Between-subjects statistics are used when comparing independent groups on an outcome. Independent groups means that the groups are "different" or "independent" from each other according to some characteristic. With between-subjects designs, participants can only be part of one group (independence) and only observed once (independence of observations, IOO).

One chooses a between-subjects statistical test based on the following:

1. Number of independent groups being compared (one group, two groups, or three or more groups)

2. Scale of measurement of the outcome (categorical, ordinal, or continuous)

3. Meeting statistical assumptions (independence of observations, normality, and homogeneity of variance)

Here is a list of between-subjects statistical tests and when they are utilized in applied quantitative research:

1. Chi-square Goodness-of-fit - One group, categorical outcome, a priori hypothesis for dispersal of outcome

2. One-sample median test - One group, ordinal outcome, a priori hypothesis for median value

3. One-sample t-test - One group, continuous outcome, meet the assumption of IOO and normality, a priori hypothesis for mean value

4. Chi-square - Two independent groups, categorical outcome, and chi-square assumption (more than five observations in each cell)

5. Fisher's Exact test - Two independent groups, categorical outcome, and when the chi-square assumption is not met

6. Mann-Whitney U - Two independent groups, ordinal outcome, and when the assumption of homogeneity of variance for independent samples t-test is violated

7. Independent samples t-test - Two independent groups, continuous outcome, meet the assumption of IOO, normality (skewness and kurtosis statistics), and homogeneity of variance (also known as homoscedasticity, tested with Levene's test)

8. Unadjusted odds ratio - Three or more independent groups, categorical outcome, chi-square assumption, choose a reference category and compare each independent group to the reference

9. Kruskal-Wallis - Three or more independent groups, ordinal outcome, and when the assumption of homogeneity of variance is violated

10. ANOVA - Three or more independent groups, continuous outcome, meet the assumption of IOO, normality, and homogeneity of variance

When it comes to running statistics on one sample, an a priori hypothesis is a necessity for making proper inferences. Many survey instruments lack fundamental psychometric evidence to back up the constructs and/or items it is intended to measure.  When a Likert-scale type response set or an ordinal outcome is to be assessed statistically in one sample, the one-sample median test is used.

In order to run the test, researchers must specify where along the ordinal continuum that they hypothesize that the population mean exists.  The one-sample median test then compares the observed median to the hypothesized median and the p-value is interpreted.

Due to limited precision, accuracy, and variability in ordinal outcomes, it behooves researchers to use either 5-point, 7-point, or higher level Likert scales.  With more options, more unique variance can be accounted for the in the analysis and statistical power is increased.  One-sampled tests possess more statistical power than other between-subjects statistics because there is only one group being analyzed, no other independent groups are included.

As we continue our journey to break through the barriers associated with statistical lexicons, here is another dichotomy of popular statistical terms that are spoken commonly but not always understood by everyone.  

Parametric statistics are used to assess differences and effects for continuous outcomes. These statistical tests include one-sample t-tests, independent samples t-tests, one-way ANOVA, repeated-measures ANOVA, ANCOVA, factorial ANOVA, multiple regression, MANOVA, and MANCOVA. 

Non-parametric statistics are used to assess differences and effects for:

1. Ordinal outcomes - One-sample median tests, Mann-Whitney U, Wilcoxon, Kruskal-Wallis, Friedman's ANOVA, Proportional odds regression

2. Categorical outcomes - Chi-square, Chi-square Goodness-of-fit, odds ratio, relative risk, McNemar's, Cochran's Q, Kaplan-Meier, log-rank test, Cochran-Mantel-Haenszel, Cox regression, logistic regression, multinomial logistic regression

3. Small sample sizes (n < 30) - Smaller sample sizes make it harder to meet the statistical assumptions associated with parametric statistics.  Non-parametric statistics can generate valid statistical inferences in these situations.

4. Violations of statistical assumptions for parametric tests - Normality, Homogeneity of variance, Normality of difference scores