| 🍌 Banana |
🍬 Candy |
|---|---|
| 9 | 3 |
| 11 | 5 |
| 13 | 4 |
| \(M = 11\) | \(M = 4\) |
15|ANOVA part 1
Overview
Intro to ANOVA
Comparing groups: \(t\) statistic
\(t = \dfrac{ \textrm{difference between groups}} {\textrm{difference expected due to chance}}\)
\[\begin{align} t &= \dfrac{(M_1-M_2)-(\mu_1-\mu_2)}{s_{(M_1-M_2)}} \\ &= \dfrac{11 - 4}{1.29} \\ &= 5.42 \end{align}\]
Comparing groups: \(t\) statistic
Manipulation
|
|
|---|---|
| 🍌 Banana |
🍬 Candy |
| 9 | 3 |
| 11 | 5 |
| 13 | 4 |
| \(M = 11\) | \(M = 4\) |
\(t = 5.42\)
\(t = \dfrac{\text{difference between groups}}{\text{difference expected due to chance}}\)
is analogous to…
\(\dfrac{treatment \cdot chance}{chance}\)
Comparing groups: Variances
Manipulation
|
|
|---|---|
| 🍌 Banana |
🍬 Candy |
| 9 | 3 |
| 11 | 5 |
| 13 | 4 |
| \(M = 11\) | \(M = 4\) |
\(t = 5.42\)
Total
variability in data
Variability
between groups
- Chance
- Treatment effect
Variability
within groups
- Chance
\(\dfrac{\text{variability between groups}}{\text{variability within groups}}\)
is analogous to…
\(\dfrac{treatment \cdot chance}{chance}\)
Total variance
Manipulation
|
|
|---|---|
| 🍌 Banana |
🍬 Candy |
| 9 | 3 |
| 11 | 5 |
| 13 | 4 |
| \(M = 11\) | \(M = 4\) |
\(t = 5.42\)
- Total variance
- \(SS_{total}\)
- Find sum of squared deviations of all scores (ignoring different groups) from grand mean (mean of all scores)
- \(df_{total} = N - 1\)
\[\begin{align} \text{Variance}_{total} &= \dfrac{SS_{total}}{df_{total}} \\ &= \dfrac{83.5}{5} \\ &= 16.7 \end{align}\]
Within groups variance
Manipulation
|
|
|---|---|
| 🍌 Banana |
🍬 Candy |
| 9 | 3 |
| 11 | 5 |
| 13 | 4 |
| \(M = 11\) | \(M = 4\) |
\(t = 5.42\)
\(\text{Variance}_{total} = 16.7\)
- Variance within groups
- \(SS_{within} = \Sigma SS_{each \ treatment}\)
- Find \(SS\) for each individual group from respective group mean
- Then add \(SS\) values
- \(df_{within} = \Sigma df_{each \ treatment}\)
\[\begin{align} \text{Variance}_{within} &= \dfrac{SS_{within}}{df_{within}} \\ &= \dfrac{8 + 2}{2 + 2} \\ &= 2.5 \end{align}\]
Between groups variance
Manipulation
|
|
|---|---|
| 🍌 Banana |
🍬 Candy |
| 9 | 3 |
| 11 | 5 |
| 13 | 4 |
| \(M = 11\) | \(M = 4\) |
\(t = 5.42\)
\(\text{Variance}_{total} = 16.7\)
\(\text{Variance}_{within} = 2.5\)
- Total variability in data = Var between groups + Var within groups
- Therefore…
\(SS_{between} = SS_{total} – SS_{within}\)
\(df_{between} = df_{total} – df_{within}\)
\[\begin{align} \text{Variance}_{between} &= \dfrac{SS_{between}}{df_{between}} \\ &= \dfrac{83.5 - 10}{5-4} \\ &= 73.5 \end{align}\]
Ratio of variances
Manipulation
|
|
|---|---|
| 🍌 Banana |
🍬 Candy |
| 9 | 3 |
| 11 | 5 |
| 13 | 4 |
| \(M = 11\) | \(M = 4\) |
\(t = 5.42\)
\(\text{Variance}_{total} = 16.7\)
\(\text{Variance}_{within} = 2.5\) \(\text{Variance}_{between} = 73.5\)
Total
variability in data
Variability
between groups
- Chance
- Treatment effect
Variability
within groups
- Chance
\(\dfrac{\text{variance between groups}}{\text{variance within groups}}\)
\(\dfrac{73.5}{2.5} = 29.4\)
The \(F\) ratio
Manipulation
|
|
|---|---|
| 🍌 Banana |
🍬 Candy |
| 9 | 3 |
| 11 | 5 |
| 13 | 4 |
| \(M = 11\) | \(M = 4\) |
\(t = 5.42\)
\(F = 29.4\)
Total
variability in data
Variability
between groups
- Chance
- Treatment effect
Variability
within groups
- Chance
\(\dfrac{\text{variance between groups}}{\text{variance within groups}}\)
is analogous to…
\(\dfrac{treatment \cdot chance}{chance}\)
Uses of ANOVA
More complicated design
Manipulation
|
||
|---|---|---|
| 🍌 Banana |
🍬 Candy |
😐 Control |
| 9 | 3 | 5 |
| 11 | 5 | 6 |
| 13 | 4 | 7 |
| \(M = 11\) | \(M = 4\) | \(M = 6\) |
- Differences among 3 means
- Did banana improve scores? Candy bar harm scores? Both? Neither?
Limitation of \(t\) test
- Can only compare two populations
- \(H_0\): \(\mu_1 - \mu_2 = 0\)
Advantage of ANOVA
- ANOVA is a tool for the general case
- Comparing any number of populations
- \(H_0\): \(\mu_1 = \mu_2 = \mu_3 = \dots = \mu_n\)
Learning checks
- If you calculate both the \(t\) and \(F\) statistics for a two-group, independent-samples design, ____ equals ____ squared. Why?
- True/False
- A \(t\) statistic can be either positive or negative
- An \(F\) statistic can be either positive or negative
- Which statistic, \(t\) or \(F\), is more ‘flexible’, i.e. applicable in a wider range of research contexts?
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