Introduction to Nonparametric Methods

Not all real world situations yield interval- or ratio-level data that meet the assumptions made by parametric statistics about the distribution underlying the data. For such situations, nonparametric statistical techniques are often available that will enable one to do hypothesis testing and draw inferences from the data. Although nonparametric statistics have several advantages, however, they are not without disadvantages as well. Some of the more commonly used nonparametric statistics include nonparametric equivalents of the t-tests, coefficients of correlation, analysis of variance, as well as ways to test whether or not data are random. Although these tests are invaluable in certain situations, in the end, for situations where parametric techniques are available and the assumptions of these techniques are met, it is always preferable to use parametric analysis rather than nonparametric analysis.

Keywords Analysis of Variance; Correlation; Distribution; Hypothesis; Inferential Statistics; Nonparametric Statistics; Normal Distribution; Parametric Statistics; Population; Sample; Statistical Significance; Statistics; Variable

Statistics > Introduction to Nonparametric Methods

Overview

Popular Uses & Shortcomings of Parametric Methods

Most inferential statistics that are commonly used in applied settings are parametric and make certain assumptions about the parameters of the data and the distribution of the underlying population from which a sample is drawn. Commonly used inferential statistics such as t-tests, analyses of variance, and Pearson product moment correlation coefficients assume that the data being analyzed have been randomly selected from a population that has a normal distribution. In addition, parametric statistics require data that are interval or ratio in nature. That is, not only do the rank orders of the data have meaning (e.g., a value of 6 is greater than a value of 5) but the intervals between the values also have meaning. For example, it is clear that the difference between 1 gram of a chemical compound and 2 grams of a chemical compound is the same as the difference between 100 grams of the compound and 101 grams of the compound. These measurements have meaning because the weight scale has a true zero (i.e., we know what it means to have 0 grams of the compound) and the intervals between values is equal. On the other hand, it may not be quite as clear that the difference between 0 and 1 on a 100 point rating scale of quality of a widget is the same as the difference between 50 and 51 or between 98 and 99. These are value judgments and the scale may not have a true zero. For example, the scale may go from 1 to 100 and not include a 0. Similarly, even if the scale does start at 0, it may be difficult to define what this value means. Does a 0 on this scale differ significantly form a score of 1? Both scores indicate that the rater did not like the widget at all. However, one cannot tell from these scores why the rater assigned the values. Harvey may dislike on particular feature of the widget and, therefore, dismiss the entire product as unacceptable. Mathilde may like the gizmo better than the widget and, therefore, dislikes the widget in comparison so rates it low as a comparative rating rather than an absolute one. Even if the various points on the scale were well-defined, Harvey may think that a widget that he does not like should be given a rating of 20 while Mathilde may think that a widget that she does not like should be given a value of 2. Ratings are subjective and although numerical values may be assigned to them, these do not necessarily meet the requirement of parametric statistics that the data be at the interval or ratio level. Similarly, the real world puts practical limits on the quality of data that can be gathered. For example, if performing a quick survey in a shopping mall, one may only have time to ask people which of two soft drinks they prefer over the other. Although the resultant data do show that one is beverage is preferred over the other, they do not give the analyst any idea the degree to which this is true.

Uses of Nonparametric Statistics

Fortunately, one need not rely on parametric statistics or forego statistical analysis completely in situations where data do not meet the assumptions of parametric statistics. A number of nonparametric procedures are available that correspond to common tests used when the shape and parameters of a distribution are known. Nonparametric tests are so-called because they make no assumptions about the underlying distribution. To deal with data that are neither interval or ratio in nature or where the assumptions about the underlying distribution cannot be reasonably made, one needs to use nonparametric rather than parametric statistics. Nonparametric statistical techniques are used in situations where it is not possible to estimate or test the values of the parameters (e.g., mean, standard deviation) of the distribution or where the shape of the underlying distribution is unknown. In addition, nonparametric statistics often can be used in situations where only ordinal or ranked data are available (i.e., where the intervals between the data points may be uneven). Some nonparametric statistical techniques are available for use with nominal data. Although nonparametric statistical techniques are not as powerful as standard parametric statistics, nonparametric statistics do allow the analyst to derive meaningful information from a less than perfect data set.

Advantages to Nonparametric Statistics

Although these characteristics may seem to imply that nonparametric statistics are somehow inferior to parametric statistics, there are, in fact, several advantages to using nonparametric statistics.

