Quant: Parametric and Non-Parametric Stats

Parametric statistics is inferential and based on random sampling from a well-defined population, and that the sample data is making strict inferences about the population’s parameters. Thus tests like t-tests, chi-square, f-tests (ANOVA) can be used (Huck, 2011; Schumacker, 2014).  Nonparametric statistics, “assumption-free tests”, is used for tests that are using ranked data like Mann-Whitney U-test, Wilcoxon Signed-Rank test, Kruskal-Wallis H-test, and chi-square (Field, 2013; Huck, 2011).

First, there is a need to define the types of data.  Continuous data is interval/ratio data, and categorical data is nominal/ordinal data.  Modified from Schumacker (2014) with data added from Huck (2011):

Statistic	Dependent Variable	Independent Variable
Analysis of Variance (ANOVA)
One way	Continuous	Categorical
t-Tests
Single Sample	Continuous
Independent groups	Continuous	Categorical
Dependent (paired groups)	Continuous	Categorical
Chi-square	Categorical	Categorical
Mann-Whitney U-test	Ordinal	Ordinal
Wilcoxon	Ordinal	Ordinal
Kruskal-Wallis H-test	Ordinal	Ordinal

ANOVAs (or F-tests) are used to analyze the differences in a group of three or more means, through studying the variation between the groups, and tests the null hypothesis to see if the means between the groups are equal (Huck, 2011). Student t-tests, or t-tests, test as a null hypothesis that the mean of a population has some specified number and is used when the sample size is relatively small compared to the population size (Field, 2013; Huck, 2011; Schumacker, 2014).  The test assumes a normal distribution (Huck, 2011). With large sample sizes, t-test/values are the same as z-tests/values, the same can happen with chi-square, as t and chi-square are distributions with samples size in their function (Schumacker, 2014).  In other words, at large sample sizes the t-distribution and chi-square distribution begin to look like a normal curve.  Chi-square is related to the variance of a sample, and the chi-square tests are used for testing the null hypothesis, which is the sample mean is part of a normal distribution (Schumacker, 2014).  Chi-square tests are so versatile it can be used as a parametric and non-parametric test (Field, 2013; Huck, 2011; Schumacker, 2014).

The Mann-Whiteney U-test and Wilcox signed-rank test are both equivalent, since they are the non-parametric equivalent of the t-tests and the samples don’t even have to be of the same sample length (Field, 2013).

The nonparametric Mann-Whitney U-test can be substituted for a t-test when the normal distribution cannot be assumed and was designed for two independent samples that do not have repeated measures (Field, 2013; Huck, 2011). Thus, this makes this a great substitution for the independent group’s t-test (Field, 2013). A benefit of choosing the Mann-Whitney U test is that it probably will not produce type II error-false negative (Huck, 2011). The null hypothesis is that the two independent samples come from the same population (Field, 2013; Huck, 2011).

The nonparametric Wilcoxon signed-rank test is best for distributions that are skewed, where variance homogeneity cannot be assumed, and a normal distribution cannot be assumed (Field, 2013; Huck, 2011).  Wilcoxon signed test can help compare two related/correlated samples from the same population (Huck, 2011). Each pair of data is chosen randomly and independently and not repeating between the pairs (Huck, 2011).  This is a great substitution for the dependent t-tests (Field, 2013; Huck, 2011).  The null hypothesis is that the central tendency is 0 (Huck, 2011).

The nonparametric Kruskal-Wallis H-test can be used to compare two or more independent samples from the same distribution, which is considered to be like a one-way analysis of variance (ANOVA) and focuses on central tendencies (Huck, 2011).  It is usually an extension of the Mann-Whitney U-test (Huck, 2011). The null hypothesis is that the medians in all groups are equal (Huck, 2011).

References

• Field, A. (2013) Discovering Statistics Using IBM SPSS Statistics (4^th ed.). UK: Sage Publications Ltd. VitalBook file.
• Huck, S. W. (2011) Reading Statistics and Research (6^th ed.). Pearson Learning Solutions. VitalBook file.
• Schumacker, R. E. (2014) Learning statistics using R. California, SAGE Publications, Inc, VitalBook file.