Multiple Regression & Fundamentals of Causal Inference

class: center, middle, inverse, title-slide

.title[
# Multiple Regression & <br><br> Fundamentals of Causal Inference
]
.subtitle[
## Interaction/Conditional effects: <br> <br>When the relation between <span class="math inline"><em>y</em></span> and <span class="math inline"><em>x</em><sub>1</sub></span> <br> depends on <span class="math inline"><em>x</em><sub>2</sub></span>
]
.author[
### Merlin Schaeffer<br> Department of Sociology
]
.date[
### 2025-11-19
]

---

# Goal of empirical sociology

.font130[.center[Use data to .alert[discover patterns], <br> and the social mechanisms that bring them about.]]

---
class: inverse middle
# Today's schedule

1. Application of the day: Xenophobia and education in Denmark and Bulgaria.
  + OLS is additive.
  + Conditional relationships are multiplicative.
  + Interaction terms between a continuous and a categorical variable.

2. Application 2: Do socialism and citizenship rights interact?
  + Interaction terms between two continuous variables.

---
class: inverse

.push-left[
<iframe src='https://www.dw.com/en/denmark-tells-syrian-refugees-to-return-to-damascus/a-57174584' width='700' height='430' frameborder='0' scrolling='yes'></iframe>
]

.push-right[
<iframe src='https://www.dw.com/en/why-do-so-many-refugees-avoid-bulgaria/a-18707897' width='700' height='430' frameborder='0' scrolling='yes'></iframe>
]
---
# Preparations

.panelset[
.panel[.panel-name[Get the ESS data]
.left-column[
Download the "ESS9e03_1.sav" data from Absalon. Place them into the folder for this course and where you have your R-project.
]
.right-column[
.font90[

``` r
pacman::p_load(
  tidyverse, # Data manipulation,
  haven, # Handle labelled data.
  ggplot2, # beautiful figures,
  estimatr, # Regression for weighted data,
  modelsummary) # regression tables with nice layout.

# Read the ESS round 9 data
ESS <- read_spss("../../../assets/ESS9e03_1.sav") %>%
  # Keep only Danes and Bulgarians
  filter(cntry == "DK" | cntry == "BG") %>% 
  mutate( # Define variables as categorical and continuous
    cntry = as_factor(cntry) %>% fct_relevel("Denmark"),
    gndr = as_factor(gndr),
    eduyrs = case_when( # Recode education to sensible levels
      eduyrs < 9 ~ 9,
      eduyrs > 21 ~ 21,
      TRUE ~zap_labels(eduyrs)),
    agea = zap_labels(agea),
    # Outcome: Subtract from maximum to turn the scale around
    imwbcnt = 10 - zap_labels(imwbcnt)) %>%
  # Keep only a minimum set of variables we need today,
  select(idno, cntry, pspwght, gndr, eduyrs, agea, imwbcnt) %>%
  drop_na() # Delete cases with missing values.
```
]]]
.panel[.panel-name[The resulting data]

``` r
ESS
# # A tibble: 3,330 × 7
#     idno cntry    pspwght gndr   eduyrs  agea imwbcnt
#    <dbl> <fct>      <dbl> <fct>   <dbl> <dbl>   <dbl>
#  1     2 Bulgaria   0.933 Female     12    72       9
#  2    27 Bulgaria   0.455 Male       17    63       8
#  3    37 Bulgaria   1.02  Male        9    74       8
#  4    50 Bulgaria   0.845 Female     12    39       7
#  5    70 Bulgaria   0.740 Female     12    70       5
#  6    76 Bulgaria   0.704 Female     12    52       3
#  7   137 Bulgaria   1.86  Male        9    24       5
#  8   182 Bulgaria   0.503 Male       17    58       4
#  9   208 Bulgaria   1.32  Male       12    52       1
# 10   234 Bulgaria   1.49  Male       17    76       7
# # ℹ 3,320 more rows
```
]]

---
# Multiple OLS

.panelset[
.panel[.panel-name[Denmark vis-á-vis Bulgaria]

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#echcrxpuyq div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
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</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="a-"> </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno">Xeno</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno-">Xeno </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno-">Xeno  </th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
<td headers="Xeno" class="gt_row gt_center">4.326***</td>
<td headers="Xeno " class="gt_row gt_center">4.909***</td>
<td headers="Xeno  " class="gt_row gt_center">4.206***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.071)</td>
<td headers="Xeno " class="gt_row gt_center">(0.213)</td>
<td headers="Xeno  " class="gt_row gt_center">(0.237)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria</td>
<td headers="Xeno" class="gt_row gt_center">1.680***</td>
<td headers="Xeno " class="gt_row gt_center">1.635***</td>
<td headers="Xeno  " class="gt_row gt_center">1.608***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.098)</td>
<td headers="Xeno " class="gt_row gt_center">(0.100)</td>
<td headers="Xeno  " class="gt_row gt_center">(0.100)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">eduyrs</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">-0.043**</td>
<td headers="Xeno  " class="gt_row gt_center">-0.040**</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">(0.014)</td>
<td headers="Xeno  " class="gt_row gt_center">(0.014)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">agea</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center"></td>
<td headers="Xeno  " class="gt_row gt_center">0.014***</td></tr>
    <tr><td headers=" " class="gt_row gt_left" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno" class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno  " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.003)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">Num.Obs.</td>
<td headers="Xeno" class="gt_row gt_center">3330</td>
<td headers="Xeno " class="gt_row gt_center">3330</td>
<td headers="Xeno  " class="gt_row gt_center">3330</td></tr>
    <tr><td headers=" " class="gt_row gt_left">R2</td>
<td headers="Xeno" class="gt_row gt_center">0.113</td>
<td headers="Xeno " class="gt_row gt_center">0.116</td>
<td headers="Xeno  " class="gt_row gt_center">0.128</td></tr>
  </tbody>
  <tfoot>
    <tr class="gt_sourcenotes">
      <td class="gt_sourcenote" colspan="4">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td>
    </tr>
  </tfoot>
</table>
</div>
]]

