Comparing means of multiple groups

Lecture 4

Iain R. Moodie

BIOB11 - Experimental design and analysis for biologists

Department of Biology, Lund University

2025-04-01

Comparing means between groups

Two groups

Comparing means between groups

Three or more groups

Comparing means between groups

Why not just do pairwise comparisons of means?

Comparing means between groups

Why not just do pairwise comparisons of means?

Comparing means between groups

Analysis of Variance (ANOVA)

Null hypothesis
- The means of all groups are the same
- Mean A == Mean B == Mean C
Alternative hypothesis
- The means of at least one group differ from the global mean

F-statistic

The ratio of variances

\[ F = \frac{\text{Variance 1}}{\text{Variance 2}} \]

Can (in general) be used to test if the variances are equal between sources of variance
- \(F=1\) if variances are equal
- \(F\) is very low if variance 1 < variance 2
- \(F\) is very high if variance 1 > variance 2

F-statistic

The ratio of between-group variance to within-group variance

\[ F = \frac{\text{Between-group variance}}{\text{Within-group variance}} \]

\[ F = \frac{\text{Between-treatment variance}}{\text{Within-treatment variance}} \]

\[ F = \frac{\text{Explained variance}}{\text{Unexplained variance}} \]

\[ F = \frac{\text{Explained variance}}{\text{Residual variance}} \]

\[ F = \frac{\text{Explained}}{\text{Error}} \]

F-statistic

The ratio of between-group variance to within-group variance

Comparing means between groups

Analysis of Variance (ANOVA)

\[ F = \frac{\text{Mean variance between-group}}{\text{Mean variance within-group}} \]

If \(F\) is very big, more likely that means are different
How do we calculate it?
- Whiteboard
- Using R

Comparing means between groups

Analysis of Variance (ANOVA)

Comparing means between groups

Analysis of Variance (ANOVA)

Calculate observed \(F\) statistic:

observed_f <-
  penguins_modified |>
  specify(response = bill_length_mm, explanatory = species) |>
  calculate(stat = "F")

observed_f

Response: bill_length_mm (numeric)
Explanatory: species (factor)
# A tibble: 1 × 1
   stat
  <dbl>
1  7.60

Comparing means between groups

Analysis of Variance (ANOVA)

Generate null \(F\) distribution:

null_f <-
  penguins_modified |>
  specify(response = bill_length_mm, explanatory = species) |>
  hypothesize(null = "independence") |>
  generate(reps = 5000, type = "permute") |>
  calculate(stat = "F")

visualise(null_f) +
  labs(x = "F statistic")

Comparing means between groups

Analysis of Variance (ANOVA)

Generate null \(F\) distribution:

Comparing means between groups

Analysis of Variance (ANOVA)

Compare observed \(F\) against null \(F\) distribution:

visualise(null_f) +
  shade_p_value(observed_f, direction = "greater") +
  labs(x = "F statistic")

Comparing means between groups

Analysis of Variance (ANOVA)

Compare observed \(F\) against null \(F\) distribution:

null_f |>
  get_p_value(obs_stat = observed_f, direction = "greater")

# A tibble: 1 × 1
  p_value
    <dbl>
1  0.0004