Nested Models & F-Tests

Notes and in-class exercises

Notes

  • You can download a template file to work with here.

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:

Exercises

Exercise 1: Nested Models

  1. 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: \(E[A \mid D] = \beta_0 + \beta_1 D\)
  1. 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: \(E[B \mid A, C] = \beta_0 + \beta_1 A + \beta_2 C + \beta_3 AC\)

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

  • Model 1 is nested in Model 2
  • Model 1 is nested in Model 3
  • Model 1 is nested in Model 4
  • Model 2 is nested in Model 3
  • Model 3 is nested in Model 2
  • Model 2 is nested in Model 6
  1. What is one (numeric) way to compare nested models? Explain how you would determine which model is “better” based on this metric.

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)
# 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.

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

Response Put your response here

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

  2. 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.

  3. 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.

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).

  1. State the null and alternative hypotheses associated with the research question in part (d).

\[ H_0: \]

\[ H_1: \]

  1. 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?

  2. 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.

  3. State the null and alternative hypotheses associated with the research question in part (g).

\[ H_0: \]

\[ H_1: \]

  1. 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?
  1. 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.

Exercise 3: More F-tests

  1. 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?

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

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

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?

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.

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

H0:

H1:

  1. 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!).

  2. 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)?







Solutions

Exercise 1: Nested Models

  1. Models 1, 2, 4 and 6.

  • Model 1 is nested in Model 2 TRUE
  • Model 1 is nested in Model 3 TRUE
  • Model 1 is nested in Model 4 FALSE
  • Model 2 is nested in Model 3 TRUE
  • Model 3 is nested in Model 2 FALSE
  • Model 2 is nested in Model 6 FALSE
  1. You could compare the Adjusted \(R^2\) values from each model, and note that the one with a higher adjusted \(R^2\) is better by this metric. Multiple \(R^2\) would not be a good metric, because the larger model (within the nesting structure) will always have a higher \(R^2\) value.

Exercise 2: F-Tests

  1. It is reasonable to assume that course grade varies by department as well as course level and instructor. Certain instructors may grade more strictly (or curve more) than others, and similarly, this can vary across department due to cultural norms within the department. As the level of a course gets higher, I would expect grades to perhaps get lower, since courses with higher numbers are expected to be more difficult. Then again, students perhaps “care” more about such courses, and may put in more effort to get a higher grade. I doubt semester plays a significant role in determining course grades, though it is possible that Fall semester first-years or Spring semester seniors have worse grades, on average. We don’t have course year as a variable in our data, so we would be unable to examine this relationship. As enrollment in a course goes up, I would expect grades to decrease, since professors have less time to dedicate to individual students when course enrollment is high.

  2. Explore the relationship between course grades and other variables in the data. Make at least four visualizations, and describe any patterns you observe.

library(stringr)
# Exploratory plots

# course grade vs. enrollment
MacGrades %>%
  ggplot(aes(enroll, grade)) +
  geom_jitter() +
  theme_classic() +
  ggtitle("Course grades by enrollment numbers")

# course grade vs. level
MacGrades %>%
  mutate(level = factor(level)) %>%
  ggplot(aes(y = grade, x = level)) +
  geom_boxplot() +
  ggtitle("Course grades by course level")

# course grade vs. level (treating grade as categorical)
MacGrades %>%
  filter(!is.na(grade)) %>%
  mutate(level = factor(level),
         grade = factor(grade)) %>%
  ggplot(aes(x = level, fill = grade)) +
  geom_bar(position = "fill") +
  scale_fill_viridis_d(option = "H") +
  theme_classic() +
  ggtitle("Course grades by course level")

# course grade vs. semester
MacGrades %>%
  filter(!is.na(grade)) %>%
  mutate(grade = factor(grade)) %>%
  ggplot(aes(x = sem, fill = grade)) +
  geom_bar(position = "fill") +
  scale_fill_viridis_d(option = "H") +
  theme_classic() +
  ggtitle("Course grades by semester") +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

# Let's do something fancy and check out how grades have changed over time... this will require
# some string manipulation

MacGrades$year <- MacGrades$sem %>%
  str_replace("FA", "") %>%
  str_replace("SP", "") %>%
  str_replace("S1", "") %>%
  str_replace("S2", "") %>% 
  str_replace("IT", "") %>% as.numeric()

MacGrades %>%
  ggplot(aes(year, grade)) +
  geom_jitter() +
  geom_smooth(method = "lm", se = FALSE) +
  theme_classic() +
  ggtitle("Course grades by year, with least-squares line")

In general, course grades seem to be associated with enrollment numbers. Specifically, when enrollments are greater than 50, we see very few students receiving a course grade lower than a 2.0, which is different than when enrollments are fewer than 50 students. There does not appear to be a clear relationship between course grade and course level, with the exception of 600-level courses. In these cases, every student received either an A or A-. It does seem like the proportion of students who received lower than a 2.67 is greater for 100-level courses than the other course levels.