.push-left[
<img src="11-Conditionals_files/figure-html/Coefplot-1.png" width="100%" style="display: block; margin: auto;" />
]]
.panel[.panel-name[Code]
.push-left[
.font90[

``` r
bind_rows( # Combine estimate results into one tibble,
  ols_bi %>% tidy(),
  ols_mult_1 %>% tidy(),
  ols_mult_2 %>% tidy(), .id = "model") %>%
  filter(term == "cntryBulgaria") %>% # Keep country estimates,
  mutate( # Rename the model variable,
    term = case_when(
      model == 1 ~ "Bivariate",
      model == 2 ~ "Controlled for \n education",
      model == 3 ~ "Controlled for \n education and age") %>%
      fct_relevel("Controlled for \n education and age", 
                  "Controlled for \n education",
                  "Bivariate")) %>%
  ggplot(aes(y = estimate, x = term, # Plot the results,
             ymin = conf.low, ymax = conf.high)) +
  geom_pointrange() +
  geom_hline(yintercept = 0, lty = "dashed", color = "red") +
  coord_flip() +
  labs(y = "Difference in xenophobia between
       Denmark (reference) and Bulgaria
       (10-point Likert Scale)",
       x = "") +
  theme_minimal()
```
]]
..push-right[
.font90[

``` r
ols_bi <- lm_robust(imwbcnt ~ cntry, data = ESS, 
                    weights = pspwght)
ols_mult_1 <- lm_robust(imwbcnt ~ cntry + eduyrs, 
                        data = ESS, weights = pspwght)
ols_mult_2 <- lm_robust(imwbcnt ~ cntry + eduyrs + agea, 
                        data = ESS, weights = pspwght)

modelsummary(list("Xeno" = ols_bi, "Xeno" = ols_mult_1, "Xeno" = ols_mult_2),
             gof_map = c("nobs", "r.squared"), 
             stars = TRUE, output = "gt")
```
]]]
.panel[.panel-name[Multiple OLS is additive]
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<td headers="Xeno" class="gt_row gt_center"></td>
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<td headers="Xeno " class="gt_row gt_center">3330</td>
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]]
.left-column[
**Multiple OLS, so far:** Allows us to express an outcome as linear function of several continuous and categorical predictor variables. The predictors are adjusted/controlled for each other .alert[and additive].

.font70[
$$
\operatorname{\widehat{imwbcnt}} = 4.21 + 1.61(\operatorname{cntry}_{\operatorname{Bulgaria}}) - 0.04(\operatorname{eduyrs}) + 0.01(\operatorname{agea})
$$

`$\rightarrow$` For a Bulgarian who is 20 and has 10 years of education, Model 3 predicts: 
`$4.21 + 1.61 + (10\times -0.04) + (20 \times 0.01) = 5.62 \text{ Xeno}$`

]]]
.panel[.panel-name[Additive implies linear]

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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
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      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno">Xeno</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno-">Xeno </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno-">Xeno  </th>
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    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
<td headers="Xeno" class="gt_row gt_center">4.326***</td>
<td headers="Xeno " class="gt_row gt_center">4.909***</td>
<td headers="Xeno  " class="gt_row gt_center">4.206***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.071)</td>
<td headers="Xeno " class="gt_row gt_center">(0.213)</td>
<td headers="Xeno  " class="gt_row gt_center">(0.237)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria</td>
<td headers="Xeno" class="gt_row gt_center">1.680***</td>
<td headers="Xeno " class="gt_row gt_center">1.635***</td>
<td headers="Xeno  " class="gt_row gt_center">1.608***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.098)</td>
<td headers="Xeno " class="gt_row gt_center">(0.100)</td>
<td headers="Xeno  " class="gt_row gt_center">(0.100)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">eduyrs</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">-0.043**</td>
<td headers="Xeno  " class="gt_row gt_center">-0.040**</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">(0.014)</td>
<td headers="Xeno  " class="gt_row gt_center">(0.014)</td></tr>
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<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center"></td>
<td headers="Xeno  " class="gt_row gt_center">0.014***</td></tr>
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<td headers="Xeno" class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno  " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.003)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">Num.Obs.</td>
<td headers="Xeno" class="gt_row gt_center">3330</td>
<td headers="Xeno " class="gt_row gt_center">3330</td>
<td headers="Xeno  " class="gt_row gt_center">3330</td></tr>
    <tr><td headers=" " class="gt_row gt_left">R2</td>
<td headers="Xeno" class="gt_row gt_center">0.113</td>
<td headers="Xeno " class="gt_row gt_center">0.116</td>
<td headers="Xeno  " class="gt_row gt_center">0.128</td></tr>
  </tbody>
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]]]

---
# But what if relations are conditional?

.panelset[
.panel[.panel-name[Plot]
.push-left[

.font90[
.content-box-blue[
Two predictors `$X_{1i}$` and `$X_{2i}$` are **"conditional on each other"** or **"interact"**, when the association between the outcome `$Y_{i}$` and one of the predictors depends on values of the other.
]

- The association between xenophobia and education is more negative in Denmark than in Bulgaria. 
  + .backgrnote[That is, the association between education and xenophobia depends on the country people live in.]

- The difference in average xenophobia between people living in Denmark or Bulgarian is larger among better educated people.
  + .backgrnote[That is, the association between country and xenophobia depends on people's level of education.]
]]

.push-right[
<img src="11-Conditionals_files/figure-html/conditional1-1.png" width="90%" style="display: block; margin: auto;" />
]]

.panel[.panel-name[Code]

``` r
ggplot(data = ESS, aes(y = imwbcnt, x = eduyrs, color = cntry)) +
  geom_jitter(aes(size = pspwght), alpha = 1/8) +
  geom_smooth(method = "lm") +
  labs(y = "Immigration makes DK/BG \n a worse place to live",
       x = "Years of education") +
  theme_minimal() +
  theme(legend.position = "bottom") +
  guides(size = FALSE)
```
]]

---
# Then specify them as *multiplicative*

`$$Y_{i} = \alpha + \beta_{1} X_{1} + \beta_{2}X_{2} + \color{red}{\beta_{3}(X_{1}\times X_{2})} + e_{i}$$`

.push-left[
.font80[

``` r
*ols_inter <- lm_robust(imwbcnt ~ cntry*eduyrs,
                       data = ESS, weights = pspwght)
# Report results in a regression table
modelsummary(list("Xeno" = ols_mult_1, "Xeno" = ols_inter),
             gof_map = c("nobs", "r.squared"), stars = TRUE, output = "gt")
```

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<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
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      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno">Xeno</th>
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    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
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<td headers="Xeno " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.027)</td></tr>
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    <tr><td headers=" " class="gt_row gt_left">R2</td>
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</div>
]]

.push-right[
<img src="11-Conditionals_files/figure-html/unnamed-chunk-11-1.png" width="75%" style="display: block; margin: auto;" />
- R *adds* an "interaction term", technically the product of the two interacted variables (i.e. `$\text{Bulgaria} \times \text{Education}$`).

