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# Multiple Regression Basics

## Motivation

### Introductory Example: tree dataset from R

```{julia}
using Statistics
using Plots
using RDatasets

trees = dataset("datasets", "trees")

scatter(trees.Volume, trees.Girth,
        legend=false, xlabel="Girth", ylabel="Volume")
```

*Aim:* Find relationship between the *response variable* `volume` and
the *explanatory variable/covariate* `girth`? Can we predict the volume
of a tree given its girth?

```{julia}
scatter(trees.Girth, trees.Volume,
        legend=false, xlabel="Girth", ylabel="Volume")
plot!(x -> -37 + 5*x)
```

First Guess: There is a linear relation!

## Simple Linear Regression

Main assumption: up to some error term, each measurement of the response
variable $y_i$ depends linearly on the corresponding value $x_i$ of the
covariate

$\leadsto$ **(Simple) Linear Model:**
$$y_i = \beta_0 + \beta_1 x_i + \varepsilon_i,   \qquad  i=1,...,n,$$
where $\varepsilon_i \sim \mathcal{N}(0,\sigma^2)$ are independent
normally distributed errors with unknown variance $\sigma^2$.

*Task:* Find the straight line that fits best, i.e., find the *optimal*
estimators for $\beta_0$ and $\beta_1$.

*Typical choice*: Least squares estimator (= maximum likelihood
estimator for normal errors)

$$ (\hat \beta_0, \hat \beta_1) = \mathrm{argmin} \ \| \mathbf{y} - \mathbf{1} \beta_0 - \mathbf{x} \beta_1\|^2 $$

where $\mathbf{y}$ is the vector of responses, $\mathbf{x}$ is the
vector of covariates and $\mathbf{1}$ is a vector of ones.

Written in matrix style:

$$
 (\hat \beta_0, \hat \beta_1) = \mathrm{argmin} \ \left\| \mathbf{y} - (\mathbf{1},\mathbf{x}) \left( \begin{array}{c} \beta_0\\ \beta_1\end{array}\right) \right\|^2
$$

Note: There is a closed-form expression for
$(\hat \beta_0, \hat \beta_1)$. We will not make use of it here, but
rather use Julia to solve the problem.

\[use Julia code (existing package) to perform linear regression for
`volume ~ girth`\]

```{julia}
lm(@formula(Volume ~ Girth), trees)
```

*Interpretation of the Julia output:*

-   column `estimate` : least square estimates for $\hat \beta_0$ and
    $\hat \beta_1$

-   column `Std. Error` : estimated standard deviation
    $\hat s_{\hat \beta_i}$ of the estimator $\hat \beta_i$

-   column `t value` : value of the $t$-statistics

    $$ t_i = {\hat \beta_i \over \hat s_{\hat \beta_i}}, \quad i=0,1, $$

    Under the hypothesis $\beta_i=0$, the test statistics $t_i$ would
    follow a $t$-distribution.

-   column `Pr(>|t|)`: $p$-values for the hyptheses $\beta_i=0$ for
    $i=0,1$

**Task 1**: Generate a random set of covariates $\mathbf{x}$. Given
these covariates and true parameters $\beta_0$, $\beta_1$ and $\sigma^2$
(you can choose them)), simulate responses from a linear model and
estimate the coefficients $\beta_0$ and $\beta_1$. Play with different
choices of the parameters to see the effects on the parameter estimates
and the $p$-values.

## Multiple Regression Model

*Idea*: Generalize the simple linear regression model to multiple
covariates, w.g., predict `volume` using `girth` and \`height\`\`.

$\leadsto$ **Linear Model:**
$$y_i = \beta_0 + \beta_1 x_{i1} + \ldots + \beta_p x_{ip} + \varepsilon_i,   \qquad  i=1,...,n,$$where

-   $y_i$: $i$-th measurement of the response,

-   $x_{i1}$: $i$ th value of first covariate,

-   ...

-   $x_{ip}$: $i$-th value of $p$-th covariate,

-   $\varepsilon_i \sim \mathcal{N}(0,\sigma^2)$: independent normally
    distributed errors with unknown variance $\sigma^2$.

*Task:* Find the *optimal* estimators for
$\mathbf{\beta} = (\beta_0, \beta_1, \ldots, \beta_p)$.

*Our choice again:* Least squares estimator (= maximum likelihood
estimator for normal errors)

$$
 \hat \beta = \mathrm{argmin} \ \| \mathbf{y} - \mathbf{1} \beta_0 - \mathbf{x}_1 \beta_1 - \ldots - \mathbf{x}_p \beta_p\|^2
$$

where $\mathbf{y}$ is the vector of responses, $\mathbf{x}$\_j is the
vector of the $j$ th covariate and $\mathbf{1}$ is a vector of ones.

