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Further elaboration of exxercises.

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Marco Oesting
2023-10-09 16:56:18 +02:00
parent 93a30acb76
commit 19d4618a53
1 changed files with 41 additions and 36 deletions
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29
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 $(\hat \beta_0, \hat \beta_1)$. We will not make use of it here, but
rather use Julia to solve the problem. rather use Julia to solve the problem.
``` julia ``` julia
lm(@formula(Volume ~ Girth), trees) lm(@formula(Volume ~ Girth), trees)
``` ```
@@ -88,8 +87,11 @@ lm(@formula(Volume ~ Girth), trees)
- column `Pr(>|t|)`: $p$-values for the hypotheses $\beta_i=0$ for - column `Pr(>|t|)`: $p$-values for the hypotheses $\beta_i=0$ for
$i=0,1$ $i=0,1$
:::callout.tip ::: callout-tip
The command `rand(n)` generates a sample of `n` "random" (i.e., uniformly distributed) random numbers. 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 **Task 1**: Generate a random set of covariates $\mathbf{x}$. Given
@@ -234,35 +236,38 @@ $$
For the models above, these are: For the models above, these are:
+----------------+------------------+-----------------+ +----------------+-----------------+----------------+
| Type of Data | Distribution | Link Function | | Type of Data | Distribution | Link Function |
| | Family | | | | Family | |
+================+==================+=================+ +================+=================+================+
| continuous | Normal | identity: | | continuous | Normal | identity: |
| | | | | | | |
| | | $$ | | | | $$ |
| | | g(x)=x | | | | g(x)=x |
| | | $$ | | | | $$ |
+----------------+------------------+-----------------+ +----------------+-----------------+----------------+
| count | Poisson | log: | | count | Poisson | log: |
| | | | | | | |
| | | $$ | | | | $$ |
| | | |
| | | g(x) = \log(x) | | | | g(x) = \log(x) |
| | | $$ | | | | $$ |
+----------------+------------------+-----------------+ +----------------+-----------------+----------------+
| binary | Bernoulli | logit: | | binary | Bernoulli | logit: |
| | | | | | | |
| | | $$ | | | | $$ |
| | | g | | | | g |
| | | (x) = \log\left | | | | ( |
| | | x) = \log\left |
| | | ( | | | | ( |
| | | \ | | | | \ |
| | | f | | | | f |
| | | ra | | | | ra |
| | | c | | | | c |
| | | {x}{1-x}\right) | | | | { |
| | | x}{1-x}\right) |
| | | $$ | | | | $$ |
+----------------+------------------+-----------------+ +----------------+-----------------+----------------+
In general, the parameter vector $\beta$ is estimated via maximizing the In general, the parameter vector $\beta$ is estimated via maximizing the
likelihood, i.e., likelihood, i.e.,
@@ -296,7 +301,7 @@ model = glm(@formula(participation ~ age^2),
::: callout-task ::: callout-task
**Task 3**: **Task 3**:
1. Reproduce the results of our data analysis of the `tree` data set using 1. Reproduce the results of our data analysis of the `tree` data set
a generalized linear model with normal distribution family. using a generalized linear model with normal distribution family.
2. Generate 2. Generate
::: :::