4 Further extensions

Jan A. van den Brakel

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So far, experimental designs are considered where the ultimate sampling units of the sampling design are randomized over the treatments. Owing to restrictions in the field work there might be practical reasons to randomize clusters of sampling units over the different treatments, at the cost of reduced power for testing hypotheses about treatment effects. It might for example be attractive to assign the sampling units that belong to the same household or are assigned to the same interviewer to the same treatment combination. In van den Brakel (2008) a design-based analysis procedure is developed for single-factor experiments designed as CRDs and RBDs where clusters of sampling units are randomized over the treatments. These methods directly extend to the analysis of the factorial designs that are considered in this paper.

Consider the general case of a $M_1\times M_2\times \mathrm{...}\times M_G$  factorial design. The clusters of sampling units in the initial sample are randomized over the different treatment combinations. The conditional probability that a sampling unit is assigned to a subsample is now derived from the fractions of clusters that are assigned to the different treatment combinations within the sample or within each block. See van den Brakel (2008) for details. The GREG estimator for ${\overline{Y}}_{a_1\mathrm{...}{a}_G}$ is defined analogously to expression (2.18). Design-based estimators for the covariance matrices of the contrasts between the elements of ${\widehat{\overline{Y}}}_{GREG}$ are defined by (2.25), where the diagonal elements of $\widehat{D}$ are defined analogously to expression (4.6) in van den Brakel (2008), which is based on the variance between the estimated cluster totals.

The target parameters of a survey are often defined as a ratio of two population totals. In van den Brakel (2008) a design-based analysis procedure is developed to test hypotheses about ratios in single-factor experiments designed as a CRD or an RBD. These results can be extended to the analysis factorial designs treated in this paper. Based on each subsample a ratio of two GREG estimators can be constructed for each treatment combination. Design-based estimators for the covariance matrices of the contrasts between the ratios are defined by (2.25), where the diagonal elements of $\widehat{D}$ are defined analogously to expression (4.11) in van den Brakel (2008), which is an estimator for the variance of the ratio of two GREG estimators. Hypotheses about main effects and interactions are tested with the Wald statistic (2.28).

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