Complete factorial designs

# Complete factorial designs

**Session 5**

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# Outline

.box-4.medium.sp-after-half[Factorial designs and interactions]

.box-5.medium.sp-after-half[Tests for two-way ANOVA]

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name: factorial-interaction
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# Factorial designs and interactions

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# Complete factorial designs?

.box-inv-4.sp-after[
.large[**Factorial design**]<br> study with multiple factors (subgroups)
]

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# Motivating example

.box-inv-4.align-left[**experimental factor 1**:<br> ending (happy or sad)]

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# Setup of design

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    <tr><td class="gt_row gt_left">complicated</td>
<td class="gt_row gt_right" style="background-color: rgba(255,179,186,0.8); color: #000000;"><img src="data:image/png;base64,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" style="height:80px;"></td>
<td class="gt_row gt_right" style="background-color: rgba(255,179,186,0.8); color: #000000;"><img src="data:image/png;base64,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" style="height:80px;"></td></tr>
    <tr><td class="gt_row gt_left">average</td>
<td class="gt_row gt_right" style="background-color: rgba(255,223,186,0.8); color: #000000;"><img src="data:image/png;base64,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" style="height:80px;"></td>
<td class="gt_row gt_right" style="background-color: rgba(255,223,186,0.8); color: #000000;"><img src="data:image/png;base64,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" style="height:80px;"></td></tr>
    <tr><td class="gt_row gt_left">easy</td>
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  </tbody>
  
  
</table>
</div>

???

These factors are crossed, meaning that you can have participants in each subconditions.

---

# Efficiency of factorial design

.box-inv-4.sp-after.medium[Cast problem<br>as a series of one-way ANOVA <br> vs simultaneous estimation]

.box-4.medium.sp-before[Factorial designs requires<br> **fewer overall observations**]

.box-4.medium.sp-before[Can study **interactions**]

???

To study each interaction (complexity, story book ending) we would need to make three group for each comparison in rows, and one in each column. So a total of 3 one-way ANOVA each with  2 groups and 2 one-way anova with 3 groups. The two-way ANOVA will lead to 6 groups instead of 12.

---

# Interaction

.box-inv-4.sp-after.medium[
**Definition**: when the effect of one factor<br> depends on the levels of another factor.
]

---

# Interaction or profile plot
.box-inv-4.large.sp-after[Graphical display: <br>plot sample mean per category]

.box-4.sp-after-half[with uncertainty measure<br>(1 std. error for mean<br>confidence interval, etc.)]

---

# Interaction plots and parallel lines

---

# Interaction plots for 2 by 2 designs

---

# Cell means for 2 by 2 designs

???

Line graph for example patterns for means for each of the possible kinds of general outcomes in a 2 by 2 design. Illustration adapted from Figure 10.2 of Crump, Navarro and Suzuki (2019) by Matthew Crump (CC BY-SA 4.0 license)

---
 
# Example 1 : loans versus credit

.pull-left[
[Sharma, Tully, and Cryder (2021)](https://doi.org/10.1177/0022243721993816) Supplementary study 5 consists of a `$2 \times 2$` between-subject ANOVA with factors

- debt type (`debttype`), either "loan" or "credit"
- `purchase` type, either `discretionary` or not (`need`)

]

<img src="05-slides_files/figure-html/unnamed-chunk-5-1.png" width="100%" style="display: block; margin: auto;" />
No evidence of interaction

]
---

# Example 2 - psychological distance

.pull-left[
[Maglio and Polman (2014)](https://doi.org/10.1177/0956797614530571) Study 1 uses a `$4 \times 2$` between-subject ANOVA with factors

- subway `station`, one of Spadina, St. George, Bloor-Yonge and Sherbourne
- `direction` of travel, either east or west

]

<img src="05-slides_files/figure-html/unnamed-chunk-6-1.png" width="100%" style="display: block; margin: auto;" />
Clear evidence of interaction (symmetry?)

]

---

layout: false
name: formulation
class: center middle section-title section-title-5 animated fadeIn

# Tests for two-way ANOVA

---

---
# Analysis of variance = regression

An analysis of variance model is simply a **linear regression** with categorical covariate(s).

