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Todd D. Little University of Kansas Director, Quantitative Training Program

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- On the Merits of Planning and Planning for Missing Data*
- *You’re a fool for not using planned missing data design

Todd D. Little

University of Kansas

Director, Quantitative Training Program

Director, Center for Research Methods and Data Analysis

Director, Undergraduate Social and Behavioral Sciences Methodology Minor

Member, Developmental Psychology Training Program

crmda.KU.edu

Workshop presented 05-21-2012 @

Max Planck Institute for Human Development in Berlin, Germany

Very Special Thanks to: Mijke Rhemtulla & Wei Wu

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University of Kansas

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University of Kansas

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University of Kansas

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University of Kansas

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University of Kansas

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- Learn about the different types of missing data
- Learn about ways in which the missing data process can be recovered
- Understand why imputing missing data is not cheating
- Learn why NOT imputing missing data is more likely to lead to errors in generalization!

- Learn about intentionally missing designs
- Discuss imputation with large longitudinal datasets
- Introduce a simple method for significance testing

Road Map

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- Recoverability
- Is it possible to recover what the sufficient statistics would have been if there was no missing data?
- (sufficient statistics = means, variances, and covariances)

- Is it possible to recover what the parameter estimates of a model would have been if there was no missing data.

- Is it possible to recover what the sufficient statistics would have been if there was no missing data?
- Bias
- Are the sufficient statistics/parameter estimates systematically different than what they would have been had there not been any missing data?

- Power
- Do we have the same or similar rates of power (1 – Type II error rate) as we would without missing data?

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- Missing Completely at Random (MCAR)
- No association with unobserved variables (selective process) and no association with observed variables

- Missing at Random (MAR)
- No association with unobserved variables, but maybe related to observed variables
- Random in the statistical sense of predictable

- No association with unobserved variables, but maybe related to observed variables
- Non-random (Selective) Missing (MNAR)
- Some association with unobserved variables and maybe with observed variables

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Statistical Power: Will always be greater when missing data is imputed!

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- List-wise Deletion
- If a single data point is missing, delete subject
- N is uniform but small
- Variances biased, means biased
- Acceptable only if power is not an issue and the incomplete data is MCAR

- Pair-wise Deletion
- If a data point is missing, delete paired data points when calculating the correlation
- N varies per correlation
- Variances biased, means biased
- Matrix often non-positive definite
- Acceptable only if power is not an issue and the incomplete data is MCAR

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- Sample-wise Mean Substitution
- Use the mean of the sample for any missing value of a given individual
- Variances reduced
- Correlations biased

- Subject-wise Mean Substitution
- Use the mean score of other items for a given missing value
- Depends on the homogeneity of the items used
- Is like regression imputation with regression weights fixed at 1.0

- Use the mean score of other items for a given missing value

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- Regression Imputation – Focal Item Pool
- Regress the variable with missing data on to other items selected for a given analysis
- Variances reduced
- Assumes MCAR and MAR

- Regression Imputation – Full Item Pool
- Variances reduced
- Attempts to account for NMAR in as much as items in the pool correlate with the unobserved variables responsible for the missingness

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MI or FIML

- In 1978, Rubin proposed Multiple Imputation (MI)
- An approach especially well suited for use with large public-use databases.
- First suggested in 1978 and developed more fully in 1987.
- MI primarily uses the Expectation Maximization (EM) algorithm and/or the Markov Chain Monte Carlo (MCMC) algorithm.

- Beginning in the 1980’s, likelihood approaches developed.
- Multiple group SEM
- Full Information Maximum Likelihood (FIML).
- An approach well suited to more circumscribed models

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- FIML maximizes the casewise -2loglikelihood of the available data to compute an individual mean vector and covariance matrix for every observation.
- Since each observation’s mean vector and covariance matrix is based on its own unique response pattern, there is no need to fill in the missing data.

- Each individual likelihood function is then summed to create a combined likelihood function for the whole data frame.
- Individual likelihood functions with greater amounts of missing are given less weight in the final combined likelihood function than those will a more complete response pattern, thus controlling for the loss of information.

- Formally, the function that FIML is maximizing is
where

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- Multiple imputation involves generating m imputed datasets (usually between 20 and 100), running the analysis model on each of these datasets, and combining the m sets of results to make inferences.
- By filling in m separate estimates for each missing value we can account for the uncertainty in that datum’s true population value.

