Introduction to statistical modelling in R

# Introduction to statistical modelling in R
## Online Workshop - Section C - SLAAS 
### Priyanga Dilini Talagala 
### 29/08/2021 <img src="fig/42_broom.png" width="201" />

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# Tidy Workflow

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# Tidy Workflow
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# Tidy Workflow
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# What is the purpose of a model?
--

# How can I fit models in R?

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# What is a Model?

---

].pull-right[

]

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# What is a Model?
--

## An Approximation of reality
--

## ... with a purpose

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# What is the purpose of  a statistical Model?

---
## What is the purpose of  a statistical Model?

- Separate **signal** from **noise**

- Understand **trends** and **patterns**

- Identify which variables are **related** to a response variable

- Quantify this relationship

- Make **predictions** about future observations

- Understand underlying **unexplained variability** in the patterns

- **Compare results** against other statistical models
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---

```r
set.seed(4343)
data <- tibble(x = rnorm(1000,0,10),
 y = -(3*x^2) + rnorm(100, 0, 45))
data1 <- data %>% filter(x<=-4)
model1 <- lm(y ~x, data = data1)
summary(model1)
```

```
## 
## Call:
## lm(formula = y ~ x, data = data1)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1862.39 -74.65 18.06 91.52 230.87 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 485.289 18.955 25.60 <2e-16 ***
## x 84.220 1.674 50.31 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 152.5 on 339 degrees of freedom
## Multiple R-squared: 0.8819,	Adjusted R-squared: 0.8815 
## F-statistic: 2531 on 1 and 339 DF, p-value: < 2.2e-16
```
].pull-right[

<img src="SLAAS-stmodelling_files/figure-html/unnamed-chunk-7-1.png" style="display: block; margin: auto;" />
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---

```r
set.seed(4343)
data <- tibble(x = rnorm(1000,0,10),
 y = -(3*x^2) + rnorm(100, 0, 45))
data1 <- data %>% filter(x<=-4)
model1 <- lm(y ~x, data = data1)
summary(model1)
```

<img src="SLAAS-stmodelling_files/figure-html/unnamed-chunk-9-1.png" style="display: block; margin: auto;" />
]

---
.pull-left[

```r
set.seed(4343)
data <- tibble(x = rnorm(1000,0,10),
 y = -(3*x^2) + rnorm(100, 0, 45))
model2 <- lm(y ~x, data = data)
summary(model2)
```

```
## 
## Call:
## lm(formula = y ~ x, data = data)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -4651.7 -91.9 142.2 248.3 413.9 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) -282.646 13.470 -20.983 < 2e-16 ***
## x -4.497 1.366 -3.292 0.00103 ** 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 425.7 on 998 degrees of freedom
## Multiple R-squared: 0.01074,	Adjusted R-squared: 0.009753 
## F-statistic: 10.84 on 1 and 998 DF, p-value: 0.001029
```
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<img src="SLAAS-stmodelling_files/figure-html/unnamed-chunk-11-1.png" style="display: block; margin: auto;" />
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---

```r
set.seed(4343)
data <- tibble(x = rnorm(1000,0,10),
 y = -(3*x^2) + rnorm(100, 0, 45))
model2 <- lm(y ~ poly(x,2), data = data)
summary(model2)
```

```
## 
## Call:
## lm(formula = y ~ poly(x, 2), data = data)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -131.035 -36.163 -0.415 39.876 123.578 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) -284.295 1.676 -169.58 <2e-16 ***
## poly(x, 2)1 -1401.415 53.015 -26.43 <2e-16 ***
## poly(x, 2)2 -13342.884 53.015 -251.68 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 53.02 on 997 degrees of freedom
## Multiple R-squared: 0.9847,	Adjusted R-squared: 0.9846 
## F-statistic: 3.202e+04 on 2 and 997 DF, p-value: < 2.2e-16
```
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<img src="SLAAS-stmodelling_files/figure-html/unnamed-chunk-13-1.png" style="display: block; margin: auto;" />
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---

