Organisms on Earth commonly exhibit a circadian rhythm, which is synchronized to the 24-hour day-night (diurnal) cycle of the planet. However, if isolated from strong environmental time cues (e.g., light-dark, temperature, etc.), many organisms revert to a "free-running" rhythm that is close to, but significantly different from, the diurnal cycle. Such a free-running rhythm is a distinct biological feature, as it requires an endogenous pacemaker that is not just passively driven by rhythms in the environment. On Mars, a free-running rhythm (i.e., significantly different from the Martian diurnal cycle of 24.66 hours) would constitute independent proof of the presence of living organisms. Evidence for such a circadian biosignature from Mars has been sought in the data sent by the 1976 Viking Labeled Release (LR) life detection experiment . In the search for circadian rhythmicity, oscillatory fluctuations in the amount of radiolabeled gas in the headspace of the LR test cell of Viking Lander 2, test cycle 3, were studied. The cycle duration of the LR oscillations examined did not differ significantly from that of the daily cell temperature oscillations controlled ultimately by the Martian diurnal cycle. Thus, these specific LR oscillations produced no independent evidence for an endogenous biological origin. However, it was found that the amplitudes of the oscillations in the gas (presumably CO2) were greater than could be accounted for by the most likely non-biological mechanism (i.e., temperature-induced changes in soil solubility of CO2). The possibility thus remained that biological activity, synchronized to the Martian diurnal cycle, could be responsible for at least part of the oscillatory activity in the LR signals. We now propose to consider all data from the nine active and control cycles of the Martian LR experiment. A comprehensive set of null and alternative hypotheses is proposed for statistical testing using the digitized data. Advanced, statistically rigorous methods of circadian rhythm analysis are laid out to determine whether an endogenous circadian rhythm was present. The data will be analyzed for any free-running rhythm deviating from the Martian diurnal cycle. The possibility that nutrient administration altered the phase (i.e., timing) of the LR oscillations (as has been observed in terrestrial microorganisms) will also be examined. Any indication that the signal may be of biological origin will be tested against the hypothesis that it was caused solely by temperature-induced changes (e.g., temperature-dependent changes in soil physical chemistry). The focus of this paper is to develop broadly accepted methodology to determine definitively whether the LR data exhibit circadian characteristics that imply the involvement of Martian biology.