Parametric statistics

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 $X_{1},\ldots ,X_{n}$ are $x_{1},\ldots ,x_{n}$ realizations of random variables In parametric statistics, it is additionally assumed that the random variables $X_{i}$ come from a family of given probability distributions (often: the normal distribution) whose elements are uniquely determined except for one (finite-dimensional) parameter. Most known statistical analysis methods are parametric methods.

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

Example

In order to test a new therapy for lowering cholesterol, the cholesterol levels of ten test persons are determined before and after 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	17	7	12	−10	23	21	−3	−5	15

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

Parametric method

Typically, one would use the two-sample t-test for dependent samples here (null hypothesis: the mean of the difference is zero). However, a prerequisite for this test 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 of the measured values have arithmetic mean $\bar d = 8{,}2$ and sample standard deviation $s_d=11{,}3867$ (rounded). This gives as test value

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

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

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 agree, but this is not necessarily the case for 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 n=10 and p=0{,}5 . The non-rejection range of the null hypothesis at a significance level of α $\alpha =5\,\%$ results in [2; 8] . Since three and seven are within the non-rejection range of the null hypothesis, it cannot be rejected.

Advantages and disadvantages

In contrast to methods of non-parametric statistics, the methods of parametric statistics are based on additional distribution assumptions. If these assumptions are correct, they generally result in more accurate and precise estimates. If they are not correct, parametric methods often produce poor estimates; the parametric concept is then not robust to the violation of the distributional assumptions. On the other hand, parametric procedures are often easier and faster to compute. Sometimes fast computation is more important than non-robustness, especially when this is taken into account in the interpretation of statistics.

Questions and Answers

Q: What is parametric statistics?

A: Parametric statistics is a branch of statistics that assumes the observations in the unknown population follow a probability distribution, with most of the parameters of the distribution being known.

Q: What is the main assumption of parametric statistics?

A: The main assumption of parametric statistics is that the observations in the unknown population follow a probability distribution.

Q: Who was the first to use the term "parametric statistics"?

A: Jacob Wolfowitz was the first to use the term "parametric statistics".

Q: What is the difference between parametric and non-parametric statistics?

A: The main difference between parametric and non-parametric statistics is that parametric statistics assume that the observations follow a probability distribution with known parameters, while non-parametric statistics make no assumptions about the population distribution.

Q: What are some examples of parametric statistical methods?

A: Examples of parametric statistical methods include t-tests, ANOVA, regression analysis, and Bayesian inference.

Q: Why are parametric statistical methods commonly used?

A: Parametric statistical methods are commonly used because they can provide more precise estimates of population parameters and can be more powerful than non-parametric methods if the assumptions are met.

Q: What are some limitations of parametric statistical methods?

A: Limitations of parametric statistical methods include the need for assumptions about the population distribution, sensitivity to outliers, and the potential for bias if the assumptions are not met.