Throughout an entire clinical trial, study investigators are ethically obligated to monitor participant safety, as well as accumulating evidence concerning the effectiveness of treatments. However, current statistical tools discourage continuous monitoring of study data. For example, it is well know that when conducting repeated significance tests on accumulating data the probability of a type one error quickly approaches unity (Armitage's repeated significance testing paradox). Sequential, Group Sequential, and Bayesian methods have failed to fill this void in current practice for a variety of reasons. An Evidential Paradigm, based on the Law of Likelihood, is examined in the context of constant monitoring. This paradigm uses (1) likelihood ratios, not p-values, to measure the strength of statistical evidence and (2) provides a bound on and control over the frequency of both misleading and weak evidence. Instead of representing evidence against a null hypothesis, the likelihood function measures relative evidence supporting one simple hypothesis over another. Re-examination of accumulating evidence does not diminish its strength, because the likelihood function is unaffected by the number of examinations. A procedure fashioned after the Law of Likelihood is proposed to accommodate composite hypotheses. Brownian Motion techniques (Siegmund, 1985) can be used to approximate and demonstrate control over the probability of misleading evidence. This procedure allows constant monitoring for evidence of a clinically significant treatment effect over no treatment effect, while maintaining a low probability of misleading evidence.