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title: "Nested Models & F-Tests"
subtitle: "Notes and in-class exercises"
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# Notes

## Nested Models

"Nested models" are models where the covariates of one model are a subset of another model.
As an example, consider the following models for estimating the association between forced expiratory volume (FEV) and smoking status:

**Model 1:**

$$
E[FEV \mid smoke] = \beta_0 + \beta_1 smoke
$$

**Model 2:**

$$
E[FEV \mid smoke, age] = \beta_0 + \beta_1 smoke + \beta_2 age
$$
Here, Model 1 is "nested" inside Model 2, since the covariates included in Model 1 (only *smoke*) are a subset of those in Model 2 (both *smoke* and *age*).

An example of non-nested models are...

**Model 3:**

$$
E[FEV \mid smoke, height] = \beta_0 + \beta_1 smoke + \beta_2 height
$$

**Model 4:**

$$
E[FEV \mid smoke, sex] = \beta_0 + \beta_1 smoke + \beta_2 sex
$$

Here, even though Model 3 and Model 4 both contain *smoke* as explanatory variables, neither is nested in the other, since *sex* is not a part of Model 3, and *height* is not a part of Model 4.

## Learning goals

By the end of this activity, you should be able to:

-   Determine if one model is nested within another
-   Determine which null and alternative hypotheses require an f-test
-   Determine which f-tests require the use of the `anova` function in R vs. the overall f-test given in regular regression output
-   Interpret the results of an f-test in context

## Readings and videos

Please read the following notes *and* watch the following video **before** class:

-   Reading: Section 7.3.4 in the [STAT 155 Notes](https://mac-stat.github.io/Stat155Notes/)

-   Video: [F-Tests](https://youtu.be/XIHtAGXhlzQ) ([script](https://drive.google.com/file/d/1sH3q8GwektjDrC9vouV1NP-RyufhN3fC/view?usp=sharing))

# Exercises

## Exercise 1: Nested Models

a. Which of the following models are nested in the model $E[A \mid B, C, D] = \beta_0 + \beta_1 D + \beta_2 B + \beta_3 C + \beta_4 B * C$?

- Model 1: $E[A \mid B] = \beta_0 + \beta_1 B$
- Model 2: $E[A \mid B, D] = \beta_0 + \beta_1 B + \beta_2 D$
- Model 3: $E[B \mid C] = \beta_0 + \beta_1 C$
- Model 4: $E[A \mid B, C, D] = \beta_0 + \beta_1 B + \beta_2 C + \beta_3 D$
- Model 5: $E[A \mid B, C, D] = \beta_0 + \beta_1 C + \beta_2 B + \beta_3 D + \beta_4 B * D$
- Model 6: $E[A \mid D] = \beta_0 + \beta_1 D$

> Your response here

b. Consider the following models involving variables A, B, C, and D:

- Model 1: $E[A \mid B] = \beta_0 + \beta_1 B$
- Model 2: $E[A \mid B, C] = \beta_0 + \beta_1 B + \beta_2 C$
- Model 3: $E[A \mid B, C] = \beta_0 + \beta_1 B + \beta_2 C + \beta_3 BC$
- Model 4: $E[A \mid C, D] = \beta_0 + \beta_1 C + \beta_2 D$
- Model 5: $E[B \mid A] = \beta_0 + \beta_1 A$
- Model 6: $E[B \mid A, C] = \beta_0 + \beta_1 A + \beta_2 C + \beta_3 AC$

> Your response here

Determine for each of the following statements whether that statement is True or False.

- Model 1 is nested in Model 2

> Your response here

- Model 1 is nested in Model 3

> Your response here

- Model 1 is nested in Model 4

> Your response here

- Model 2 is nested in Model 3

> Your response here

- Model 3 is nested in Model 2

> Your response here

- Model 2 is nested in Model 6

> Your response here




c. What is one (numeric) way to compare nested models? Explain how you would determine which model is "better" based on this metric.

