Caution in exploring the effects of distant past outcomes on sequential choices

We sometimes make sequential decisions depending solely on the immediate past outcomes, e.g., according to the Win-Stay-Lose-Switch rule. In other occasions, we make decisions depending also on the distant past outcomes. It appears of interest to distinguish these two cases based on the generated choice sequences. At first glance, it may seem straightforward to distinguish the two cases by examining whether the rate of reselecting the same option that was chosen in the distant past, for example, at two trials before, depends on the outcome obtained there. However, such naive analysis can theoretically lead to detection of spurious dependence of three different types. Whereas two of them can easily be avoided by calculating the rate of reselection separately for each case sorted by the choice and outcome at the previous trial, the third type of spurious dependence appears after being sorted by the previous choice and outcome. Here we show how such spurious dependence appears. This article exemplifies the need for caution in analyzing the limited number of sequential choices.