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Update Exercises 2 and 3 (Multiple Regression).

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Marco Oesting 2023-10-09 17:48:50 +02:00
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material/3_wed/regression/MultipleRegressionBasics.qmd
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@ -100,9 +100,10 @@ normal distribution with mean 0.0 and standard deviation 2.0, and sample
## 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 $\beta_1$.
2. Given these covariates and the true parameters $\beta_0=-3$,
$\beta_1=2$ and $\sigma=0.5$, simulate responses from a linear model
(with normally distributed errors) 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.
@ -189,10 +190,18 @@ regression model, but we provide explicit formulas now:
p\text{-value} = \mathbb{P}(|T| > t_i), \quad \text{where } T \sim t_{n-p}
$$
**Task 2**: Implement functions that estimate the $\beta$-parameters,
the corresponding standard errors and the $t$-statistics. Test your
functions with the \`\`\`tree''' data set and try to reproduce the
::: {.callout-caution collapse="false"}
## Task 2
1. Implement functions that estimate the $\beta$-parameters,
the corresponding standard errors and the $t$-statistics.
2. Test your functions with the `tree' data set and try to reproduce the
output above.
:::
Which model is the best? For linear models, one often uses the $R^2$ characteristic.
Roughly speaking, it gives the percentage (between 0 and 1) of the variance that can be explained by the linear model.
``` julia
r2(linmod1)
@ -305,10 +314,11 @@ model = glm(@formula(participation ~ age^2),
SwissLabor, Binomial(), ProbitLink())
```
::: callout-task
**Task 3**:
::: {.callout-caution collapse="false"}
##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
2. Generate $n=20$ random covariates $\mathbf{x}$ and Poisson-distributed counting data with parameters $\beta_0 + \beta_1 x_i$. Re-estimate the parameters by a generalized linear model.
:::