for linear mixed effects model are:
1) Linearity
2) Homoskedasticity
3) Normality of residuals
4) Absence of collinearity
5) Absence of influential outliers
6) (Note: Independence is NOT an assumption due to repeated measures)
An SPSS manual providing instructions for this analysis is found here:
http://www.spss.ch/upload/1126184451_Linear%20Mixed%20Effects%20Modeling%20in%20SPSS.pdf
3.2 Linear Regression Model to Assess Pre-Post Comparison
A linear regression model for pre-post comparison may be used to determine potentially
influential covariates for change-in-test-scores, run a series of linear regression models cycling through
our covariates of interest may be conducted. We suggest to center the covariate in order to provide an
easier interpretation and to avoid extrapolation.
The regression model is:
ΔScorei=B1 +B 2(covariate*i) + ei , where:
i=subject 1,2,….,30
B1 = mean change score
B 2 = covariate effect
1)
2)
3)
4)
The assumptions for the linear regression model are:
Linearity
Homoskedasticity
Normality of residuals
Independence of observations
Demographic information and test scores were simulated using R. The details of the data
generation process may be found in the supplementary file. The following tables detail the results of the
simulated study, including the estimated effects, true effects, and p-values. p-values less than 0.05 are
considered significant.
Table 2: Mindfulness Results
Covariate
Estimated Value
Mean
1.8667
Baseline Mindfulness
0.0762
Age
-0.4994
Gender
~0
Prior Knowledge
0.6667
Dose
0.0395
*correct conclusion; !incorrect conclusion
True Value
0
0
0
0.2
0.02
p-value
3.27x10^-9 *
0.533 *
0.0234 !
~1 *
0.155 !
0.0002 *
Client Report 5
Table 3: Stress Results
Covariate
Estimated Value
Mean
-1.9667
Baseline Mindfulness
0.0360
Age
-0.4994
Gender
0.6
Prior Knowledge
-0.6825
Dose
-0.02
*correct conclusion; !incorrect conclusion
Table 4: Eating Behavior Results
Covariate
Estimated Value
Mean
0.90
Baseline Mindfulness
-0.0346
Age
0.4413
Gender
0.2
Prior Knowledge
0.1429
Dose
0.0082
*correct conclusion; !incorrect conclusion
Table 5: Motivation Results
Covariate
Estimated Value
Mean
0.6333
Baseline Mindfulness
0.0968
Age
0.0729
Gender
0.8667
Prior Knowledge
0.8413
Dose
0.0082
*correct conclusion; !incorrect conclusion
True Value
0
0
0
-0.2
-0.02
True Value
0
0
0
0.1
0.01
p-value
3.05x10^-9 *
0.783 *
0.924 *
0.192 *
0.173 !
0.119 !
p-value
6.76x10^-4 *
0.7953 *
0.0700 *
0.674 *
0.7833 !
0.5310 !
True Value
0
0
0
0.1
0.01
p-value
0.0072 *
0.4360 *
0.314 *
0.0427 !
0.0750 !
0.6387 !
This model does identify differences between post- and pre- measurements, however, it does
not seem to be accurate at determining important covariates. This is likely due to the variance in the
data overshadowing the effects.
An SPSS manual providing instructions for this analysis is found here:
https://statistics.laerd.com/spss-tutorials/linear-regression-using-spss-statistics.php
3.3 Further Analyses to Assess Effect of Intervention Data
The intervention can also be evaluated by “Mindfulness State Time” which is obtained based on
bellowing equations:
𝑀𝑆𝑡𝑖𝑚𝑒 =
𝑇𝑜𝑡𝑎𝑙 𝑇𝑖𝑚𝑒
𝑁𝑜𝑛 − 𝑀𝑖𝑛𝑑𝑓𝑢𝑙𝑛𝑒𝑠𝑠 𝑐𝑜𝑢𝑛𝑡
The above equation is reasonable to represents the intervention in the experiment. However,
sometimes the self-reported Non-Mindfulness count might be 0, and the MStime will goes to infinity.
