The three conditions necessary for determining causality are empirical association, appropriate time order, and nonspuriousness.

• Empirical association means that there is a correlation between the cause and effect. In the experiment you described, there is an empirical association between taking pain aides and experiencing relief from headaches.
• Appropriate time order means that the cause must come before the effect. In the experiment you described, the participants took the pain aides before they experienced relief from their headaches.
• Nonspuriousness means that there is no other variable that is responsible for the observed effect. In the experiment you described, it is possible that some of the participants who took placebos experienced relief from their headaches due to the placebo effect.

To establish causality, all three of these conditions must be met. In the experiment you described, the first two conditions are met, but the third condition is not. This is because there is another possible explanation for the observed effect (the placebo effect).

Therefore, the experiment you described does not establish causality.

Here are some ways to strengthen the experiment to establish causality:

• Increase the sample size. This will help to reduce the chance that the results are due to chance.
• Use a randomized controlled trial. This means that the participants are randomly assigned to either the treatment group (pain aides) or the control group (placebos). This helps to ensure that the two groups are similar in all other respects, except for the treatment they receive.
• Measure the outcome variable multiple times. This will help to reduce the chance that the results are due to random variation.
• Control for other variables that could affect the outcome variable. For example, the researchers could control for the participants’ age, gender, and other medical conditions.

By taking these steps, the researchers can increase their confidence that the observed effect is due to the treatment (pain aides) and not to other factors.

Here are two academic sources that discuss the three conditions necessary for determining causality:

• Hernán, Miguel A., and James M. Robins. Causal inference: What it is and how to do it. 2nd ed. Boca Raton, FL: Chapman & Hall/CRC, 2020.
• Holland, Paul W. “Statistics and causal inference.” Journal of the American Statistical Association 81.393 (1986): 459-466.