You start with the null hypothesis. If consumption of substance X increases significantly but there is almost no evidence of a significant increase in the canc

You start with the null hypothesis. If consumption of substance X increases significantly but there is almost no evidence of a significant increase in the cancer rate over that time period then you have a large hill to climb out of in terms of proving causality. Cancer rates over the past 100 years have been mostly dropping. If aspartame really was what some claim then I would expect to see a bump like the one you see in lung cancer starting in the late 40s and climbing hard and fast up until the late-80s/early-90s whereupon it the rate turns negative and starts dropping fast.