Choosing a statistical test can be a daunting task for those starting out in the analysis of experiments. This chapter provides a table of tests and models covered in this book, as well as some general advice for approaching the analysis of your data.
Plan your experimental design before you collect data
It is important to have an experimental design planned out before you start collecting data, and to have some an idea of how you plan on analyzing the data. One of the most common mistakes people make in doing research is collecting a bunch of data without having thought through what questions they are trying to answer, what specific hypotheses they want to test, and what statistical tests they can use to test these hypotheses.
What is the hypothesis?
The most important consideration in choosing a statistical test is determining what hypothesis you want to test. Or, more generally, what question are you are trying to answer.
Often people have a notion about the purpose of the research they are conducting, but haven’t formulated a specific hypothesis. It is possible to begin with exploratory data analysis, to see what interesting secrets the data wish to say. But ultimately, choosing a statistical test relies on having in mind a specific hypothesis to test.
For example, we may know that our goal is to determine if one curriculum works better than another. But then we must be more specific in our hypothesis. Perhaps we wish to compare the mean of scores that students get on an exam across the different curricula. Then a specific null hypothesis is, There is no difference among the mean of student scores across curricula.
In this example, we identified the dependent variable as Student scores, and the independent variable as Curriculum.
Of course, we might make things more complicated. For example, if the curricula were used in different classrooms, we might want to include Classroom as an independent blocking variable.
What number and type of variables do you have?
To a large extent, the appropriate statistical test for your data will depend upon the number and types of variables you wish to include in the analysis.
Consider the type of dependent variable you wish
to include.
• If it is of interval/ratio type, you can consider
parametric tests or nonparametric tests.
• However, if it is an ordinal variable, you
would look toward ordinal regression models, permutation tests,
nonparametric tests, or tests for ordinal tables.
• Nominal variables arranged in contingency
tables can be analyzed with chi-square and similar tests. Nominal
dependent variables can be related to independent variables with logistic
regression.
•
Count data dependent variables can be related to independent
variables with Poisson regression and related models.
If the dependent variable is a proportion or percentage, beta
regression might be appropriate.
The number and type of independent variables will also be taken into account. As will whether there are paired observations or random blocking variables.
The table below lists the tests in this book according to their number and types of variables.
Note that each test has its own set of assumptions for appropriate data, which should be assessed before proceeding with the analysis.
Also note that the tests in this book cover cases with a single dependent variable only. There are other statistical tests, included under the umbrella of multivariate statistics that can analyze multiple dependent variables simultaneously. These include multivariate analysis of variance (MANOVA), canonical correlation, and discriminant function analysis.
The “References” and “Optional readings” sections of this chapter includes a few other guides to choosing statistical tests.
Test |
DV type, or variable type when there is no DV |
DV |
IV type |
Number of IV |
Levels in IV |
Test type |
One-sample Wilcoxon |
Ordinal or interval/ratio |
Independent |
Single default value |
N/A |
N/A |
Nonparametric |
Sign test for one-sample |
Ordinal or interval/ratio |
Independent |
Single default value |
N/A |
N/A |
Nonparametric |
Two-sample Mann–Whitney |
Ordinal or interval/ratio |
Independent |
Nominal |
1 |
2 |
Nonparametric |
Mood’s median test for two-sample |
Ordinal or interval/ratio |
Independent |
Nominal |
1 |
2 |
Nonparametric |
Two-sample paired rank-sum |
Ordinal or interval/ratio |
Paired |
Nominal |
1, or 2 when one is blocking |
2 |
Nonparametric |
Sign test for two-sample paired |
Ordinal or interval/ratio |
Paired |
Nominal |
1, or 2 when one is blocking |
2 |
Nonparametric |
Kruskal–Wallis |
Ordinal or interval/ratio |
Independent |
Nominal |
1 |
2 or more |
Nonparametric |
Mood’s median |
Ordinal or interval/ratio |
Independent |
Nominal |
1 |
2 or more |
Nonparametric |
Friedman |
Ordinal or interval/ratio |
Independent blocked, or paired |
Nominal |
2 when one is blocking, in unreplicated complete block design |
2 or more |
Nonparametric |
Quade |
Ordinal or interval/ratio |
Independent blocked, or paired |
Nominal |
2 when one is blocking, in unreplicated complete block design |
2 or more |
Nonparametric |
One-way Permutation Test of Independence |
Ordinal or interval/ratio |
Independent |
Nominal |
1 |
2 or more |
Permutation |
One-way Permutation Test of Symmetry |
Ordinal or interval/ratio |
Independent blocked, or paired |
Nominal |
2 when one is blocking |
2 or more |
Permutation |
Two-sample CLM |
Ordinal |
