Lecture 1: Outline of the planned work + linear regression

class: center
background-image: url("mainFinal2.png")

---

## Press record

---
## Logistics

- What are we planning to talk about : https://nemanjavaci.netlify.app/advanced-stats-course/course-handbook/ 
 a) Generalized linear models 
 b) Structural equation modelling: path models and confirmatory factor analysis 
 c) Mixed-effects models: cross-sectional and longitudinal data 
- Theory and practice (1 + 1 hour): 
a) Theoretical aspect – why and when would we want to use a certain statistical model 
b) Mathematical aspect – the mathematical basis of the model and how can we transform the data and parameters 
c) Practical aspect – using R to analyse the data and build the statistical models on real-world data
 
 
- R statistical environment
 
 
- Materials: 
a) Presentations (press __p__ for additional content) 
b) Commented R code
([link](https://nemanjavaci.netlify.app/advanced-stats-course/psy6210-r-code/) for this lecture) 
c) Slack channel to discuss statistical questions 
d) Course glossary: [link](https://nemanjavaci.netlify.app/advanced-stats-course/psy6210-glossary/)
???
General overview of the course 
 
Focus on theory and applications, but also on how to use these models in the R environment 
 
Each lecture is planned to be supported by HTML presentation with main slide content and lecturer's notes + commented code that produced outputs in the presentation. Often, I will include additional content that you can explore, but it is not going to be used for any subsequent knowledge assessments. 
---
## Knowledge assessment

####The application of statistical methods to the existing data
A series of research questions for which you will have to propose a statistical model that tests hypothetical assumptions, motivate your models, build them in the R statistical environment, and interpret results (parameters, fit and critique of the model)

Homework: 10% of the final mark 
Final essay type exam: 90% of the final mark

The exam will focus on the: 
a) Theoretical aspect 
b) Mathematical aspect 
c) Practical aspect

---
## Opportunities for feedback:

Discussions during the classes 
Feedback on the homework 
Slack channel and Feedback form

Course communication: n.vaci@sheffield.ac.uk 
Office hours: Friday from 1 to 2 pm (https://tinyurl.com/y2ptqmwx) 
Post your questions on the spreadsheet queries

---

## Today: linear regression

Intended learning outcomes:

Explain and motivate the linear regression model and how it relates to other statistical methods

Build a model; analyse and interpret the coefficients (model with one predictor, categorical predictors, multiple predictors, and interactions)

Interpret other information that linear regression reports: determination coefficient, F-test, residuals

Evaluate fit and assumptions of the linear regression model

---
class: center, inverse
background-image: url("main.gif")
---
## Why do we use statistical models?
--
<img src="Distributions.jpeg" width="40%" style="display: block; margin: auto;" />

Full image: [link](https://miro.medium.com/max/1400/1*NhLlwFMzN0yWSvhipqMgfw.jpeg)
---

## Gaussian (normal) distribution

`$$y_ \sim \mathcal{N}(\mu,\sigma)$$`

???
Gaussian distribution is described with two parameters: mean (measure of central tendency) and dispersion around the mean (standard deviation).

Central limit theorem: [link](https://en.wikipedia.org/wiki/Central_limit_theorem)

What can we do with one distribution: [Moments](https://en.wikipedia.org/wiki/Moment_(mathematics), [Normality](https://en.wikipedia.org/wiki/Normality_test#:~:text=In%20statistics%2C%20normality%20tests%20are,set%20to%20be%20normally%20distributed.) 
What can we do with two? With three or more? 
---

## Linear regression

Method that summarises how the average values of numerical outcome variable vary over values defined by linear functions of predictors ([visualisation](https://seeing-theory.brown.edu/regression-analysis/index.html#section1))

$$ y = \alpha + \beta * x + \epsilon $$ 
--
<img src="Week1_files/figure-html/unnamed-chunk-3-1.png" style="display: block; margin: auto;" />
???
The mathematical equation that estimates a dependent variable Y from a set of predictor variables or regressors X.

Each regressor gets a weight/coefficient that is used to estimate Y. This coefficient is estimated according to some criteria, eg. minimises the sum of squared errors or maximises the logarithm of the likelihood function. Additional reading: [Estimation of the parameters](https://data.princeton.edu/wws509/notes/c2s2)

`$\epsilon$` - measure of accuracy of the model 
`$SS_{residual}=\sum_{i=1}^{n}(Y_{i}-\hat{Y}_{i})^2=\sum_{i=1}^{n}e_{i}^2$` 
 
Other options: https://en.wikipedia.org/wiki/Deming_regression
---

## Example

Predict the height (cm) of babies from their age (months), weight (grams) and sex:

```r
par(mfrow=c(1,3), bty='n',mar = c(5, 4, .1, .1), cex=1.1, pch=16)
plot(Babies$Age, Babies$Height, xlab='Age (months)', ylab='Height (cm)')
plot(Babies$Weight, Babies$Height, xlab='Weight (grams)', ylab='Height (cm)')
boxplot(Babies$Height~Babies$Sex,xlab='Sex', ylab='Height (cm)')
```

---

## Interpretation: one predictor

$$ y = \alpha + \beta * Age + \epsilon $$

```r
lm1<-lm(Height~Age, data=Babies)
lm1$coefficients
```

```
## (Intercept)         Age 
##   57.025804    0.143174
```

Counterfactual interpretation (causal): Increase of 1 unit in predictor value - Age (being one month older) changes the outcome by the `$\beta$` (model estimated value)

Predictive interpretation (descriptive): If we compare babies that differ in their age by 1 month, we expect to see that older babies are taller by `$\beta$` on average

???
Model summarises the difference in average Height between the children as a function of their Age. 
Intercept is the average (predicted) score for children at birht (0 - months)

Type of interpretation usually depends on the strength of the theoretical framework and methodological design. If you have well developed theoretical assumptions and experimental design, this perhaps allows you to use counterfactual interpretation.

