Repetition Multiple imputation

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RepetitionMultiple imputation

• Ziad Taib
• Biostatistics, AZ
• May 20, 2009

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Data set with missing values
Result
Completed set
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General principles
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Informal justification
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The algorithm (Estimation)
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Pooling information
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MI in practice
A simulation-based approach to missing data
1. Generate M gt 1 plausible versions of .
Complete Cases
2. Analyze each of the M datasets by standard
complete-data methods.





3. Combine the results across the M datasets (M
3-5 is usually OK).
imputation for Mth dataset
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Software
2. SAS software (experimental) It is part of
SAS/STAT version 8.02 SAS institute paper on
multiple imputation, gives an example and SAS
code http//www.sas.com/rnd/app/papers/multipleim
putation.pdf SAS documentation on PROC
MI http//www.sas.com/rnd/app/papers/miv802.pdf S
AS documentation on PROC MIANALYZE http//www.sas.
com/rnd/app/papers/mianalyzev802.pdf
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Software
1. Joe Schafers software from his web site.
(0) http//www.stat.psu.edu/7Ejls/misoftwa.html
top Schafer has written publicly available
software primarily for S-plus. There is a
stand-alone Windows package for data that is
multivariate normal. This web site contains much
useful information regarding multiple imputation.
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Software
3. SOLAS version 3.0 (1K) http//www.statsol.ie/s
olas/solas.htm Windows based software that
performs different types of imputation
Hot-deck imputation Predictive
OLS/discriminant regression Nonparametric
based on propensity scores Last value carried
forward Will also combine parameter results
across the M analyses.
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MI in SAS
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MI Analysis of the Orthodontic Growth Data
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RepetitionPower and sample size estimation

• Ziad Taib
• Biostatistics, AZ
• May 20, 2009

Name, department
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Example Estimating the sample size needed in a
trial for chronic pulmonary diseases

• Chronic pulmonary diseases (such as Chronic
  Obstructive Pulmonary Disease COPD) concern the
  development of emphysema.
• A clinical trial using lung densitometry
  (measuring the lung density through CT scan) as
  an endpoint is typically designed as a
  longitudinal study with repeated measurements at
  fixed time intervals.
• Since lung density measurements are closely
  correlated with lung volume (inspiration level),
  it is important to include lung volume
  measurements in statistical analyses as a
  longitudinal covariate.
• Lung volume is normally measured at the same time
  as the lung density is measured.

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• The clinical efficacy can be assessed by
  comparing the progression of lung density loss
  between two treatment groups (active vs. placebo)
  using a random coefficient model a longitudinal
  linear mixed model with a random intercept and
  slope.
• In planning the clinical trial with such complex
  statistical analyses, the calculation of the
  sample size required to achieve a given power to
  detect a specified treatment difference is an
  important, often complex issue.
• In this example, an empirical approach is used to
  calculate the sample size by simulating
  trajectories of lung density and lung volume
  using SAS. We present step-by-step details for
  sample size calculation through simulation, and
  discuss the pros and cons of this approach.

(1)
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• Yij is the efficacy endpoint (i.e. lung density)
  measurement for subject i 1, 2,, n, at fixed
  time point j 1, 2, , K.
• TRT is an indicator of subject is treatment
  group (i.e. TRT1 for active drug TRT0 for
  placebo).
• COVij is a longitudinal covariate (i.e. logarithm
  of lung volume) for subject i 1, 2,, n, at
  fixed time point j 1, 2, , K.
• b0 and b2 are subject-specific random effects for
  the intercept and slope, respectively, which are
  from a normal distribution with mean 0 and
  variance s02 and s02, respectively.
• eij is the random error from a normal
  distribution with mean 0 and variance s2 .
• ß0, ß1, ß2, ß3, and ß4 are the fixed effects for
  intercept, treatment, time, covariate and
  interaction of treatment and time respectively.
• Here we assume that the benefits can be assessed
  quantitatively by comparing the slopes of lung
  density trajectories for the two treatment
  groups. This quantity is captured by ß4.

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Sample Size Estimation Using Simulations

• In the model, ß4 is typically our interest, which
  is the difference in slope of time between two
  treatment groups (active vs. placebo).
• There is no direct mathematical formula to
  calculate the sample size for a given statistical
  power (i.e. 80) to test the null hypothesis
  ß40 with a specified type I error (i.e. a0.05).
  
• One approach to calculate the sample size for a
  given power is through the simulation.

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Methods used

• Assume we know the parameters ß0, ß1, ß2, ß3, and
  ß4 , and s02 and s02 from either history data,
  previous clinical trials or meaningful clinical
  differences.
• We want to test, the study design in terms of
  number of time points (K) and fixed time
  intervals (TIME), and the longitudinal covariate
  COVij.
• For a fixed equal sample size n for each
  treatment, the trajectories of efficacy
  measurement Yij (i.e. lung density) for the n
  subjects can be simulated through the model for
  each treatment group.
• Then, perform a statistical test on ß4 0 by
  using the SAS Proc MIXED on the simulated data
  set, and record whether the p-value lt 0.05.

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• 5. The sample code to perform the test is as
  follow
• proc mixed data data
• class id trt
• model y trt time trttime cov / solution
• random intercept time/ subject id type un
• run
• For the fixed sample size n per treatment group,
  simulate M (i.e. M1000) times and the proportion
  of significance tests of ß4 0 among the total M
  simulations is the statistical power () for the
  sample size n per treatment group.
• Then, adjust the sample size n to achieve
  desirable statistical power.

