04-nonlinear.rmd

---
title: "HW - Nonlinearity"
author: 
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
require(rms)
require(splines2)
```

# Q1

From RMS section 2.9 question 5.

In “The Class of 1988: A Statistical Portrait,” the College Board reported
mean SAT scores for each state.  The data is available online.

```{r, echo = FALSE}
d1 <- read.csv("https://biostat.app.vumc.org/wiki/pub/Main/MsciBiostatIIAssignments/sat.csv")
```

a. Fit a linear spline model.  See `?lsp` in the `rms` package to learn how to specify the knots.

```{r}
m1 <- ols(score ~ lsp(pct,???), data = d1)
```

b. Describe the strategy you used to determine the number and placement of the knots.

c. Create a partial effect plot of the model.

d. Write down the model.  (You don't need to plug-in the values of $\beta$.)

$$
\begin{aligned}
E[??|??] &= ??\\
V[??|??] &= ??
\end{aligned}
$$

e. What is the interpretation of $\beta_2$ in your model?

f. Fit the same data, but with a natural cubic splines.  Generate a partial effect plot of the model.  See `?nsp` in the `splines2` package to learn how to specify the knots.

```{r}
m2 <- lm(score ~ nsp(pct,???), data = d1)
```

g. Describe how you selected the location and number of knots in part f.

h. Using the function you estimated in part f, create a plot of derivates similar to the [one posted here](https://hbiostat.org/rmsc/genreg#cubic-spline-functions). 

i. Fit the same data, but with a restricted cubic splines.  Generate a partial effect plot of the model.

```{r}
m3 <- ols(score ~ rcs(pct,???), data = d1)
```

j. Describe how you selected the location and number of knots in part i.

k. Using the function you estimated in part i, create a plot of derivates similar to the [one posted here](https://hbiostat.org/rmsc/genreg#cubic-spline-functions). 

l. Using the plots from h and k, how are models `m2` and `m3` different?

m. Create 3 plots, one for each model, which shows the basis functions you used in the model.

# Submission instructions

1.  Within your course repo, create a folder called `04-nonlinear`
1.  Within the folder, create an .html or .pdf file with your solutions.
1.  Be prepared to share your solutions with the class when the deliverable is due
1.  The deliverable should be your own work.  You may **discuss**
    concepts with classmates, and you may **share** code or text.