Biostatistics Case Studies 2007 Peter D. Christenson Biostatistician Session 3: Incomplete Data in Longitudinal Studies

Case Study

Study Design

Study Results 1 2 3

Enrolled and Completed Subjects Completer Analysis:N= LOCF Analysis:N= =63 were imputed. MMRM Analysis:N= None were imputed. When?

General Reasoning Completer Analysis: Biased; completers may differ from randomized in Phase III study. May be preferred in Phase II. LOCF: Proposed as a neutral method to implement analysis on an intent-to-treat population. MMRM: Uses all data, unlike completer analysis, but doesn’t impute unobserved data as in LOCF. ? Reasoning for this particular study ?

Imputation with LOCF Completer 30 HAM-A Score Week 0 Ignores potential progression; conservative; usually attenuates likely changes and ↑ standard deviations. No correction for using unobserved data as if real. Individual Subjects denotes imputed: N=63/260 Use all 260 values as if observed here.

Change from Baseline Baseline Final Visit Intermediate Visit 0 Change from Baseline Intermediate Visit Final VisitBaseline 0 LOCF: Ignore Potential Progression LRCF: Maintain Expected Relative Progression Individual Subjects One Alternative: Last Rank Carried Forward

Completer vs. LOCF Analysis LOCF Analysis Δ b/w groups = 1.8 N=260: 197 actual, 63 imputed Completer Analysis Δ b/w groups = 2.5 N=197: 197 actual Δ from baseline =~ 10 Clinically relevant Δ=? (Week 8 or earlier)

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* *Brown, Applied Mixed Models in Medicine, Wiley Consider a crossover (paired) study with 6 subjects. Subject 5 missed treatment A and subject 6 missed B. Completer analysis would use IDs 1-4; trt diff=4.25. Strict LOCF analysis would impute 22,17; trt diff=2.83. LOCF Difference

MMRM Example Cont’d Δ W =4.25 Paired Δ B =5 Unpaired Mixed model gets the better* estimate of the A-B difference from the 4 completers paired mean Δ w =4.25. It gets a poorer unpaired estimate from the other 2 subjects Δ B = = 5. How are these two “sub-studies” combined? *Why better?

MMRM Example Cont’d Δ W =4.25 Paired Δ B =5 Unpaired The overall estimated Δ is a weighted average of the separate Δs, inversely weighting by their variances: Δ = [Δ W /SE 2 (Δ W ) + Δ B /SE 2 (Δ B )]/K = [4.25/ /43.1]/(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 Covariance Parameter Estimates CS Residual Standard Effect trt Estimate Error DF t Value Pr > |t| Intercept trt A-B Diff Within Subjects (Paired) Among Subjects SE 2 for N=4+4 Paired Δ W =4.25: 8.90(1/4 + 1/4) = 4.45 SE 2 for N=1+1 Unpaired Δ B =43.1: ( )(1/1 + 1/1) = 43.1 Also 4.45 and 43.1 are used to get SE(Δ) = 2.01

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 8 estimate.

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 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 Compound Symmetry Estimated Covariance Pattern: Week (7.06) (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 Unstructured Estimated Covariance Pattern: Week (5.21) (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 Remaining slides put MMRM in context with other major methods. Mixed models are more general; MMRM is special case.

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 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 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” 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 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.