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Biostatistics Case Studies 2007. Session 3: Incomplete Data in Longitudinal Studies. Peter D. Christenson Biostatistician Case Study. Study Design. Study Results. 1. 2. 3. Enrolled and Completed Subjects. When?. Completer Analysis: N= 97+100.

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biostatistics case studies 2007
Biostatistics Case Studies 2007

Session 3:

Incomplete Data in Longitudinal Studies

Peter D. Christenson


enrolled and completed subjects
Enrolled and Completed Subjects


Completer Analysis: N= 97+100.

LOCF Analysis: N=130+130. 33+30=63 were imputed.

MMRM Analysis: N=130+130. None were imputed.

general reasoning
General Reasoning

Completer Analysis:

Biased; completers may differ from randomized in Phase III study. May be preferred in Phase II.


Proposed as a neutral method to implement analysis on an intent-to-treat population.


Uses all data, unlike completer analysis, but doesn’t impute unobserved data as in LOCF.

? Reasoning for this particular study ?

imputation with locf
Imputation with LOCF
  • Ignores potential progression; conservative; usually attenuates likely changes and ↑ standard deviations.
  • No correction for using unobserved data as if real.


denotes imputed: N=63/260

HAM-A Score


Individual Subjects


0 1 2 3 4 6 8


Use all 260 values as if observed here.

One Alternative: Last Rank Carried Forward


Ignore Potential Progression


Change from Baseline

Individual Subjects


Intermediate Visit

Final Visit


Maintain Expected Relative Progression


Change from Baseline

Intermediate Visit


Final Visit

completer vs locf analysis
Completer vs. LOCF Analysis

LOCF Analysis

Δ b/w groups = 1.8


197 actual, 63 imputed

Completer Analysis

Δ b/w groups = 2.5


197 actual

(Week 8 or earlier)

Δ from baseline =~ 10

Clinically relevant Δ=?

mixed model approach
Mixed Model Approach
  • The completer analysis removes some early data.
  • The LOCF method adds unobserved later data.
  • E.g., remove week 0 or add week 8.
  • Mixed Model for Repeated Measures MMRM:
  • Performs completer analysis.
  • Makes valid, but less preferred, comparison using data omitted from completer analysis.
  • Combines these two results.
  • This paper only gives p-values for results.

Example: Next Slide

mmrm example
Consider a crossover (paired) study with 6 subjects. Subject 5 missed treatment A and subject 6 missed B.MMRM Example*

LOCF Difference








Completer analysis would use IDs 1-4; trt diff=4.25.

Strict LOCF analysis would impute 22,17; trt diff=2.83.

*Brown, Applied Mixed Models in Medicine, Wiley 1999.

mmrm example cont d
MMRM Example Cont’d

ΔW=4.25 Paired

ΔB=5 Unpaired

Mixed model gets the better* estimate of the A-B difference from the 4 completers paired mean Δw=4.25.

It gets a poorer unpaired estimate from the other 2 subjects ΔB = 22-17 = 5.

How are these two “sub-studies” combined?

*Why better?

mmrm example cont d1
MMRM Example Cont’d

ΔW=4.25 Paired

ΔB=5 Unpaired

The overall estimated Δ is a weighted average of the separate Δs, inversely weighting by their variances:

Δ = [ΔW/SE2(ΔW) + ΔB/SE2(ΔB)]/K

= [4.25/4.45 + 5.0/43.1]/(1/4.45 + 1/43.1) = 4.32

The 4.45 and 43.1 incorporate the Ns and whether data is paired or unpaired: How are they found?

mmrm example sas output
MMRM Example - SAS Output


Effect trt Estimate Error DF t Value Pr > |t|

Intercept 18.1454 2.0162 5 9.00 0.000

trt 1 4.3203 2.0082 3 2.15 0.1206

A-B Diff

Covariance Parameter Estimates

CS 12.6264

Residual 8.8996

Among Subjects

Within Subjects (Paired)

SE2 for N=4+4 Paired ΔW=4.25:

8.90(1/4 + 1/4) = 4.45

SE2 for N=1+1 Unpaired ΔB=43.1:

(8.90+12.63)(1/1 + 1/1) = 43.1

Also 4.45 and 43.1 are used to get SE(Δ) = 2.01

mmrm more general i
MMRM - More General I

The example was “balanced” in missing data, with information from both treatments A and B in the unpaired data.