- Thereby, multiple OLS processes the conditional relation in an additive way (aka "linearization").
]

---
# Then specify them as *multiplicative*

.push-left[

``` r
ESS <- ESS %>% mutate(
    cntry_num = case_when(
      # Numerical cntry dummy variable,
      cntry == "Denmark" ~ 0,
      cntry == "Bulgaria" ~ 1),
    # Generate interaction term by hand
    intr_trm = cntry_num * eduyrs)

ESS %>% 
  select(idno, cntry, cntry_num, eduyrs, intr_trm)
# # A tibble: 3,330 × 5
#     idno cntry    cntry_num eduyrs intr_trm
#    <dbl> <fct>        <dbl>  <dbl>    <dbl>
#  1     2 Bulgaria         1     12       12
#  2    27 Bulgaria         1     17       17
#  3    37 Bulgaria         1      9        9
#  4    50 Bulgaria         1     12       12
#  5    70 Bulgaria         1     12       12
#  6    76 Bulgaria         1     12       12
#  7   137 Bulgaria         1      9        9
#  8   182 Bulgaria         1     17       17
#  9   208 Bulgaria         1     12       12
# 10   234 Bulgaria         1     17       17
# # ℹ 3,320 more rows
```
]

.push-right[
<img src="11-Conditionals_files/figure-html/unnamed-chunk-13-1.png" width="75%" style="display: block; margin: auto;" />
- R *adds* an "interaction term", technically the product of the two interacted variables (i.e. `$\text{Bulgaria} \times \text{Education}$`).

- Thereby, multiple OLS processes the conditional relation in an additive way (aka "linearization").
]

---
# Then specify them as *multiplicative*

.push-left[
.font90[

``` r
ols_inter2 <- lm_robust(imwbcnt ~ cntry + eduyrs + intr_trm,
                       data = ESS, weights = pspwght)
# Report results in a regression table
modelsummary(list("Xeno" = ols_inter, "Xeno" = ols_inter2),
             gof_map = c("nobs", "r.squared"), stars = TRUE, output = "gt")
```

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    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
<td headers="Xeno" class="gt_row gt_center">5.873***</td>
<td headers="Xeno " class="gt_row gt_center">5.873***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
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<td headers="Xeno " class="gt_row gt_center">(0.279)</td></tr>
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<td headers="Xeno" class="gt_row gt_center">-0.375</td>
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    <tr><td headers=" " class="gt_row gt_left"></td>
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<td headers="Xeno " class="gt_row gt_center">(0.385)</td></tr>
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<td headers="Xeno" class="gt_row gt_center">-0.113***</td>
<td headers="Xeno " class="gt_row gt_center">-0.113***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.019)</td>
<td headers="Xeno " class="gt_row gt_center">(0.019)</td></tr>
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<td headers="Xeno " class="gt_row gt_center"></td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
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<td headers="Xeno " class="gt_row gt_center"></td></tr>
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<td headers="Xeno" class="gt_row gt_center"></td>
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<td headers="Xeno " class="gt_row gt_center">3330</td></tr>
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</div>
]]

.push-right[
<img src="11-Conditionals_files/figure-html/unnamed-chunk-15-1.png" width="75%" style="display: block; margin: auto;" />
- R *adds* an "interaction term", technically the product of the two interacted variables (i.e. `$\text{Bulgaria} \times \text{Education}$`).

- Thereby, multiple OLS processes the conditional relation in an additive way (aka "linearization").
]

---
layout: true
class: clear
# Tricky interpretation .font70[The two conditional "Main terms"]

---
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<td headers="Xeno " class="gt_row gt_center">-0.375</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.100)</td>
<td headers="Xeno " class="gt_row gt_center">(0.385)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">-0.043**</td>
<td headers="Xeno " class="gt_row gt_center">-0.113***</td></tr>
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<td headers="Xeno " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.027)</td></tr>
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]]

.push-right[
<img src="11-Conditionals_files/figure-html/unnamed-chunk-17-1.png" width="75%" style="display: block; margin: auto;" />
<br>

`$$\beta_1\color{red}{|x_2=0} \text{ & } \beta_2\color{red}{|x_1=0}$$`

`$\rightarrow \beta_{\text{eduyrs}}\color{red}{|\text{Bulgaria} = 0},$`

`$\rightarrow \beta_{\text{Bulgaria}}\color{red}{|\text{eduyrs} = 0}.$`
]

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]]

.push-right[
<img src="11-Conditionals_files/figure-html/unnamed-chunk-19-1.png" width="75%" style="display: block; margin: auto;" />
<br>

`$$\beta_1\color{red}{|x_2=0} \text{ & } \beta_2\color{red}{|x_1=0}$$`

`$\rightarrow \beta_{\text{eduyrs}}\color{red}{|\text{Bulgaria} = 0}$`: A year increase in education goes along with a decline of xenophobia by -0.113 scale points, .alert[if Bulgaria = 0]. `$\rightarrow$` .alert[In Denmark]

`$\rightarrow \beta_{\text{Bulgaria}}\color{red}{|\text{eduyrs} = 0}.$`
]

---

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  overflow-x: hidden;
}

#qdtywonseu .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

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

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

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

#qdtywonseu .gt_spanner_row {
  border-bottom-style: hidden;
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#qdtywonseu .gt_group_heading {
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  border-left-width: 1px;
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  border-right-style: none;
  border-right-width: 1px;
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  vertical-align: middle;
  text-align: left;
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  color: #333333;
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  border-top-color: #D3D3D3;
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#qdtywonseu .gt_from_md > :first-child {
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#qdtywonseu .gt_from_md > :last-child {
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}

#qdtywonseu .gt_row {
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  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;
}

#qdtywonseu .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;
}

#qdtywonseu .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
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  border-right-width: 2px;
  border-right-color: #D3D3D3;
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  padding-right: 5px;
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}

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

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

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

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

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

#qdtywonseu .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
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}