- Data sets can be generated in a number of ways, but the two most common approaches are through an MCMC simulation technique such as Tanner & Wong’s (1987) Data Augmentation algorithm or through bootstrapping likelihood estimates, such as the bootstrapped EM algorithm used by Amelia II.
- SAS uses data augmentation to pull random draws from a specified posterior distribution (i.e., stationary distribution of EM estimates).

- After m data sets have been created and the analysis model has been run on each separately, the resulting estimates are commonly combined with Rubin’s Rules (Rubin, 1987).

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- (But only if variables related to missingness are included in analysis, or missingness is MCAR)
- EM Imputation
- Imputes the missing data values a number of times starting with the E step
- The E(stimate)-step is a stochastic regression-based imputation
- The M(aximize)-step is to calculate a complete covariance matrix based on the estimated values.
- The E-step is repeated for each variable but the regression is now on the covariance matrix estimated from the first E-step.
- The M-step is repeated until the imputed estimates don’t differ from one iteration to the other

- MCMC imputation is a more flexible (but computer-intensive) algorithm.

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- (But only if variables related to missingness are included in analysis, or missingness is MCAR)
- Multiple (EM or MCMC) Imputation
- Impute N (say 20) datasets
- Each data set is based on a resampling plan of the original sample
- Mimics a random selection of another sample from the population

- Run your analyses N times
- Calculate the mean and standard deviation of the N analyses

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- Fraction Missing is a measure of efficiency lost due to missing data. It is the extent to which parameter estimates have greater standard errors than they would have had all data been observed.
- It is a ratio of variances:
Estimated parameter variance in the complete data set

Between-imputation variance

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- Fraction of Missing Information (asymptotic formula)
- Varies by parameter in the model
- Is typically smaller for MCAR than MAR data

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Obs BADL0 BADL1 BADL3 BADL6 MMSE0 MMSE1 MMSE3 MMSE6

1 65 95 95 100 23 25 25 27

2 10 10 40 25 25 27 28 27

3 95 100 100 100 27 29 29 28

4 90 100 100 100 30 30 27 29

5 30 80 90 100 23 29 29 30

6 40 50 . . 28 27 3 3

7 40 70 100 95 29 29 30 30

8 95 100 100 100 28 30 29 30

9 50 80 75 85 26 29 27 25

10 55 100 100 100 30 30 30 30

11 50 100 100 100 30 27 30 24

12 70 95 100 100 28 28 28 29

13 100 100 100 100 30 30 30 30

14 75 90 100 100 30 30 29 30

15 0 5 10 . 3 3 3 .

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PROC MI data=sample out=outmi

seed = 37851 nimpute=100

EM maxiter = 1000;

MCMC initial=em (maxiter=1000);

Var BADL0 BADL1 BADL3 BADL6 MMSE0 MMSE1 MMSE3 MMSE6;

run;

out=

Designates output file for imputed data

nimpute =

# of imputed datasets

Default is 5

Var

Variables to use in imputation

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Obs _Imputation_ BADL0 BADL1 BADL3 BADL6 MMSE0 MMSE1 MMSE3 MMSE6

1 1 65 95 95 100 23 25 25 27

2 1 10 10 40 25 25 27 28 27

3 1 95 100 100 100 27 29 29 28

4 1 90 100 100 100 30 30 27 29

5 1 30 80 90 100 23 29 29 30

6 1 40 50 21 12 28 27 3 3

7 1 40 70 100 95 29 29 30 30

8 1 95 100 100 100 28 30 29 30

9 1 50 80 75 85 26 29 27 25

10 1 55 100 100 100 30 30 30 30

11 1 50 100 100 100 30 27 30 24

12 1 70 95 100 100 28 28 28 29

13 1 100 100 100 100 30 30 30 30

14 1 75 90 100 100 30 30 29 30

15 1 0 5 10 8 3 3 3 2

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- I pity the fool who does not impute
- Mr. T

- If you compute you must impute
- Johnny Cochran

- Go forth and impute with impunity
- Todd Little

- If math is God’s poetry, then statistics are God’s elegantly reasoned prose
- Bill Bukowski

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- In planned missing data designs, participants are randomly assigned to conditions in which they do not respond to all items, all measures, and/or all measurement occasions
- Why would you want to do this?
- Long assessments can reduce data quality
- Repeated assessments can induce practice effects
- Collecting data can be time- and cost-intensive
- Less taxing assessments may reduce unplanned missingness

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- Cross-Sectional Designs
- Matrix sampling (brief)
- Three-Form Design (and Variations)
- Two-Method Measurement (very cool)

- Longitudinal Designs
- Developmental Time-Lag
- Wave- to Age-based designs
- Monotonic Sample Reduction
- Growth-Curve Planned Missing

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Test a few participants on full item bank

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Or, randomly sample items and people…

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- Assumptions
- The K items are a random sample from a population of items (just as N participants are a random sample from a population)

- Limitations
- Properties of individual items or relations between items are not of interest

- Not used much outside of ability testing domain.