What if all missing data are at week 8, and none at week 0, as in our paper?

The unpaired week 0 mean is compared with the combined paired week 0 and week 8 mean, giving an estimate of half of the week 0 to week 8 difference. It is appropriately weighted with the paired week 0 to week 8estimate.

mmrm more general ii
MMRM - More General II

Can the intervening week 1 to week 6 data be used to improve further the week 0 to week 8 comparison?

That information could be used to better estimate the variances and covariances, if we are willing to make assumptions, e.g., a consistency of variability at each time.

Are these just “just so”, common-sense results?

Mixed model estimates satisfy certain statistical optimality criteria, provided that the model assumptions hold.

mmrm warning
MMRM - Warning

Software has many options since mixed models are general and flexible. Defaults may not be appropriate.

Requires specifying model structure; assumptions needed; should check assumptions.

More experience needed than typical methods. Start by comparing ANOVA for a no-missing-data study with mixed model.

See next slide for some modeling needed.

some covariance patterns
Some Covariance Patterns

Compound Symmetry

Estimated Covariance Pattern:

Week 0 8

0 (7.06)2 12.4

8 12.4 (7.06)2

Correlation = 12.4/7.1*7.1=0.25

This model forces the SD among subjects to be the same at each week.

But: Week 0 SD = 5.2

Week 8 SD = 8.8


Estimated Covariance Pattern:

Week 0 8

0 (5.21)2 12.4

8 12.4 (8.79)2

Correlation = 12.4/5.2*8.8=0.27

This model allows different SDs among subjects at each week.

mmrm role in major analysis methods
MMRM Role in Major Analysis Methods

Remaining slides put MMRM in context with other major methods.

Mixed models are more general; MMRM is special case.

big picture multiple data lingo
Big Picture: “Multiple” Data Lingo

Multiple Regression:

Outcome: Single value, say HAM-A at 8 weeks.

Predictors: Multiple - treatment, covariates (age, baseline disease severity, other meds, etc.)

Multivariate ANOVA (MANOVA):

Outcome: Multiple, say (HAM-A, SDS, Dizziness) at 8 weeks, as a pattern or profile.

Predictors: Single, say only treatment, or multiple.

Repeated Measures: (longitudinal, as in this paper)

Outcome: Single quantity, say HAM-A.

Predictors: Time, and others (treatment, covariates).

repeated measures studies
Repeated Measures Studies

The same subjects are measured repeatedly on the same outcome, usually at different times or body sites to be compared.

Does not apply to only replicated measurements, e.g. multiple histology slices that are averaged.

Time is usually relative, such as from start of treatment, or may be calendar time as in epidemiological studies.

Usually have fixed time intervals, but times may be different for different subjects, e.g., retrospective series of clinic visits.

Study goals will dictate type of analysis - next slide.

goals in repeated measures studies
Goals in Repeated Measures Studies
  • Some study objectives:
    • Compare overall time-averaged treatment.
    • Specific features of pattern, as in pharmacokinetic studies of AUC, peak, half- life, etc.
    • Compare treatments at every time point.
    • Compare treatments on rate of change over time.
    • Compare treatments at end of study.
mixed models in general
Mixed Models in General

“Mixed” means combination of fixed effects (e.g., drugs; want info on those particular drugs) and random effects (e.g., centers or patients; not interested in the particular ones in the study).

AKA multilevel models, hierarchical models.

Very flexible, incorporate unequal patient variability, correlation, pairing, repeated values at multiple levels, subject clustering e.g., from the same family, and data missing at random.

More specifications required than typical analyses.

summary mixed models repeated measures
Summary: Mixed Models Repeated Measures
  • Currently one of the preferred methods for missing data.
  • Does not resolve bias if missingness is related to treatment.
  • Requires more model specifications than is typical.
  • Mild deviations from assumed covariance pattern do not usually have a large influence.
  • May be difficult to apply objectively in clinical trials where the primary analysis needs to be detailed a priori.
  • Can be intimidating; need experience with modeling; software has many options to be general and flexible.