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

#qdtywonseu .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
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#qdtywonseu .gt_table_body {
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  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#qdtywonseu .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;
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#qdtywonseu .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding-top: 4px;
  padding-bottom: 4px;
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  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

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

#qdtywonseu .gt_left {
  text-align: left;
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#qdtywonseu .gt_center {
  text-align: center;
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  text-align: right;
  font-variant-numeric: tabular-nums;
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#qdtywonseu .gt_font_normal {
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#qdtywonseu .gt_font_bold {
  font-weight: bold;
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#qdtywonseu .gt_font_italic {
  font-style: italic;
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#qdtywonseu .gt_super {
  font-size: 65%;
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  position: initial;
}

#qdtywonseu .gt_asterisk {
  font-size: 100%;
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#qdtywonseu .gt_indent_1 {
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#qdtywonseu .gt_indent_2 {
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#qdtywonseu .gt_indent_3 {
  text-indent: 15px;
}

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

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

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

#qdtywonseu 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" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="a-"> </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno">Xeno</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno-">Xeno </th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
<td headers="Xeno" class="gt_row gt_center">4.909***</td>
<td headers="Xeno " class="gt_row gt_center">5.873***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.213)</td>
<td headers="Xeno " class="gt_row gt_center">(0.279)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria</td>
<td headers="Xeno" class="gt_row gt_center">1.635***</td>
<td headers="Xeno " class="gt_row gt_center">-0.375</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.100)</td>
<td headers="Xeno " class="gt_row gt_center">(0.385)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">-0.043**</td>
<td headers="Xeno " class="gt_row gt_center">-0.113***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.014)</td>
<td headers="Xeno " class="gt_row gt_center">(0.019)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria × eduyrs</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">0.153***</td></tr>
    <tr><td headers=" " class="gt_row gt_left" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno" class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.027)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">Num.Obs.</td>
<td headers="Xeno" class="gt_row gt_center">3330</td>
<td headers="Xeno " class="gt_row gt_center">3330</td></tr>
    <tr><td headers=" " class="gt_row gt_left">R2</td>
<td headers="Xeno" class="gt_row gt_center">0.116</td>
<td headers="Xeno " class="gt_row gt_center">0.127</td></tr>
  </tbody>
  <tfoot>
    <tr class="gt_sourcenotes">
      <td class="gt_sourcenote" colspan="3">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td>
    </tr>
  </tfoot>
</table>
</div>
]]

.push-right[
<img src="11-Conditionals_files/figure-html/conditional2-1.png" width="75%" style="display: block; margin: auto;" />
<br>

`$$\beta_1\color{red}{|x_2=0} \text{ & } \beta_2\color{red}{|x_1=0}$$`

`$\rightarrow \beta_{\text{Bulgaria}}\color{red}{|\text{eduyrs} = 0}.$`
]

---
layout: false
class: clear
# Solution: Rescale .font70[(e.g. mean center, or z-standardize)]

.push-left[

``` r
ESS <- ESS %>% mutate(
    # Center education around the mean of education.
    mc_eduyrs = eduyrs - mean(eduyrs, na.rm = TRUE))

summary(ESS$mc_eduyrs)
#    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#   -4.24   -2.24   -1.24    0.00    2.76    7.76
```
]
.push-right[
<img src="11-Conditionals_files/figure-html/conditional3-1.png" width="100%" style="display: block; margin: auto;" />
]

---
layout: true
class: clear
# Improved interpretation .font70[The two conditional "Main terms"]

---

.push-left[
<div id="ryctdcouqt" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
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#ryctdcouqt p {
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  border-right-width: 2px;
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  border-bottom-width: 2px;
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#ryctdcouqt .gt_caption {
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}

#ryctdcouqt .gt_title {
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}

#ryctdcouqt .gt_bottom_border {
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  border-right-width: 1px;
  border-right-color: #D3D3D3;
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  padding-left: 5px;
  padding-right: 5px;
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#ryctdcouqt .gt_column_spanner_outer {
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  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
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#ryctdcouqt .gt_column_spanner_outer:first-child {
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#ryctdcouqt .gt_column_spanner_outer:last-child {
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#ryctdcouqt .gt_column_spanner {
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#ryctdcouqt .gt_spanner_row {
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  text-align: left;
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#ryctdcouqt .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;
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#ryctdcouqt .gt_from_md > :first-child {
  margin-top: 0;
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#ryctdcouqt .gt_from_md > :last-child {
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}

#ryctdcouqt .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
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  padding-right: 5px;
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  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
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#ryctdcouqt .gt_stub_row_group {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
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}

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#ryctdcouqt .gt_first_summary_row.thick {
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}

#ryctdcouqt .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
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  border-top-style: double;
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  border-left-style: none;
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  <tbody class="gt_table_body">
    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
<td headers="Xeno" class="gt_row gt_center">5.873***</td>
<td headers="Xeno " class="gt_row gt_center">4.378***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.279)</td>
<td headers="Xeno " class="gt_row gt_center">(0.072)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria</td>
<td headers="Xeno" class="gt_row gt_center">-0.375</td>
<td headers="Xeno " class="gt_row gt_center">1.650***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.385)</td>
<td headers="Xeno " class="gt_row gt_center">(0.099)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">-0.113***</td>
<td headers="Xeno " class="gt_row gt_center"></td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.019)</td>
<td headers="Xeno " class="gt_row gt_center"></td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria × eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">0.153***</td>
<td headers="Xeno " class="gt_row gt_center"></td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.027)</td>
<td headers="Xeno " class="gt_row gt_center"></td></tr>
    <tr><td headers=" " class="gt_row gt_left">mc_eduyrs</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">-0.113***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">(0.019)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria × mc_eduyrs</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">0.153***</td></tr>
    <tr><td headers=" " class="gt_row gt_left" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno" class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.027)</td></tr>
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<td headers="Xeno" class="gt_row gt_center">3330</td>
<td headers="Xeno " class="gt_row gt_center">3330</td></tr>
    <tr><td headers=" " class="gt_row gt_left">R2</td>
<td headers="Xeno" class="gt_row gt_center">0.127</td>
<td headers="Xeno " class="gt_row gt_center">0.127</td></tr>
  </tbody>
  <tfoot>
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</div>
]