---

# Modelling in R

---

**Explore data** using ggplot before modelling

```
ggplot(...) +
 geom_point(...) +
 geom_smooth(...)

```

```r
ggplot(data, aes(x,y))+
  geom_point() +
  geom_smooth(method = "lm", 
*             formula = y ~ x,
              col = "blue", se = FALSE) +
  geom_smooth(method = "lm", 
*             formula = y ~ poly(x,2),
              col = "red", se = FALSE) + 
  theme(aspect.ratio = 1) 
```

]
.pull-right[

]

---

`lm(y ~ x, data)`
 
 
- Assign the linear model to be an **object**

`model <- lm(y ~ x, data)`

```r
model1 <- lm(
* body_mass_g ~ flipper_length_mm,
 data = penguins)
```

].pull-right[

<img src="SLAAS-stmodelling_files/figure-html/unnamed-chunk-18-1.png" style="display: block; margin: auto;" />
]

---

`lm(y ~ x, data)`
 
 
- Assign the linear model to be an **object**

`model <- lm(y ~ x,`
` data)`

```r
model1 <- lm(
 body_mass_g ~ flipper_length_mm, 
 data = penguins)
*summary(model1)
```

]

```

Call:
lm(formula = flipper_length_mm ~ body_mass_g, data = penguins)

Residuals:
     Min       1Q   Median       3Q      Max 
-23.7626  -4.9138   0.9891   5.1166  16.6392

Coefficients:
 Estimate Std. Error t value Pr(>|t|) 
(Intercept) 1.367e+02 1.997e+00 68.47 <2e-16 ***
body_mass_g 1.528e-02 4.668e-04 32.72 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.913 on 340 degrees of freedom
 (2 observations deleted due to missingness)
Multiple R-squared: 0.759,	Adjusted R-squared: 0.7583 
F-statistic: 1071 on 1 and 340 DF, p-value: < 2.2e-16
```
]

--

```r
typeof(model1)
```

```
## [1] "list"
```
]

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# broom

## let's tidy up a bit

---
## `Broom` Toolkit

- The `broom` package takes the **messy output** of built-in functions in R, such as `lm`,  `t.test`, and turns them into tidy **tibbles**.

- This package provides three methods that do three distinct kinds of tidying.

- `tidy` 
     - `augment`
     - `glance`

---

## `broom::tidy`

- Returns the statistical findings of the model

- This includes coefficients and p-values for each term in a regression

```r
library(broom)

model1 <- lm( body_mass_g ~ flipper_length_mm, data = penguins)

*tidy(model1)
```

```
# A tibble: 2 × 5
 term estimate std.error statistic p.value
 <chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) -5781. 306. -18.9 5.59e- 55
2 flipper_length_mm 49.7 1.52 32.7 4.37e-107
```

---
## `broom::augment`

<!--
https://cran.r-project.org/web/packages/broom/vignettes/broom.html

Instead of viewing the coefficients, you might be interested in the fitted values and residuals for each of the original points in the regression. For this, use augment, which augments the original data with information from the model:
-->

-  Add columns to the original data that was modeled.

- This includes predictions, residuals, and cluster assignments.

```r
library(broom)

model1 <- lm( body_mass_g ~ flipper_length_mm, data = penguins)

*augment(model1)
```

```
# A tibble: 342 × 9
 .rownames body_mass_g flipper_length_… .fitted .resid .hat .sigma .cooksd
 <chr> <int> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
 1 1 3750 181 3212. 538. 0.00881 394. 8.34e-3
 2 2 3800 186 3461. 339. 0.00622 394. 2.33e-3
 3 3 3250 195 3908. -658. 0.00344 393. 4.83e-3
 4 5 3450 193 3808. -358. 0.00385 394. 1.60e-3
 5 6 3650 190 3659. -9.43 0.00469 395. 1.35e-6
 6 7 3625 181 3212. 413. 0.00881 394. 4.91e-3
 7 8 4675 195 3908. 767. 0.00344 393. 6.56e-3
 8 9 3475 193 3808. -333. 0.00385 394. 1.39e-3
 9 10 4250 190 3659. 591. 0.00469 394. 5.31e-3
10 11 3300 186 3461. -161. 0.00622 395. 5.23e-4
# … with 332 more rows, and 1 more variable: .std.resid <dbl>
```

<!--
augment creates a data frame that combines the original data with related information from the model
fit.

