- 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 2 COMMENTS:

## By ozman on July 29th, 2023

Hi Ruben,

I enjoy reading your tutorials and always look forward to the latest instalment.

I have one small (trivial) comment on the tutorial titled "which statistical test...".

In the last 2 tables (Factor, Cluster) you have "Madelyn" listed twice as a first name. I realize that there may be two Madelyns with different last names, but in the absence of information on last name, one concludes that this is the same person (a repeated measure), which may cause some confusion for the reader. Perhaps a different first name is warranted.

Thanks and keep up the good work.

regards,

John

## By Ruben Geert van den Berg on July 29th, 2023

Dear John,

Thanks for your suggestion, I'll implement it in the next update.

These are simulated data but "Madelyn" appearing twice is indeed somewhat unfortunate.

Great catch, really. If all analysts had this level of "eye for detail", there wouldn't be much of a replication crisis in the social sciences.

Kind regards from Amsterdam!

Ruben

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