The role of inferential statistical tests in psychological research

In psychological research, inferential statistical tests help determine whether observed patterns in data are likely due to a real effect or just chance. These tests analyse relationships between variables, providing evidence to support or reject hypotheses. They calculate values that indicate the strength, direction, and significance of findings, allowing researchers to draw reliable conclusions from sample data about larger populations.

Key concepts in inferential testing

• Variables - These are measurable factors in a study. An independent variable (IV) is manipulated by the researcher, while a dependent variable (DV) is measured to see the effect. In correlations, co-variables are studied without manipulation.
• Test of difference - Used in experiments to compare groups or conditions, identifying if the IV affects the DV.
• Test of relationship - Used in correlational studies to examine links between co-variables, without implying causation.
• Significance levels - Results are often tested at p < 0.05 (5% chance of error) or p < 0.01 (1% chance of error). If the observed value meets or exceeds a critical value, the result is significant.
• One-tailed and two-tailed tests - A one-tailed test predicts direction (e.g., positive correlation), while a two-tailed test does not.

Using a decision tree to select the appropriate statistical test

A decision tree guides researchers in choosing the right inferential test by considering the study's aim, data type, and design. Start by identifying if the study tests differences (experiments with IV and DV) or relationships (correlations between co-variables). Then, consider the level of measurement for the data.

Levels of measurement for data

• Nominal data - Categories without order, such as yes/no responses or group labels.
• Ordinal data - Ranked data with order but unequal intervals, like satisfaction ratings (low to high).
• Interval or ratio data - Numerical data with equal intervals, such as test scores (interval) or time measurements (ratio, which includes a true zero).

Decision tree steps

1. Determine the type of test - Is it an experiment (test of difference) or a correlation (test of relationship)?
2. Identify the data type - Nominal, ordinal, or interval/ratio?
3. For experiments with ordinal data, check the experimental design - Repeated measures (same participants in all conditions) or independent groups (different participants per condition)?

Spearman's rho test for correlations

Spearman's rho is a non-parametric inferential test used to measure the strength and direction of the relationship between two ordinal-level co-variables. It produces a correlation coefficient (r_s) ranging from -1 to +1. A value close to +1 indicates a strong positive correlation (as one variable increases, so does the other), while close to -1 shows a strong negative correlation (as one increases, the other decreases). A value near 0 suggests no correlation.

When to use Spearman's rho

• Apply it for correlational studies with ordinal data.
• The test assesses if the observed correlation is significant by comparing r_s to critical values from tables, based on sample size (n) and significance level.

Formula for Spearman's rho

$r_s=1-\frac{6\sum d^2}{n(n^2-1)}$

Where:

• $\sum$ d² = Sum of the squared differences between ranks
• n = Number of participants or pairs of data

After calculating r_s, compare it to critical values: If r_s meets or exceeds the critical value for the chosen p-level and test type (one- or two-tailed), the correlation is significant.

Worked example - Calculating Spearman's rho

An investigation explores the relationship between students' scores on a memory test (out of 50) and the number of practice sessions they completed. Data for 7 students is as follows:

Calculate Spearman's rho for a one-tailed test at p < 0.05 and p < 0.01, and determine significance.

Step 1: Rank the data

Rank memory scores (lowest to highest: 1 = lowest score) and practice sessions similarly.

Step 2: Calculate sum of squared differences

$\sum$ d² = 1 + 4 + 0 + 4 + 0 + 0 + 25 = 34

Step 3: Apply the formula

For n=7, $\sum$ d² = 34:

$r_s=1-\frac{6\times 34}{7(7^2-1)}$

$r_s=1-\frac{204}{7(49-1)}$

$r_s=1-\frac{204}{7\times 48}$

$r_s=1-\frac{204}{336}$

$r_s=1-\mathrm{0.60714...}$

$r_s\approx 0.393$

Step 4: Determine significance

For n=7, one-tailed test: Critical value at p < 0.05 is 0.714; at p < 0.01 is 0.893.

Observed r_s = 0.393 is less than both critical values.

Therefore, the correlation is not significant at either p < 0.05 or p < 0.01. This suggests there is no statistically significant link between practice sessions and memory scores in this sample.