.push-right[
<img src="11-Conditionals_files/figure-html/unnamed-chunk-22-1.png" width="75%" style="display: block; margin: auto;" />
<br>

`$$\beta_1\color{red}{|x_2=0} \text{ & } \beta_2\color{red}{|x_1=0}$$`

`$\rightarrow \beta_{\text{Bulgaria}}\color{red}{|\text{eduyrs} = 0}.$`
]

---

.push-left[

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  <tbody class="gt_table_body">
    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
<td headers="Xeno" class="gt_row gt_center">5.873***</td>
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    <tr><td headers=" " class="gt_row gt_left"></td>
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<td headers="Xeno" class="gt_row gt_center">-0.375</td>
<td headers="Xeno " class="gt_row gt_center">1.650***</td></tr>
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<td headers="Xeno" class="gt_row gt_center">(0.385)</td>
<td headers="Xeno " class="gt_row gt_center">(0.099)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">-0.113***</td>
<td headers="Xeno " class="gt_row gt_center"></td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.019)</td>
<td headers="Xeno " class="gt_row gt_center"></td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria × eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">0.153***</td>
<td headers="Xeno " class="gt_row gt_center"></td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.027)</td>
<td headers="Xeno " class="gt_row gt_center"></td></tr>
    <tr><td headers=" " class="gt_row gt_left">mc_eduyrs</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">-0.113***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">(0.019)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria × mc_eduyrs</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">0.153***</td></tr>
    <tr><td headers=" " class="gt_row gt_left" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno" class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.027)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">Num.Obs.</td>
<td headers="Xeno" class="gt_row gt_center">3330</td>
<td headers="Xeno " class="gt_row gt_center">3330</td></tr>
    <tr><td headers=" " class="gt_row gt_left">R2</td>
<td headers="Xeno" class="gt_row gt_center">0.127</td>
<td headers="Xeno " class="gt_row gt_center">0.127</td></tr>
  </tbody>
  <tfoot>
    <tr class="gt_sourcenotes">
      <td class="gt_sourcenote" colspan="3">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td>
    </tr>
  </tfoot>
</table>
</div>
]

.push-right[
<img src="11-Conditionals_files/figure-html/unnamed-chunk-24-1.png" width="50%" style="display: block; margin: auto;" />
<br>

`$$\beta_1\color{red}{|x_2=0} \text{ & } \beta_2\color{red}{|x_1=0}$$`

`$\rightarrow \beta_{\text{Bulgaria}}\color{red}{|\text{eduyrs} = 0}$`: Bulgarians are, on average, 1.65 scale points more xenophobic than Danes, .alert[if education = 0]. `$\rightarrow$` .alert[Among persons with an average level of education].

]

---
layout: true
class: clear
# Interpretation .font70[The "Interaction term" *alters* the main terms]

.push-left[
Interaction terms, `$\beta_3(X_{1i}\times X_{2i})$`, have two interpretations!
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#vbdynmnfww .gt_column_spanner {
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#vbdynmnfww 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" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="a-"> </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno">Xeno</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
<td headers="Xeno" class="gt_row gt_center">4.378***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.072)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria</td>
<td headers="Xeno" class="gt_row gt_center">1.650***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.099)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">mc_eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">-0.113***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.019)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria × mc_eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">0.153***</td></tr>
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<td headers="Xeno" class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.027)</td></tr>
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<td headers="Xeno" class="gt_row gt_center">3330</td></tr>
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<td headers="Xeno" class="gt_row gt_center">0.127</td></tr>
  </tbody>
  <tfoot>
    <tr class="gt_sourcenotes">
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    </tr>
  </tfoot>
</table>
</div>
]

---

.push-right[
<img src="11-Conditionals_files/figure-html/unnamed-chunk-25-1.png" width="66%" style="display: block; margin: auto;" />
<br>

1) When the variable `cntryBulgaria` increases by one unit, `$\beta_{\text{Educ}}$` increases by `$\beta_{3}$`. 
  + In Bulgaria, the association between education and xenophobia is `$-0.113 + 0.153 = 0.04$`.
  + The difference in `$\beta_{\text{Educ}}$` between Denmark and Bulgaria is 0.153.

]

---

.push-right[
<img src="11-Conditionals_files/figure-html/unnamed-chunk-26-1.png" width="66%" style="display: block; margin: auto;" />
<br>

2) When the variable `mc_eduyrs` increases by one unit, `$\beta_{\text{cntryBulgaria}}$` increases by `$\beta_{3}$`.
  + Among the best educated (i.e. 8 years more than the average), the xenophobia gap between Danes and Bulgarians is `$1.65 + 8 \times 0.153 = 2.874$`
  + With every year of education, the average xenophobia gap between Danes and Bulgarians changes by 0.153.

]

---
layout: false
# Another example

.panelset[
.panel[.panel-name[Age and gender]
.left-column[
.font90[
<div id="jaoyybbbzr" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
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  -moz-osx-font-smoothing: grayscale;
}

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}

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  border-top-style: solid;
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  border-top-color: #A8A8A8;
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  border-right-color: #D3D3D3;
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}

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  color: #333333;
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]]
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<td headers="Xeno" class="gt_row gt_center">(0.021)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">gndrFemale</td>
<td headers="Xeno" class="gt_row gt_center">-0.097</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.102)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">mc_eduyrs × gndrFemale</td>
<td headers="Xeno" class="gt_row gt_center">-0.037</td></tr>
    <tr><td headers=" " class="gt_row gt_left" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno" class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.029)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">Num.Obs.</td>
<td headers="Xeno" class="gt_row gt_center">3330</td></tr>
    <tr><td headers=" " class="gt_row gt_left">R2</td>
<td headers="Xeno" class="gt_row gt_center">0.013</td></tr>
  </tbody>
  <tfoot>
    <tr class="gt_sourcenotes">
      <td class="gt_sourcenote" colspan="2">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td>
    </tr>
  </tfoot>
</table>
</div>
]]
.right-column[
<img src="11-Conditionals_files/figure-html/unnamed-chunk-28-1.png" width="45%" style="display: block; margin: auto;" />
`$\rightarrow \beta_{Educ}|\text{Female}=0$`: Among men, an additional year of education is significantly associated with -0.06 scale points less xenophobia.

