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Marco Oesting
2023-10-09 16:14:00 +02:00
parent 08951f610d
commit 39892ad1c1
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material/3_wed/regression/MultipleRegressionBasics.qmd
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@@ -10,7 +10,7 @@ editor:
### Introductory Example: tree dataset from R ### Introductory Example: tree dataset from R
``` julia ``` julia
using Statistics using Statistics
using Plots using Plots
using RDatasets using RDatasets
@@ -25,7 +25,7 @@ scatter(trees.Volume, trees.Girth,
the *explanatory variable/covariate* `girth`? Can we predict the volume the *explanatory variable/covariate* `girth`? Can we predict the volume
of a tree given its girth? of a tree given its girth?
``` julia ``` julia
scatter(trees.Girth, trees.Volume, scatter(trees.Girth, trees.Volume,
legend=false, xlabel="Girth", ylabel="Volume") legend=false, xlabel="Girth", ylabel="Volume")
plot!(x -> -37 + 5*x) plot!(x -> -37 + 5*x)
@@ -68,7 +68,7 @@ rather use Julia to solve the problem.
\[use Julia code (existing package) to perform linear regression for \[use Julia code (existing package) to perform linear regression for
`volume ~ girth`\] `volume ~ girth`\]
``` julia ``` julia
lm(@formula(Volume ~ Girth), trees) lm(@formula(Volume ~ Girth), trees)
``` ```
@@ -183,7 +183,7 @@ the corresponding standard errors and the $t$-statistics. Test your
functions with the \`\`\`tree''' data set and try to reproduce the functions with the \`\`\`tree''' data set and try to reproduce the
output above. output above.
``` julia ``` julia
r2(linmod1) r2(linmod1)
r2(linmod2) r2(linmod2)
@@ -232,31 +232,33 @@ $$
For the models above, these are: For the models above, these are:
+--------------+---------------------+--------------------+ +---------------+-------------------+------------------+
| Type of Data | Distribution Family | Link Function | | Type of Data | Distribution | Link Function |
+==============+=====================+====================+ | | Family | |
| continuous | Normal | identity: | +===============+===================+==================+
| | | | | continuous | Normal | identity: |
| | | $$ | | | | |
| | | 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(x) = \log\left | | | | $$ |
| | | ( | | | | g(x) = \log\left |
| | | \ | | | | ( |
| | | f | | | | \ |
| | | rac{x}{1-x}\right) | | | | f |
| | | $$ | | | | ra |
+--------------+---------------------+--------------------+ | | | c{x}{1-x}\right) |
| | | $$ |
+---------------+-------------------+------------------+
In general, the parameter vector $\beta$ is estimated via maximizing the In general, the parameter vector $\beta$ is estimated via maximizing the
likelihood, i.e., likelihood, i.e.,
@@ -274,7 +276,7 @@ $$
In the Gaussian case, the maximum likelihood estimator is identical to In the Gaussian case, the maximum likelihood estimator is identical to
the least squares estimator considered above. the least squares estimator considered above.
``` julia ``` julia
using CSV using CSV
using HTTP using HTTP