- 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.

For each type and measurement level, this tutorial immediately points out the right statistical test. We'll also briefly define the 6 basic types of tests and illustrate them with simple examples.

## 1. 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 |

## Univariate Tests - Quick Definition

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).

## 2. Overview Within-Subjects Tests

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

DICHOTOMOUS | McNemar test Z-test for dependent proportions | Cochran Q test |

NOMINAL | Marginal homogeneity test | (Not available) |

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

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

## Within-Subjects Tests - Quick Definition

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.

## 3. 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) |

## Between-Subjects Tests - Quick Definition

Between-subjects tests examine 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.

## 4. 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 |

## Association Measures - Quick Definition

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.

## 5. 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 |

## Prediction Analyses - Quick Definition

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.

## 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 15 COMMENTS:

## By N M on May 18th, 2021

Hi could you do a post on point biserial and biserial please ? Thank you so much for the work you do

## By Maxine Duncan on July 6th, 2022

This article was a valuable source of information to me.

## By Roger ILUNGA NTABALA on July 18th, 2022

Verry good

## By YY on July 19th, 2022

A comprehensive guide. Very useful!

Can the McNemar-Bowker test provided by SPSS be used when there are two related nominal variables?

## By Ruben Geert van den Berg on July 20th, 2022

Yes. We haven't examined this procedure very thoroughly but I believe that's correct.