Now, in addition to the proportionBlack and ageInYears variables of the original data, we have columns like .fitted (value of Y predicted by the model for the corresponding value of X), .resid (difference between the actual Y and the predicted value), and a variety of other information for evalulating model uncertainty.

One thing we can do with this ???augmented??? data frame is to use it to better visualize and explore the model. For example, if we wanted to generate a figure highlighting the deviations from the model using vertical lines emanating from the regression line, we could do something like this:

-->
---

## Plotting Augmented Data
<!--One thing we can do with this ???augmented??? data frame is to use it to better visualize and explore the model. For example, if we wanted to generate a figure highlighting the deviations from the model using vertical lines emanating from the regression line, we could do something like this

-->

```r
model1 <- lm(
 body_mass_g ~ flipper_length_mm, 
 data = penguins)

*mod1_arg <- augment(model1)

mod1_arg %>%
  ggplot( aes(flipper_length_mm, .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, 
             color = "red",
             linetype='dashed') +
  labs(y = "Residuals",
       title = "Residual plot")
```

]

<img src="SLAAS-stmodelling_files/figure-html/unnamed-chunk-28-1.png" style="display: block; margin: auto;" />
]

---
## Plotting Augmented Data: qq-plots

```r
model1 <- lm(
 body_mass_g ~ flipper_length_mm, 
 data = penguins)

mod1_arg <- augment(model1)

mod1_arg %>%
* ggplot(aes(sample = .std.resid)) +
* geom_qq() +
* geom_qq_line(color="firebrick")
```

]

]

---
## `broom::glance`

<!--
https://cran.r-project.org/web/packages/broom/vignettes/broom.html

Finally, several summary statistics are computed for the entire regression, such as R^2 and the F-statistic. These can be accessed with the glance function:
-->

-  Returns a concise one-row summary of the model.

- This typically contains values such as `\(R^2\)`, adjusted `\(R^2\)`, and residual standard error that are computed once for the entire model.

```r
library(broom)

model1 <- lm( body_mass_g ~ flipper_length_mm, data = penguins)

*glance(model1)
```

```
# A tibble: 1 × 12
 r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
 <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.759 0.758 394. 1071. 4.37e-107 1 -2528. 5063. 5074.
# … with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
```

---
# Multiple regression

Multiple regression is also possible using the same function

## `lm(y ~ x, data)`
 
--
 
## `lm(y ~ x1 + x2, data)`

## `lm(y ~ x1 * x2, data)`

---

Similarly, **transformations** are also quite easy to apply

## `lm(log(y) ~ x, data)`

## `lm(y ~ log(x), data)`

## `lm(y ~ poly(x, 2), data)`

---
##  What about all the other kinds of model?

## `lm(y ~ x, data)`

X is **categorical**

].pull-right[

### Linear Regression
## `lm(y ~ x, data)`

X is **numerical**
]

---

### Common statistical tests are linear models (or: how to teach stats): https://lindeloev.github.io/tests-as-linear/

---

# Other kinds of models

- **Not all** tests are linear models

- For **non-normal** distributions of residuals, we have **generalised linear model**

## `glm(y ~ x, data, family = "")`

```
glm(y ~ x, data, family = "binomial")

glm(y ~ x, data, family = "poisson")

glm(y ~ x, data, family = "Gamma")
```

---
## CRAN Task Views

- The CRAN task view provides an overview of the packages available for **various research fields** and **methodologies**

https://cran.r-project.org/web/views/

---
# Key points

- **Know your data** - explore it before jumping into a formal model.

- If exploration brings up something interesting or unexpected, try to **incorporate this** into the model.

- Always consider the **trade-off** between something which is over-simplified and something which is over-complicated.

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pridiltal and thiyangt

This work was supported in part by RETINA research lab funded by the OWSD, a program unit of United Nations Educational, Scientific and Cultural Organization (UNESCO).