---

## Interpretation: multiple predictors

`$$y = \alpha + \beta_1 * Age + \beta_2 * Weight + \epsilon$$` 
Interpretation becomes contingent on other variables in the model

```r
lm2<-lm(Height~Age+Weight, data=Babies)
lm2$coefficients
```

```
## (Intercept)         Age      Weight 
##  40.1134418   0.1327346   0.0042808
```

Age: Comparing babies that have same Weight, but differ in their Age by one month, the model predicts difference by a value of `$\beta_1$` in their Height on average

Weight: Comparing babies that have same Age, but differ on their Weight by one gram, the model predicts difference by a value of `$\beta_2$` in their Height on average

Regression coefficient is a [partial correlation](https://en.wikipedia.org/wiki/Partial_correlation) estimate

???

```r
AgeRes=residuals(lm(Age~Weight, data=Babies))
HeightRes=residuals(lm(Height~Weight, data=Babies))
lm(HeightRes~AgeRes)$coefficients
```

```
##  (Intercept)       AgeRes 
## 1.808264e-17 1.327346e-01
```
---

## Interpretation: multiple predictors

`$$y = \alpha + \beta_1 * Age + \beta_2 * Sex + \epsilon$$`

Interpretation becomes contingent on other variables in the model

```r
lm2<-lm(Height~Age+Sex, data=Babies)
lm2$coefficients
```

```
## (Intercept)         Age     SexBoys 
##  57.5038799   0.1403955  -0.8309449
```

Age: Comparing babies that have identical Sex, but differ in their Age by one month, the model predicts difference by a value of `$\beta_1$` in their Height on average

Sex: Comparing babies that have same Age, but have different Sex, the model predicts difference by a value of `$\beta_2$` in their Height on average

???
The model summarises difference in average Height between the children as a function of their Age and Sex.

Intercept is the average (prediction) score for girls (0 on Sex variable) and that have 0 months (birth).

---

## Categorical predictors

`$$y = \alpha + \beta_1 * Age + \beta_2 * Sex_{Boys} + \epsilon$$`

What is the model doing:
.center[
<img src="CatPred.png", width = "30%">
]

Each level is assigned a value: Girls - 0, Boys - 1 
The slope coefficient `$\beta_2$` models the difference between the two levels

???
Additional reading on type of dummy coding in R: [link](https://stats.idre.ucla.edu/r/modules/coding-for-categorical-variables-in-regression-models/)
---

## Interpretation: interactions

`$$y = \alpha + \beta_1 * Age + \beta_2 * Sex_{Boys} + \beta_3 * Age*Sex_{Boys} + \epsilon$$`

We allow the slope of age to linearly change across the subgroups of Sex variable

```r
lm3<-lm(Height~Age*Sex, data=Babies)
lm3$coefficients
```

```
## (Intercept)         Age     SexBoys Age:SexBoys 
##  56.1448603   0.2202400   1.8315868  -0.1607307
```

Age: In the case of girls, comparing difference in babies older by month, the model predicts average difference by `$\beta_1$` coefficient

Sex: Expected difference between girls at birth and boys at birth is `$\beta_2$` coefficient

Age*Sex: Difference in the Age slope for Girls and Boys
???
Intercept is the predicted Girls Hight at birth 
Predicted Height for Boys, when Age is 0: Intercept + `$Sex_{boys}$` 
Slope for the Boys: Age + Age: `$Sex_{Boys}$`
---

## Interpretation: interactions

What about by-linear continous interactions?

`$$y = \alpha + \beta_1 * Age + \beta_2 * Weight + \beta_3 * Age*Weight + \epsilon$$`

```r
lm4<-lm(Height~Age*Weight, data=Babies)
lm4$coefficients
```

```
##   (Intercept)           Age        Weight    Age:Weight 
## 30.7244904854  0.6913148965  0.0066360329 -0.0001397745
```

Age: Keeping weight at value of 0 and comparing babies that differ by 1 month in their age, the model predicts average difference of `$\beta_1$`

Weight: Keeping age at value of 0 (birth) and comparing babies that differ by 1 gram in their weight, the model predicts average difference of `$\beta_2$`

Age*Weight: The average difference between babies that differ by 1 month in their age, changes by `$\beta_3$` with every 1 gram change in babies weight

???
By including an interaction we allow a model to be fit differently to subsets of data. 
When to test the interactions: does the theory/clinical practice/previous studies postulate possibilities of interaction existing between your main effects? If yes, then proceed with caution. 
+ a good sign is when the main effects are relatively strong and large and they explain a large amount of variation in the data
---

## What other information do we get from linear model?

```r
lm1<-lm(Height~Age, data=Babies)
summary(lm1)
```

```
## 
## Call:
## lm(formula = Height ~ Age, data = Babies)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -14.4765 -4.1601 -0.3703 3.9198 12.3842 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 57.02580 1.18751 48.021 <2e-16 ***
## Age 0.14317 0.06426 2.228 0.0282 * 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.283 on 98 degrees of freedom
## Multiple R-squared: 0.04821,	Adjusted R-squared: 0.0385 
## F-statistic: 4.964 on 1 and 98 DF, p-value: 0.02817
```

???
Residuals - estimate of the error of the model 
Coefficients with standard errors, as well as t-test and significance values. Test statistic (t-test) is used to test the significance of the predictor against 0. The reason why is it approximated with t distribution: [Link](https://stats.stackexchange.com/a/344008) 
Residual standard error - estimate of the fit of our model: `$\sqrt(\frac{SS_{residual}}{df})$`
---

## Determination coefficient

`$R^2 = 1 - \frac{SS_{residual}}{SS_{total}}$`

```r
ggplot()+geom_linerange(data=Babies,aes(x=Age, ymin=Height,ymax=lm1,colour=diff), size=1.2)+geom_line(data=Babies,aes(x=Age, y=lm1), size=1.2)+geom_point(data=Babies, aes(Age, y=Height), size=2)+ylab('Height')+xlab('Age')+ggtitle('SS_residual')+theme(axis.title=element_text(size=14), axis.text =element_text(size=12))
```

