Improving The Efficiency of The Case-Crossover Design

The case-crossover design has been used by many researchers to study the transient effect of an exposure on the risk of a rare outcome. In a case-crossover design, only cases are sampled and each case will act as his/her own control. The time of failure acts as the case and non failure times act as the controls. Case-crossover designs have frequently been used to study the effect of environmental exposures on rare diseases or mortality. Time trends and seasonal confounding may be present in environmental studies and thus need to be controlled for by the sampling design. Several sampling methods are available for this purpose. In time-stratified sampling, disjoint strata of equal size are formed and the control times within the case stratum are used for comparison. The random semi-symmetric sampling design randomly selects a control time for comparison from two possible control times. The fixed semi-symmetric sampling design is a modified version of the random semi-symmetric sampling design that removes the random selection. Simulations show that the fixed semi-symmetric sampling design improves the variance of the random semi-symmetric sampling estimator by at least 35% for the exposures we studied. We derive expressions for the asymptotic variance of risk estimators for these designs, and show, that while the designs are not theoretically equivalent, in many realistic situations, the random semi-symmetric sampling design has similar efficiency to a time-stratified sampling design of size two and the fixed semi-symmetric sampling design has similar efficiency to a time-stratified sampling design of size three.     

The Demand for Lotto: The Role of Conscious Selection

This article presents estimates of the elasticity of demand for lottery tickets using time series data in which there is variation in the expected value of a lottery ticket induced by rollovers. An important feature of our data is that there are far more rollovers than expected given the lottery design. We find strong evidence that individuals do not choose their lottery numbers uniformly from a uniform distribution—that is, conscious selection. We use our estimates to derive the inverse supply function for the industry, and this enables us to identify the demand elasticity. We find the price elasticity to be close to unity, which implies that the operator is revenue maximizing—which is the regulator's objective.