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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 ::: callout-tip
The command `rand(n)` generates a sample of `n` "random" (i.e., 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 uniformly distributed) random numbers. If you want to sample from
with mean 0.0 and standard deviation 2.0, and sample `n` times from this another distribution, use the `Distributions` package, define an object
distribution by `rand(d, n)`. 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 ::: {.callout-caution collapse="false"}
these covariates and true parameters $\beta_0$, $\beta_1$ and $\sigma^2$ ## Task 1
(you can choose them)), simulate responses from a linear model and
estimate the coefficients $\beta_0$ and $\beta_1$. Play with different 1. Generate $n=20$ covariates $\mathbf{x}$ randomly.
choices of the parameters to see the effects on the parameter estimates 2. Given these covariates and true parameters $\beta_0=-3$, $\beta_1=2$
and the $p$-values. 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 ## Multiple Regression Model
@ -236,38 +243,38 @@ $$
For the models above, these are: For the models above, these are:
+----------------+-----------------+----------------+ +----------------+----------------+----------------+
| Type of Data | Distribution | Link Function | | Type of Data | Distribution | Link Function |
| | Family | | | | Family | |
+================+=================+================+ +================+================+================+
| continuous | Normal | identity: | | continuous | Normal | identity: |
| | | | | | | |
| | | $$ | | | | $$ |
| | | g(x)=x | | | | g(x)=x |
| | | $$ | | | | $$ |
+----------------+-----------------+----------------+ +----------------+----------------+----------------+
| count | Poisson | log: | | count | Poisson | log: |
| | | | | | | |
| | | $$ | | | | \$\$ |
| | | | | | | |
| | | 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.,