Thus instead of using MStime directly, Here a new parameter is defined:
Client Report 6
λ(t): average non-mindfulness count per mins at week t. t ϵ{1,2,3,4,5,6}
It is further assumed that individual’s Non-mindfulness count follows a Poisson distribution:
𝐾𝑖 (𝑡)~𝑃𝑜𝑖𝑠𝑠𝑜𝑛 (𝜆(𝑡) ∗ 𝜏𝑖 (𝑡)), t ϵ{1,2,3,4,5,6}, i = 1,2, … , N
where 𝐾𝑖 (𝑡) is individual i’s Non-mindfulness count at experiment on week t, 𝜏𝑖 (𝑡) is total time the
individual spent on the intervention at week t. Particularly in this case study, 𝜏𝑖 (𝑡) is same for each
individual i, i.e. 𝜏1 (1) = 𝜏2 (1) = ⋯ = 𝜏30 (1) = 3. In other words, individual 1 to 30 spent equal time
watching the intervention videos. And further 𝜏𝑖 (1)=3, 𝜏𝑖 (2)=5, 𝜏𝑖 (3)=8, 𝜏𝑖 (4)=10, 𝜏𝑖 (5)=12, 𝜏𝑖 (6)=15.
Based on the Poisson assumption, the maximum-likelihood estimator of 𝜆(𝑡) can be obtained
by:
𝜆̂(𝑡) =
∑𝑁
𝑖=1 𝐾𝑖 (𝑡)
∑𝑁
𝑖=1 𝜏𝑖 (𝑡)
Furthermore, because 𝜏𝑖 (𝑡) is fixed for each individual. The estimator can be rewrite as:
𝜆̂(𝑡) =
∑𝑁
𝑖=1 𝐾𝑖 (𝑡)
𝑁𝜏(𝑡)
Based on factorization theorem, it can be proved that this estimator is sufficient and complete.
In addition the estimator is unbiased and the variance achieves Cramér–Rao lower bound. Thus it is a
best unbiased estimator.
The confidence interval for this estimator can be obtained from chi-square distribution:
𝛼
𝜒 2 ( 2 , 2 ∑𝑁
𝑖=1 𝐾𝑖 (𝑡))
2𝑁𝜏(𝑡)
≤ 𝜆(𝑡) ≤
𝛼
𝜒 2 (1 − 2 , 2 ∑𝑁
𝑖=1 𝐾𝑖 (𝑡) + 2)
2𝑁𝜏(𝑡)
Based on this confidence interval and point estimator, the estimation of the simulation study is
shown in the bellowing figure:
Client Report 7
From the figure, the average non-mindfulness count shows significant decrease along the time.
Further, based on the Poisson assumption, the “Mindfulness State Time” follows an exponential
distribution as:
1
𝑀𝑠𝑡𝑖𝑚𝑒 ~ 𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙(
)
𝜆(𝑡)
Then the confidence interval of MStime is shown as bellowing:
The results show that MStime increases from week 1 to week 6 significantly.
Correlation between intervention and questionnaire result
In the above section, the intervention data is analyzed and it is concluded that the average
mindfulness performance increased along time. The previous section focused on population scope
(average), while this section focuses on individual scope. For one particular individual, what is concerned
is that, the correlation between increase of intervention performance and questionnaire scores. The
increase of questionnaire scores can be modeled easily by post questionnaire scores minus baseline
questionnaire scores. While the increase of intervention performance can be measured by a linear
model:
𝜆𝑖 (𝑡) = −𝑘𝑖 𝑡 + 𝜆𝑖,0
where 𝜆𝑖 (𝑡) is the average non-mindfulness count for individual i at time t. 𝑘𝑖 is the increasing rate of
intervention performance. If 𝑘𝑖 is high, it means the non-mindfulness count will decrease along time
significantly.
Based on this model, the increase of intervention performance for each individual can be
modeled by 𝑘𝑖 . Furthermore, denote the change of mindfulness questionnaire score as 𝑚𝑖 . The
correlation between 𝑘𝑖 and 𝑚𝑖 can be modeled by:
𝐶𝑜𝑟 =
𝐶𝑜𝑣(𝑚𝑖 , 𝑘𝑖 )
√𝑉𝑎𝑟(𝑘𝑖 ) 𝑉𝑎𝑟(𝑚𝑖 )
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Based on the simulation data, cor = 0.12503. It is concluded that the increase of intervention
performance and increase of questionnaire scores are positively correlated. In addition, the correlation
is not significantly high. It indicates in this sample, the major variation of mindfulness questionnaire
score does not comes from intervention.
4. Conclusion and Discussion
The above analytical methods will provide robust statistical analyses of both the effects of
intervention as well as change in mindfulness score over time. However, the study design could be
strengthened by a larger sample size and/or comparison group.