<img src="Week1_files/figure-html/unnamed-chunk-14-1.png" style="display: block; margin: auto;" />
???
Residual sum of squares: calculating vertical distances between observed data points and fitted regression line, raising them to the power of 2 (squaring) and summing them (addition). THe residual sum of squares informs us on the unexplained variance in our data - variance in the error term. 
---

## Determination coefficient

`$R^2 = 1 - \frac{SS_{residual}}{SS_{total}}$`

```r
lm0<-lm(Height~1, data=Babies)
summary(lm0)
```

```
## 
## Call:
## lm(formula = Height ~ 1, data = Babies)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -16.4165 -4.2284 -0.2062 3.6744 13.5940 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 59.3953 0.5388 110.2 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.388 on 99 degrees of freedom
```

???
We can fit only intercept model - only mean level of the dependent variable
---

## Determination coefficient

`$R^2 = 1 - \frac{SS_{residual}}{SS_{total}}$`

```r
ggplot()+geom_linerange(data=Babies,aes(x=Age, ymin=Height,ymax=lm0,colour=diff2), size=1.2)+geom_line(data=Babies,aes(x=Age, y=lm0), size=1.2)+geom_point(data=Babies, aes(Age, y=Height), size=2)+ylab('Height')+xlab('Age')+ggtitle('SS_total')+theme(axis.title=element_text(size=14), axis.text =element_text(size=12))
```

???
Total sum of squares: calculating vertical distances between observed data points and null model regression line (only intercept - mean of dependent variable), raising them to the power of 2 (squaring) and summing them (addition). The total sum of squares informs us on the total variance in our outcome/dependent variable. 
---

## Improvement in our prediction?

`$F = \frac{SS_m/df_m}{SS_r/df_r}$`

```r
ggplot()+geom_linerange(data=Babies,aes(x=Age, ymin=lm1,ymax=lm0,colour=diff3), size=1.2)+geom_line(data=Babies,aes(x=Age, y=lm0), size=1.2)+geom_line(data=Babies, aes(Age, y=lm1), size=1.3, linetype='dotdash')+geom_point(data=Babies, aes(x=Age, y=Height), size=0.9)+ylab('Height')+xlab('Age')+ggtitle('Improvement')+theme(axis.title=element_text(size=14), axis.text =element_text(size=12))
```

<img src="Week1_files/figure-html/unnamed-chunk-19-1.png" style="display: block; margin: auto;" />
???
We can also calculate an F-test by hand. First we would need to calculate the amount of variance in the data that was explained by our model. We would get that by calculating distances between our fitted data points (fitted line) and null model line (mean level of dependent variable). We would square those distances and sum them up to get model/explained sum of squares. 
To get an F-test, we would divide model sum of squares by the degrees of freedom spent on estimating the model (dfmodel - number of predictors). We would divide that by another unexplained amount of variance (residual sum of squares divided by residual degrees of freedom -> n - k +1 ).
---

## Why is this important?

The general linear model is "basis" for all other models covered by this course

In particular, more complex statistical models are often just generalisations of the general linear model

Other statistical model, such as correlations, t-test, ANOVA, ANCOVA can be expressed and estimated within the general linear model

```r
cor(Babies$Height,Babies$Weight)
```

```
## [1] 0.3879701
```

```r
Babies$HeightStand=scale(Babies$Height, scale=TRUE, center=TRUE)
Babies$WeightStand=scale(Babies$Weight, scale=TRUE, center=TRUE)
lmCorr<-lm(HeightStand~-1+WeightStand, data=Babies)
lmCorr$coefficients
```

```
## WeightStand 
##   0.3879701
```
???
General linear model is a special case of the broad family of models (Generalized Linear Models - more next week)
---

## Asumptions

a) Error distribution: `$\mathcal{N}^{iid}(0,\sigma^2)$` 
b) Linearity and additivity 
c) Validity 
d) Lack of perfect multicolinearity

???
0 being mean of residuals is a by-product of OLS when we include intercept. Assumptioms of IID ~ independently and identically distributed residuals, are often problematic in psychology, when we have clustered answers.

Linearity - outcome can be modelled as a linear function of separate predictors 
Additivity - postulated linear function should not vary across values of other variables, if it does - we need to add interactions 
Validity - data that you are measuring should reflect the phenomenon you are interested in

---
class: inverse, middle, center
# Practical aspect

---

#Practical work
Example dataset: Income inequality and rates of hate crimes -
[Article](https://fivethirtyeight.com/features/higher-rates-of-hate-crimes-are-tied-to-income-inequality/) and 
[Data](https://github.com/fivethirtyeight/data/tree/master/hate-crimes)

Reading local files to R:

```r
inequality<-read.table('inequality.txt',sep=',', header=T)
knitr::kable(head(inequality[,c(1,2,8,12)]), format = 'html')
```

<table>
 <thead>
 <tr>
 <th style="text-align:left;"> state </th>
 <th style="text-align:right;"> median_household_income </th>
 <th style="text-align:right;"> gini_index </th>
 <th style="text-align:right;"> avg_hatecrimes_per_100k_fbi </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Alabama </td>
 <td style="text-align:right;"> 42278 </td>
 <td style="text-align:right;"> 0.472 </td>
 <td style="text-align:right;"> 1.8064105 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Alaska </td>
 <td style="text-align:right;"> 67629 </td>
 <td style="text-align:right;"> 0.422 </td>
 <td style="text-align:right;"> 1.6567001 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Arizona </td>
 <td style="text-align:right;"> 49254 </td>
 <td style="text-align:right;"> 0.455 </td>
 <td style="text-align:right;"> 3.4139280 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Arkansas </td>
 <td style="text-align:right;"> 44922 </td>
 <td style="text-align:right;"> 0.458 </td>
 <td style="text-align:right;"> 0.8692089 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> California </td>
 <td style="text-align:right;"> 60487 </td>
 <td style="text-align:right;"> 0.471 </td>
 <td style="text-align:right;"> 2.3979859 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Colorado </td>
 <td style="text-align:right;"> 60940 </td>
 <td style="text-align:right;"> 0.457 </td>
 <td style="text-align:right;"> 2.8046888 </td>
 </tr>
</tbody>
</table>

