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Update Exercise 1 (Multiple Regression).

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Marco Oesting 2023-10-09 17:30:26 +02:00
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material/3_wed/regression/MultipleRegressionBasics.qmd
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@ -89,17 +89,24 @@ lm(@formula(Volume ~ Girth), trees)
::: 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)`.
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
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.
::: {.callout-caution collapse="false"}
## Task 1
1. Generate $n=20$ covariates $\mathbf{x}$ randomly.
2. Given these covariates and true parameters $\beta_0=-3$, $\beta_1=2$
and $\sigma=0.5$, simulate responses from a linear model and
estimate the coefficients $\beta_0$ and $\beta_1$.
3. Play with different choices of the parameters above (including the
sample size $n$) to see the effects on the parameter estimates and
the $p$-values.
:::
## Multiple Regression Model
@ -236,23 +243,23 @@ $$
For the models above, these are:
+----------------+-----------------+----------------+
+----------------+----------------+----------------+
| Type of Data | Distribution | Link Function |
| | Family | |
+================+=================+================+
+================+================+================+
| continuous | Normal | identity: |
| | | |
| | | $$ |
| | | g(x)=x |
| | | $$ |
+----------------+-----------------+----------------+
+----------------+----------------+----------------+
| count | Poisson | log: |
| | | |
| | | $$ |
| | | \$\$ |
| | | |
| | | g(x) = \log(x) |
| | | $$ |
+----------------+-----------------+----------------+
| | | \$\$ |
+----------------+----------------+----------------+
| binary | Bernoulli | logit: |
| | | |
| | | $$ |
@ -267,7 +274,7 @@ For the models above, these are:
| | | { |
| | | x}{1-x}\right) |
| | | $$ |
+----------------+-----------------+----------------+
+----------------+----------------+----------------+
In general, the parameter vector $\beta$ is estimated via maximizing the
likelihood, i.e.,