Causality comes form experiment manipulation or statistical calculation?

Question

It is known that "Regression can only imply correlation but not causality"

But whether a research can draw their variables to causality relationship is somewhat determined by whether the researcher designed a experiment with well variable manipulation and measure.

Then, what if I use regression to analyze the experiment result？ Can this explain the causality between variable？

And on the other hand, though usually an factorial experimental design will be analyzed by ANOVA, but ANOVA is also kind of regression？

Everything seems contradict, do I misunderstand any information？

1. "Regression can only imply correlation but not causality"
2. Causality comes from experimental design
3. ANOVA is kind of regression

Answer 1

Regression and ANOVA both estimate associations (i.e., correlations). Those associations can be interpreted causally when the assumptions for causality are met, which they are in a randomized experiment. There are other situations where the assumptions for causality are met, but associations cannot be interpreted causally when they are not met. The claim that correlation does not imply causation is used to prevent people from interpreting associations causally when the causal assumptions are not met, but it does not mean that no correlations are not causal (i.e., some correlations are causal).

Causal inference is a fairly technical field, but there are some excellent resources out there to learn more about it, such as Pearl's Book of Why.