Independent |
Nominal |
1 |
2 |
Ordinal regression |
Two-sample paired CLMM |
Ordinal |
Paired |
Nominal |
2 when one is blocking |
2 |
Ordinal regression |
One-way ordinal ANOVA CLM |
Ordinal |
Independent |
Nominal |
1 |
2 or more |
Ordinal regression |
One-way repeated ordinal ANOVA CLMM |
Ordinal |
Independent |
Nominal |
2 when one is blocking |
2 or more |
Ordinal regression |
Two-way ordinal ANOVA CLM |
Ordinal |
Independent |
Nominal |
2 |
2 or more |
Ordinal regression |
Two-way repeated ordinal ANOVA CLMM |
Ordinal |
Independent |
Nominal |
3 when one is blocking |
2 or more |
Ordinal regression |
Goodness-of-fit tests for nominal variables • binomial test • multinomial test • G-test goodness-of-fit • Chi-square test goodness-of-fit |
Nominal |
Independent |
Expected counts |
N/A |
Overall: vector of counts and expected proportions |
Nominal |
Association tests for nominal variables • Fisher exact test of association • G-test of association • Chi-square test of association |
Nominal |
Independent |
Nominal |
N/A |
Overall: 2-way contingency table |
Nominal |
Tests for paired nominal data • McNemar • McNemar–Bowker |
Nominal |
Paired |
Nominal |
N/A |
Overall: 2-way marginal contingency table |
Nominal |
Cochran–Mantel–Haenszel |
Nominal |
Independent |
Nominal |
N/A |
Overall: 3-way contingency table |
Nominal |
Cochran’s Q |
Nominal (2 levels only) |
Paired |
Nominal |
2 when one is blocking |
2 or more |
Nominal |
Linear-by-linear |
Ordered nominal (ordinal) |
Independent |
Ordered nominal (ordinal) |
N/A |
Overall: 2-way or 3-way contingency table |
Nominal |
Cochran–Armitage (extended) |
Ordered nominal (ordinal) |
Independent |
Nominal |
N/A |
Overall: 2-way or 3-way contingency table |
Nominal |
Log-linear model (multiway frequency analysis) |
Nominal |
Independent |
Nominal |
N/A |
Overall: contingency table with 2 or dimensions |
Generalized linear model |
Logistic regression (standard) |
Nominal with 2 levels |
Independent |
Interval/ratio or nominal |
1 or more |
2 or more |
Generalized linear model |
Multinomial logistic regression |
Nominal with 2 or more levels |
Independent |
Interval/ratio or nominal |
1 or more |
2 or more |
Generalized linear model |
Mixed-effects logistic regression |
Nominal with 2 levels |
Independent or paired |
Interval/ratio or nominal |
1 or more when one is blocking or random |
2 or more |
Generalized linear model |
One-sample t-test |
Interval/ratio |
Independent |
Single default value |
N/A |
N/A |
Parametric |
Two-sample t-test |
Interval/ratio |
Independent |
Nominal |
1 |
2 |
Parametric |
Paired t-test |
Interval/ratio |
Paired |
Nominal |
1, or 2 when one is blocking |
2 |
Parametric |
One-way ANOVA |
Interval/ratio |
Independent |
Nominal |
1 |
2 or more |
Parametric |
One-way ANOVA with blocks |
Interval/ratio |
Independent |
Nominal |
2 when one is blocking |
2 or more |
Parametric |
One-way ANOVA with random blocks |
Interval/ratio |
Independent |
Nominal |
2 when one is blocking |
2 or more |
Parametric |
Two-way ANOVA |
Interval/ratio |
Independent |
Nominal |
2 |
2 or more |
Parametric |
Repeated measures ANOVA |
Interval/ratio |
Paired across time |
Nominal |
2 or more when one is time effect |
2 or more |
Parametric |
Multiple correlation |
Interval/ratio or ordinal, depending on type selected |
Independent |
Interval/ratio or ordinal, depending on type selected |
1 or more |
Overall: multiple vectors of interval/ratio or ordinal data |
Parametric or nonparametric depending on type selected |
Pearson correlation |
Interval/ratio |
Independent |
Interval/ratio |
1 |
Overall: two vectors of interval/ratio data |
Parametric |
Kendall correlation |
Interval/ratio or ordinal |
Independent |
Interval/ratio or ordinal |
1 |
Overall: two vectors of interval/ratio or ordinal data |
Nonparametric |
Spearman correlation |
Interval/ratio or ordinal |
Independent |
Interval/ratio or ordinal |
1 |
Overall: two vectors of interval/ratio or ordinal data |
Nonparametric |
Linear regression |
Interval/ratio |
Independent |
Interval/ratio |
1 |
N/A |
Parametric |
Polynomial regression |
Interval/ratio |
Independent |
Interval/ratio |
2 or more that are polynomial terms |
N/A |
Parametric |
Nonlinear regression and curvilinear regression |
Interval/ratio |
Independent |
Interval/ratio |
1 |
N/A |
Parametric |
Multiple regression |
Interval/ratio |
Independent |
Interval/ratio |
2 or more |
N/A |
Parametric |
Robust linear regression |
Interval/ratio |
Independent |
Interval/ratio |
1 |
N/A |
Robust parametric |
Kendall–Theil regression |
Interval/ratio |
Independent |
Interval/ratio |
1 |
N/A |
Nonparametric |
Linear plateau and quadratic plateau models |
Interval/ratio |
Independent |
Interval/ratio |
1 |
N/A |
Parametric |
Cate–Nelson analysis |
Interval/ratio |
Independent |
Interval/ratio |
1 |
N/A |
Mostly nonparametric |
Hermite and Poisson regression • Hermite regression • Poisson regression • Negative binomial regression • Zero-inflated regression |
Count |
Independent |
Interval/ratio or nominal |
1 or more |
2 or more |
Generalized linear model |
Beta regression |
Proportion or percentage |
Independent |
Interval/ratio or nominal |
1 or more |
2 or more |
Generalized linear model |
Optional discussion: Sometimes it’s all about the hypothesis
Tests that have analogous purposes, like comparing a measurement variable across two groups, may test very different hypotheses.