???
In order to read/load the data in R, you need to specify where R is supposed to look for the files. One way you can achieve this is by setting up the working directory. You can click on the tab __Session__ then __Set working directory__ then __Choose directory__ and then navigate to folder where your data is. Then you can just follow the code outlined in the presentations and/or in the R code for this lecture. 
---

##Dataset: outcome measures

Probability density plots: probability of the random variable falling within a range of values

```r
par(mfrow=c(1,2), bty='n',mar = c(5, 4, 1, .1), cex=1.1, pch=16)
plot(density(inequality$hate_crimes_per_100k_splc, na.rm = T), main='Crimes per 100k')
plot(density(inequality$avg_hatecrimes_per_100k_fbi, na.rm = T), main='Average Crimes')
```

<img src="Week1_files/figure-html/unnamed-chunk-23-1.png" style="display: block; margin: auto;" />
???
Shape of the distribution, the most likely range of values, the spread of values, modality etc. 
---

##Dataset: some of the predictors

```r
par(mfrow=c(1,2), bty='n',mar = c(5, 4, 1, .1), cex=1.1, pch=16)
plot(density(inequality$median_household_income, na.rm = T), main='Income')
plot(density(inequality$gini_index, na.rm = T), main='Gini')
```

<img src="Week1_files/figure-html/unnamed-chunk-24-1.png" style="display: block; margin: auto;" />
???
We do not have distributional expectations in the case of predictors, however it is always a good practice to see density functions, histograms, scatter plots, and descriptive statistics as they might reveal mistakes in the data, non-meaningful data points, or just outliers and leverage values. 
---

##Bivariate plots

Scatter plots:

```r
par(bty='n',mar = c(5, 4, .1, .1), cex=1.1, pch=16)
plot(inequality$median_household_income, inequality$avg_hatecrimes_per_100k_fbi, xlab='Median household income',ylab='Avg hatecrimes')
```

---

##Bivariate plots

Scatter plots:

```r
par(bty='n',mar = c(5, 4, .1, .1), cex=1.1, pch=16)
plot(inequality$gini_index, inequality$avg_hatecrimes_per_100k_fbi,xlab='Gini index',ylab='Avg hatecrimes')
```

---

##Correlations in the data

```r
names(inequality)[c(2,8,12)]=c('Median_income','Gini','Avg_hate')
cor(inequality[,c(2,8,12)], use="complete.obs")
```

```
##               Median_income       Gini  Avg_hate
## Median_income     1.0000000 -0.1497444 0.3182464
## Gini             -0.1497444  1.0000000 0.4212719
## Avg_hate          0.3182464  0.4212719 1.0000000
```

???
In this situation, I am primarily using correlation to summarise collected data. High correlations might prepare us for collinearity issues in regressions, unexpected changes in the coefficients etc.  
---

## Linear regression in R

```r
mod1<-lm(Avg_hate~Median_income, data=inequality)
summary(mod1)
```

```
## 
## Call:
## lm(formula = Avg_hate ~ Median_income, data = inequality)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.3300 -1.1183 -0.1656  0.7827  7.7762 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   -9.564e-01  1.448e+00  -0.661   0.5121  
## Median_income  6.054e-05  2.603e-05   2.326   0.0243 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.642 on 48 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.1013,	Adjusted R-squared:  0.08256 
## F-statistic: 5.409 on 1 and 48 DF,  p-value: 0.0243
```

???
In this model, we analyse average hate reported by fbi as a function of median income of the population. We are asking does knowing median income of the population help us to say what is average hate better than just knowing mean level of hate.
---

## Adding more predictors

Model comparison works beyond null - intercept only model and models with one or more predictors

```r
mod2<-lm(Avg_hate~Median_income+Gini, data=inequality)
anova(mod2)
```

```
## Analysis of Variance Table
## 
## Response: Avg_hate
##               Df Sum Sq Mean Sq F value    Pr(>F)    
## Median_income  1 14.584  14.584  7.0649 0.0107093 *  
## Gini           1 32.389  32.389 15.6906 0.0002517 ***
## Residuals     47 97.020   2.064                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
???

We can add a second predictor - Gini index and calculate anova table using __anova__ function. We are comparing the model that we have build with the null model (only intercept model)
---

## Adding more predictors

We could use stepwise approach and build our models by sequentially adding the predictors

This allows us to see relative improvement that we get by adding gini index to the model with median household income

```r
anova(mod1,mod2)
```

```
## Analysis of Variance Table
## 
## Model 1: Avg_hate ~ Median_income
## Model 2: Avg_hate ~ Median_income + Gini
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1     48 129.41                                  
## 2     47  97.02  1    32.389 15.691 0.0002517 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
???
When given more than one model __anova__ functions compares the models agains each other, in order specified.
---

## Main effects for two predictors

```r
summary(mod2)
```

```
## 
## Call:
## lm(formula = Avg_hate ~ Median_income + Gini, data = inequality)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.2251 -0.9284 -0.2918  0.9861  4.5301 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -1.959e+01  4.871e+00  -4.021 0.000208 ***
## Median_income  7.421e-05  2.304e-05   3.221 0.002319 ** 
## Gini           3.936e+01  9.938e+00   3.961 0.000252 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.437 on 47 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.3262,	Adjusted R-squared:  0.2975 
## F-statistic: 11.38 on 2 and 47 DF,  p-value: 9.337e-05
```