- Typically, the parametrization is chosen so that parameters reflect differences to the global mean (sum-to-zero parametrization).
- The full model includes interactions between all combinations of factors
   - one average for each subcategory
   - one-way ANOVA!

---

# Formulation of the two-way ANOVA

Two factors: `$A$` (complexity) and `$B$` (ending) with `$n_a=3$` and `$n_b=2$` levels, and their interaction.

Write the average response `$Y_{ijr}$` of the `$r$`th measurement in group `$(a_i, b_j)$` as
`\begin{align*}
\underset{\text{average response}\vphantom{b}}{\mathsf{E}(Y_{ijr})} = \underset{\text{subgroup mean}}{\mu_{ij}}
\end{align*}`
where `$Y_{ijr}$` are independent observations with a common std. deviation `$\sigma$`.

- We estimate `$\mu_{ij}$` by the sample mean of the subgroup `$(i,j)$`, say `$\widehat{\mu}_{ij}$`.
- The fitted values are `$\widehat{y}_{ijr} = \widehat{\mu}_{ij}$`.

---

# One average for each subgroup

| `$\qquad B$` `ending`<br> `$A$` `complexity` `$\qquad$` | `$b_1$` (`happy`) | `$b_2$` (`sad`)| *row mean* |
|------------|:----------:|:-----:|:-----:|
| `$a_1$` (`complicated`) | `$\mu_{11}$` | `$\mu_{12}$` | `$\mu_{1.}$` |
|  `$a_2$` (`average`)  | `$\mu_{21}$` | `$\mu_{22}$` | `$\mu_{2.}$` |
| `$a_3$` (`easy`) |  `$\mu_{31}$` | `$\mu_{32}$` | `$\mu_{3.}$` |
|*column mean* | `$\mu_{.1}$` | `$\mu_{.2}$` | `$\mu$` |

---

# Row, column and overall average

- Mean of `$A_i$` (average of row `$i$`): 
`$$\mu_{i.} = \frac{\mu_{i1} + \cdots + \mu_{in_b}}{n_b}$$`

- Mean of `$B_j$` (average of column `$j$`):
`$$\mu_{.j} = \frac{\mu_{1j} + \cdots + \mu_{n_aj}}{n_a}$$`

]
.pull-right[
- Overall average:
`$$\mu = \frac{\sum_{i=1}^{n_a} \sum_{j=1}^{n_b} \mu_{ij}}{n_an_b}$$`

.small[
- Row, column and overall averages are **equiweighted** combinations of the cell means `$\mu_{ij}$`.
- Estimates are obtained by replacing `$\mu_{ij}$` in formulas by subgroup sample mean.
]
]

---

# Vocabulary of effects

- .color-5[**simple effects**]: difference between levels of one in a fixed combination of others (change in difficulty for happy ending)
- .color-5[**main effects**]: differences relative to average for each condition of a factor (happy vs sad ending)
- .color-5[**interaction effects**]: when simple effects differ depending on levels of another factor

---

# Main effects

Obtained by *marginalization*, i.e., averaging over the other dimension.

Main effects are not of interest if there is an interaction.

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  border-bottom-style: hidden;
}

#xdduhtxcdz .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  text-align: left;
}

#xdduhtxcdz .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#xdduhtxcdz .gt_from_md > :first-child {
  margin-top: 0;
}

#xdduhtxcdz .gt_from_md > :last-child {
  margin-bottom: 0;
}

#xdduhtxcdz .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#xdduhtxcdz .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#xdduhtxcdz .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#xdduhtxcdz .gt_row_group_first td {
  border-top-width: 2px;
}

#xdduhtxcdz .gt_row_group_first th {
  border-top-width: 2px;
}

#xdduhtxcdz .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#xdduhtxcdz .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#xdduhtxcdz .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#xdduhtxcdz .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#xdduhtxcdz .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#xdduhtxcdz .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#xdduhtxcdz .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#xdduhtxcdz .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#xdduhtxcdz .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#xdduhtxcdz .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#xdduhtxcdz .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#xdduhtxcdz .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#xdduhtxcdz .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#xdduhtxcdz .gt_left {
  text-align: left;
}