For example, imagine you are investigating the income of two towns. Let’s say the income of Town A is normally distributed about a mean and median of $48,000. The income of Town B has a similar median, but has right skew, with some observations close to $1 million.
What test or statistic would you use to compare the income of these two towns?
You might be tempted to compare the means of the two towns with a t-test. In this case, however, means may not be the best statistic for skewed data, and this data may not meet the assumptions of the t-test.
You might be interested in comparing the median of the income of the two towns, for example with Mood’s median test. This might make sense for some regulatory purpose that is concerned with medians.
On the other hand, looking for a systemic change in the income across the two towns may make more sense. For example, the higher incomes in Town B may give the town a different character, for example, some streets with larger homes or upscale stores. For this, you might use the Mann–Whitney test.
Another approach is to use a permutation test.
Or you might compare the overall distributions of incomes for the two towns using the Kolmogorov–Smirnov test.
Finally, you might want to compare at the 75^{th} percentile of income for the two towns. This could be done using quantile regression.
Example
The following code compares some of these results for a hypothetical data set of income in two towns.
Note that the assumptions and pitfalls of these tests are not discussed here, but should be considered in real situations.
### load required packages
if(!require(FSA)){install.packages("FSA")}
if(!require(psych)){install.packages("psych")}
if(!require(RVAideMemoire)){install.packages("RVAideMemoire")}
if(!require(coin)){install.packages("coin")}
if(!require(quantreg)){install.packages("quantreg")}
### Read the data frame
TwoTowns = read.table("http://rcompanion.org/documents/TwoTowns.csv",
header=TRUE, sep=",")
### Check the data frame
library(psych)
headTail(TwoTowns)
summary(TwoTowns)
### Summarize the data
library(FSA)
Summarize(Income ~ Town,
data=TwoTowns,
digits=3)
Town n mean sd min Q1 median Q3 max
1 Town.A 101 48146.43 10851.67 23560 40970 48010 56420 77770
2 Town.B 101 115275.22 163878.17 29050 34140 47220 108200 880000
boxplot(Income ~ Town,
data=TwoTowns)
### Mood’s median test
library(RVAideMemoire)
mood.medtest(Income ~ Town,
data = TwoTowns)
Mood's median test
X-squared = 0, df = 1, p-value = 1
### Mann–Whitney test
wilcox.test(Income ~ Town,
data=TwoTowns)
Wilcoxon rank sum test with continuity correction
W = 4672, p-value = 0.3029
alternative hypothesis: true location shift is not equal to 0
### Permutation test
library(coin)
independence_test(Income ~ Town,
data = TwoTowns)
Asymptotic General Independence Test
Z = -3.9545, p-value = 7.669e-05
### Kolmogorov–Smirnov test
library(FSA)
ksTest(Income ~ Town,
data = TwoTowns)
Two-sample Kolmogorov-Smirnov test
D = 0.35644, p-value = 5.349e-06
### quantile regression considering the
75^{th} percentile
library(quantreg)
model.q = rq(Income ~ Town,
data = TwoTowns,
tau = 0.75)
model.null = rq(Income ~ 1,
data = TwoTowns,
tau = 0.75)
anova(model.q, model.null)
Quantile Regression Analysis of Deviance Table
Df Resid Df F value Pr(>F)
1 1 200 5.7342 0.01756 *
References
[IDRE] Institute for Digital Research and Education. 2015. What statistical analysis should I use? UCLA. www.ats.ucla.edu/stat/stata/whatstat/.
“Choosing a statistical test” in McDonald, J.H. 2014. Handbook of Biological Statistics. www.biostathandbook.com/testchoice.html.
Optional readings
[Video] “Choosing which statistical test to use” from Statistics Learning Center (Dr. Nic). 2014. www.youtube.com/watch?v=rulIUAN0U3w.