() In reality ß40.7 gt0 (in our simulations) so
the proportion of times we reject the hypothesis
ß4 0 of the power.
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Simulating the response

• In order to simulate the trajectories of Yij, it
  is necessary to simulate the trajectories of
  longitudinal covariate COVij. Similarly, assume
  COVij is from a linear model regressing against
  time with a random intercept
• Where g0 and g1 are the fixed intercept and slope
  respectively r0 and eij are from a normal
  distribution with mean 0 and variance d12 and
  d22, respectively. If we know the parameters (g0,
  g1 , d12 and d22 ) from history data or previous
  clinical trials for the study population, it will
  be simple to simulate the trajectories of the
  longitudinal covariate COVij by using SAS random
  generating functions

(2)
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• Summary
• 1. Obtain the pre-specified parameters through
  either history data, previous clinical trials or
  meaningful clinical difference to be tested from
  clinicians
• 2. Specify a desired statistical power (i.e. 80)
  and a type-1 error rate (i.e. 5)
• 3. Simulate trajectories of efficacy measurement
  (i.e. lung density) and longitudinal covariate
  (i.e. logarithm of lung volume) for a fixed
  sample size (n) of subjects within each treatment
  arm
• A. Trajectories of longitudinal covariate (i.e.
  logarithm of lung volume) are simulated through
  model (2)
• B. Trajectories of efficacy measurement (i.e.
  lung density) are simulated through model (1)

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• 4. Perform the statistical test on ß40 based on
  the simulated data set. Record whether a p-value
  lt 0.05 was obtained
• 5. Repeat steps 3 and 4 M (i.e. M1000) times and
  calculate the statistical power for the fixed
  sample size
• 6. Repeat steps 3 - 5 for various values of n.
  Stop when desired statistical power is obtained

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Results Example of a Simulation

• Assume there are two treatment groups (active vs.
  placebo) in a study design. The efficacy endpoint
  along with the longitudinal covariate will be
  measured at K4 time points at baseline, 1 year,
  2 years and 3 years. All corresponding parameters
  specified in model (1) and (2) could be obtained
  either through history data, previous clinical
  trials or meaningful clinical difference to be
  tested from clinicians. For purpose of
  simulation, they are randomly selected and
  specified as below

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• The summary of statistical power for a given
  sample size per treatment based on M 1000
  simulated data sets is listed below
• Therefore, a sample size 45 per treatment arm has
  an estimated statistical 80 power to detect the
  treatment slope difference of 0.7 in a random
  coefficient model for the study design above.

n
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Conclusions and Discussion

• In practice, it is rarely the case that all
  subjects have the complete data for all visits in
  the study because of missing certain study
  visits, drop out or other reasons. Since our
  simulation framework assumes there are no missing
  observations, we recommend that the implemented
  sample size for the designed trial include more
  subjects than the number estimated from the
  simulation. In most cases an increase of 5 or
  10 should suffice, but depending on the
  characteristics of the designed trial such as the
  study population, difficulty of study procedure,
  difficulty of study measurement etc to cause the
  subjects drop out or missing of study
  measurements. The appropriate percentage could
  vary.

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Post-Hoc Power (also known as observed power or
retrospective power)

• You have collected the data, ran an appropriate
  statistical analysis, and did not observe
  statistical significance as indicated by a
  relatively large p-value. So you decide to
  compute post-hoc power to see how powerful the
  test was, which, by itself is essentially an
  empty, meaningless result.
• Post-hoc power is merely a one-to-one
  transformation of the p-value (based on the
  F-statistic and degrees of freedom as illustrated
  above).
• In this situation power was computed based only
  on what this particular sample data showed the
  observed difference in means, the computed
  standard error, and the actual sample sizes of
  the groups all contributed to the observed
  power exactly as they did to the p-value.

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• So, power calculations can only be considered
  as a prospective or an "a priori" concept. Power
  calculations should be directed towards planning
  a study, not an after-theexperiment review of the
  results.
• None of the SAS statistical procedures (e.g.,
  PROCs REG, TTEST, GLM, or MIXED and others)
  provide retrospective (post hoc) power
  calculations. (However, through saving results
  from PROC MIXED with the ODS and following
  through with a few basic SAS functions, it is
  quite simple to compute them in a DATA step or
  with the inputs to PROC POWER or PROC GLMPOWER.)

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Sample Questions for the Final Exam
Ziad Taib Biostatistics, AZ MV, CTH
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Question 1

• Formulate the general LMM and
• State its underlying assumptions.
• Explain/interpret its ingredients
• Explain what is meant by the marginal model
• Explain what is meant by the hierarchical model.
• Explain the difference between the marginal- and
  the hierarchical model

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Question 2

• Explain how and why the predicted values of the
  random effects in a linear mixed model can be
  used to identify outliers

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Question 3

• Formulate the generalized linear mixed model and
  give an example of your choice of its use.

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Question 4

• Skiss for how we can obtain a useful formula for
  the prediction of the random effects. Only a
  principle description is needed.

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Question 5

• Define missing at random and missing completely
  at random. Argue why the the direct maximum
  likelihood method can be used under a suitable
  ignorability condition.

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Question 6

• Explain what is meant by Multiple Imputation
  (MI). Describe the main algorithm used in MI and
  give a informal argument for why it is valid.

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Question 7

• Give two different candidates for the definition
  of residual in a general LMM.

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Answers to samplequestion 1

• Ziad Taib
• Biostatistics, AZ
• May 20, 2009

Name, department
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mij P(Yij1) p EYij
1-P(Yij1)
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Answers to samplequestions 2

• Ziad Taib
• Biostatistics, AZ
• May 20, 2009

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Lund0 Göteborg1
Lund
Göteborg
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