#xdduhtxcdz .gt_center {
  text-align: center;
}

#xdduhtxcdz .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#xdduhtxcdz .gt_font_normal {
  font-weight: normal;
}

#xdduhtxcdz .gt_font_bold {
  font-weight: bold;
}

#xdduhtxcdz .gt_font_italic {
  font-style: italic;
}

#xdduhtxcdz .gt_super {
  font-size: 65%;
}

#xdduhtxcdz .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#xdduhtxcdz .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#xdduhtxcdz .gt_indent_1 {
  text-indent: 5px;
}

#xdduhtxcdz .gt_indent_2 {
  text-indent: 10px;
}

#xdduhtxcdz .gt_indent_3 {
  text-indent: 15px;
}

#xdduhtxcdz .gt_indent_4 {
  text-indent: 20px;
}

#xdduhtxcdz .gt_indent_5 {
  text-indent: 25px;
}

#xdduhtxcdz .katex-display {
  display: inline-flex !important;
  margin-bottom: 0.75em !important;
}

#xdduhtxcdz div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
  height: 0px !important;
}
</style>
<table class="gt_table" style="table-layout: fixed;" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <colgroup>
    <col/>
    <col style="width:120px;"/>
  </colgroup>
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="complexity">complexity</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="row means">row means</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="complexity" class="gt_row gt_left">complicated</td>
<td headers="response" class="gt_row gt_left" style="background-color: rgba(255,179,186,0.8); color: #000000;">$$\mu_{1.}$$</td></tr>
    <tr><td headers="complexity" class="gt_row gt_left">average</td>
<td headers="response" class="gt_row gt_left" style="background-color: rgba(255,223,186,0.8); color: #000000;">$$\mu_{2.}$$</td></tr>
    <tr><td headers="complexity" class="gt_row gt_left">easy</td>
<td headers="response" class="gt_row gt_left" style="background-color: rgba(186,225,255,0.8); color: #000000;">$$\mu_{3.}$$</td></tr>
  </tbody>
  
  
</table>
</div>

]

---

# Simple effects

<div id="ebohtplcwi" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>#ebohtplcwi table {
  font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji';
  -webkit-font-smoothing: antialiased;
  -moz-osx-font-smoothing: grayscale;
}

#ebohtplcwi thead, #ebohtplcwi tbody, #ebohtplcwi tfoot, #ebohtplcwi tr, #ebohtplcwi td, #ebohtplcwi th {
  border-style: none;
}

#ebohtplcwi p {
  margin: 0;
  padding: 0;
}

#ebohtplcwi .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 40px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #A8A8A8;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#ebohtplcwi .gt_caption {
  padding-top: 4px;
  padding-bottom: 4px;
}

#ebohtplcwi .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#ebohtplcwi .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 3px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#ebohtplcwi .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#ebohtplcwi .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#ebohtplcwi .gt_col_headings {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#ebohtplcwi .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 40px;
  font-weight: normal;
  text-transform: inherit;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#ebohtplcwi .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 40px;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#ebohtplcwi .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#ebohtplcwi .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#ebohtplcwi .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#ebohtplcwi .gt_spanner_row {
  border-bottom-style: hidden;
}

#ebohtplcwi .gt_group_heading {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  text-align: left;
}

#ebohtplcwi .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#ebohtplcwi .gt_from_md > :first-child {
  margin-top: 0;
}

#ebohtplcwi .gt_from_md > :last-child {
  margin-bottom: 0;
}

#ebohtplcwi .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#ebohtplcwi .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
}

#ebohtplcwi .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
  padding-right: 5px;
  vertical-align: top;
}

#ebohtplcwi .gt_row_group_first td {
  border-top-width: 2px;
}

#ebohtplcwi .gt_row_group_first th {
  border-top-width: 2px;
}

#ebohtplcwi .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#ebohtplcwi .gt_first_summary_row {
  border-top-style: solid;
  border-top-color: #D3D3D3;
}

