Parametric statistics

The parametric statistics is a branch of inductive statistics . In order to derive statements about an unknown population with the help of data from a sample , it is assumed in inductive statistics that the observation data are realizations of random variables . In parametric statistics it is also assumed that the random variables come from a family of predetermined probability distributions (often: the normal distribution ), the elements of which are uniquely determined except for one (finite-dimensional) parameter. Most of the known statistical analysis methods are parametric methods. ${\displaystyle x_{1},\ldots ,x_{n}}$ ${\displaystyle X_{1},\ldots ,X_{n}}$ ${\displaystyle X_{i}}$

This is in contrast to the non-parametric statistics . Since their methods do not require a distribution assumption with regard to the random variables , they are also called distribution-free . ${\displaystyle X_{i}}$

example

In order to test a new therapy for lowering the cholesterol level, the cholesterol levels are determined in ten test subjects before and after the treatment. The following measurement results are obtained:

Before treatment:	223	259	248	220	287	191	229	270	245	201
After treatment:	218	242	241	208	297	168	208	273	250	186
Difference:	5	17th	7th	12	−10	23	21st	−3	−5	15th

If the new therapy has an effect, then the mean of the differences should deviate significantly from zero. The parametric test rejects the null hypothesis, while the nonparametric test cannot reject it. In practice, one- sided tests would of course be carried out here.

Parametric method

Usually one would use the two-sample t-test for dependent samples (null hypothesis: the mean of the difference is zero). A prerequisite for this test, however, is that either the sample size is larger 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 between the measured values ​​have the arithmetic mean and the sample standard deviation (rounded). That results as a test value ${\displaystyle {\bar {d}}=8{,}2}$ ${\displaystyle s_{d}=11{,}3867}$

${\displaystyle t={\sqrt {10}}{\frac {8{,}2}{11{,}3867}}=2{,}281}$ (rounded).

The non-rejection range of the null hypothesis at a significance level of results in . Since the test value is outside the non-rejection range of the null hypothesis, it must be rejected. ${\displaystyle \alpha =5\,\%}$ ${\displaystyle [-2{,}262;+2{,}262]}$

Non-parametric method

The non-parametric alternative to this is the sign test . Here the null hypothesis is that the median is zero. In the normal distribution, the median and mean always match, but this is not necessarily the case with other probability distributions. Here exactly three differences are less than zero and seven are greater than zero. The test statistic follows a binomial distribution with and . The non-rejection range of the null hypothesis at a significance level of results in . Since three and seven are within the non-rejection range of the null hypothesis, it cannot be rejected. ${\displaystyle n=10}$ ${\displaystyle p=0{,}5}$ ${\displaystyle \alpha =5\,\%}$ ${\displaystyle [2;8]}$

advantages and disadvantages

In contrast to methods of nonparametric statistics, the methods of parametric statistics are based on additional distribution assumptions. If these assumptions are correct, more accurate and precise estimates will usually result. If they are not correct, parametric methods often give poor estimates; the parametric concept is then not robust against the violation of the distribution assumptions. On the other hand, parametric methods are often easier and faster to calculate. Sometimes a quick computation is more important than non-robustness, especially when this is taken into account when interpreting statistics.

Concept history

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

“Most of these developments have this feature in common, that the distribution functions of the various stochastic variables which enter into their problems are assumed to be of known functional form, and the theories of estimation and of testing hypotheses are theories of estimation of and of testing hypotheses about, one or more parameters. . ., the knowledge of which would completely determine 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. "

- Jacob Wolfowitz

Individual evidence

1. ^ Seymour Geisser, Wesley O. Johnson: Modes of Parametric Statistical Inference . Wiley, 2006, ISBN 978-0-471-74313-2 .
2. ^ DR Cox: Principles of Statistical Inference . Cambridge University Press, 2006, ISBN 978-0-521-68567-2 .
3. ↑ David C. Hoaglin, John Tukey, Frederick Mosteller: Understanding Robust and Exploratory Data Analysis . John Wiley & Sons, 2000, ISBN 978-0-471-38491-5 .
4. ^ Gregory W. Corder and Dale I. Foreman: Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach . John Wiley & Sons, 2009, ISBN 978-0-470-45461-9 .
5. ^ David A. Freedman: Statistical Models: Theory and Practice . 2nd Edition. Cambridge University Press, 2009, ISBN 978-0-521-74385-3 .
6. ^ Jacob Wolfowitz: Additive Partition Functions and a Class of Statistical Hypotheses . In: Annals of Mathematical Statistics . tape 13 , 1942, pp. 264 .