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This question already has an answer here:

what does it mean for a variable to be statistically significant??

Could anyone give a complete answer to this question?

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marked as duplicate by gung, Glen_b, Nick Cox, Momo, Scortchi Feb 11 '14 at 10:09

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Usually, some hypothesis test relating to that variable is rejected (e.g. when a test of whether a regression coefficient is zero is rejected). A complete answer would require a book, since there are too many contexts in which a person might use such a phrase (not always correctly) $\endgroup$ – Glen_b Feb 11 '14 at 6:31
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It means that it is a very unlikely result IF the variable in question had no effect; hence the "no effect" hypothesis is cast into doubt and the alternative hypothesis becomes more plausible.

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I'm going to get a little technical in my explanation here, but it's for your benefit.

The concept of "significance" in hypothesis testing has a technical definition that often doesn't get explained (unfortunately) when the notion of "statistical signficance" is first learned, usually in the science classroom. Specifically, the "significance level" of a test is the probability that the test will produce a "Type I error." A Type I error occurs when we reject the null hypothesis when the null hypothesis is actually correct. A result is deemed "statistically significant" when we calculate that the probability that we observe a result at least as extreme as that which we calculated (assuming the null hypothesis is correct) is lower than the significance level.

Now, in layman's terms: a result is statistically significant when the result we observe is, under the null hypothesis, less probable than the event that our test produces a false positive in favor of rejecting the null hypothesis.

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Colloquially (stat.), if somebody tell you that he/she found a statistically significant effect in some regression, he/she usually means that some variable x causally influences some variable y with a high probability.

Whether such a statement is appropriate is another matter entirely.

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    $\begingroup$ I don't think this is correct: it focuses on regression when nothing in the question signals that; if "usually" means "most or almost all of the time", it is empirically wrong, as I would say that significance testing and causal inference are often, but not usually, mixed. $\endgroup$ – Nick Cox Feb 11 '14 at 10:17

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