#ebohtplcwi .gt_first_summary_row.thick {
  border-top-width: 2px;
}

#ebohtplcwi .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#ebohtplcwi .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#ebohtplcwi .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#ebohtplcwi .gt_last_grand_summary_row_top {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: double;
  border-bottom-width: 6px;
  border-bottom-color: #D3D3D3;
}

#ebohtplcwi .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#ebohtplcwi .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#ebohtplcwi .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#ebohtplcwi .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#ebohtplcwi .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#ebohtplcwi .gt_sourcenote {
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
  padding-left: 5px;
  padding-right: 5px;
}

#ebohtplcwi .gt_left {
  text-align: left;
}

#ebohtplcwi .gt_center {
  text-align: center;
}

#ebohtplcwi .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#ebohtplcwi .gt_font_normal {
  font-weight: normal;
}

#ebohtplcwi .gt_font_bold {
  font-weight: bold;
}

#ebohtplcwi .gt_font_italic {
  font-style: italic;
}

#ebohtplcwi .gt_super {
  font-size: 65%;
}

#ebohtplcwi .gt_footnote_marks {
  font-size: 75%;
  vertical-align: 0.4em;
  position: initial;
}

#ebohtplcwi .gt_asterisk {
  font-size: 100%;
  vertical-align: 0;
}

#ebohtplcwi .gt_indent_1 {
  text-indent: 5px;
}

#ebohtplcwi .gt_indent_2 {
  text-indent: 10px;
}

#ebohtplcwi .gt_indent_3 {
  text-indent: 15px;
}

#ebohtplcwi .gt_indent_4 {
  text-indent: 20px;
}

#ebohtplcwi .gt_indent_5 {
  text-indent: 25px;
}

#ebohtplcwi .katex-display {
  display: inline-flex !important;
  margin-bottom: 0.75em !important;
}

#ebohtplcwi div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
  height: 0px !important;
}
</style>
<table class="gt_table" style="table-layout: fixed;" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <colgroup>
    <col/>
    <col style="width:120px;"/>
    <col style="width:120px;"/>
  </colgroup>
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id=" "> </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="happy">happy</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="sad">sad</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers="X" class="gt_row gt_left">means (easy)</td>
<td headers="happy" class="gt_row gt_left" style="background-color: rgba(186,225,255,0.8); color: #000000;">$$\mu_{13}$$</td>
<td headers="sad" class="gt_row gt_left" style="background-color: rgba(186,225,255,0.8); color: #000000;">$$\mu_{23}$$</td></tr>
  </tbody>
  
  
</table>
</div>

]

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  <colgroup>
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    <col style="width:120px;"/>
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  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="complexity">complexity</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="mean (happy)">mean (happy)</th>
    </tr>
  </thead>
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    <tr><td headers="complexity" class="gt_row gt_left">complicated</td>
<td headers="response" class="gt_row gt_left" style="background-color: rgba(255,179,186,0.8); color: #000000;">$$\mu_{11}$$</td></tr>
    <tr><td headers="complexity" class="gt_row gt_left">average</td>
<td headers="response" class="gt_row gt_left" style="background-color: rgba(255,223,186,0.8); color: #000000;">$$\mu_{21}$$</td></tr>
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<td headers="response" class="gt_row gt_left" style="background-color: rgba(186,225,255,0.8); color: #000000;">$$\mu_{31}$$</td></tr>
  </tbody>
  
  
</table>
</div>

]

---

# Contrasts

We collapse categories to obtain a one-way ANOVA with categories `$A$` (complexity) and `$B$` (ending).

Q: How would you write the weights for contrasts for testing the

- main effect of `$A$`: complicated vs average, or complicated vs easy.
- main effect of `$B$`: happy vs sad.
- interaction `$A$` and `$B$`: difference between complicated and average, for happy versus sad?

The order of the categories is `$(a_1, b_1)$`, `$(a_1, b_2)$`, `$\ldots$`, `$(a_3, b_2)$`.

