In simple terms the p-value expresses how surprised you are with the data, assuming there is no effect. The lower the p-value, the more incompatible the data seems with your model (i.e. the assumption that there is no effect).
Treatment A is compared to treatment B, you assume there is no effect or no difference; you expect the null hypothesis to be correct. You perform the test and get a p-value of 0.02. That means that the data you gathered is pretty surprising, considering that you assumed the groups would not differ.
The p-value exists to protect yourself from randomness. If you perform a study, chances are that the effects you see are just random— or data noise, as we call it. That’s why you might see noticeable differences in the mean values between groups, but no statistically significant effect. It can go the other way around as well. A study might show a non-significant result, but there might be a true effect; perhaps because the sample size is just too small.
P-values are influenced by a few different factors: sample size, effect size, and the type of test with its assumptions.
Sample size: the bigger the group, the faster you’ll get statistically significant results with small differences— and vice versaEffect size: the bigger the effect size, the faster you’ll get statistically significant results, even with smaller groups— and vice versaType of test: a test gets more sensitive to differences with certain assumptions about for example the data distribution, independence of measures, homoscedasticity, one-sided vs two-sided, between-group vs within-group, etc.
A huge study can find statistically significant results with even the smallest of effects. These effects might not mean a thing. This is where clinical significance comes into play.The original penicillin study used a tiny sample to make the data show that there are huge effects on eliminating bacteria.
The threshold for statistical significance most researchers use (i.e. p < 0.05) is just arbitrary. All things considered, it should change based on your study setup. If you really do not want false positive results (eg. a decision to undergo a life-threatening operation), you need a low threshold number. If you really don’t want false negatives (eg. diagnosing aggressive tumors), you need a high-powered study with subsequently a higher p-value threshold number. This illustrates the give-and-take relation between type 1 (α) and type 2 (ß) errors.
Do note that the p-value is derived from the data, not the theory. You cannot ‘prove’ your theory with a statistically significant effect. The only thing you can do is try to refute your theory with different studies, if it holds, your theory stands. This is falsification.
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