• First, nonparametric statistics are less demanding about the characteristics of the data and their underlying distribution. Parametric statistics can only be validly used in situations where certain underlying assumptions are met, particularly if the sample sizes are small. For example, the one-sample Student's t-test requires that the underlying distribution for the population be normally distributed. Further, for independent samples, there is the additional requirement that the standard deviations be equal. If these assumptions are not true and the statistical technique is used, the results of the analysis cannot be trusted. The nonparametric equivalents of these tests, on the other hand, do not make these assumptions.
• Second, nonparametric statistics frequently require less time and effort to calculate, particularly for small sample sizes. For example, the nonparametric sign test provides the analyst with a quick test of whether or not two treatments are equally effective just by counting the number of times one treatment is better than the other.
• Third, nonparametric statistics can be used to provide some objectivity in situations were there is no reliable underlying scale for the data or where the use of parametric statistics would depend on an artificial metric. In fact, some nonparametric statistics are available for use with both nominal data (i.e., data that only indicate in which category or class a data point belongs but which indicates nothing about the relative intervals between data points) and ordinal data (i.e., using a scale on which data can be rank ordered but which indicates nothing about the intervals between the data points).
• Fourth, in some situations, even though sufficient interval or ratio data are available, they have not been randomly sampled from a larger population and there is no way to acquire a random sample. This often occurs in real world situations where analysis needs to be performed on existing data and the analyst or researcher cannot collect data that meet the requirements of parametric statistics. When this situation occurs, standard parametric statistics cannot be used. However, the data can sometimes be analyzed using nonparametric statistics.
• Finally, in some situations, nonparametric statistics offer the only choice for analyzing data.

Disadvantages of Nonparametric Statistics

Although these advantages make the use choice of nonparametric statistics over parametric statistics tempting, it must be borne in mind that parametric statistics are typically more powerful tools for data analysis than are nonparametric statistics. Despite the fact that in many cases nonparametric statistics are quicker and easier to use and require less assumptions about the way that data are collected, nonparametric statistics are not without their disadvantages.

• First, because by definition there are no parameters to describe nonparametric data, it is difficult to make quantitative statements about the actual differences between populations.
• Second, nonparametric statistics can be wasteful and disregard valuable information. For example, the sign test mentioned previously completely disregards the values of the data and only examines whether or not the differences between values are positive or negative. To take the best advantage of the data, therefore, it is advisable to use parametric statistics wherever the data allow.
• Third, although many nonparametric techniques are available, they are not available for as wide a range of analytic problems as are parametric statistics.
• Fourth, although calculating nonparametric statistics for small sample sizes may be easier than for parametric statistics, for large sample sizes the calculation of nonparametric statistics can be tedious. In the end, for situations where parametric techniques are available and the assumptions of these techniques are met, it is always preferable to use parametric analysis rather than nonparametric analysis.

Applications

As discussed above, a wide range of nonparametric statistical techniques is available for drawing inferences from sets of data that do not meet the assumptions for parametric statistical tests. Many of these statistical methods are nonparametric parallels of parametric tests. Although typically less powerful than their parametric cousins, nonparametric statistics can even be used for hypothesis testing. A thorough discussion of all the nonparametric tests available is well beyond the scope of this article. However, the following paragraphs summarize some of the more commonly used nonparametric statistics including nonparametric equivalents of t-tests, coefficients of correlation, analysis of variance, as well as uniquely nonparametric methods.

Nonparametric Equivalents of the Student's -Test

One of the fundamental inferential statistics techniques is the Student's t-test. This test is useful for comparing the means of two independent samples. For example, a marketer might want to know if there is any difference in reaction of people who responded to a proposed new packaging for a widget to those who responded to the existing packaging. However, one cannot always estimate the means of the two independent samples.

Mann-Whitney U Test

For such situations, the nonparametric Mann-Whitney U test (also know as the Wilcoxon rank-sum test) can be used instead. The Mann-Whitney U test enables the analyst to make meaningful comparisons between two independent nonparametric samples. This test enables the analyst to test whether or not two samples were drawn from the same population. The only assumptions made by the Man-Whitney U test are that the two samples are independent and that the observations are ordinal or continuous so that one can determine which of a value in a pair of measurements is greater.

Wilcoxon Signed Rank Test

Sometimes one needs to examine a set of differences. In parametric situations, this is typically done using a paired t-test. The nonparametric equivalent of this test is the Wilcoxon signed rank test. This test is a viable alternative to the t-test for those situations where the assumptions about the distribution made by the t-test cannot be met. This test is an alternative to the paired Student's t-test for cases where there are related samples or repeated measurements on a single sample and can be used to compare the differences between the measurements in nonparametric data. Although the Wilcoxon does not make assumptions about the shape of the underlying distribution, it does require interval data.

