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Elaborate regression exercises.

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Marco Oesting 2023-10-09 16:39:16 +02:00
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74
material/3_wed/regression/MultipleRegressionBasics.qmd
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@ -65,8 +65,6 @@ 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)
@ -87,9 +85,13 @@ lm(@formula(Volume ~ Girth), trees)
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
- column `Pr(>|t|)`: $p$-values for the hypotheses $\beta_i=0$ for
$i=0,1$
:::callout.tip
The command `rand(n)` generates a sample of `n` "random" (i.e., uniformly distributed) random numbers.
:::
**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
@ -232,33 +234,35 @@ $$
For the models above, these are:
+---------------+-------------------+------------------+
| Type of Data | Distribution | Link Function |
| | Family | |
+===============+===================+==================+
| continuous | Normal | identity: |
| | | |
| | | $$ |
| | | g(x)=x |
| | | $$ |
+---------------+-------------------+------------------+
| count | Poisson | log: |
| | | |
| | | $$ |
| | | g(x) = \log(x) |
| | | $$ |
+---------------+-------------------+------------------+
| binary | Bernoulli | logit: |
| | | |
| | | $$ |
| | | g(x) = \log\left |
| | | ( |
| | | \ |
| | | f |
| | | ra |
| | | c{x}{1-x}\right) |
| | | $$ |
+---------------+-------------------+------------------+
+----------------+------------------+-----------------+
| Type of Data | Distribution | Link Function |
| | Family | |
+================+==================+=================+
| continuous | Normal | identity: |
| | | |
| | | $$ |
| | | g(x)=x |
| | | $$ |
+----------------+------------------+-----------------+
| count | Poisson | log: |
| | | |
| | | $$ |
| | | g(x) = \log(x) |
| | | $$ |
+----------------+------------------+-----------------+
| binary | Bernoulli | logit: |
| | | |
| | | $$ |
| | | g |
| | | (x) = \log\left |
| | | ( |
| | | \ |
| | | f |
| | | ra |
| | | c |
| | | {x}{1-x}\right) |
| | | $$ |
+----------------+------------------+-----------------+
In general, the parameter vector $\beta$ is estimated via maximizing the
likelihood, i.e.,
@ -289,6 +293,10 @@ 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.
::: callout-task
**Task 3**:
1. Reproduce the results of our data analysis of the `tree` data set using
a generalized linear model with normal distribution family.
2. Generate
:::