Written in matrix style:

$$
\mathbf{\hat \beta} = \mathrm{argmin} \ \left\| \mathbf{y} - (\mathbf{1},\mathbf{x}_1,\ldots,\mathbf{x}_p) \left( \begin{array}{c} \beta_0 \\ \beta_1 \\ \vdots \\ \beta_p\end{array} \right) \right\|^2
$$

Defining the *design matrix*

$$ \mathbf{X} = \left( \begin{array}{cccc}
                1 & x_{11} & \ldots & x_{1p} \\
                \vdots & \vdots & \ddots & \vdots \\
                1 & x_{11} & \ldots & x_{1p} 
                \end{array}\right) \qquad 
  (\text{size } n \times (p+1)), $$

we get the short form

$$
\mathbf{\hat \beta} = \mathrm{argmin} \ \| \mathbf{y} - \mathbf{X} \mathbf{\beta}  \|^2 = (\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{y}
$$

\[use Julia code (existing package) to perform linear regression for
`volume ~ girth + height`\]

The interpretation of the Julia output is similar to the simple linear
regression model, but we provide explicit formulas now:

-   parameter estimates:

    $$
     (\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{y}
    $$

-   estimated standard errors:

    $$
     \hat s_{\beta_i} = \sqrt{([\mathbf{X}^\top \mathbf{X}]^{-1})_{ii} \frac 1 {n-p} \|\mathbf{y} - \mathbf{X} \beta\|^2}
    $$

-   $t$-statistics:

    $$ t_i = \frac{\hat \beta_i}{\hat s_{\hat \beta_i}}, \qquad i=0,\ldots,p. $$

-   $p$-values:

    $$
    p\text{-value} = \mathbb{P}(|T| > t_i), \quad \text{where } T \sim t_{n-p}
    $$

**Task 2**: Implement functions that estimate the $\beta$-parameters,
the corresponding standard errors and the $t$-statistics. Test your
functions with the \`\`\`tree''' data set and try to reproduce the
output above.

```{julia}
r2(linmod1)
r2(linmod2)

linmod3 = lm(@formula(Volume ~ Girth + Height + Girth*Height), trees)

r2(linmod3)
```

## Generalized Linear Models

Classical linear model

$$
 \mathbf{y} = \mathbf{X} \beta + \varepsilon 
$$

implies that
$$ \mathbf{y} \mid \mathbf{X} \sim \mathcal{N}(\mathbf{X} \mathbf{\beta}, \sigma^2\mathbf{I}).$$

In particular, the conditional expectation satisfies
$\mathbb{E}(\mathbf{y} \mid \mathbf{X}) = \mathbf{X} \beta$.

However, the assumption of a normal distribution is unrealistic for
non-continuous data. Popular alternatives include:

-   for counting data: $$
       \mathbf{y} \mid \mathbf{X} \sim \mathrm{Poisson}(\exp(\mathbf{X}\mathbf{\beta})) \qquad \leadsto  \mathbb{E}(\mathbf{y} \mid \mathbf{X}) = \exp(\mathbf{X} \beta)
      $$

    Here, the components are considered to be independent and the
    exponential function is applied componentwise.

-   for binary data: $$
       \mathbf{y} \mid \mathbf{X} \sim \mathrm{Bernoulli}\left( \frac{\exp(\mathbf{X}\mathbf{\beta})}{1 + \exp(\mathbf{X}\mathbf{\beta})} \right) \qquad \leadsto  \mathbb{E}(\mathbf{y} \mid \mathbf{X}) = \frac{\exp(\mathbf{X}\mathbf{\beta})}{1 + \exp(\mathbf{X}\mathbf{\beta})}
      $$

    Again, the components are considered to be independent and all the
    operations are applied componentwise.

All these models are defined by the choice of a family of distributions
and a function $g$ (the so-called *link function*) such that

$$
 \mathbb{E}(\mathbf{y} \mid \mathbf{X}) = g^{-1}(\mathbf{X} \beta).
$$

For the models above, these are:

+--------------+---------------------+--------------------+
| Type of Data | Distribution Family | Link Function      |
+==============+=====================+====================+
| continuous   | Normal              | identity:          |
|              |                     |                    |
|              |                     | $$                 |
|              |                     | g(x)=x             |
|              |                     | $$                 |
+--------------+---------------------+--------------------+
| count        | Poisson             | log:               |
|              |                     |                    |
|              |                     | $$                 |
|              |                     |  g(x) = \log(x)    |
|              |                     | $$                 |
+--------------+---------------------+--------------------+
| binary       | Bernoulli           | logit:             |
|              |                     |                    |
|              |                     | $$                 |
|              |                     | g(x) = \log\left   |
|              |                     | (                  |
|              |                     | \                  |
|              |                     | f                  |
|              |                     | rac{x}{1-x}\right) |
|              |                     | $$                 |
+--------------+---------------------+--------------------+

In general, the parameter vector $\beta$ is estimated via maximizing the
likelihood, i.e.,

$$
\hat \beta = \mathrm{argmax} \prod_{i=1}^n f(y_i \mid \mathbf{X}_{\cdot i}),
$$

which is equivalent to the maximization of the log-likelihood, i.e.,

$$
\hat \beta = \mathrm{argmax} \sum_{i=1}^n \log f(y_i \mid \mathbf{X}_{\cdot i}),
$$

In the Gaussian case, the maximum likelihood estimator is identical to
the least squares estimator considered above.

```{julia}
using CSV
using HTTP

http_response = HTTP.get("https://vincentarelbundock.github.io/Rdatasets/csv/AER/SwissLabor.csv")
SwissLabor = DataFrame(CSV.File(http_response.body))

SwissLabor[!,"participation"] .= (SwissLabor.participation .== "yes")

model = glm(@formula(participation ~ age^2), 
            SwissLabor, Binomial(), ProbitLink())
```

**Task 3:** Reproduce the results of our data analysis of the `tree`
data set using a generalized linear model with normal distribution
family.