---

## Interactions?

Should median household income be adjusted across the levels of gini index (or vice versa)?

```r
mod3<-lm(Avg_hate~Median_income*Gini, data=inequality)
anova(mod1,mod2, mod3)
```

```
## Analysis of Variance Table
## 
## Model 1: Avg_hate ~ Median_income
## Model 2: Avg_hate ~ Median_income + Gini
## Model 3: Avg_hate ~ Median_income * Gini
## Res.Df RSS Df Sum of Sq F Pr(>F) 
## 1 48 129.41 
## 2 47 97.02 1 32.389 19.742 5.539e-05 ***
## 3 46 75.47 1 21.550 13.135 0.0007218 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
???
Interaction between variables is an easy thing to add. You can use multiplication (*) - Median_income * Gini will result in two main effects (Median_income and Gini index) and one interaction (between two main effects).

You can include only an interaction by using this specification of the model regressors -> Median_income:Gini. This will result in only interaction model, without main effects. However, this is not something that you would do often. 
---

## Interactions!

```
## 
## Call:
## lm(formula = Avg_hate ~ Median_income * Gini, data = inequality)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.16383 -0.96279 -0.01053  0.88008  2.75763 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         9.056e+01  3.070e+01   2.950 0.004986 ** 
## Median_income      -1.724e-03  4.965e-04  -3.472 0.001136 ** 
## Gini               -2.001e+02  6.666e+01  -3.002 0.004330 ** 
## Median_income:Gini  3.907e-03  1.078e-03   3.624 0.000722 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.281 on 46 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.4759,	Adjusted R-squared:  0.4417 
## F-statistic: 13.92 on 3 and 46 DF,  p-value: 1.367e-06
```

---

## Visualisations of the predictions

```r
require(interactions)
interact_plot(mod3, pred=Median_income, modx=Gini, plot.points = T)
```

---

## Visualisations of the predictions