---

# Contrasts

Suppose the order of the coefficients is factor `$A$` (complexity, 3 levels, complicated/average/easy) and factor `$B$` (ending, 2 levels, happy/sad).

| test | `$\mu_{11}$` | `$\mu_{12}$` | `$\mu_{21}$` | `$\mu_{22}$` | `$\mu_{31}$` | `$\mu_{32}$` |
|:--|--:|--:|--:|--:|--:|--:|
| main effect `$A$` (complicated vs average) |  `$1$` | `$1$` | `$-1$` |  `$-1$` | `$0$` | `$0$` |
| main effect `$A$` (complicated vs easy) |    `$1$` |  `$1$` | `$0$` | `$0$` | `$-1$` |  `$-1$` |
| main effect `$B$` (happy vs sad) |    `$1$` |  `$-1$` | `$1$` |  `$-1$` | `$1$` | `$-1$` |
| interaction `$AB$` (comp. vs av, happy vs sad) | `$1$` | `$-1$` | `$-1$` | `$1$` | `$0$` | `$0$` |
| interaction `$AB$` (comp. vs easy, happy vs sad) |  `$1$` | `$-1$` | `$0$` | `$0$` | `$-1$` | `$1$` |

---

# Global hypothesis tests

.box-inv-5[
Main effect of factor `$A$`
]

`$\mathscr{H}_0$`: `$\mu_{1.} = \cdots = \mu_{n_a.}$` vs `$\mathscr{H}_a$`: at least two marginal means of `$A$` are different
.box-inv-5[
Main effect of factor `$B$`
]

`$\mathscr{H}_0$`: `$\mu_{.1} = \cdots = \mu_{.n_b}$` vs `$\mathscr{H}_a$`: at least two marginal means of `$B$` are different
.box-inv-5[
Interaction
]

`$\mathscr{H}_0$`: `$\mu_{ij} = \mu_{i \cdot} + \mu_{\cdot j}$` (sum of main effects) vs `$\mathscr{H}_a$`: effect is not a combination of row/column effect.

---

# Comparing nested models

Rather than present the specifics of ANOVA, we consider a general hypothesis testing framework which is more widely applicable.

We compare two competing models, `$\mathbb{M}_a$` and `$\mathbb{M}_0$`.

- the **alternative** or full model `$\mathbb{M}_a$` under the alternative `$\mathscr{H}_a$` with `$p$` parameters for the mean
- the simpler **null** model `$\mathbb{M}_0$` under the null `$\mathscr{H}_0$`, which imposes `$\nu$` restrictions on the full model

???

The same holds for analysis of deviance for generalized linear models. The latter use likelihood ratio tests (which are equivalent to F-tests for linear models), with a `$\chi^2_{\nu}$` null distribution.

---

# Intuition behind `$F$`-test for ANOVA

The **residual sum of squares** measures how much variability is leftover,
`$$\mathsf{RSS}_a = \sum_{i=1}^n \left(y_i - \widehat{y}_i^{\mathbb{M}_a}\right)^2$$`
where `$\widehat{y}_i$` is the estimated mean under model `$\mathbb{M}_a$` for the observation `$y_i$`.

The more complex fits better (it is necessarily more flexible), but requires estimation of more parameters.

- We wish to assess the improvement that would occur by chance, if the null model was correct.

---

# Testing linear restrictions in linear models

If the alternative model has `$p$` parameters for the mean, and we impose `$\nu$` linear restrictions under the null hypothesis to the model estimated based on `$n$` independent observations, the test statistic is

`\begin{align*}
F = \frac{(\mathsf{RSS}_0 - \mathsf{RSS}_a)/\nu}{\mathsf{RSS}_a/(n-p)}
\end{align*}`

- The numerator is the difference in residuals sum of squares, denoted `$\mathsf{RSS}$`, from models fitted under `$\mathscr{H}_0$` and `$\mathscr{H}_a$`, divided by degrees of freedom `$\nu$`.
- The denominator is an estimator of the variance, obtained under `$\mathscr{H}_a$` (termed mean squared error of residuals)
- The benchmark for tests in linear models is Fisher's `$\mathsf{F}(\nu, n-p)$`.

