Non-parametric Alternatives in Psychological Research

Parametric Requirements: The primary requirement for parametric data is a continuous Dependent Variable ( $DV$ ). When this is met, parametric assumptions (e.g., Normality) can be tested.
Variance and Information: Continuous data provides the highest potential for variance, which translates to more information for researchers.
- Continuous: Real numbers/decimals (e.g., averaged rankings from $1$ to $7$ ).
- Ordinal: Ranks without decimals (e.g., ratings of $1, 2, 3, 4, 5, 6, 7$ ).
- Nominal: Categories with no numerical variance (e.g., high/low groups).
Statistical Power: Non-parametric tests are generally less powerful than parametric tests for detecting significance and are used primarily when data is naturally ordinal/nominal or when continuous data violates assumptions.

Normality: This is the primary parametric assumption and can only be statistically measured for continuous data. It is tested using the Shapiro-Wilk test, graphical methods (Stem and Leaf plots, $Q-Q$ plots), or statistics like skewness and kurtosis.
Outliers: Defined as values more than $3$ standard deviations from the mean. Outliers can bias central tendencies (mean) and distributions (standard deviation).
Handling Non-normality:
- Removing or "bringing in" outliers (adjusting them to exactly $3$ standard deviations from the mean).
- Mathematical data transformation.
- Using Welch’s t-test for violations of homogeneity of variance.
- Switching to non-parametric alternatives where continuous data is treated as ordinal.

Research designs and data types determine the appropriate analysis:

Between-Subjects (Independent Samples):
- Continuous: Independent samples t-test.
- Ordinal: Mann-Whitney U test.
- Nominal: Chi-square test of contingencies.
Within-Subjects (Paired Samples):
- Continuous: Paired samples t-test.
- Ordinal: Wilcoxon signed-rank test.
- Nominal: McNemar test of change.

The Mann-Whitney U Test: Example comparing Australian Millennials and Zoomers on communication rankings.
- Results: $Mean Rank_{diff} = 1.00$ , $N = 16$ , $r = .45$ , $U = 49.00$ , $p = .083$ (one-tailed).
- Observation: A medium effect size ( $r = .45$ ) can still be non-significant if the sample size is low, as the analysis is statistically underpowered.
The Wilcoxon Signed-Rank Test: Example measuring changes in grocery purchasing considerations (pre- vs. post-lecture).
The Chi-square Test of Contingencies: Example comparing depression symptom reduction (Yes/No) based on Mental Health Care Plan ( $MHCP$ ) session attendance (6-or-less vs. all 10).
The McNemar Test of Change: Example evaluating changes in belief (Yes/No) regarding psychology as a science before and after a unit.

While less restrictive, non-parametric tests still require:

Level of Measurement: DV must be at least ordinal for Mann-Whitney and Wilcoxon.
Distribution Shape: Groups should have roughly equivalent distribution shapes for Mann-Whitney and Wilcoxon.
Minimum Expected Frequencies: For nominal tests (Chi-square, McNemar), each "cell" in the table should have at least $5$ cases.