Abstract
Many observational studies assess the impact of a treatment on an outcome that has several dimensions. In the observational study that we discuss, physical abuse of children may affect the degree to which the child exhibits depression, withdrawal or aggression. A treatment may affect all, some or none of these dimensions. In addition to the scientific interest in learning the effect on each dimension, it is also known that an appropriate combination of dimensions may increase power, efficiency and insensitivity to unmeasured biases; however, finding this appropriate combination requires corrections for multiple testing that erode power. We explore this trade-off by developing a new formula for the power of a sensitivity analysis in a simple situation with several dimensions. The methodology is applied to study the effects of physical abuse in early childhood and its possible effects on several dimensions of subsequent behavioral problems. Also, a general method is proposed for converting any signed rank test for matched pairs into an analogous test for matching each treated individual to several controls, and the performance of this extension is examined. The proposed method aids in studying the relative magnitude of the effect on different dimensions. A second evidence factor considers the dose or intensity of physical abuse.
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