Biostatistics Case Studies 2008

Peter D. Christenson

Biostatistician

http://gcrc.labiomed.org/biostat

Session 5:

Choices for Longitudinal Data Analysis

Case Study

Study Goal - General

Specific Primary Aim

The “ANCOVA” would be a t-test, if we ignored the baseline values and the different centers.

The outcome is change in HAM-A. The groups are drug and placebo. The signal:noise ratio is ……

Comparison of Change Means with t-test

Strength of Treatment Effect:

Signal:Noise Ratio t=

Observed Δ

SDΔ √(1/N1 + 1/N2)

Δ = Drug - Placebo Mean (Final-Base) Diff in HAM-A changes

SD = Std Dev of within group HAM-A changes

N1 = N2 = Group size

| t | > ~1.96 ↔ p<0.05

Comparison of Change Means with t-test

Strength of Treatment Effect:

Signal:Noise Ratio t=

Observed Δ

SDΔ √(1/N1 + 1/N2)

-11.8 - (-10.2)

(?) √(1/134 + 1/132)

= -1.10 → p=0.27

=

(Actually adjusted for baseline and center)

More than Two Visits

How can we get one

signal:noise ratio

incorporating all visits?

Perhaps we want to detect

treatment effect at any

visit.

Suppose Only Three Visits - Weeks 0, 4, 8

Two Treatment Differences in Changes:

Δ1 = D1 - P1

Δ2 = D2 - P2

D1

D2

P1

P2

Total Effect:

Δ12 + Δ2

2

Comparison of Change Means with ANOVA

Strength of Treatment Effect:

Signal:Noise Ratio F=

Observed (Δ12 + Δ2

2 )

√V

V involves SDΔ1 and SDΔ2 and the 1/Ns.

Large F ↔ Δ12 + Δ2

2 too large to be random

↔ p<0.05

Repeated Measures ANOVA• The previous slide is “classical” repeated measures

ANOVA.

• Could have many groups and many time points.

• If the overall “total” effect is significant, then we would examine which Δs are the cause.

• Same conclusions if changes from baseline, not sequential changes were used.

Since the signal or effect Δ12+Δ2

2 equally weights the two Δ, we must know all changes for a subject. If we do not (missing data), then that subject is completely removed from the analysis.

Mixed Models for Repeated Measures (MMRM)

• “Classical” repeated measures ANOVA uses only subjects with no missing visits.

• MMRM overcomes that limitation by making a signal:noise ratio as the weighted average of signals or effects from sets of subjects with the same missing visit pattern.

• MMRM still provides the overall ratio, as in the classical ANOVA that cannot handle missing visits.

Mixed Models for Repeated Measures (MMRM)

The next four slides use a simpler example to give the idea of how the weighting is done in MMRM.

These four slides can be skipped to get to the bigger picture of longitudinal analyses.

MMRM Example*

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

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

8

2

-1

8

0

0

2.83

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’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.

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 for A, and none for B?

The unpaired A mean is compared with the combined A and B mean, giving an estimate of half of the A - B difference. It is appropriately weighted with the paired A - B estimate.

Competing Conclusions

The next three slides show differences obtained by using different repeated measures approaches.

These three slides can be skipped to get to other approaches for longitudinal analyses.

Competing Conclusions

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

0 1 2 3 4 6 8

denotes imputed: N=63/260

Use all 260 values as if observed here.

Completer vs. LOCF vs. MMRM 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

MMRM uses all available visits for all 260. No imputation

(Week 8 or earlier)

MMRM vs. Classical: Why Distinguish?

Doesn’t distinguishing MMRM and classical seem to be about a minor technical point about weighting? Why make such a big deal?

The MMRM is not in many basic software packages.

It is not obvious how to perform it in software that does have it.

So, it is not user-friendly yet.

If you have missing data, ask a statistician to set it up in software correctly.

Other Approaches to Longitudinal Data

So far, we have considered all sequential changes or changes from baseline.

What other outcomes could be of interest?

Some Other Goals with Longitudinal Data

Use one visit at a time:

• Compare treatments at each time separately - doesn’t look at changes in individuals.

• Compare treatments at end of study.

Create summary over time:

• Compare average over time - trends unimportant.

• Specific pattern features, as in pharmacokinetic studies of AUC, peak, half-life, etc.

• Compare treatments on rate of change over time.

Average over Time - Trends Unimportant

AUC

Area Under the Curve (AUC),

divided by total length of time, is an average outcome, weighted for time.

Larger weights are given to the larger

time intervals, since AUC is just a sum

of trapezoids.

. . . . . .

“Growth” Curves

Parabola or line or equation based on theory describes

time trend.

The idea is to compare

treatment groups on a parameter describing the pattern, e.g.,

slope.

“Growth” Curves

The logic is to compare treatment groups by finding

means over subjects in each group for a

parameter describing the pattern, e.g.,

slope.

Next slide for correct method.

“Growth” Curves

The idea is to compare treatment groups on a parameter describing the pattern, e.g., slope.

Conceptually, we could just fit a separate regression line for each subject, get the slopes, and compare mean slopes between groups with a t-test.

But subjects may have different numbers of visits, and the slope might be correlated with the intercept (e.g., start off higher → smaller slope).

So, another form of “mixed models” is more accurate: “random coefficient” models. They give slopes also. Like MMRM, they are not very user-friendly in software, so ask a statistician to set up.

Summary on 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.