3.10 Missing Data
4.1.1 Procedure
Suppose we have a random variable Yjki that describes a Gaussian outcome measured from independent individuals i = 1, . . . , nk in randomized treatment groups k = 1, . . . , q at repeated occasions j = 1, . . . , m. The realizations of Yjki are denoted by yjki. There are n = Pq
k=1nk independent units, and the total number of observations is N , which equals mn if all subjects are measured at all occasions (i.e., no missing values). The combinations of occasions and treatment groups are indexed with b = 1, . . . , s where s = mq.
To reflect the data’s mean structure, variability, and dependencies among repeated mea- surements in a model framework, we may take either of two basic modeling approaches: joint modeling with an LMM (as outlined in 3.1), or a combination of marginal linear models (as in 3.5). Both approaches provide us with estimates of β and Σ that we can use for simultaneous inference about the fixed-effects parameters in our longitudinal MCTs.
Joint modeling: We focus on two basic types of joint models that are special cases of our general LMM notion:
• an extended linear model (ELM) where the residuals may exhibit correlation and heteroskedasticity over time and possibly also across treatment groups, or
• a conditional independence model (CIM) that assumes constant residual variance and independent errors, conditional on the random effects.
Whichever LMM we fit, the fixed effects will be parameterized in cell means style i.e., one mean parameter for every combination of treatment group and occasion. This entails maximum flexibility for all shapes of means across time and treatments. The cell means design matrix (for complete and balanced data without additional covariates) is
X = Imq⊗ 1nq
where I is an identity matrix and 1 is a column vector of ones. Handling incompleteness and including covariates is straightforward.
Considerably harder is the choice of a covariance structure for the joint model. The CIM requires to model random effects that reflect the dependencies among repeated measurements. With the simplest random subject effects, the random-effects design matrix (for complete and balanced data without additional random factors) is
Z = 1q⊗ In.
If we allow for occasion-specific random subject effects, it becomes
Z = 1q⊗ e1 .. . 1q⊗ em
where ej is the jth unit vector of size m. Likewise, we could make the random sub- ject effects both occasion- and treatment-specific, but this might already run us into computational trouble when actually trying to fit the model.
Many practical experiments are more complex and contain additional hierarchies e.g., the subjects might be hospitalized in different clinics (if they are patients), live in different cages (if they are lab animals), or belong to different spatial blocks (if they are plants in the field). The CIM can smoothly incorporate the hierarchical architecture of the experiment via random effects.
In the ELM framework we are required to make two choices as the residual covariance matrix is assembled from a variance portion and a correlation portion; we described a variety of possible choices in 3.1. In case we decide to fit an LMM that includes both random effects and some residual covariance structure, the agony of choice intensifies. If unsure about which is the most appropriate model—and this should really be the standard situation—a reasonable option is to let an information criterion do the job. So having assembled several plausible models for the random effects and/or residual covari- ances, we entrust AICc (see 3.2) with picking the “best” of them. We must keep in mind, however, that AICc (or any other information-based criterion) judges the “goodness” of a model relative to its competitors in the candidate set. It says nothing about whether it is good in an absolute sense—it might just be the best in a set of terrible models.
Multiple marginal models: The other strategy is to fit one linear model separately for every measurement occasion j = 1, . . . , m. The jth of these marginal models is
y(j) = X(j)β(j)+ (j)
where the superscript index signalizes belonging to occasion j. The design matrix X(j) needs to be arranged such that there is one parameter for each of the treatments’ means, and possible covariates4. The dependency structure across time points is established from
−E 1 σ2X
(j)X(j)T −1
X(j)y(j)− X(j)Tβ(j) as described in 3.5 and Pipper et al. (2012).
Multiple Contrast Tests: Whatever modeling approach (ELM, CIM, some other LMM, or marginal models) has provided us with estimates of β and Σ, we continue with simultaneous inference under the assumption of the ˆβ being multivariate normal and ˆΣ a consistent estimator of Σ.
