Statistics and Quantitative Methods (S1)

.title[
# Statistics and Quantitative Methods (S1)
]
.subtitle[
## Week 3
]
.author[
### Dr Stefano Coretta
]
.institute[
### University of Edinburgh
]
.date[
### 2022/10/04
]

---

# Slido quiz 1

<br>

???

Slido poll. <https://app.sli.do/event/krtv5FmoQy4ESjd45RhHQF>

---

class: inverse
background-image: url(../../img/time-travel.jpg)
background-size: contain

# Time travel...

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# Given the .green[formula], find the .orange[points]

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## You have:

## Find:

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---

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# Given the .orange[points], find the .green[formula]

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## You have:

## Find:

---

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# But...

Measurements are .red[noisy].
]

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# Given the .orange[points], find the .green[formula]

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# If `$x$` changes, what happens to `$y$`?

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# Now enter Linear Models

???

[via GIPHY](https://giphy.com/gifs/starwars-movie-star-wars-3owzVYjZSzuFivWpHi)

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# Linear model: the basics

---

---

---

---

<br>

<br>

We know `$x$` and `$y$` and we need to .green[estimate] `$\beta_0$` and `$\beta_1$`.

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???

<https://stefanocoretta.shinyapps.io/lines/>

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# Slido quiz 2

<br>

???

Slido poll. <https://app.sli.do/event/krtv5FmoQy4ESjd45RhHQF>

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# Linear model: the basics

---

---

```r
line_lm <- lm(y ~ x, data = line_2)
summary(line_lm)
```

```
## 
## Call:
## lm(formula = y ~ x, data = line_2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -25.2197  -5.2243   0.3434   6.2411  20.7275 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  1.99182    2.66626   0.747    0.459
## x            3.02079    0.04762  63.432   <2e-16
## 
## Residual standard error: 10.07 on 48 degrees of freedom
## Multiple R-squared:  0.9882,	Adjusted R-squared:  0.988 
## F-statistic:  4024 on 1 and 48 DF,  p-value: < 2.2e-16
```

---

---

# Standard error

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```r
summary(line_lm)
```

???

The **standard error** is a measure of (lack of) precision of the estimates.

The greater the standard error the least precise the estimate is.

Standard errors tend to decrease with more data.

---

???

In this plot you see a set of lines that represent some of the possible lines that are a good fit for the data (according to the model).

These lines are a random selection from all the possible lines that are compatible with the estimates and standard errors calculated by the model.

In other words, any of these could be the actual line that "generated" the data.
**We cannot be sure because of the error each estimate comes with!**

---

Let's try now with more data (N = 300).

```
## 
## Call:
## lm(formula = y ~ x, data = line_3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -29.718  -6.762  -0.188   6.479  40.657 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  2.10313    1.18137    1.78   0.0761
## x            2.99189    0.02079  143.94   <2e-16
## 
## Residual standard error: 10.05 on 298 degrees of freedom
## Multiple R-squared:  0.9858,	Adjusted R-squared:  0.9858 
## F-statistic: 2.072e+04 on 1 and 298 DF,  p-value: < 2.2e-16
```

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???

Compare this to the previous plot.

Now the lines are much less variable, because the error of the estimates is smaller.
(Of course it is, there is much more data to estimate those numbers with!)

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# Residuals

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---

???

A **residual** is the difference between the *y* value of a raw data point and the predicted value for the *x* value of that point.

One way to estimate linear models is to find the line that minimises the residuals across all data points.
There are other methods, but for the purpose of applied data analysis, it doesn't matter which one is used and specific R packages use specific methods.

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???

See how the residuals here are much larger, because the line is "off" relative to the data points?

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???

Another example of bad "fit" of the line (or model) to the data.