Statistical analysis for ecological design

Context

• This is not a statistics module — stats are used here as a tool to support survey design.

• The aim is to design and implement a scientifically sound survey and report, not to perform advanced mathematical analysis.

• Statistical methods are used to test hypotheses and interpret ecological data.

1. Statistical Tools for Ecological Analysis

Allowed Techniques

You’ll use familiar and straightforward methods:

• ANOVA (Analysis of Variance) – compare means between groups.

• Correlation – assess strength of relationship between two variables.

• Regression – model relationships (one variable predicts another).

You can combine these methods if appropriate.

• One-way ANOVA

• Two-way ANOVA (multi-factor)

• Linear regression

• Correlation (Pearson/Spearman)

2. Single-Factor vs Multi-Factor Designs

Single-Factor (One-Way)

• One predictor variable (factor).

• Example:
  Predictor = Shore height (3 levels: low, mid, high).
  Response = Algal diversity or abundance.

Goal → Compare means between levels (e.g. low vs mid vs high shore).

Multi-Factor (Two-Way)

• Two predictor variables.

• Example:

  • Factor 1 = Shore height (low, mid, high).

  • Factor 2 = Rock pools (inside vs outside).

Now the survey examines:

• Main effects of each factor (shore height, pool presence).

• Interaction effects (how one factor modifies the effect of the other).

→ Analysed using Two-Way ANOVA.

3. Comparing Averages

Examples of response variables:

• Mean abundance of a species.

• Mean size (e.g., of limpets).

• Mean diversity index (e.g., Shannon diversity).

Examples of predictor variables:

• Habitat type (sheltered vs exposed).

• Shore height (low, mid, high).

• Substrate heterogeneity.

When to Use Which Test

4. Parametric vs Non-Parametric Tests

Parametric Tests

• Require assumptions:

  1. Normal distribution of data.

  2. Homogenous variances (equal variance between groups).

  3. Independence of samples.

• Examples: t-test, ANOVA.

• Compare means.

• More powerful (detect smaller effects).

Non-Parametric Tests

• No assumption about data distribution (“distribution-free”).

• Compare medians (use data ranks).

• Examples:

  • Mann–Whitney U test (instead of t-test)

  • Kruskal–Wallis test (instead of one-way ANOVA)

• Less powerful, but often more appropriate for biological data (which is messy).

5. Assumptions of ANOVA

1. Independence

   • Each sample must be independent.

   • Example: don’t place quadrats right next to each other.

   • Violating independence invalidates tests.

2. Normality

   • Data should roughly follow a normal distribution.

   • Lecturer notes: “I’ve never tested for normality — ANOVA is robust to slight non-normality” (following advice of Tony Underwood).

   • With small sample sizes (n < 100), focus on homogeneity rather than strict normality.

3. Homogeneity of Variance

   • Variances between groups must be similar.

   • Test using:

     • Cochran’s test

     • Levene’s test (simpler)

   • If variances unequal → transform data (square root, log, or arcsine).

   • Retest. If still unequal → use non-parametric test.

6. What ANOVA Actually Does

Conceptual Breakdown

• ANOVA compares variation between group means vs variation within groups (residuals).

Example:

• Two treatments:

  • Group A mean = 10

  • Group B mean = 15

• If little variation within groups → differences between groups are likely real.

• If large variation within groups → may not be a true difference.

ANOVA calculates an F statistic:

F=Mean Square (factor)Mean Square (residual)F = \frac{\text{Mean Square (factor)}}{\text{Mean Square (residual)}}F=Mean Square (residual)Mean Square (factor)​

• High F → greater difference between means relative to internal variation → significant result.

Interpreting Output

ANOVA table includes:

• Significant p (<0.05) → factor affects response.

• If more than 2 levels, use post-hoc tests (e.g. Tukey’s test) to identify which groups differ.

7. Two-Way (Multi-Factor) ANOVA

Example

• Factors:

  • Shore height (low, mid, high)

  • Wave exposure (sheltered vs exposed)

• Response: Limpet density (number/m²)

Analysis

• Tests:

  1. Effect of shore height

  2. Effect of wave exposure

  3. Interaction effect (does effect of exposure depend on height?)

Interpretation:

• If an interaction is significant, it means:

  The effect of one variable depends on the other.

  e.g. Wave exposure affects limpet density only on the low shore, not mid/high.

Always report interactions first in your results section.

Example phrasing for report:

8. Continuous Relationships – Correlation & Regression

When to Use

Instead of comparing categories, sometimes you study relationships between continuous variables.

Examples:

• % algal cover vs limpet density

• Limpet size vs detachment strength

• Rock pool size vs algal diversity

• Seaweed length vs bladder number

Choosing Between Correlation and Regression

In regression:

• X = independent/predictor variable

• Y = dependent/response variable

Example:

Regression Output

Report:

1. Significance (p-value) – tests if slope ≠ 0

2. Equation (e.g. Y = 3.2X + 5)

3. R² value – proportion of variation in Y explained by X (0–1 scale)

Rules:

• Only report regression equation if significant.

• Higher R² → stronger predictive relationship.

9. Multi-Factor Regressions (Advanced Concepts)

ANCOVA (Analysis of Covariance)

• Tests if two regression lines differ significantly in slope or intercept.

• Used when comparing relationships across categories.

Example:

• Independent variable: pool size (continuous)

• Covariate/factor: exposure (categorical)

• Response: grazer density

Simplified approach for coursework:
Plot both regressions (e.g. exposed vs sheltered) and compare qualitatively — no formal ANCOVA test required.

10. Practical & Reporting Guidance

Design Advice

• Choose two predictor variables where possible for a richer analysis.

• Plan for ANOVA or regression during survey design.

• Ensure units are always labelled on axes (e.g. “Density (ind/m²)”).

Interpretation

• Always check for interactions in multi-factor designs.

• Avoid overcomplicating analysis — aim for clarity and coherence.

Key Phrases to Use

• “Significant effect of [factor] on [response variable] (ANOVA, p<0.05).”

• “No significant interaction between [factors] (Two-Way ANOVA, p>0.05).”

• “Regression between [X] and [Y] was significant (p<0.05, R²=0.62).”

11. Summary Table

Lecturer’s Key Advice

• Focus on designing good surveys, not on complex maths.

• Always think about your analysis plan before collecting data.

• Keep methods simple but robust.

• Avoid pseudoreplication and label all axes with units.

• Two-factor designs and visual interpretation of interactions earn stronger marks.