```r
p<-ggplot(simulatedD, aes(Median_income, Avg_hate_pred, color=Gini,frame=Gini))+geom_point()
plotly::ggplotly(p)
```

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5.7884495622 Gini: 0.53 Gini: 0.53","Median_income: 61400 Avg_hate_pred: 5.8231433691 Gini: 0.53 Gini: 0.53","Median_income: 61500 Avg_hate_pred: 5.8578371759 Gini: 0.53 Gini: 0.53","Median_income: 61600 Avg_hate_pred: 5.8925309827 Gini: 0.53 Gini: 0.53","Median_income: 61700 Avg_hate_pred: 5.9272247896 Gini: 0.53 Gini: 0.53","Median_income: 61800 Avg_hate_pred: 5.9619185964 Gini: 0.53 Gini: 0.53","Median_income: 61900 Avg_hate_pred: 5.9966124033 Gini: 0.53 Gini: 0.53","Median_income: 62000 Avg_hate_pred: 6.0313062101 Gini: 0.53 Gini: 0.53","Median_income: 62100 Avg_hate_pred: 6.0660000169 Gini: 0.53 Gini: 0.53","Median_income: 62200 Avg_hate_pred: 6.1006938238 Gini: 0.53 Gini: 0.53","Median_income: 62300 Avg_hate_pred: 6.1353876306 Gini: 0.53 Gini: 0.53","Median_income: 62400 Avg_hate_pred: 6.1700814375 Gini: 0.53 Gini: 0.53","Median_income: 62500 Avg_hate_pred: 6.2047752443 Gini: 0.53 Gini: 0.53","Median_income: 62600 Avg_hate_pred: 6.2394690511 Gini: 0.53 Gini: 0.53","Median_income: 62700 Avg_hate_pred: 6.2741628580 Gini: 0.53 Gini: 0.53","Median_income: 62800 Avg_hate_pred: 6.3088566648 Gini: 0.53 Gini: 0.53","Median_income: 62900 Avg_hate_pred: 6.3435504717 Gini: 0.53 Gini: 0.53","Median_income: 63000 Avg_hate_pred: 6.3782442785 Gini: 0.53 Gini: 0.53","Median_income: 63100 Avg_hate_pred: 6.4129380853 Gini: 0.53 Gini: 0.53","Median_income: 63200 Avg_hate_pred: 6.4476318922 Gini: 0.53 Gini: 0.53","Median_income: 63300 Avg_hate_pred: 6.4823256990 Gini: 0.53 Gini: 0.53","Median_income: 63400 Avg_hate_pred: 6.5170195058 Gini: 0.53 Gini: 0.53","Median_income: 63500 Avg_hate_pred: 6.5517133127 Gini: 0.53 Gini: 0.53","Median_income: 63600 Avg_hate_pred: 6.5864071195 Gini: 0.53 Gini: 0.53","Median_income: 63700 Avg_hate_pred: 6.6211009264 Gini: 0.53 Gini: 0.53","Median_income: 63800 Avg_hate_pred: 6.6557947332 Gini: 0.53 Gini: 0.53","Median_income: 63900 Avg_hate_pred: 6.6904885400 Gini: 0.53 Gini: 0.53","Median_income: 64000 Avg_hate_pred: 6.7251823469 Gini: 0.53 Gini: 0.53","Median_income: 64100 Avg_hate_pred: 6.7598761537 Gini: 0.53 Gini: 0.53","Median_income: 64200 Avg_hate_pred: 6.7945699606 Gini: 0.53 Gini: 0.53","Median_income: 64300 Avg_hate_pred: 6.8292637674 Gini: 0.53 Gini: 0.53","Median_income: 64400 Avg_hate_pred: 6.8639575742 Gini: 0.53 Gini: 0.53","Median_income: 64500 Avg_hate_pred: 6.8986513811 Gini: 0.53 Gini: 0.53","Median_income: 64600 Avg_hate_pred: 6.9333451879 Gini: 0.53 Gini: 0.53","Median_income: 64700 Avg_hate_pred: 6.9680389948 Gini: 0.53 Gini: 0.53","Median_income: 64800 Avg_hate_pred: 7.0027328016 Gini: 0.53 Gini: 0.53","Median_income: 64900 Avg_hate_pred: 7.0374266084 Gini: 0.53 Gini: 0.53","Median_income: 65000 Avg_hate_pred: 7.0721204153 Gini: 0.53 Gini: 0.53","Median_income: 65100 Avg_hate_pred: 7.1068142221 Gini: 0.53 Gini: 0.53","Median_income: 65200 Avg_hate_pred: 7.1415080290 Gini: 0.53 Gini: 0.53","Median_income: 65300 Avg_hate_pred: 7.1762018358 Gini: 0.53 Gini: 0.53","Median_income: 65400 Avg_hate_pred: 7.2108956426 Gini: 0.53 Gini: 0.53","Median_income: 65500 Avg_hate_pred: 7.2455894495 Gini: 0.53 Gini: 0.53","Median_income: 65600 Avg_hate_pred: 7.2802832563 Gini: 0.53 Gini: 0.53","Median_income: 65700 Avg_hate_pred: 7.3149770632 Gini: 0.53 Gini: 0.53","Median_income: 65800 Avg_hate_pred: 7.3496708700 Gini: 0.53 Gini: 0.53","Median_income: 65900 Avg_hate_pred: 7.3843646768 Gini: 0.53 Gini: 0.53","Median_income: 66000 Avg_hate_pred: 7.4190584837 Gini: 0.53 Gini: 0.53","Median_income: 66100 Avg_hate_pred: 7.4537522905 Gini: 0.53 Gini: 0.53","Median_income: 66200 Avg_hate_pred: 7.4884460974 Gini: 0.53 Gini: 0.53","Median_income: 66300 Avg_hate_pred: 7.5231399042 Gini: 0.53 Gini: 0.53","Median_income: 66400 Avg_hate_pred: 7.5578337110 Gini: 0.53 Gini: 0.53","Median_income: 66500 Avg_hate_pred: 7.5925275179 Gini: 0.53 Gini: 0.53","Median_income: 66600 Avg_hate_pred: 7.6272213247 Gini: 0.53 Gini: 0.53","Median_income: 66700 Avg_hate_pred: 7.6619151316 Gini: 0.53 Gini: 0.53","Median_income: 66800 Avg_hate_pred: 7.6966089384 Gini: 0.53 Gini: 0.53","Median_income: 66900 Avg_hate_pred: 7.7313027452 Gini: 0.53 Gini: 0.53","Median_income: 67000 Avg_hate_pred: 7.7659965521 Gini: 0.53 Gini: 0.53","Median_income: 67100 Avg_hate_pred: 7.8006903589 Gini: 0.53 Gini: 0.53","Median_income: 67200 Avg_hate_pred: 7.8353841657 Gini: 0.53 Gini: 0.53","Median_income: 67300 Avg_hate_pred: 7.8700779726 Gini: 0.53 Gini: 0.53","Median_income: 67400 Avg_hate_pred: 7.9047717794 Gini: 0.53 Gini: 0.53","Median_income: 67500 Avg_hate_pred: 7.9394655863 Gini: 0.53 Gini: 0.53","Median_income: 67600 Avg_hate_pred: 7.9741593931 Gini: 0.53 Gini: 0.53","Median_income: 67700 Avg_hate_pred: 8.0088531999 Gini: 0.53 Gini: 0.53","Median_income: 67800 Avg_hate_pred: 8.0435470068 Gini: 0.53 Gini: 0.53","Median_income: 67900 Avg_hate_pred: 8.0782408136 Gini: 0.53 Gini: 0.53","Median_income: 68000 Avg_hate_pred: 8.1129346205 Gini: 0.53 Gini: 0.53","Median_income: 