Nonparametric Coefficients of Correlation

The Pearson product moment coefficient of correlation is used to determine degree to which two events or variables are consistently related. Correlation may be positive (i.e., as the value of one variable increases the value of the other variable increases), negative (i.e., as the value of one variable increases the value of the other variable decreases), or zero (i.e., the values of the two variables are unrelated). The Pearson product moment cannot be used on nonparametric data. However, for these situations the Spearman rank correlation coefficient can be used instead to determine the degree of association between the two variables. Unlike the Pearson coefficient of correlation which requires interval or ratio level data, the Spearman can be used with ordinal level (i.e., ranked) data. In addition, the Spearman does not require interval data nor does it assume that there is a linear relationship between the variables. The Spearman rank correlation can be used in situations such as when one wanted to determine if supervisory ratings done by two different supervisors were close enough to be pooled (i.e., to determine whether or not both supervisors were using the same subjective criteria when rating employees).

Nonparametric Analysis of Variance

Parametric analysis of variance techniques are a family of statistical tests that are used on parametric data that analyze the joint and separate effects of multiple independent variables on a single dependent variable and determine the statistical significance of the effect. For situations were the data do not meet the assumptions for performing the parametric tests, two techniques are available for performing similar analyses on nonparametric data.

The Kruskal-Wallis Analysis

The Kruskal-Wallis analysis of variance by ranks (also referred to as the H test) is used to test whether three or more samples come from the same population. As opposed to the one-way analysis of variance (ANOVA), the Kruskal-Wallis does not assume that the underlying populations are normally distributed, that the groups are independent, or that the population variances are equal. Further, the Kruskal-Wallis only requires ordinal (i.e., ranked) data and makes the assumptions that the groups are independent and that the individual items are randomly selected.

Friedman Two-Way Analysis of Variance Test

Another nonparametric alternative to parametric analysis of variance tests is the Friedman two-way analysis of variance test. This statistic is used for the comparison of groups that are classified by two different factors. The parametric test of the randomized block design analysis of variance assumes that the sample is drawn from a normal distributed population and requires interval or ratio data. The Friedman test, on the other hand, only assumes that the blocks are independent, that no interaction is present between the blocks and the treatments, and that the observations within each block can be rank ordered. An example of where this test might be used would be in a situation where one wanted to know whether the quality of parts received from multiple shipments over multiple days was consistent over the five days of the week that the shipments are received.

Other Nonparametric Statistical Methods

The One-Sample Runs Test

The one-sample runs test is a nonparametric method used to determine whether the order of observations in a data set is random. For situations where the data may only have one of two values (e.g., heads or tails, true or false, agree or disagree), the runs test examines whether or not it is likely that a succession of observations that all have the same characteristic (i.e., a "run") is due to chance. For example, suppose an engineer took a sample of 50 observations of widgets coming off a production line to determine whether or not the widgets were within tolerance and met quality control standards for the product. If the first 30 widgets observed were all within tolerance (a run of 30) and the last 20 all were not (a run of 20), it is not highly probable that these results are due to chance alone. Rather, the engineer would probably look for some other factor such as a break down in the machinery, a different production worker running the line, raw materials coming from a different shipment, and so forth. The runs test examines the data in order to test the null hypothesis that the observations in the sample are randomly generated. Different versions of the test are available for use with small and large sample sizes.

Chi-Square (χ²) Test for Qualitative Data

Another commonly used nonparametric test is the chi-square (χ²) test for qualitative data. This statistical procedure is used for hypothesis testing when the data are nominal and qualitative rather than quantitative. The chi-square test could be used to test whether distributions with categorical information differ from one another. For example, the chi-square test would be appropriate if one wanted to know whether or not there was a difference between the number of widgets produced off assembly line A and assembly line B. Both variables in this case are qualitative, nominal data: The name of the assembly line is merely a category as is whether or not a given widget passed a quality control check. The chi-square test can help one determine whether any difference in the number of rejected widgets between the two assembly lines is due to a real and persistent difference between the two lines or if it is merely due to randomness ("noise").

Terms & Concepts

Analysis of Variance (ANOVA): A family of statistical techniques that analyze the joint and separate effects of multiple independent variables on a single dependent variable and determine the statistical significance of the effect.

