- Univariate Tests
- Within-Subjects Tests
- Between-Subjects Tests
- Association Measures
- Prediction Analyses
- Classification Analyses

## Summary

Finding the appropriate statistical test is easy if you're aware of

- the basic
**type**of test you're looking for and - the
**measurement levels**of the variables involved.

This tutorial briefly defines the 6 basic types of tests and illustrates them with simple examples. We'll then present full overviews of all tests belonging to each type.

## 1. Univariate Tests

Univariate tests are tests that involve only 1 variable. Univariate tests either test if

- some population
**parameter**-usually a mean or median- is equal to some hypothesized value or - some population
**distribution**is equal to some function, often the normal distribution.

A textbook example is a one sample t-test: it tests if a population mean -a parameter- is equal to some value *x*. This test involves only 1 variable (even if there's many more in your data file).

## Overview Univariate Tests

MEASUREMENT LEVEL | NULL HYPOTHESIS | TEST |
---|---|---|

Dichotomous | Population proportion = x? | Binomial test Z-test for 1 proportion |

Categorical | Population distribution = f(x)? | Chi-square goodness-of-fit test |

Quantitative | Population mean = x? | One-sample t-test |

Population median = x? | Sign test for 1 median | |

Population distribution = f(x)? | Kolmogorov-Smirnov test Shapiro-Wilk test |

## 2. Within-Subjects Tests

Within-subjects tests compare 2+ variables

measured on the same subjects (often people).
An example is repeated measures ANOVA: it tests if 3+ variables measured on the same subjects have equal population means.

**Within-subjects** tests are also known as

**paired samples**tests (as in a paired samples t-test) or**related samples**tests.

## Overview Within-Subjects Tests

MEASUREMENT LEVEL | 2 VARIABLES | 3+ VARIABLES |
---|---|---|

DICHOTOMOUS | McNemar test | Cochran Q test |

ORDINAL | Wilcoxon signed-ranks test Sign test for 2 related medians | Friedman test |

QUANTITATIVE | Paired samples t-test | Repeated measures ANOVA |

## 3. Between-Subjects Tests

Between-subjects tests examines if 2+ subpopulations

are identical with regard to

- a
**parameter**(population mean, standard deviation or proportion) or - a
**distribution**.

The best known example is a one-way ANOVA as illustrated below. Note that the subpopulations are represented by subsamples -groups of observations indicated by some categorical variable.

“Between-subjects” tests are also known as “**independent samples**” tests, such as the independent samples t-test. “Independent samples” means that subsamples don't overlap: each observation belongs to only 1 subsample.

## Overview Between-Subjects Tests

OUTCOME VARIABLE | 2 SUBPOPULATIONS | 3+ SUBPOPULATIONS |
---|---|---|

Dichotomous | Z-test for 2 independent proportions | Chi-square independence test |

Nominal | Chi-square independence test | Chi-square independence test |

Ordinal | Mann-Whitney test (mean ranks) Median test for 2+ independent medians | Kruskal-Wallis test (mean ranks) Median test for 2+ independent medians |

Quantitative | Independent samples t-test (means) Levene's test (variances) | One-way ANOVA (means) Levene's test (variances) |

## 4. Association Measures

Association measures are numbers that indicate

to what extent 2 variables are associated.
The best known association measure is the Pearson correlation: a number that tells us to what extent 2 quantitative variables are linearly related. The illustration below visualizes correlations as scatterplots.

## Overview Association Measures

(VARIABLES ARE) | QUANTITATIVE | ORDINAL | NOMINAL | DICHOTOMOUS |
---|---|---|---|---|

QUANTITATIVE | Pearson correlation | |||

ORDINAL | Spearman correlation Kendall’s tau Polychoric correlation | Spearman correlation Kendall’s tau Polychoric correlation | ||

NOMINAL | Eta squared | Cramér’s V | Cramér’s V | |

DICHOTOMOUS | Point-biserial correlation Biserial correlation | Spearman correlation Kendall’s tau Polychoric correlation | Cramér’s V | Phi-coefficient Tetrachoric correlation |

## 5. Prediction Analyses

Prediction tests examine how and to what extent

a variable can be predicted from 1+ other variables.
The simplest example is simple linear regression as illustrated below.

Prediction analyses sometimes quietly assume **causality**: whatever *predicts* some variable is often thought to *affect* this variable. Depending on the contents of an analysis,
causality may or may not be plausible.
Keep in mind, however, that the analyses listed below don't *prove* causality.

## Overview Prediction Analyses

OUTCOME VARIABLE | ANALYSIS |
---|---|

Quantitative | (Multiple) linear regression analysis |

Ordinal | Discriminant analysis or ordinal regression analysis |

Nominal | Discriminant analysis or nominal regression analysis |

Dichotomous | Logistic regression |

## 6. Classification Analyses

Classification analyses attempt to identify and

describe groups of observations or variables.
The 2 main types of classification analysis are

**factor analysis**for finding groups of**variables**(“factors”) and**cluster analysis**for finding groups of**observations**(“clusters”).

Factor analysis is based on correlations or covariances. Groups of variables that correlate strongly are assumed to measure similar underlying factors -sometimes called “constructs”. The basic idea is illustrated below.

Cluster analysis is based on distances among observations -often people. Groups of observations with small distances among them are assumed to represent clusters such as market segments.

Right. So that'll do for a basic overview. Hope you found this guide helpful! And last but not least,

**thanks for reading!**

## THIS TUTORIAL HAS 9 COMMENTS:

## By Julio on February 5th, 2020

Excelent for beginners !

## By Ruben Geert van den Berg on February 6th, 2020

Thanks for the compliment! Happy to hear you liked it!

## By Jon K Peck on February 6th, 2020

Don't forget polyserial correlation for continuous (quantitative) with ordinal.

And for classification, there are lots of other good choices such as trees and support vector machines

## By Ruben Geert van den Berg on February 6th, 2020

Thanks for the suggestions!

I was planning to write a separate overview on association measures in order to cover them a little more in-depth but it'll take at least some more weeks.

With "trees", are you referring to CHAID and similar procedures? IMHO, these fall more into prediction than classification.

P.s. "quantitative" is not always "continuous" as in number of children/cars in a household. Strictly, even amounts of dollars are not continuous as there's no value between $0.00 and $0.01. I kinda feel "continuous" should only be used for truly continuous variables such as length/weight and so on.

## By Jon K Peck on February 6th, 2020

Statistics calls "quantitative" "scale". And what do you have against 2.5 children? :-)

Trees - CHAID et al, is a classification procedure with a categorical dependent variable, but, of course, it's also doing prediction as also with SVM or logistic regression.