68100 Avg_hate_pred: 8.1476284273 Gini: 0.53 Gini: 0.53","Median_income: 68200 Avg_hate_pred: 8.1823222341 Gini: 0.53 Gini: 0.53","Median_income: 68300 Avg_hate_pred: 8.2170160410 Gini: 0.53 Gini: 0.53","Median_income: 68400 Avg_hate_pred: 8.2517098478 Gini: 0.53 Gini: 0.53","Median_income: 68500 Avg_hate_pred: 8.2864036547 Gini: 0.53 Gini: 0.53","Median_income: 68600 Avg_hate_pred: 8.3210974615 Gini: 0.53 Gini: 0.53","Median_income: 68700 Avg_hate_pred: 8.3557912683 Gini: 0.53 Gini: 0.53","Median_income: 68800 Avg_hate_pred: 8.3904850752 Gini: 0.53 Gini: 0.53","Median_income: 68900 Avg_hate_pred: 8.4251788820 Gini: 0.53 Gini: 0.53","Median_income: 69000 Avg_hate_pred: 8.4598726889 Gini: 0.53 Gini: 0.53","Median_income: 69100 Avg_hate_pred: 8.4945664957 Gini: 0.53 Gini: 0.53","Median_income: 69200 Avg_hate_pred: 8.5292603025 Gini: 0.53 Gini: 0.53","Median_income: 69300 Avg_hate_pred: 8.5639541094 Gini: 0.53 Gini: 0.53","Median_income: 69400 Avg_hate_pred: 8.5986479162 Gini: 0.53 Gini: 0.53","Median_income: 69500 Avg_hate_pred: 8.6333417231 Gini: 0.53 Gini: 0.53","Median_income: 69600 Avg_hate_pred: 8.6680355299 Gini: 0.53 Gini: 0.53","Median_income: 69700 Avg_hate_pred: 8.7027293367 Gini: 0.53 Gini: 0.53","Median_income: 69800 Avg_hate_pred: 8.7374231436 Gini: 0.53 Gini: 0.53","Median_income: 69900 Avg_hate_pred: 8.7721169504 Gini: 0.53 Gini: 0.53","Median_income: 70000 Avg_hate_pred: 8.8068107573 Gini: 0.53 Gini: 0.53","Median_income: 70100 Avg_hate_pred: 8.8415045641 Gini: 0.53 Gini: 0.53","Median_income: 70200 Avg_hate_pred: 8.8761983709 Gini: 0.53 Gini: 0.53","Median_income: 70300 Avg_hate_pred: 8.9108921778 Gini: 0.53 Gini: 0.53","Median_income: 70400 Avg_hate_pred: 8.9455859846 Gini: 0.53 Gini: 0.53","Median_income: 70500 Avg_hate_pred: 8.9802797915 Gini: 0.53 Gini: 0.53","Median_income: 70600 Avg_hate_pred: 9.0149735983 Gini: 0.53 Gini: 0.53","Median_income: 70700 Avg_hate_pred: 9.0496674051 Gini: 0.53 Gini: 0.53","Median_income: 70800 Avg_hate_pred: 9.0843612120 Gini: 0.53 Gini: 0.53","Median_income: 70900 Avg_hate_pred: 9.1190550188 Gini: 0.53 Gini: 0.53","Median_income: 71000 Avg_hate_pred: 9.1537488256 Gini: 0.53 Gini: 0.53","Median_income: 71100 Avg_hate_pred: 9.1884426325 Gini: 0.53 Gini: 0.53","Median_income: 71200 Avg_hate_pred: 9.2231364393 Gini: 0.53 Gini: 0.53","Median_income: 71300 Avg_hate_pred: 9.2578302462 Gini: 0.53 Gini: 0.53","Median_income: 71400 Avg_hate_pred: 9.2925240530 Gini: 0.53 Gini: 0.53","Median_income: 71500 Avg_hate_pred: 9.3272178598 Gini: 0.53 Gini: 0.53","Median_income: 71600 Avg_hate_pred: 9.3619116667 Gini: 0.53 Gini: 0.53","Median_income: 71700 Avg_hate_pred: 9.3966054735 Gini: 0.53 Gini: 0.53","Median_income: 71800 Avg_hate_pred: 9.4312992804 Gini: 0.53 Gini: 0.53","Median_income: 71900 Avg_hate_pred: 9.4659930872 Gini: 0.53 Gini: 0.53","Median_income: 72000 Avg_hate_pred: 9.5006868940 Gini: 0.53 Gini: 0.53","Median_income: 72100 Avg_hate_pred: 9.5353807009 Gini: 0.53 Gini: 0.53","Median_income: 72200 Avg_hate_pred: 9.5700745077 Gini: 0.53 Gini: 0.53","Median_income: 72300 Avg_hate_pred: 9.6047683146 Gini: 0.53 Gini: 0.53","Median_income: 72400 Avg_hate_pred: 9.6394621214 Gini: 0.53 Gini: 0.53","Median_income: 72500 Avg_hate_pred: 9.6741559282 Gini: 0.53 Gini: 0.53","Median_income: 72600 Avg_hate_pred: 9.7088497351 Gini: 0.53 Gini: 0.53","Median_income: 72700 Avg_hate_pred: 9.7435435419 Gini: 0.53 Gini: 0.53","Median_income: 72800 Avg_hate_pred: 9.7782373488 Gini: 0.53 Gini: 0.53","Median_income: 72900 Avg_hate_pred: 9.8129311556 Gini: 0.53 Gini: 0.53","Median_income: 73000 Avg_hate_pred: 9.8476249624 Gini: 0.53 Gini: 0.53","Median_income: 73100 Avg_hate_pred: 9.8823187693 Gini: 0.53 Gini: 0.53","Median_income: 73200 Avg_hate_pred: 9.9170125761 Gini: 0.53 Gini: 0.53","Median_income: 73300 Avg_hate_pred: 9.9517063830 Gini: 0.53 Gini: 0.53","Median_income: 73400 Avg_hate_pred: 9.9864001898 Gini: 0.53 Gini: 0.53","Median_income: 73500 Avg_hate_pred: 10.0210939966 Gini: 0.53 Gini: 0.53","Median_income: 73600 Avg_hate_pred: 10.0557878035 Gini: 0.53 Gini: 0.53","Median_income: 73700 Avg_hate_pred: 10.0904816103 Gini: 0.53 Gini: 0.53","Median_income: 73800 Avg_hate_pred: 10.1251754172 Gini: 0.53 Gini: 0.53","Median_income: 73900 Avg_hate_pred: 10.1598692240 Gini: 0.53 Gini: 0.53","Median_income: 74000 Avg_hate_pred: 10.1945630308 Gini: 0.53 Gini: 0.53","Median_income: 74100 Avg_hate_pred: 10.2292568377 Gini: 0.53 Gini: 0.53","Median_income: 74200 Avg_hate_pred: 10.2639506445 Gini: 0.53 Gini: 0.53","Median_income: 74300 Avg_hate_pred: 10.2986444513 Gini: 0.53 Gini: 0.53","Median_income: 74400 Avg_hate_pred: 10.3333382582 Gini: 0.53 Gini: 0.53","Median_income: 74500 Avg_hate_pred: 10.3680320650 Gini: 0.53 Gini: 0.53","Median_income: 74600 Avg_hate_pred: 10.4027258719 Gini: 0.53 Gini: 0.53","Median_income: 74700 Avg_hate_pred: 10.4374196787 Gini: 0.53 Gini: 0.53","Median_income: 74800 Avg_hate_pred: 10.4721134855 Gini: 0.53 Gini: 0.53","Median_income: 74900 Avg_hate_pred: 10.5068072924 Gini: 0.53 Gini: 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---

## Model diagnostics