`$\rightarrow \beta_{Female}|\text{Educ}=0$`: At average levels of education, the gender gap in xenophobia is -0.097 but insignificant.
]]
.panel[.panel-name[Interpretation Interaction term]

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#glhnttsnqa div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
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</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="a-"> </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno">Xeno</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
<td headers="Xeno" class="gt_row gt_center">5.278***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.075)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">mc_eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">-0.060**</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.021)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">gndrFemale</td>
<td headers="Xeno" class="gt_row gt_center">-0.097</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.102)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">mc_eduyrs × gndrFemale</td>
<td headers="Xeno" class="gt_row gt_center">-0.037</td></tr>
    <tr><td headers=" " class="gt_row gt_left" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno" class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.029)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">Num.Obs.</td>
<td headers="Xeno" class="gt_row gt_center">3330</td></tr>
    <tr><td headers=" " class="gt_row gt_left">R2</td>
<td headers="Xeno" class="gt_row gt_center">0.013</td></tr>
  </tbody>
  <tfoot>
    <tr class="gt_sourcenotes">
      <td class="gt_sourcenote" colspan="2">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td>
    </tr>
  </tfoot>
</table>
</div>
]]
.right-column[
<img src="11-Conditionals_files/figure-html/unnamed-chunk-30-1.png" width="45%" style="display: block; margin: auto;" />

1) The association between education and xenophobia among women is `$-0.06 + -0.037= -0.097$`. This is -0.037 more negative than the association among men, but this different in slopes is not significant.

2) With every year increase of education, the gender gap in xenophobia declines by -0.037. At two years more education than the average citizen, the gender gap is `$-0.097 + 2\times -0.037= -0.171$` scale points, but (probably) insignificant.

]]
.panel[.panel-name[Code]
.push-left[

``` r
ols_inter <- lm_robust(imwbcnt ~ mc_eduyrs*gndr,
                       data = ESS, weights = pspwght)
# Report results in a regression table
modelsummary(list("Xeno" = ols_inter),
             gof_map = c("nobs", "r.squared"), 
             stars = TRUE, output = "gt")
```
]

.push-right[

``` r
ggplot(data = ESS, aes(y = imwbcnt, x = mc_eduyrs, color = gndr)) +
  geom_jitter(aes(size = pspwght), alpha = 1/8) +
  geom_vline(xintercept = 0, color = "red") +
  geom_smooth(method = "lm", fullrange=TRUE) +
  labs(y = "Immigration makes DK/BG \n a worse place to live",
       x = "Years of education \n (centered around the mean)") +
  theme_minimal() +
  scale_x_continuous(limits = c(-5, 8), breaks = c(-4, -2, 0, 2, 4, 6, 8)) +
  theme(legend.position = "bottom") +
  guides(size = FALSE)
```
]]]

---
# Control for potential confounders .font70[Easy ...]

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#rxagzcalrw div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after {
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</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="a-"> </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno">Xeno</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno-">Xeno </th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
<td headers="Xeno" class="gt_row gt_center">4.378***</td>
<td headers="Xeno " class="gt_row gt_center">3.789***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.072)</td>
<td headers="Xeno " class="gt_row gt_center">(0.150)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria</td>
<td headers="Xeno" class="gt_row gt_center">1.650***</td>
<td headers="Xeno " class="gt_row gt_center">1.625***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.099)</td>
<td headers="Xeno " class="gt_row gt_center">(0.099)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">mc_eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">-0.113***</td>
<td headers="Xeno " class="gt_row gt_center">-0.108***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.019)</td>
<td headers="Xeno " class="gt_row gt_center">(0.018)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria × mc_eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">0.153***</td>
<td headers="Xeno " class="gt_row gt_center">0.150***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.027)</td>
<td headers="Xeno " class="gt_row gt_center">(0.027)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">gndrFemale</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">-0.135</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">(0.096)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">agea</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">0.014***</td></tr>
    <tr><td headers=" " class="gt_row gt_left" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno" class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.003)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">Num.Obs.</td>
<td headers="Xeno" class="gt_row gt_center">3330</td>
<td headers="Xeno " class="gt_row gt_center">3330</td></tr>
    <tr><td headers=" " class="gt_row gt_left">R2</td>
<td headers="Xeno" class="gt_row gt_center">0.127</td>
<td headers="Xeno " class="gt_row gt_center">0.139</td></tr>
  </tbody>
  <tfoot>
    <tr class="gt_sourcenotes">
      <td class="gt_sourcenote" colspan="3">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td>
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  </tfoot>
</table>
</div>
]

---
# Visualize!

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

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

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

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

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

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

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

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

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

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

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

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

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

#oheecowzcg 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" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="a-"> </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno">Xeno</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Xeno-">Xeno </th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
<td headers="Xeno" class="gt_row gt_center">4.378***</td>
<td headers="Xeno " class="gt_row gt_center">3.789***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.072)</td>
<td headers="Xeno " class="gt_row gt_center">(0.150)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria</td>
<td headers="Xeno" class="gt_row gt_center">1.650***</td>
<td headers="Xeno " class="gt_row gt_center">1.625***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.099)</td>
<td headers="Xeno " class="gt_row gt_center">(0.099)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">mc_eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">-0.113***</td>
<td headers="Xeno " class="gt_row gt_center">-0.108***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.019)</td>
<td headers="Xeno " class="gt_row gt_center">(0.018)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">cntryBulgaria × mc_eduyrs</td>
<td headers="Xeno" class="gt_row gt_center">0.153***</td>
<td headers="Xeno " class="gt_row gt_center">0.150***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center">(0.027)</td>
<td headers="Xeno " class="gt_row gt_center">(0.027)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">gndrFemale</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">-0.135</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">(0.096)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">agea</td>
<td headers="Xeno" class="gt_row gt_center"></td>
<td headers="Xeno " class="gt_row gt_center">0.014***</td></tr>
    <tr><td headers=" " class="gt_row gt_left" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno" class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Xeno " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.003)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">Num.Obs.</td>
<td headers="Xeno" class="gt_row gt_center">3330</td>
<td headers="Xeno " class="gt_row gt_center">3330</td></tr>
    <tr><td headers=" " class="gt_row gt_left">R2</td>
<td headers="Xeno" class="gt_row gt_center">0.127</td>
<td headers="Xeno " class="gt_row gt_center">0.139</td></tr>
  </tbody>
  <tfoot>
    <tr class="gt_sourcenotes">
      <td class="gt_sourcenote" colspan="3">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td>
    </tr>
  </tfoot>
</table>
</div>
]