# Make level categorical
MacGrades <- MacGrades %>%
  mutate(level = factor(level))
mod <- lm(grade ~ level, data = MacGrades)
summary(mod)

Call:
lm(formula = grade ~ level, data = MacGrades)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.4776 -0.3492  0.2089  0.5224  0.6508 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.34924    0.01208 277.166  < 2e-16 ***
level200     0.11183    0.01995   5.606 2.17e-08 ***
level300     0.09078    0.01949   4.659 3.25e-06 ***
level400     0.12835    0.03168   4.052 5.15e-05 ***
level600     0.63339    0.13624   4.649 3.41e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.5915 on 5704 degrees of freedom
  (437 observations deleted due to missingness)
Multiple R-squared:  0.01085,   Adjusted R-squared:  0.01016 
F-statistic: 15.65 on 4 and 5704 DF,  p-value: 9.713e-13

We observe that as course level goes up, course grades also tend to increase on average.

  1. State the null and alternative hypotheses associated with the research question in part (d).

\[ H_0: \beta_1 = \beta_2 = \beta_3 = \beta_4 = 0 \]

\[ H_1: \text{One of } \beta_1, \beta_2, \beta_3, \beta_4 \neq 0 \] In words, the null is that there is no relationship between course level and course grades, and the alternative is that there is some relationship (either positive or negative) between course level and course grades.

  1. The p-value associated with this hypothesis test is 9.713 x \(10^{-13}\). We do have enough evidence to reject the null hypothesis.

mod <- lm(grade ~ enroll, data = MacGrades)
summary(mod)

Call:
lm(formula = grade ~ enroll, data = MacGrades)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.4529 -0.3871  0.2265  0.5534  0.8448 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.4842350  0.0173790 200.485  < 2e-16 ***
enroll      -0.0031336  0.0006683  -4.689 2.81e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.5934 on 5707 degrees of freedom
  (437 observations deleted due to missingness)
Multiple R-squared:  0.003838,  Adjusted R-squared:  0.003664 
F-statistic: 21.99 on 1 and 5707 DF,  p-value: 2.806e-06

\[ H_0: \beta_1 = 0 \]

\[ H_1: \beta_1 \neq 0 \]

  1. The p-value associated with this hypothesis test is 2.806 x \(10^{-06}\). We do have enough evidence to reject the null hypothesis. Note that this p-value could be obtained from either the overall model fit or from the individual coefficient for enroll (they are the same). They may be ever so slightly different when there are few observations in your dataset, but when there are a lot, they will be exactly identical.
  1. We do not need to conduct an F-test, because our hypothesis test involves only a single regression coefficient, and therefore is readily obtained from the summary output of our linear model in R.

Exercise 3: More F-tests

\[ E[grade | enroll, level] = \beta_0 + \beta_1 enroll + \beta_2 level200 + \beta_3 level300 + \beta_4 level400 + \beta_5 level600 \]

The relevant coefficient that answers our scientific question is \(\beta_1\), or the coefficient that corresponds to enrollment.

The relevant null and alternative hypotheses are:

\[ H_0: \beta_1 = 0 \] \[ H_1: \beta_1 \neq 0 \] We do not need to conduct an F-test to complete this hypothesis testing procedure, since our hypothesis involves only a single regression coefficient.

mod <- lm(grade ~ enroll + level, data = MacGrades)
summary(mod)

Call:
lm(formula = grade ~ enroll + level, data = MacGrades)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.4916 -0.3481  0.1907  0.5162  0.7764 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.4091632  0.0219143 155.568  < 2e-16 ***
enroll      -0.0022628  0.0006907  -3.276 0.001058 ** 
level200     0.1040873  0.0200701   5.186 2.22e-07 ***
level300     0.0743387  0.0201066   3.697 0.000220 ***
level400     0.1118660  0.0320492   3.490 0.000486 ***
level600     0.6182478  0.1361974   4.539 5.76e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.591 on 5703 degrees of freedom
  (437 observations deleted due to missingness)
Multiple R-squared:  0.01271,   Adjusted R-squared:  0.01185 
F-statistic: 14.69 on 5 and 5703 DF,  p-value: 2.453e-14

We have statistically significant evidence of a relationship between enrollment and course grade, for courses of the same level (p = 0.001058). We reject the null hypothesis that there is no relationship between enrollment and course grade, adjusting for course level.

Exercise 4

Our model statement is identical to that in Exercise 3, but the relevant coefficients are \(\beta_2, \beta_3, \beta_4\), and \(\beta_5\).