---

# Analysis of variance table

.small[
  
| term | degrees of freedom | mean square | `$F$` | 
|------|--------|------|--------|
| `$A$`  | `$n_a-1$`   | `$\mathsf{MS}_{A}=\mathsf{SS}_A/(n_a-1)$` | `$\mathsf{MS}_{A}/\mathsf{MS}_{\text{res}}$` |
| `$B$`  | `$n_b-1$`   | `$\mathsf{MS}_{B}=\mathsf{SS}_B/(n_b-1)$` | `$\mathsf{MS}_{B}/\mathsf{MS}_{\text{res}}$` |
| `$AB$` | `$(n_a-1)(n_b-1)$` | `$\mathsf{MS}_{AB}=\mathsf{SS}_{AB}/\{(n_a-1)(n_b-1)\}$` | `$\mathsf{MS}_{AB}/\mathsf{MS}_{\text{res}}$` |
| residuals | `$n-n_an_b$` | `$\mathsf{MS}_{\text{resid}}=\mathsf{RSS}_{a}/ (n-n_an_b)$` | |
| total | `$n-1$` | | 
    
]

Read the table backward (starting with the interaction).
- If there is a significant interaction, the main effects are **not** of interest and potentially misleading.

---

# Intuition behind degrees of freedom
.small[

The model always includes an overall average `$\mu$`.  There are

- `$n_a-1$` free row means since `$n_a\mu = \mu_{1.} + \cdots + \mu_{n_a.}$` 
- `$n_b-1$` free column means as `$n_b\mu = \mu_{.1} + \cdots + \mu_{.n_b}$`
- `$n_an_b-(n_a-1)-(n_b-1)-1$` interaction terms

| `$\qquad B$` `ending`<br> `$A$` `complexity` `$\qquad$` | `$b_1$` (`happy`) | `$b_2$` (`sad`)| *row mean* |
|------------|:----------:|:-----:|:-----:|
| `$a_1$` (`complicated`) | `$\mu_{11}$` | `$\mathsf{X}$` | `$\mu_{1.}$` |
|  `$a_2$` (`average`)  | `$\mu_{21}$` | `$\mathsf{X}$`  | `$\mu_{2.}$` |
| `$a_3$` (`easy`) |  `$\mathsf{X}$`  | `$\mathsf{X}$`  | `$\mathsf{X}$` |
|*column mean* | `$\mu_{.1}$` | `$\mathsf{X}$` | `$\mu$` |
 
Terms with `$\mathsf{X}$` are fully determined by row/column/total averages

]

---

# Example 1

The interaction plot suggested that the two-way interaction wasn't significant. The `$F$` test confirms this.

There is a significant main effect of both `purchase` and `debttype`.

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> SS </th>
   <th style="text-align:right;"> df </th>
   <th style="text-align:right;"> F stat </th>
   <th style="text-align:left;"> p-value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> purchase </td>
   <td style="text-align:right;"> 752.3 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 98.21 </td>
   <td style="text-align:left;"> &lt; .001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> debttype </td>
   <td style="text-align:right;"> 92.2 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 12.04 </td>
   <td style="text-align:left;"> &lt; .001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> purchase:debttype </td>
   <td style="text-align:right;"> 13.7 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1.79 </td>
   <td style="text-align:left;"> .182 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 11467.4 </td>
   <td style="text-align:right;"> 1497 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:left;">  </td>
  </tr>
</tbody>
</table>

---

# Example 2

There is a significant interaction between `station` and `direction`, so follow-up by looking at simple effects or contrasts.

The tests for the main effects are not of interest! Disregard other entries of the ANOVA table

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> SS </th>
   <th style="text-align:right;"> df </th>
   <th style="text-align:right;"> F stat </th>
   <th style="text-align:left;"> p-value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> station </td>
   <td style="text-align:right;"> 75.2 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 23.35 </td>
   <td style="text-align:left;"> &lt; .001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> direction </td>
   <td style="text-align:right;"> 0.4 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0.38 </td>
   <td style="text-align:left;"> .541 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> station:direction </td>
   <td style="text-align:right;"> 52.4 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 16.28 </td>
   <td style="text-align:left;"> &lt; .001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 208.2 </td>
   <td style="text-align:right;"> 194 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:left;">  </td>
  </tr>
</tbody>
</table>

---

# Main effects for Example 1

We consider differences between debt type labels.

