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In statistical inference, or hypothesis testing, the traditional tests are called .. Conover, W. J. (), Practical Nonparametric Statistics, 3rd Edition, New York: . CONCOVER, W. J.: Practical nonparametric statistics. J. Wiley and Sons Inc., New York , S., £ P. Mager · Search for more papers by this author. Get this from a library! Practical nonparametric statistics. [W J Conover].
Definitions[ edit ] The term "nonparametric statistics" has been imprecisely defined in the following two ways, among others. The first meaning of nonparametric covers techniques that do not rely on data belonging to any particular parametric family of probability distributions. These include, among others: distribution free methods, which do not rely on assumptions that the data are drawn from a given parametric family of probability distributions.
As such it is the opposite of parametric statistics. Order statistics , which are based on the ranks of observations, is one example of such statistics. The following discussion is taken from Kendall's. For example, the hypothesis a that a normal distribution has a specified mean and variance is statistical; so is the hypothesis b that it has a given mean but unspecified variance; so is the hypothesis c that a distribution is of normal form with both mean and variance unspecified; finally, so is the hypothesis d that two unspecified continuous distributions are identical.
It will have been noticed that in the examples a and b the distribution underlying the observations was taken to be of a certain form the normal and the hypothesis was concerned entirely with the value of one or both of its parameters.
Such a hypothesis, for obvious reasons, is called parametric. Hypothesis c was of a different nature, as no parameter values are specified in the statement of the hypothesis; we might reasonably call such a hypothesis non-parametric. Hypothesis d is also non-parametric but, in addition, it does not even specify the underlying form of the distribution and may now be reasonably termed distribution-free.
Practical nonparametric statistics
Notwithstanding these distinctions, the statistical literature now commonly applies the label "non-parametric" to test procedures that we have just termed "distribution-free", thereby losing a useful classification. The second meaning of non-parametric covers techniques that do not assume that the structure of a model is fixed.
Typically, the model grows in size to accommodate the complexity of the data. In these techniques, individual variables are typically assumed to belong to parametric distributions, and assumptions about the types of connections among variables are also made. These techniques include, among others: non-parametric regression , which is modeling whereby the structure of the relationship between variables is treated non-parametrically, but where nevertheless there may be parametric assumptions about the distribution of model residuals.
Applications and purpose[ edit ] Non-parametric methods are widely used for studying populations that take on a ranked order such as movie reviews receiving one to four stars. The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences.
Scanlan, EdD, RRT Parametric statistics assume 1 that the distribution characteristics of a sample's population are known e. Frequently, however, these assumptions cannot be met.
Commonly, this occurs with nominal- or ordinal-level data for which there are no measures of means or standard deviations. Alternatively, continuous data may be so severely skewed from normal that it cannot be analyzed using regular parametric methods. In these cases we cannot perform analyses based on means or standard deviations.
Instead, we must use nonparametric methods.
Unlike their parametric counterparts, non-parametric tests make no assumptions about the distribution of the data nor do they rely on estimates of population parameters such as the mean in order to describe a variable's distribution. For this reason, nonparametric tests often are called 'distribution-free' or ' parameter-free' statistics. Given that nonparametric methods make less stringent demands on the data, one might wonder why they are not used more often.
There are several reasons. First, nonparametric statistics cannot provide definitive measures of actual differences between population samples. A nonparametric test may tell you that two interventions are different, but it cannot provide a confidence interval for the difference or even a simple mean difference between the two. Second, nonparametric procedures discard information. For example, if we convert severely skewed interval data into ranks, we are discarding the actual values and only retaining their order.
Because vital information is discarded, nonparametric tests are less powerful more prone to Type II errors than parametric methods. This also means that nonparametric tests typically require comparably larger sample sizes in order to demonstrate an effect when it is present.
Last, there are certain types of information that only parametric statistical tests can provide. A good example is independent variable interaction, as provided by factorial analysis of variance. There is simply no equivalent nonparametric method to analysis such interactions.
For these reasons, you will see nonparametric analysis used primarily on an as-needed basis, either 1 to analyze nominal or ordinal data or 2 to substitute for parametric tests when their assumptions are grossly violated, e.
Discussion here will be limited to the analysis of nominal or ordinal data. Nominal Categorical Data Analysis We previously have learned that the Pearson product-moment correlation coefficient r is commonly used to assess the relationship between two continuous variables. If instead the two variables are measured at the nominal level categorical in nature , we assess their relationship by crosstabulating the data in a contingency table.
A contingency table is a two-dimensional rows x columns table formed by 'cross-classifying' subjects or events on two categorical variables. One variable's categories define the rows while the other variable's categories define the columns.
The intersection crosstabulation of each row and column forms a cell, which 1 displays the count frequency of cases classified as being in the applicable category of both variables.