```r
summary(mod3$residuals)
```

```
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -2.16383 -0.96279 -0.01053  0.00000  0.88008  2.75763
```

Studentized residuals

`$\epsilon_{Ti}=\frac{\epsilon_i}{\sigma_{(-i)}\sqrt{1-h_i}}$`

???
Studentized residual: division of a residual by an estimate of its standard devitation weighted by an estimate of leverage (hat values)

Delete the observations one at the time and fit the model on the n-1 observations
---
class: center, inverse
background-image: url("overfit.jpg")
---

## Normality

```r
car::qqPlot(mod3)
```

```
## [1]  9 18
```

[Cross-validated on residuals](https://stats.stackexchange.com/a/101290) and 
[Cross-validated shiny](https://xiongge.shinyapps.io/QQplots/)
???
We would expect that our residuals follow the diagonal line and do not deviate from it considerably - outside of the intervals 
---

## Homoscedasticity

```r
par(bty='n',mar = c(5, 4, .1, .1), cex=1.1, pch=16)
car:: residualPlot(mod3, type='rstudent')
```

<img src="Week1_files/figure-html/unnamed-chunk-39-1.png" style="display: block; margin: auto;" />
???
We would equal spread of residuals across the fitted values - to follow horizontal line and are equidistant from it. 
---

## Outliers and leverage points

```r
par(bty='n',mar = c(5, 4, .1, .1), cex=1.1, pch=16)
car:: influenceIndexPlot(mod3, vars=c("Cook","studentized"))
```

???
Cook's Distance of observation __i__ is defined as a sum of all the changes in the regression model when observation __i__ is removed from it - > indicate data points that change the behaviour of the model when removed (influential data points).

The studentized residuals estimate varying of residuals when the Y data point is deleted and they provide us with information on outliers. 
---

## Outliers and leverage points

```r
par(bty='n',mar = c(5, 4, .1, .1), cex=1.1, pch=16)
car:: influenceIndexPlot(mod3, vars=c("Bonf","hat"))
```

<img src="Week1_files/figure-html/unnamed-chunk-41-1.png" style="display: block; margin: auto;" />
---

## Autocorrelation between the residuals

```r
par(bty='n',mar = c(5, 4, .1, .1), cex=1.1, pch=16)
stats::acf(resid(mod3))
```

???

We also want to check the autocorrelation between our residuals. This plot shows lagged autocorrelation, where at position 0 is correlation of residual with itself, and then with subsequent residuals.
---

## Observed versus predicted values

```r
par(bty='n',mar = c(5, 4, .1, .1), cex=1.1, pch=16)
plot(predict(mod3),mod3$model$Avg_hate, xlab='Predicted values (model 3)', ylab='Observed values (avg hatecrimes)')
```

???
[Further reading](https://www.sagepub.com/sites/default/files/upm-binaries/38503_Chapter6.pdf)

For maybe better package when checking the model assumptions try: 
require(performance) 
check_model(mod3) 
---

## Model evaluation

Rerun the model excluding leverage values
.tiny[

```r
mod3.cor<-lm(Avg_hate~Median_income*Gini, data=inequality, subset=-9)
summary(mod3.cor)
```

```
## 
## Call:
## lm(formula = Avg_hate ~ Median_income * Gini, data = inequality, 
##     subset = -9)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.90964 -0.94966 -0.08526  0.73470  2.55257 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)
## (Intercept)         4.429e+01  3.080e+01   1.438    0.157
## Median_income      -7.992e-04  5.228e-04  -1.529    0.133
## Gini               -9.668e+01  6.725e+01  -1.438    0.157
## Median_income:Gini  1.837e-03  1.145e-03   1.604    0.116
## 
## Residual standard error: 1.154 on 45 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.1285,	Adjusted R-squared:  0.07041 
## F-statistic: 2.212 on 3 and 45 DF,  p-value: 0.09974
```
]
---

## Model evaluation

Interaction plot
.tiny[

```r
mod2.cor<-lm(Avg_hate~Median_income+Gini, data=inequality, subset=-9)
summary(mod2.cor)
```

```
## 
## Call:
## lm(formula = Avg_hate ~ Median_income + Gini, data = inequality, 
##     subset = -9)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.86717 -0.97806 -0.05752  1.05394  2.50740 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   -4.491e+00  5.015e+00  -0.895   0.3752  
## Median_income  3.906e-05  2.012e-05   1.942   0.0583 .
## Gini           1.005e+01  1.005e+01   1.000   0.3226  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.174 on 46 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.07866,	Adjusted R-squared:  0.0386 
## F-statistic: 1.964 on 2 and 46 DF,  p-value: 0.152
```
]
---

## Take away message (important bits)

Linear model - continuous and categorical predictors 
Interpreting estimated coefficients 
Interpreting results (significance, determination coefficient, F-test, residuals) 
Understanding assumptions of linear model 
Using linear models in R: overall and stepwise approach 
Model diagnostics based on pre and post-modelling procedures covered by the lecture and beyond

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## Literature

First and second chapter of "Data Analysis Using Regression and Multilevel/Hierarchical Models" by Andrew Gelman and Jennifer Hill

First three chapter of "Regression Analysis and Linear Models: Concepts, Applications, and Implementation" by Richard B. Darlington and Andrew F. Hayes

van Rij, J., Vaci, N., Wurm, L. H., & Feldman, L. B. (2020). Alternative quantitative methods in psycholinguistics: Implications for theory and design. Word Knowledge and Word Usage, 83-126.

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## Exercise for the next week

1. Fill the reflection and feedback form by Monday: https://forms.gle/DoeBNKZqCeRvSmZ46

2. Go over the practical aspect of the lecture: read the data in R, plot uni- and bi-variate plots, calculate descriptive statistics, make a regression model, and evaluate it's fit

3. Think about how you could make a model where you have high or a perfect multicolinearity between two predictors

4. How would you create a model where you have `$R^2=1$`

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# Thank you for your attention

[Song for the weekend](https://youtu.be/pZTLFu79UbY)