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- Graham Graham, Taylor, Olchowski, & Cumsille(2006)
- Raghunathan & Grizzle (1995) “split questionnaire design”
- Wacholder et al. (1994) “partial questionnaire design”

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- What goes in the Common Set?

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- 21 questions made up of 7 3-question subtests

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- Common Set (X)

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- Common Set (X)

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- Set A

I start conversations.

I get stressed out easily.

I am always prepared.

I have a rich vocabulary.

I am interested in people.

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- Set B

I am the life of the party.

I get irritated easily.

I like order.

I have excellent ideas.

I have a soft heart.

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- Set C

I am comfortable around people.

I have frequent mood swings.

I pay attention to details.

I have a vivid imagination.

I take time out for others.

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- (Graham, Taylor, Olchowski, & Cumsille, 2006)

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- (Graham, Taylor, Olchowski, & Cumsille, 2006)

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- Expensive Measure 1
- Gold standard– highly valid (unbiased) measure of the construct under investigation
- Problem: Measure 1 is time-consuming and/or costly to collect, so it is not feasible to collect from a large sample

- Inexpenseive Measure 2
- Practical– inexpensive and/or quick to collect on a large sample
- Problem: Measure 2 is systematically biased so not ideal

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- e.g., measuring stress
- Expensive Measure 1 = collect spit samples, measure cortisol
- Inexpensive Measure 2 = survey querying stressful thoughts

- e.g., measuring intelligence
- Expensive Measure 1 = WAIS IQ scale
- Inexpensive Measure 2 = multiple choice IQ test

- e.g., measuring smoking
- Expensive Measure 1 = carbon monoxide measure
- Inexpensive Measure 2 = self-report

- e.g., Student Attention
- Expensive Measure 1 = Classroom observations
- Inexpensive Measure 2 = Teacher report

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- How it works
- ALL participants receive Measure 2 (the cheap one)
- A subset of participants also receive Measure 1 (the gold standard)
- Using both measures (on a subset of participants) enables us to estimate and remove the bias from the inexpensive measure (for all participants) using a latent variable model

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Self-Report

Bias

Self-

Report 1

Self-

Report 2

CO

Cotinine

Smoking

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- Example
- Does child’s level of classroom attention in Grade 1 predict math ability in Grade 3?
- Attention Measures
- 1) Direct Classroom Assessment (2 items, N = 60)
- 2) Teacher Report (2 items, N = 200)

- Math Ability Measure, 1 item (test score, N = 200)

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Attention

(Grade 1)

Math Score

(Grade 3)

Teacher

Report 1

(N = 200)

Math Score

(Grade 3)

(N = 200)

Teacher

Report 2

(N = 200)

Direct

Assessment 1

(N = 60)

Direct

Assessment 2

(N = 60)

Teacher

Bias

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Attention

(Grade 1)

Math Score

(Grade 3)

Teacher

Report 1

(N = 200)

Math Score

(Grade 3)

(N = 200)

Teacher

Report 2

(N = 200)

Direct

Assessment 1

(N = 60)

Direct

Assessment 2

(N = 60)

Teacher

Bias

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Attention

(Grade 1)

Math Score

(Grade 3)

Teacher

Report 1

(N = 200)

Math Score

(Grade 3)

(N = 200)

Teacher

Report 2

(N = 200)

Direct

Assessment 1

(N = 60)

Direct

Assessment 2

(N = 60)

Teacher

Bias

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Attention

(Grade 1)

Math Score

(Grade 3)

Teacher

Report 1

(N = 200)

Math Score

(Grade 3)

(N = 200)

Teacher

Report 2

(N = 200)

Direct

Assessment 1

(N = 60)

Direct

Assessment 2

(N = 60)

Teacher

Bias

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Attention

(Grade 1)

Math Score

(Grade 3)

Teacher

Report 1

(N = 200)

Math Score

(Grade 3)

(N = 200)

Teacher

Report 2

(N = 200)

Direct

Assessment 1

(N = 60)

Direct

Assessment 2

(N = 60)

Teacher

Bias

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Attention

(Grade 1)

Math Score

(Grade 3)

Teacher

Report 1

(N = 200)

Math Score

(Grade 3)

(N = 200)

Teacher

Report 2

(N = 200)

Direct

Assessment 1

(N = 60)

Direct

Assessment 2

(N = 60)

Teacher

Bias

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2-Method Planned Missing Design

- Assumptions:
- expensive measure is unbiased (i.e., valid)
- inexpensive measure is systematically biased
- both measures access the same construct

- Goals
- Optimize cost
- Optimize power

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- All participants get the inexpensive measure
- Only a subset get the expensive measure
- Cost:

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- Holding cost constant, as Ntotal increases, Nexpensive decreases
- As Ntotal increases, SEs begin to decrease (power increases); as Ntotal continues to increase, SEs increase again, driving power back down

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- Goal: find the sweet spot!