.push-right[
.panelset[
.panel[.panel-name[Fictional data]
.font90[

``` r
(fict_dat <- tibble(
* # Repeat the vector -5:8, each element twice
* mc_eduyrs = rep(-5:8, each = 2),
* # Repeat DK/BG 14 times
* cntry = rep(c("Denmark", "Bulgaria"), 14),
  # Set controls to constant value
  gndr = "Female",
  agea = mean(ESS$agea, na.rm = TRUE)))
# # A tibble: 28 × 4
#    mc_eduyrs cntry    gndr    agea
#        <int> <chr>    <chr>  <dbl>
#  1        -5 Denmark  Female  52.2
#  2        -5 Bulgaria Female  52.2
#  3        -4 Denmark  Female  52.2
#  4        -4 Bulgaria Female  52.2
#  5        -3 Denmark  Female  52.2
#  6        -3 Bulgaria Female  52.2
#  7        -2 Denmark  Female  52.2
#  8        -2 Bulgaria Female  52.2
#  9        -1 Denmark  Female  52.2
# 10        -1 Bulgaria Female  52.2
# # ℹ 18 more rows
```
]]
.panel[.panel-name[Predictions]
.font90[

``` r
*(fict_dat <- predict(
* ols_inter4, newdata = fict_dat,
* interval = "confidence", level = 0.95)$fit %>%
   as_tibble() %>% # Turn into a tibble, then
   bind_cols(fict_dat, .)) # Add to the synthetic data.
# # A tibble: 28 × 7
#    mc_eduyrs cntry    gndr    agea   fit   lwr   upr
#        <int> <chr>    <chr>  <dbl> <dbl> <dbl> <dbl>
#  1        -5 Denmark  Female  52.2  4.91  4.64  5.19
#  2        -5 Bulgaria Female  52.2  5.79  5.55  6.02
#  3        -4 Denmark  Female  52.2  4.81  4.56  5.05
#  4        -4 Bulgaria Female  52.2  5.83  5.63  6.04
#  5        -3 Denmark  Female  52.2  4.70  4.48  4.92
#  6        -3 Bulgaria Female  52.2  5.87  5.69  6.06
#  7        -2 Denmark  Female  52.2  4.59  4.39  4.79
#  8        -2 Bulgaria Female  52.2  5.92  5.75  6.08
#  9        -1 Denmark  Female  52.2  4.48  4.30  4.66
# 10        -1 Bulgaria Female  52.2  5.96  5.80  6.11
# # ℹ 18 more rows
```
]]
.panel[.panel-name[Plot]
<img src="11-Conditionals_files/figure-html/predictions-1.png" width="90%" style="display: block; margin: auto;" />
]
.panel[.panel-name[Code]

``` r
ggplot(data = fict_dat, aes(y = fit, x = mc_eduyrs)) +
  geom_ribbon(aes(ymin = lwr, ymax = upr, fill = cntry), alpha = 0.5) +
  geom_line(aes(color = cntry)) +
  labs(
    fill = "Country:", color = "Country:",
    title = "Predictions based on multiple OLS",
    x = "By mean-centered years of education
    (At average age and gender set to women)",
    y = "Predicted xenophobia") +
  theme_minimal() +
  theme(legend.position = "bottom")
```
]]]

---
class: inverse middle center
# Break

---
class: middle clear

.left-column[
<img src="https://www.laserfiche.com/wp-content/uploads/2014/10/femalecoder.jpg" width="80%" style="display: block; margin: auto;" />

]

.right-column[
<br>
<iframe src='exercise1.html' width='1000' height='600' frameborder='0' scrolling='yes'></iframe>
]

---
class: inverse middle center
# Break

---
class: inverse
# Revised research question

.center[.font140[
Is socialism good at reducing poverty if combined with citizenship rights,<br><br> but not good at it if under autocratic rule?
]]

<br>
.push-left[
<img src="https://miro.medium.com/max/1280/1*8Y_EPw2a67TRRos3b24YlA.jpeg" width="90%" style="display: block; margin: auto;" />
]

.push-right[
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/32/Marx_-_Engels_-_Lenin_-_Stalin.svg/984px-Marx_-_Engels_-_Lenin_-_Stalin.svg.png" width="96%" style="display: block; margin: auto;" />
]

---
class: inverse
# Revised research question

.center[.font140[
Is socialism good at reducing poverty if combined with citizenship rights,<br><br> but not good at it if under autocratic rule?
]]

<br>
.push-left[
<img src="11-Conditionals_files/figure-html/socialism-corr1-1.png" width="90%" style="display: block; margin: auto;" />
]

.push-right[
<img src="11-Conditionals_files/figure-html/citizen-corr1-1.png" width="90%" style="display: block; margin: auto;" />
]

---
class: clear
# Interaction of two continuous variables

.panelset[
.panel[.panel-name[Additive multiple OLS]
.push-left[
<div id="zllhmujytv" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
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  -moz-osx-font-smoothing: grayscale;
}

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

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

#zllhmujytv .gt_table {
  display: table;
  border-collapse: collapse;
  line-height: normal;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  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;
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  border-left-color: #D3D3D3;
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#zllhmujytv .gt_caption {
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  font-size: 85%;
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  padding-top: 3px;
  padding-bottom: 5px;
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#zllhmujytv .gt_heading {
  background-color: #FFFFFF;
  text-align: center;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
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  border-right-width: 1px;
  border-right-color: #D3D3D3;
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#zllhmujytv .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
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  border-top-width: 2px;
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  border-left-width: 1px;
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  border-right-width: 1px;
  border-right-color: #D3D3D3;
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#zllhmujytv .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
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  border-left-width: 1px;
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  border-right-style: none;
  border-right-width: 1px;
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  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
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#zllhmujytv .gt_column_spanner_outer {
  color: #333333;
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  font-size: 100%;
  font-weight: normal;
  text-transform: inherit;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

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

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

#zllhmujytv .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%;
}

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

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

#zllhmujytv .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;
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  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

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

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

#zllhmujytv .gt_row {
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  margin: 10px;
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  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
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  border-right-width: 1px;
  border-right-color: #D3D3D3;
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  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 5px;
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}

