---
title: "Lab 8: How Good Is This Model?"
author: "Your Name Here"
date: "`r Sys.Date()`"
format: docx
editor_options:
  chunk_output_type: console
  markdown:
    wrap: 72
---

```{r setup, message = FALSE, warning = FALSE}
library(tidyverse)
library(rio)
library(here)
library(broom)

coping <- import(here("data", "stress.csv"))
```

# Lab 8 — How Good Is This Model?

**Due Tuesday, October 13, 11:59 pm** (during Fall Break — submitted
online as usual). Render to Word and submit both files.

118 undergraduates reported their current `Anxiety`, their perceived
`Stress` over the past month, and their perceived social `Support`. Half
completed a brief coping workshop (`Condition`).

Answer under each **Your Answer:** line, in complete sentences.

------------------------------------------------------------------------

## Exercise 1 — Describe the data (3 points)

a)  Report n, mean and SD for `Anxiety`, `Stress`, and `Support`.
b)  Produce a correlation matrix of those three variables.
c)  Before fitting anything: based on the correlations alone, which
    variable looks like the better predictor of anxiety?

```{r}
# Your code here

```

**Your Answer:**

------------------------------------------------------------------------

## Exercise 2 — Model 1: stress alone (4 points)

Fit `Anxiety ~ Stress` and print the full summary.

```{r}
# Your code here

```

a)  Report the slope, its *t*, and its *p*-value.
b)  Report R² **and** adjusted R².
c)  The adjusted R² is **negative**. What does that mean? (It is not a
    typo and not an error.)
d)  Compare the residual standard error to the SD of `Anxiety`. What
    does that comparison tell you about this model?

**Your Answer:**

------------------------------------------------------------------------

## Exercise 3 — Model 2: support alone (4 points)

Fit `Anxiety ~ Support`.

```{r}
# Your code here

```

a)  Report the slope and interpret it in the units of the study.
b)  Report R² and the residual standard error.
c)  How much better is this model than Model 1? Use both R² and the
    residual SE in your answer.
d)  Is support a *cause* of lower anxiety? Why or why not?

**Your Answer:**

------------------------------------------------------------------------

## Exercise 4 — Partition the variance (4 points)

For **Model 2**, verify the decomposition by hand.

```{r}
# Hint: anova(model2) gives you the sums of squares,
# and augment() gives you fitted values and residuals.

```

a)  Report $SS_{model}$ and $SS_{residual}$.
b)  Add them. Show that the total equals
    $\sum(Y_i - \bar{Y})^2$ computed directly from `Anxiety`.
c)  Compute R² as $SS_{model} / SS_{total}$ and confirm it matches the
    R² the summary printed.
d)  In one sentence: what does $SS_{residual}$ represent, in terms of
    actual people in this study?

**Your Answer:**

------------------------------------------------------------------------

## Exercise 5 — Fit in real units (3 points)

A campus counseling center is deciding whether to use a support
questionnaire to flag students who may need follow-up.

a)  Using Model 2, what is the typical size of the model's error, in
    anxiety points?
b)  The anxiety scale in this sample runs from about 0.7 to 14.6. Given
    your answer to (a), would you be comfortable using this model to
    predict an individual student's anxiety? Explain.
c)  R² = .30 is a *large* effect by psychology standards. Reconcile that
    with your answer to (b).

**Your Answer:**

------------------------------------------------------------------------

## Exercise 6 — 👀 Sneak peek: both predictors (6 points)

Now fit `Anxiety ~ Stress + Support`.

```{r}
# Your code here

```

a)  Report R² and adjusted R² for this model.
b)  Compute $\Delta R^2$ relative to Model 2 (support alone).
c)  **Look at the `Stress` coefficient.** Compare it to Model 1. What
    changed — the estimate, its sign, its significance, or all three?
d)  In Model 1, stress was unrelated to anxiety. In Model 3, it is a
    significant **positive** predictor. Both results come from the same
    118 people. How is that possible?
e)  Look at the correlation between `Stress` and `Support` from Exercise
    1. How might that explain what you just saw?
f)  Which model would you report, and why?

::: callout-note
You are not expected to know the technical name for this. Week 8 is
where we unpack it. Describe what you see and make your best reasoned
guess — (d) and (e) are graded on the quality of the reasoning, not on
getting the vocabulary right.
:::

**Your Answer:**

------------------------------------------------------------------------

## Exercise 7 — Judgment (2 points)

Suppose the researcher had only run Model 1, seen *p* = .61, and
concluded: *"Perceived stress is unrelated to anxiety in college
students."*

a)  What is wrong with that conclusion, given everything you now know?
b)  What does this tell you about reporting a single-predictor model as
    if it were the whole story?

**Your Answer:**

------------------------------------------------------------------------

## Exercise 8 — Write it up (2 points)

Write **one paragraph** reporting Model 3 as it would appear in a Results
section: the overall model test, R², both coefficients, and one sentence
about what it means.

**Your Answer:**

------------------------------------------------------------------------

## Bonus (+1 point)

Add `Condition` to the model (`Anxiety ~ Stress + Support + Condition`).
Does the coping workshop predict anxiety once stress and support are
accounted for? Report the coefficient and say what you conclude.

```{r}
# Your code here

```

**Your Answer:**

------------------------------------------------------------------------

**Total: 28 points (+1 bonus)**

*Remember to render to Word before submitting.*