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- Rather than specific items missing, longitudinal planned missing designs tend to focus on whole waves missing for individual participants
- Researchers have long turned complete data into planned missing data with more time points
- e.g., data at 3 grades transformed into 8 ages

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- Use 2-time point data with variable time-lags to measure a growth trajectory + practice effects (McArdle & Woodcock, 1997)

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Time

Age

student

T1

T2

2

4

6

0

1

3

5

1

5;6

5;7

2

5;3

5;8

3

4;9

4;11

4

4;6

5;0

5

4;11

5;4

6

5;7

5;10

7

5;2

5;3

8

5;4

5;8

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T0

T1

T2

T3

T4

T5

T6

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Intercept

1

1

1

1

1

1

1

T0

T1

T2

T3

T4

T5

T6

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Linear growth

Intercept

Growth

1

0

6

1

1

5

1

2

4

3

1

1

1

1

T0

T1

T2

T3

T4

T5

T6

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Constant Practice Effect

Intercept

Growth

Practice

0

1

0

6

1

1

1

5

1

1

2

4

3

1

1

1

1

1

1

1

1

T0

T1

T2

T3

T4

T5

T6

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Exponential Practice Decline

Intercept

Growth

Practice

0

1

0

6

1

1

1

5

.87

1

2

4

3

.67

1

1

.55

1

.45

.35

1

T0

T1

T2

T3

T4

T5

T6

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The Equations for Each Time Point

Constant Practice Effect Declining Practice Effect

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- Summary
- 2 measured time points are formatted according to time-lag
- This formatting allows a growth-curve to be fit, measuring growth and practice effects

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- The idea of reformatting data to answer a different question is not limited to time-lag designs
- Wave-based data collection (e.g., data collected at Grade 1-3) can be transformed into age-based data with missingness

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age

grade

5;6-

5;11

6;6-

6;11

7;6-

7;11

4;6-

4;11

5;0-

5;5

6;0-

6;5

7;0-

7;5

2

student

K

1

1

5;6

6;7

7;3

2

5;3

6;0

7;4

3

4;9

5;11

6;10

4

4;6

5;5

6;4

5

4;11

5;9

6;10

6

5;7

6;7

7;5

7

5;2

6;1

7;3

8

5;4

6;5

7;6

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age

- Out of 3 waves, we create 7 waves of data with high missingness
- Allows for more fine-tuned age-specific growth modeling
- Even high amounts of missing data are not typically a problem for estimation

5;6-

5;11

6;6-

6;11

7;6-

7;11

4;6-

4;11

5;0-

5;5

6;0-

6;5

7;0-

7;5

5;6

6;7

7;3

5;3

6;0

7;4

4;9

5;11

6;10

4;6

5;5

6;4

4;11

5;9

6;10

5;7

6;7

7;5

5;2

6;1

7;3

5;4

6;5

7;6

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- Advantages:
- Cost reduction
- A lot of power to estimate effects at earlier waves

- Disadvantages:
- Very little power to estimate effects dependent on the last wave of data, e.g., growth curve models (may be missing 80% of data)
- It is important to be able to estimate attrition rates before beginning data collection

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- Sometimes used in large datasets (e.g., Early Childhood Longitudinal Study) to reduce costs
- At each wave, a randomly-selected subgroup of the original sample is observed again
- The remainder of the original participants do not need to be kept track of, dramatically reducing costs

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- With a particular analysis in mind, missingness may be tailored to maximize power
- In growth-curve designs, the most important parameters are the growth parameters (e.g., estimate the steepness and the shape of the curve)
- Estimation precision depends heavily on the first and last time points
- A planned missing design can take advantage of this by putting missingness in the middle

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- Purposeful missing data can address several issue in study design
- Cost of data collection
- Participant burden/fatigue
- Practice effects
- Participant dropout

- Rearranging data can turn one complete design into a more nuanced missing data design
- Developmental time-lag designs
- Wave-missing into age-missing

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- Consider the following Monte Carlo simulation:
- 60% MAR (i.e., Aux1) missing data
- 1,000 samples of N = 100

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90

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91

Simulation Results Showing the Bias Associated with Omitting a Correlate of Missingness.