Correlation: The degree to which two events or variables are consistently related. Correlation may be positive (i.e., as the value of one variable increases the value of the other variable increases), negative (i.e., as the value of one variable increases the value of the other variable decreases), or zero (i.e., the values of the two variables are unrelated). Correlation does not imply causation.

Distribution: A set of numbers collected from data and their associated frequencies.

Hypothesis: An empirically-testable declaration that certain variables and their corresponding measure are related in a specific way proposed by a theory.

Inferential Statistics: A subset of mathematical statistics used in the analysis and interpretation of data. Inferential statistics are used to make inferences such as drawing conclusions about a population from a sample and in decision making.

Nonparametric Statistics: A class of statistical procedures that is used in situations where it is not possible to estimate or test the values of the parameters (e.g., mean, standard deviation) of the distribution or where the shape of the underlying distribution is unknown.

Normal Distribution: A continuous distribution that is symmetrical about its mean and asymptotic to the horizontal axis. The area under the normal distribution is 1. The normal distribution is actually a family of curves and describes many characteristics observable in the natural world. The normal distribution is also called the Gaussian distribution or the normal curve or errors.

Parametric Statistics: A class of statistical procedures that is used in situations where it is reasonable to make certain assumptions about the underlying distribution of the data and where the values to be analyzed are either interval- or ratio-level data.

Population: The entire group of subjects belonging to a certain category (e.g., all women between the ages of 18 and 27; all dry cleaning businesses; all college students).

Sample: A subset of a population. A random sample is a sample that is chosen at random from the larger population with the assumption that such samples tend to reflect the characteristics of the larger population.

Statistical Significance: The degree to which an observed outcome is unlikely to have occurred due to chance.

Statistics: A branch of mathematics that deals with the analysis and interpretation of data. Mathematical statistics provides the theoretical underpinnings for various applied statistical disciplines, including business statistics, in which data are analyzed to find answers to quantifiable questions. Applied statistics uses these techniques to solve real world problems.

Variable: An object in a research study that can have more than one value. Independent variables are stimuli that are manipulated in order to determine their effect on the dependent variables (response). Extraneous variables are variables that affect the response but that are not related to the question under investigation in the study.

Bibliography

Black, K. (2006). Business statistics for contemporary decision making (4th ed.). New York: John Wiley & Sons.

Hollander, M. & Wolfe, D. A. (1973). Nonparametric statistical methods. New York: John Wiley and Sons.

Witt, R. S. (1980). Statistics. New York: Holt, Rinehart and Winston.

Suggested Reading

Aneiros-Pérez, G., Cao, R., & Vilar-Fernández, J. M. (2011). Functional methods for time series prediction: a nonparametric approach. Journal of Forecasting, 30, 377-392. Retrieved November 15, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=61378399&site=ehost-live

Burruss, G. W. & Furlow, M. H. (2007). Teaching statistics visually: A quasi-experimental evaluation of teaching chi-square through computer tutorials. Journal of Criminal Justice Education, 18, 209-230. Retrieved September 13, 2007, from EBSCO Online Database Academic Search Complete. http://search.ebscohost.com/login.aspx?direct=true&db=a9h&AN=25347355&site=ehost-live

Grantz, K., Rajagopalan, B., Clark, M., & Zagona, E. (2007). Seasonal shifts in the North American monsoon. Journal of Climate, 20, 1923-1935.

Heydarbeygie, A., & Ahmadi, N. (2013). Nonparametric methods for the estimation of imputation uncertainty. Journal of Applied Statistics, 40, 693-698.Retrieved November 15, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=85199838&site=ehost-live

Schneider, N., Dreier, M., Amelung, V. E., & Buser, K. (2007). Hospital stay frequency and duration of patients with advanced cancer diseases — differences between the most frequent tumour diagnoses: A secondary data analysis. European Journal of Cancer Care, 16, 172-177. Retrieved September 13, 2007, from EBSCO Online Database Academic Search Complete. http://search.ebscohost.com/login.aspx?direct=true&db=a9h&AN=24361363&site=ehost-live

Shieh, G., Jan, S.-L., & Randles, R. H. (2007). Power and sample size determinations for the Wilcoxon signed-rank test. Journal of Statistical Computation & Simulation, 77, 717-724. Retrieved September 13, 2007, from EBSCO Online Database Academic Search Complete. http://search.ebscohost.com/login.aspx?direct=true&db=a9h&AN=26055653&site=ehost-live