#zllhmujytv .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;
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  padding-left: 5px;
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  vertical-align: top;
}

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

#zllhmujytv .gt_row_group_first th {
  border-top-width: 2px;
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#zllhmujytv .gt_summary_row {
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  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
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  padding-right: 5px;
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#zllhmujytv .gt_first_summary_row {
  border-top-style: solid;
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}

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

#zllhmujytv .gt_last_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-bottom-style: solid;
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#zllhmujytv .gt_grand_summary_row {
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  text-transform: inherit;
  padding-top: 8px;
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  padding-left: 5px;
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#zllhmujytv .gt_first_grand_summary_row {
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  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
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#zllhmujytv .gt_last_grand_summary_row_top {
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  padding-left: 5px;
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  border-bottom-style: double;
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#zllhmujytv .gt_striped {
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#zllhmujytv .gt_table_body {
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  border-top-color: #D3D3D3;
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  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#zllhmujytv .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;
}

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

#zllhmujytv .gt_sourcenotes {
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  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;
}

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

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

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

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

#zllhmujytv .gt_font_normal {
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}

#zllhmujytv .gt_font_bold {
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}

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

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

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

#zllhmujytv .gt_asterisk {
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  vertical-align: 0;
}

#zllhmujytv .gt_indent_1 {
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}

#zllhmujytv .gt_indent_2 {
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#zllhmujytv .gt_indent_3 {
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#zllhmujytv .gt_indent_4 {
  text-indent: 20px;
}

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

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

#zllhmujytv 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" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
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</style>
<table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false">
  <thead>
    <tr class="gt_col_headings">
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="a-"> </th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Poverty">Poverty</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Poverty-">Poverty </th>
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    <tr><td headers=" " class="gt_row gt_left">(Intercept)</td>
<td headers="Poverty" class="gt_row gt_center">14.391***</td>
<td headers="Poverty " class="gt_row gt_center">14.765***</td></tr>
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<td headers="Poverty" class="gt_row gt_center">(1.552)</td>
<td headers="Poverty " class="gt_row gt_center">(1.592)</td></tr>
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<td headers="Poverty" class="gt_row gt_center">-0.212+</td>
<td headers="Poverty " class="gt_row gt_center">-0.107</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Poverty" class="gt_row gt_center">(0.109)</td>
<td headers="Poverty " class="gt_row gt_center">(0.126)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">mc_equal_liberty</td>
<td headers="Poverty" class="gt_row gt_center">-28.855***</td>
<td headers="Poverty " class="gt_row gt_center">-27.768***</td></tr>
    <tr><td headers=" " class="gt_row gt_left"></td>
<td headers="Poverty" class="gt_row gt_center">(6.797)</td>
<td headers="Poverty " class="gt_row gt_center">(6.662)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">mc_socialist × mc_equal_liberty</td>
<td headers="Poverty" class="gt_row gt_center"></td>
<td headers="Poverty " class="gt_row gt_center">0.828*</td></tr>
    <tr><td headers=" " class="gt_row gt_left" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Poverty" class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;"></td>
<td headers="Poverty " class="gt_row gt_center" style="border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: #000000;">(0.356)</td></tr>
    <tr><td headers=" " class="gt_row gt_left">Num.Obs.</td>
<td headers="Poverty" class="gt_row gt_center">161</td>
<td headers="Poverty " class="gt_row gt_center">161</td></tr>
    <tr><td headers=" " class="gt_row gt_left">R2</td>
<td headers="Poverty" class="gt_row gt_center">0.131</td>
<td headers="Poverty " class="gt_row gt_center">0.148</td></tr>
  </tbody>
  <tfoot>
    <tr class="gt_sourcenotes">
      <td class="gt_sourcenote" colspan="3">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td>
    </tr>
  </tfoot>
</table>
</div><img src="11-Conditionals_files/figure-html/unnamed-chunk-43-1.png" width="504" style="display: block; margin: auto;" />
]
.push-right[
`$\beta_{\text{Socialism}}|\text{Citizen Rights}=0$`: At average levels of citizenship rights, socialism is significantly (but only marginally) related to declines in poverty.

`$\beta_{\text{Citizen Rights}}|\text{Socialism}=0$`: At average levels of socialism, a unit increase in citizenship rights is significantly related to a -27.768 percentage point decline in poverty.

`$$\beta_{\text{Socialism} \times \text{Citizen Rights}}$$`
1) With every unit more citizenship rights, socialism gets 0.828 percentage points worse at being related to declines in poverty.

2) With every additional year of socialism, citizenship rights are 0.828 percentage points less associated with declines in poverty.

]]
.panel[.panel-name[Code]

``` r
# Same data as Lecture 2 
Dat <- Dat %>% mutate( # Mean-center predictors
  mc_socialist = socialist - mean(socialist, na.rm = TRUE),
  mc_equal_liberty = equal_liberty - mean(equal_liberty, na.rm = TRUE))

# Multiple OLS
ols_inter <- lm_robust(poverty ~ mc_socialist*mc_equal_liberty, data = Dat)

# Report results in a regression table
modelsummary(list("Poverty" = ols_add, "Poverty" = ols_inter),
             gof_map = c("nobs", "r.squared"), 
             stars = TRUE, output = "gt")

# The 3-D plot
library(rockchalk)
plotPlane(ols_inter, "mc_equal_liberty", "mc_socialist", pch=16, alty=1,alwd=1, theta=50, phi=10)
```
]]

---
class: inverse middle center

.font150[
Are citizenship rights off-setting the benefits of socialism?

Is socialism off-setting the benefits of citizenship rights?

`$\rightarrow$` We would need a clever *causal research design* to answer that!
]
---
layout: false
class: inverse
# Today's general lessons

1. *Interaction*: When the association between an outcome and one predictor depends on the values of another predictor, the two predictors are said to be "conditional on each other" or to "interact."

2. *Interaction Term*: A mathematical expression that captures the interaction effect between two predictors. It is typically created by multiplying the two variables involved in the interaction.

3. *Interpreting Interaction Terms*:
  + *Main Effects*: The relationship between each predictor and the outcome when the other predictor is held at zero.
  + *Interaction Effect*: The change in the relationship between a predictor and the outcome as the other predictor increases by one unit, and vice versa.

4. *Visualization*: Due to the complexity of interaction terms, visualizing them using predictions is recommended.