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Further elaboration of exxercises.

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Marco Oesting 2023-10-09 16:56:18 +02:00
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material/3_wed/regression/MultipleRegressionBasics.qmd
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@ -65,7 +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.
``` julia
lm(@formula(Volume ~ Girth), trees)
```
@ -87,9 +86,12 @@ lm(@formula(Volume ~ Girth), trees)
- 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.
::: callout-tip
The command `rand(n)` generates a sample of `n` "random" (i.e.,
uniformly distributed) random numbers. If you want to sample from another distribution, use the `Distributions` package, define an object being the distribution of interest, e.g. `d = Normal(0.0, 2.0)` for a normal distribution
with mean 0.0 and standard deviation 2.0, and sample `n` times from this
distribution by `rand(d, n)`.
:::
**Task 1**: Generate a random set of covariates $\mathbf{x}$. Given
@ -234,35 +236,38 @@ $$
For the models above, these are:
+----------------+------------------+-----------------+
| Type of Data | Distribution | Link Function |
| | Family | |
+================+==================+=================+
| continuous | Normal | identity: |
| | | |
| | | $$ |
| | | 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.,
@ -296,7 +301,7 @@ model = glm(@formula(participation ~ age^2),
::: 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
1. Reproduce the results of our data analysis of the `tree` data set
using a generalized linear model with normal distribution family.
2. Generate
:::