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Simulation Results Showing the Bias Reduction Associated with Including Auxiliary Variables in a MNAR Situation.

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Improvement in power relative to the power of a model with no auxiliary variables.

Q

Q

Simulation results showing the relative power associated with including auxiliary variables in a MCAR Situation.

Q

Q

Q

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- Use PCA to reduce the dimensionality of the auxiliary variables in a data set.
- A new smaller set of auxiliary variables are created (e.g., principal components) that contain all the useful information (both linear and non-linear) in the original data set.

- These principal component scores are then used to inform the missing data handling procedure (i.e., FIML, MI).

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96

- Consider a series of simulations:
- MCAR, MAR, MNAR (10-60%) missing data
- 1,000 samples of N = 50-1000

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97

60% MAR correlation estimates with no auxiliary variables

Simulation results showing XY correlation estimates (with 95 and 99% confidence intervals) associated with a 60% MAR Situation.

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Bias – Linear MAR process

ρAux,Y = .60; 60% MAR

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99

Non-Linear Missingness

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Bias – Non-Linear MAR process

ρAux,Y = .60; 60% non-linear MAR

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101

Bias

ρAux,Y = .60; 60% MAR

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102

Bias

ρAux,Y = .60; 60% MAR

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103

Bias

ρAux,Y = .60; 60% MAR

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104

60% MAR correlation estimates with no auxiliary variables

Simulation results showing XY correlation estimates (with 95 and 99% confidence intervals) associated with a 60% MAR Situation.

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105

60% MAR correlation estimates with all possible auxiliary variables (r = .60)

Simulation results showing XY correlation estimates (with 95 and 99% confidence intervals) associated with a 60% MAR Situation and 8 auxiliary variables.

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60% MAR correlation estimates with 1 PCA auxiliary variable (r = .60)

Simulation results showing XY correlation estimates (with 95 and 99% confidence intervals) associated with a 60% MAR Situation and 1 PCA auxiliary variable.

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107

1 PCA Auxiliary

Auxiliary Variable Power Comparison

All 8 Auxiliary Variables

1 Auxiliary

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Faster and more reliable convergence

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- Including principal component auxiliary variables in the imputation model improves parameter estimation compared to
- the absence of auxiliary variables and
- beyond the improvement of typical auxiliary variables in most cases, particularly with the non-linear MAR type of missingness.

- Improve missing data handling procedures when the number of potential auxiliary variables is beyond a practical limit.

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110

www.quant.ku.edu

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- Generate multiply imputed datasets (m).
- Calculate a single covariance matrix on all N*m observations.
- By combining information from all m datasets, this matrix should represent the best estimate of the population associations.

- Run the Analysis model on this single covariance matrix and use the resulting estimates as the basis for inference and hypothesis testing.
- The fit function from this approach should be the best basis for making inferences about model fit and significance.

- Using a Monte Carlo Simulation, we test the hypothesis that this approach is reasonable.

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Population Model

.52

Factor B

Factor A

1*

1*

.76

.75

.68

.70

.67

.72

.69

.79

.72

.75

.81

.72

.74

.70

.71

.69

.81

.73

.78

.79

A1

A2

A3

A4

A5

A6

A7

A8

A9

A10

B1

B2

B3

B4

B5

B6

B7

B8

B9

B10

.49

.43

.53

.48

.52

.38

.42

.51

.35

.49

.45

.52

.50

.38

.53

.35

.47

.44

.55

.39

Note: These are fully standardized parameter estimates

RMSEA = .047, CFI = .967, TLI = .962, SRMR = .021

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- On the Merits of Planning and Planning for Missing Data*
- *You’re a fool for not using planned missing data design

Thanks for your attention!

Questions?

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Workshop presented 05-21-2012

Max Planck Institute for Human Development, Berlin, Germany

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References

crmda.KU.edu

Enders, C. K. (2010). Applied missing data analysis. New York: Guilford Press.

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Dr. Todd Little is currently at

Texas Tech University

Director, Institute for Measurement, Methodology, Analysis and Policy (IMMAP)

Director, “Stats Camp”

Professor, Educational Psychology and Leadership

Email: [email protected]

IMMAP (immap.educ.ttu.edu)

Stats Camp (Statscamp.org)

www.Quant.KU.edu