Citation

Using simulations, a bias caused by inappropriate averaging in experiments with randomized stimuli is described. As an illustrating example, experiments aimed at demonstrating “presentiment” by showing arousal to be higher prior to arousing stimuli than prior to calm stimuli are considered. It is shown that such results could be obtained if (a) the participant believes that the likelihood of an activating stimulus being presented on the next trial increases after a calm stimulus has been presented (the gambler's fallacy) and (b) overall arousal means are calculated across individual arousal means or for a pooled sample of all possible stimulus sequences. The effect becomes very small when participants are pooled before averaging, provided that the number of trials per participant and/or the number of participants is sufficiently large. The bias decreases as the length of the stimulus sequence increases, and becomes zero in an infinitely long sequence. There are two independent sources of the bias: (1) omission of stimulus sequences consisting solely of one type of stimulus and (2) reweighting of arousal levels for calm and activating stimuli occurring when within-sequence comparisons or comparisons within subsets of all possible sequences are made. A two-way analysis of variance can both detect an effect caused by expectation and prevent it from biasing the estimate of the presentiment effect. A generalized version of the bias is outlined and discussed. It is argued that, in some form, the bias may occur in various types of experiments, both within and outside parapsychology. It is also argued that numerous previous experiments need to be checked for occurrence of the bias¹.