Restricted maximum likelihood (REML) is now well established as a method for estimating the parameters of the general Gaussian linear model with a structured covariance matrix, in particular for mixed linear models. Conventionally, estimates of precision and inference for fixed effects are based on their asymptotic distribution, which is known to be inadequate for some small-sample problems. In this paper, we present a scaled Wald statistic, together with an F approximation to its sampling distribution, that is shown to perform well in a range of small sample settings. The statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias. This approach has the advantage that it reproduces both the statistics and F distributions in those settings where the latter is exact, namely for Hotelling T2 type statistics and for analysis of variance F-ratios. The performance of the modified statistics is assessed through simulation studies of four different REML analyses and the methods are illustrated using three examples.

Protocol deviations, for example, due to early withdrawal and noncompliance, are unavoidable in clinical trials. Such deviations often result in missing data. Additional assumptions are then needed for the analysis, and these cannot be definitively verified from the data at hand. Thus, as recognized by recent regulatory guidelines and reports, clarity about these assumptions and their implications is vital for both the primary analysis and framing relevant sensitivity analysis. This article focuses on clinical trials with longitudinal quantitative outcome data. For the target population, we define two estimands, the de jure estimand, "does the treatment work under the best case scenario," and the de facto estimand, "what would be the effect seen in practice." We then carefully define the concept of a deviation from the protocol relevant to the estimand, or for short a deviation. Each patient's postrandomization data can then be divided into predeviation data and postdeviation data. We set out an accessible framework for contextually appropriate assumptions relevant to de facto and de jure estimands, that is, assumptions about the joint distribution of pre- and postdeviation data relevant to the clinical question at hand. We then show how, under these assumptions, multiple imputation provides a practical approach to estimation and inference. We illustrate with data from a longitudinal clinical trial in patients with chronic asthma.

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It is our experience that in many settings, crossover trials that have within-period baseline measurements are analyzed wrongly. A "conventional" analysis of covariance in this setting uses each baseline as a covariate for the following outcome variable in the same period but not for any other outcome. If used with random subject effects such an analysis leads to biased treatment comparisons; this is an example of cross-level bias. Using a postulated covariance structure that reflects the symmetry of the crossover setting, we quantify such bias and, at the same time, investigate potential gains and losses in efficiency through the use of the baselines. We then describe alternative methods of analysis that avoid the cross-level bias. The development is illustrated throughout with 2 example trials, one balanced and orthogonal and one highly unbalanced and nonorthogonal.

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