The relevant null and alternative hypotheses are:

\[ H_0: \beta_2 = \beta_3 = \beta_4 = \beta_5 = 0 \]

\[ H_1: \text{At least one of } \beta_2, \beta_3, \beta_4, \beta_5 \neq 0 \]

We do need to conduct an F-test to complete this hypothesis testing procedure, since our hypothesis involves more than one regression coefficient.

# Same model as in Question 10, we just now need to do an F-test!
mod <- lm(grade ~ enroll + level, data = MacGrades)
smaller_mod <- lm(grade ~ enroll, data = MacGrades)

anova(smaller_mod, mod)
Analysis of Variance Table

Model 1: grade ~ enroll
Model 2: grade ~ enroll + level
  Res.Df    RSS Df Sum of Sq      F   Pr(>F)    
1   5707 2009.8                                 
2   5703 1991.9  4    17.904 12.815 2.19e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We have statistically significant evidence of a relationship between course level and course grade, for courses of the same enrollment (\(p = 2.19 \times 10^{-10}\)). We reject the null hypothesis that there is no relationship between course level and course grade, adjusting for enrollment.

Exercise 5: Reference categories

H0: There is no relationship between course grade and department.

H1: There is some relationship between course grade and department.

mod <- lm(grade ~ dept, data = MacGrades)
summary(mod)

Call:
lm(formula = grade ~ dept, data = MacGrades)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.5268 -0.2749  0.1475  0.4515  0.9039 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.5000000  0.4088676   8.560   <2e-16 ***
deptb       -0.2536066  0.4155163  -0.610    0.542    
deptB       -0.2614286  0.4278948  -0.611    0.541    
deptC        0.0268398  0.4106338   0.065    0.948    
deptd       -0.1825995  0.4098240  -0.446    0.656    
deptD        0.0617623  0.4105399   0.150    0.880    
depte       -0.0162500  0.4122607  -0.039    0.969    
deptE        0.1391667  0.4416275   0.315    0.753    
deptF       -0.2251373  0.4104679  -0.548    0.583    
deptg        0.0956522  0.4176615   0.229    0.819    
deptG       -0.3303306  0.4105537  -0.805    0.421    
deptH       -0.0307813  0.4120495  -0.075    0.940    
depti       -0.1474011  0.4111711  -0.358    0.720    
deptI        0.0011538  0.4243020   0.003    0.998    
deptj       -0.0625248  0.4108867  -0.152    0.879    
deptJ       -0.2815172  0.4116777  -0.684    0.494    
deptk        0.0124521  0.4104311   0.030    0.976    
deptK       -0.2431624  0.4123474  -0.590    0.555    
deptL        0.0468000  0.4169648   0.112    0.911    
deptm        0.0224798  0.4099682   0.055    0.956    
deptM       -0.4039440  0.4099067  -0.985    0.324    
deptn        0.1487363  0.4111080   0.362    0.718    
deptN        0.1670000  0.4139469   0.403    0.687    
depto       -0.3616667  0.4255629  -0.850    0.395    
deptO        0.0066856  0.4100242   0.016    0.987    
deptp       -0.0139370  0.4120744  -0.034    0.973    
deptP       -0.1140000  0.4249077  -0.268    0.788    
deptq        0.0001132  0.4104076   0.000    1.000    
deptQ       -0.0644094  0.4120744  -0.156    0.876    
deptR       -0.0381579  0.4110139  -0.093    0.926    
depts       -0.0154839  0.4218507  -0.037    0.971    
deptS       -0.0247500  0.4139469  -0.060    0.952    
deptt       -0.0126923  0.4166562  -0.030    0.976    
deptT       -0.2733962  0.4165106  -0.656    0.512    
deptU       -0.1136842  0.4298486  -0.264    0.791    
deptV       -0.0242500  0.4189646  -0.058    0.954    
deptW       -0.0507164  0.4100863  -0.124    0.902    
deptX       -0.1189865  0.4116209  -0.289    0.773    
deptY        0.0814894  0.4174763   0.195    0.845    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.5782 on 5670 degrees of freedom
  (437 observations deleted due to missingness)
Multiple R-squared:  0.06037,   Adjusted R-squared:  0.05407 
F-statistic: 9.586 on 38 and 5670 DF,  p-value: < 2.2e-16

We have statistically significant evidence of a relationship between department and course grades at a significance level of 0.05 (p-value < 2.2 x \(10^{-16}\)). We reject the null hypothesis that there is no relationship between course grade and department.

None of the individual p-values for department are significant! These p-values tell us about whether or not there is a statistically significant difference in course grades between each respective department and the reference department (Department “A”). This doesn’t contradict our answer to part (b) because there are different hypothesis tests that answer different questions!