> Your response here

## Exercise 2: F-Tests

This exercise involves the `MacGrades.csv` dataset, which contains a sub-sample (to help preserve anonymity) of every grade assigned to a former Macalester graduating class. For each of the 6414 rows of data, the following information is provided (with a few missing values):

- `sessionID`: A section ID number
- `sid`: A student ID number
- `grade`: The grade obtained, as a numerical value (i.e. an A is a 4, an A- is a 3.67, etc.)
- `dept`: A department identifier (these have been made ambiguous to maintain anonymity)
- `level`: The course level (e.g. 100-, 200-, 300-, and 600-)
- `sem`: A semester identifier
- `enroll`: The section enrollment
- `iid`: An instructor identifier (these have been made ambiguous to maintain anonymity)

```{r, warning = FALSE, message = FALSE}
# load necessary packages
library(ggplot2)
library(dplyr)
library(readr)

# load datasets
MacGrades <- read_csv("https://mac-stat.github.io/data/MacGrades.csv")
```


**NOTE:** Questions (a) and (b), since they are exploratory in nature, can suck up a lot of time if you let them! For the sake of getting to the rest of the activity, please spend no more than ~5 minutes on them.

a. Hypothesize *two* relationships between the variables in the dataset (pick any two relationships you want!). Your response should be written in a paragraph form.

> Your response here

b. Explore the relationship between course grades and other variables in the data. Make *two* visualizations, *and* describe any patterns you observe. 

```{r}
# Two visualizations:

```


> Your response here


c. Note that the `level` variable is currently quantitative. For this activity, we'd like to treat it as categorical. Create a new variable `level_cat` so that we can consider level categorically in the following analysis.

```{r}

```



d. Suppose we are interested in the relationship between course level (categorical) and student grades. Using grade as your outcome variable, fit a linear regression model to investigate this question.

```{r}

```


Comment on the nature of the relationship between course level and student grades (this should not be a coefficient interpretation, but instead a description of a general trend, or lack thereof).

> Your response here

e. State the null and alternative hypotheses *in terms of $\beta$'s* associated with the research question in part (d).

$$
H_0: 
$$

$$
H_1:
$$

> Pause. Did you fill in the formulas above?

f. What is the p-value associated with this hypothesis test? Do we have enough evidence to reject the null hypothesis, using a significance threshold of 0.05?

> Your response here

g. Suppose we are interested in the relationship between course enrollment and student grades. Again, use grade as your outcome variable, and fit a linear regression model to investigate this question.

```{r}

```


h. State the null and alternative hypotheses *in terms of $\beta$'s* associated with the research question in part (g).

$$
H_0: 
$$

$$
H_1:
$$

> Pause. Did you fill in the formulas above?


i. What is the p-value associated with this hypothesis test? Do we have enough evidence to reject the null hypothesis, using a significance threshold of 0.05?

> Your response here


j. Do we need to conduct a *nested* F-test using the `anova` function to complete our hypothesis testing procedure for the research question posed in part (g)? Explain why or why not.

> Your response here

## Exercise 3: More F-tests


a. Suppose we are now interested in the association between course grade and enrollment for classes of the same level. Write a model statement in the form $E[Y | X] = ...$ that will produce a statistical model that will allow us to answer our scientific question. Replace Y and X, where appropriate, with response and predictor variables. 

$$
E[Y | X] = ___
$$

Which coefficient(s) in your model is the one that is relevant to your research question?

> Your response here

b. What are the relevant null and alternative hypotheses that address the scientific question in part (a)?

> Your response here

c. Fit the model you wrote in part (a), calculate a p-value, and report the results of the hypothesis test in part (b). 

```{r}

```


> Your response here


## Reflection

F-tests are useful when the null hypothesis you wish to test is such that *more than one* covariate is simultaneously equal to a specific number (typically zero). What scenarios, outside of those shown in this example, can you think of where a relevant scientific hypothesis you want to test involves more than one covariate being simultaneously equal to zero?

> **Response** Put your response here.

# Additional Practice

## Exercise 4

Repeat Exercise 3, supposing we are instead interested in the association between course grade and course level for classes of the same enrollment.

## Exercise 5: Reference categories

Our final research question pertains to whether or not there is a relationship between course grade and department. Again, use course grade as the outcome variable in your linear regression model.

a. State the null and alternative hypotheses *in colloquial language* associated with the relevant hypothesis test.

H0:

H1:

b. Fit a linear regression model, and conduct your hypothesis testing procedure to answer the research question posed in this Exercise. State your conclusions accordingly (you do not need to interpret any regression coefficients, just state and interpret the results of your hypothesis test!).

c. Are any of the individual department p-values significant?

What do these p-values tell us, and why is this *not* contradictory to your answer in part (b)?