We define the comparisons of interest in a coefficient matrix C. Assume the model parameterization is such that all treatment means at the first occasion come first, followed by the treatment means at the second occasion, and so on. Then the coefficient matrix is block-diagonal and can be constructed as the Kronecker product of an m-dimensional identity matrix and an “elementary” coefficient matrix C0 as
C = Im⊗ C0.
As an example, the full contrast coefficient matrix for many-to-one comparisons among q = 3 treatment groups separately and simultaneously at m = 4 occasions is
CtrtDun= I4⊗ CDun = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ⊗ −1 1 0 −1 0 1 = −1 1 0 0 0 0 0 0 0 0 0 0 −1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 0 0 0 −1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 0 0 0 −1 1 0 .
The subsequent computation of MCTs i.e., adjusted p-values and SCIs, is to a great extent identical to the standard MCT procedure described in 3.7, albeit with one cru- cial difference: the correlation matrix of the test statistics no longer contains just known quantities (sample sizes and contrast coefficients) but also covariances reflecting the longi- tudinal correlation and possible heteroscedasticity in the repeated measurements. These covariances are a priori unknown, and we get by with plugging in their estimates from MMM or a joint model, which makes our longitudinal MCT procedure approximate. By means of the coefficients in C we formulate contrasts
ηh = s X
b=1 chbβb
and use them to specify a set of elementary linear hypotheses, the hth pair of which (for two-sided inference) is
H0(h) : ηh = δh versus H (h)
A : ηh 6= δh
where more often than not δh = 0 ∀ h. The hth test statistic is computed as Th =
ˆ ηh− δh q
chΣcb Th with estimated contrast
ˆ ηh = s X b=1 chbβˆb.
The exact distribution of T under H0 is unclear but may be approximated as z-variate t with ν DF and correlation ˜Γ:
Tappr.∼ Tz(ν, ˜Γ).
It is not straightforward to see what the DF should be in general, and especially not in the presence of unbalanced data e.g., when numbers of repeated observations differ between
experimental units. A whole range of possible DF approximations for small samples were listed in 3.8.
The asymptotic distribution of T is z-variate normal with correlation ˜Γ: Tasym.∼ Nz(0, ˜Γ).
Details on the multivariate normal and t-distribution are given in Appendix A. We will explore the performances of both the asymptotic and the approximate procedure with different DF approximations in a simulation study in 4.1.3.
The covariance matrix of test statistics T = (T 1, . . . , Tz) under H0 is ˜
Σ = CΣCT
so that we obtain ˜Γ as
˜
Γ = V ˜ΣV
where V = diag( ˜Σ)−12 is the inverse of a matrix with the square root of the diagonal
elements from ˜Σ on its diagonal and all off-diagonal elements zero. As Σ is unknown in practice, we plug in a consistent estimate bΣ (from a joint model or the MMM approach) so as to get bΣ and b˜ Γ.˜
An elementary hypothesis H0(h) gets rejected if |Th| > ttwoz,1−α(ν, bΓ)˜ with ttwo
z,1−α(ν, bΓ) the two-sided equicoordinate (1 − α) quantile of T˜ z(ν, bΓ). Bounds of˜ SCIs with coverage probability 1 − α are obtained as
ˆ
ηh∓ ttwoz,1−α(ν, bΓ)˜ q
chΣcb Th. Adjusted p-values are given by
ph = 1 − Z |Thobs| −|Tobs h | . . . Z |Thobs| −|Tobs h | tz(x; ν, bΓ) dx˜ where Tobs
h designates an observed value of the test statistic Th, and tz(x; ν, bΓ) is the PDF˜ corresponding to Tz(ν, bΓ). The z-dimensional integral needs to be solved numerically e.g.,˜ using a software implementation of the Genz-Bretz algorithm (Genz and Bretz 2009). Since the global null hypothesis is the intersection of elementary nulls
H0 = z \
h=1 H0(h)
we reject the global H0if at least one of the H (h)
0 is rejected. This implies a maximum-type test with test statistic
Tmax = max h |Th| whose p-value is computed as
p = min h ph.