Below is a simple example of a hypothetical contingency table that crosstabulates patient gender against survival of chest trauma: Outcome Survives Dies Total Male 34 16 50 Female 7 43 50 Total 41 59 Testing for Independence Chi-square and Related Tests Based on simple probability, we can easily compute the expected values for each cell, i.
The greater the difference between the observed O and expected E cell counts, the less likely that the null hypothesis of independence holds true, i.
To determine whether or not the row and column categories for the table as a whole are independent of each other, i. As indicated in the formula, one first computes the differences between the observed and expected frequencies in each cell, squares this difference, and then divides the squared difference by that cell's expected frequency. In this case, an alternative is needed. If one is assessing the relationship between cause and effect, other nonparametric test would need to be considered.
To test the strength of such relationships we use correlation-like measures such as the Contingency Coefficient, the Phi coefficient or Cramer's V. These coefficients can be thought of as Pearson product-moment correlations for categorical variables. Unfortunately, the maximum value of the contingency coefficient varies with table size being larger for larger tables.
For this reason, it is difficult to compare the association among variables among different size tables using this coefficient. If we were conducting crosstabulation on contingency tables larger than 2 x 2, Cramer's V is the nominal association measure of choice. The formula for Cramer's V is: 3 where N is the total number of cases and k is the lesser of the number of rows or columns. Ordinal Ranked Data Analysis Testing for the Strength of Ordinal Ranked Relationships As with continuous and nominal data, measures exist to quantify the strength of association between variable measured at the ordinal level.
Both Spearman's rho and Kendall's tau require that the two variables, X and Y, are paired observations, with the variables measured are at least at the ordinal level. Spearman's rho. In principle, Spearman's rho is simply a special case of the Pearson product- moment coefficient in which the data are converted to ranks before calculating the coefficient.
The raw scores are converted to ranks, and the differences D between the ranks of each observation on the two variables are calculated. Kendall's tau. The main advantage of using Kendall's tau over Spearman's rho is that one can interpret its value as a direct measure of the probabilities of observing concordant and discordant pairs.
As long as one of the variables is presorted by order, Kendall's tau can be computed using the following formula: Where P is the sum, over all the cases, of cases ranked after the given item by both rankings, and n is the number of paired items.
Using the same data as we employed to compute Spearman's rho, we note that the paired observations are sorted in order of height, so we will compute P based on the weight data.
In the Weight row of this table, the first entry, 3, has five higher ranks to the right of it; so its contribution to P is 5. Moving to the second entry, 4, we see that there are four higher ranks to the right of it and its contribution to P is 4. Again, we see a positive correlation between the height and weight ranks, albeit less strong than that revealed by Spearman's rho. Alternatively, interval or ratio-level measurements on groups may be so skewed as to make regular parametric analysis impossible.
In these cases, comparable nonparametric approaches to traditional t-testing or analysis of variance ANOVA are needed. Nonparametric statistics. The Mann-Whitney U ranks all the cases for each of the two groups from the lowest to the highest value.
Then a mean rank, sum of ranks and 'U' score is computed for each group. Two U scores are computed: U1 and U2. U1 is defined as the number of times that a score from group 1 is lower in rank than a score from group 2. Likewise U2 is defined as the number of times that a score from group 2 is lower in rank that a score from group 1.
However, if the sample data are continuous and normally distributed, then nonparametric tests like the Mann-Whitney U Test should not be employed since they are less powerful than their parametric equivalents and thus more likely to miss a true difference between groups.
If the rank distributions are identical to one another, then the Z-score will equal 0. Positive Z- scores indicate that the sums of the ranks of group 2 are greater than that of group 1, while negative Z-scores indicate the opposite, i.
At the normal confidence level of 0. Note that if the observations are paired instead of independent of each other e. The observations represent kilograms of weight lost over a 3 month period. However, if the sample data are continuous and normally distributed, then nonparametric tests like the Mann-Whitney U Test should not be employed since hey are less powerful than their parametric equivalents and thus more likely to miss a true difference between groups.
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Since Inspecting the sum of ranks suggests that plans A and B are the best and nearly equivalent , whereas plan C ranks lowest in weight loss. Note that if the observations are repeated more than once e.
Introduction to categorical data analysis. NY: Wiley. Altman, D. Comparing groups — categorical data Chapter In Practical statistics for medical research. Becker, L.Use the link below to share a full-text version of this article with your friends and colleagues. New York: McGraw-Hill. Share full text access. Alternatively, continuous data may be so severely skewed from normal that it cannot be analyzed using regular parametric methods.
Nonparametric statistical methods 2nd ed. Hardcover Brand: The intersection crosstabulation of each row and column forms a cell, which 1 displays the count frequency of cases classified as being in the applicable category of both variables.