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I have seen that there are parametric and non-parametric statistical tests. What are the differences between them?

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    $\begingroup$ parametric -- defined by a particular finite number of parameters (non-parametric is, as the name suggests, not parametric). $\endgroup$ – Glen_b -Reinstate Monica Jun 9 '16 at 1:55
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Parametric tests assume that your data come from some sort of parametric model. For example, it can be as simple as assuming your data are iid draws from a normal distribution, to a complicated time series model where the variables come from some parametric distribution and possess a parametric functional relationship.

For example, we can say $Z_t=\phi_1\varepsilon_{t}+\phi_2\varepsilon_{t-1}\phi_3\varepsilon_{t-2}$ where $\varepsilon_t \sim F_{\theta}$

Here, we have two types of parameters, $\phi$ which define a functional relationship, and $\theta$, which specifies the particular member of a parametric family (e.g., $F_{\mu,\sigma}=N(\mu,\sigma)\to F_{0,2}:=N(0,2)$).

So, if you have a bunch of $Z_i$, then you'd use the above model to estimate these parameters (e.g., using maximum likelihood estimation).

Nonparametric models are somewhat of a misnomer, since they still impose structure on your problem. However, they usually only partially specify the model, often very partial. A good example is the signed rank test, which assumes only that the distribution is symmetric, and you are trying to estimate its median.

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Parametric most often means that a test assumes the tested variables are Normally distributed. Such tests often use regular nominal values, and look at differences in Averages. Non-parametric tests do not assume that the tested variables are Normally distributed. Such tests at times use value ranks instead of nominal values, and also may look at difference in Medians instead of Averages. The mentioned metrics (Ranks and Medians) are more resilient to the shape of the data being further away from being Normally distributed than nominal values and Averages.

Many variables are somewhat undetermined, meaning by that when you test them for Normality, depending on the test, you may not be able to reject that they are Normally distributed or you may be able to. There can be a lot of ambivalence regarding whether you should use a parametric or non-parametric test.

In such cases, I would recommend you use both parametric and non-parametric tests. Quite often, both tests will give you directionally the same answer. Granted their respective p values will most always be a bit different (that is the case even within tests of same types (i.e. both being parametric)). But, pretty often they are in a fairly similar ballpark. If that is the case, regardless of what test type you use, your conclusion or answer is the same. If in typically the minority of the case when the two different test types give you pretty divergent results, you have to look very closely at your data, and your objective. Given that go with the test that gives you the more cautious, prudent answer given the specific situation or objective.

Others have pointed out that the differentiation between parametric and non-parametric tests is and can be extended beyond the Normal distribution. This is a valid point, but in my view way beyond the basic question being asked. And, to its fullest extent the question would often not have a clear cut finish line. Some tests deemed non-parametric because they do not rely on the Normal distribution still rely on other parameters (i.e. the distribution being symmetric). Thus, some parties may view such a test as still being parametric. Meanwhile, the test would be classified in the literature as being non-parametric. To its fullest extend, few if any tests would be entirely non-parametric (most often they do rely on some mathematical structure assumption).

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    $\begingroup$ Logistic regression is a parametric model, & it doesn't assume any variables are normal. $\endgroup$ – gung - Reinstate Monica Jun 9 '16 at 21:00

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