Participants are more likely to consider the offer if it is branded as `credit` than loan. The difference is roughly 0.5 (on a Likert scale from 1 to 9).
.small[

```
## $emmeans
##  debttype emmean    SE   df lower.CL upper.CL
##  credit     5.12 0.101 1497     4.93     5.32
##  loan       4.63 0.101 1497     4.43     4.83
## 
## Results are averaged over the levels of: purchase 
## Confidence level used: 0.95 
## 
## $contrasts
##  contrast      estimate    SE   df t.ratio p.value
##  credit - loan    0.496 0.143 1497   3.469  0.0005
## 
## Results are averaged over the levels of: purchase
```
]
---

# Toronto subway station

<div class="figure" style="text-align: center">
<img src="img/05/Toronto_subway.png" alt="Simplified depiction of the Toronto metro stations used in the experiment, based on work by Craftwerker on Wikipedia, distributed under CC-BY-SA 4.0." width="70%" />
<p class="caption">Simplified depiction of the Toronto metro stations used in the experiment, based on work by Craftwerker on Wikipedia, distributed under CC-BY-SA 4.0.</p>
</div>

---
# Reparametrization for Example 2

Set `stdist` as `$-2$`, `$-1$`, `$+1$`, `$+2$` to indicate station distance, with negative signs indicating stations in opposite direction of travel

The ANOVA table for the reparametrized models shows no evidence against the null of symmetry (interaction).

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> SS </th>
   <th style="text-align:right;"> df </th>
   <th style="text-align:right;"> F stat </th>
   <th style="text-align:left;"> p-value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> stdist </td>
   <td style="text-align:right;"> 121.9 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 37.86 </td>
   <td style="text-align:left;"> &lt; .001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> direction </td>
   <td style="text-align:right;"> 0.4 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0.35 </td>
   <td style="text-align:left;"> .554 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> stdist:direction </td>
   <td style="text-align:right;"> 5.7 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 1.77 </td>
   <td style="text-align:left;"> .154 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 208.2 </td>
   <td style="text-align:right;"> 194 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:left;">  </td>
  </tr>
</tbody>
</table>

---
# Interaction plot for reformated data

<img src="05-slides_files/figure-html/unnamed-chunk-17-1.png" width="70%" style="display: block; margin: auto;" />
---

# Custom contrasts for Example 2

We are interested in testing the perception of distance, by looking at `$\mathscr{H}_0: \mu_{-1} = \mu_{+1}, \mu_{-2} = \mu_{+2}$`.

``` r
mod3 <- lm(distance ~ stdist * direction, data = MP14_S1) 
(emm <- emmeans(mod3, specs = "stdist"))
  # order is -2, -1, 1, 2
contrasts <- emm |> contrast(
  list("two dist" = c(-1, 0, 0, 1), 
        "one dist" = c(0, -1, 1, 0)))
contrasts # print pairwise contrasts
test(contrasts, joint = TRUE)
```
---

# Estimated marginal means and contrasts

```
##  stdist emmean    SE  df lower.CL upper.CL
##  -2       3.83 0.145 194     3.54     4.11
##  -1       2.48 0.144 194     2.20     2.76
##  +1       1.62 0.150 194     1.33     1.92
##  +2       2.70 0.145 194     2.42     2.99
## 
## Results are averaged over the levels of: direction 
## Confidence level used: 0.95
```

```
##  contrast estimate    SE  df t.ratio p.value
##  two dist   -1.122 0.205 194  -5.470  <.0001
##  one dist   -0.856 0.207 194  -4.129  0.0001
## 
## Results are averaged over the levels of: direction
```

```
##  df1 df2 F.ratio p.value
##    2 194  23.485  <.0001
```

]