Parametric statistics

Parametric statistics is a branch of inductive statistics. To derive using data from a sample statements about an unknown population, it is assumed in the inductive statistics assume that the observed data are realizations of random variables. In the parameterischen statistics is also assumed that the random variables from a family of predetermined probability distributions (often: the normal distribution ) come whose elements are uniquely determined up to a ( finite ) parameters. Most of the known statistical analysis methods are parametric methods.

In contrast, the non-parametric or nonparametric statistics. Since the method does not require the distributional assumptions regarding the random variables, they are also called distribution -free.

Example

To test a new treatment to lower cholesterol, the cholesterol values ​​are determined in ten subjects before and after treatment. This results in the following test results:

If the new therapy has an effect, then the average of the differences should be significantly different from zero. The parametric test rejects the null hypothesis, while the non-parametric test, this can not discard. In practice one would of course perform one-sided tests.

A parametric method

Usually, you would here the two-sample t-test for dependent samples used ( null hypothesis: the mean of the difference is zero ). However, is a prerequisite for this test that either the sample size is greater than 30 (rule of thumb ) or the differences are normally distributed. If the differences are normally distributed, one can show that the test statistic follows a t-distribution.

The differences in the measured values ​​are the arithmetic mean and the sample standard deviation ( rounded). This results in a test value

The non- rejection region of the null hypothesis at a significance level of is given by. Since the test value is outside the non- rejection region of the null hypothesis, it must be discarded.

Non- parametric method

The non-parametric alternative to this is the sign test. Here is the null hypothesis that the median is zero. In the normal distribution vote median and mean coincide always, but this is not necessarily the case with other probability distributions. Here are just three differences less than zero and seven greater than zero. The test statistic follows a binomial distribution with and. The non- rejection region of the null hypothesis at a significance level of is given by. Since three and seven are within the non- rejection region of the null hypothesis, it can not be discarded.

Advantages and Disadvantages

The method of parametric statistics are based as opposed to methods of non- parametric statistics on additional distributional assumptions. Are these assumptions correct, result in generally more accurate and precise estimates. If they are not correct, provide parametric methods in many cases poor estimates; the parametric approach is then not robust against the violation of distributional assumptions. On the other hand, parametric methods are often easier and faster to compute. Sometimes a fast calculation is more important than the non- robust, especially when considered in the interpretation of statistics.

Conceptual history

The statistician Jacob Wolfowitz coined the statistical concept of parametric statistics to define its opposite:

"Most of thesis Developments have this feature in common, did the distribution functions of the various stochastic variable Which enter into Their problems are Assumed to be of known functional form, and the theories of estimation and testing of hypotheses are theories of estimation of and of testing hypotheses about, one or more parameters. . . , The nominal real knowledge would completely Call deterministic mine the various distribution functions Involved. We shall refer to this situation. . . : as the parametric case, and denote the opposite case, where the functional forms of the distributions are unknown, as the non- parametric case. "