Visualizing the Model:
Beta Regression

Introduction

  • In the previous weeks, we have built up what we understand about data visualization.

    • Week 1: Visualizing models with only continuous predictors
    • Week 2: Visualizing models with only categorical predictors; visualizing models with both continuous and categorical predictors
    • Week 3: Visualizing models with interaction terms.

  • All three weeks, we were dealing with a continuous outcome that we assumed had a normal distribution.

  • In this lecture, we will focus on visualizing gamma regression models.

Lecture Example Set Up

  • Recall our data for task completion,

    • Task completion rate (0 to 1): the proportion of assigned tasks that were completed by the end of the shift
    • Character: who was leading the shift (Minnie or Daisy)
    • Location: which operations team they worked with (Main Street Operations, Toontown Crew, or Epcot Showcase)
    • Task load: the number of tasks assigned during the shift
    • Shift hours: how long the shift lasted (in hours)

  • We constructed the model,

\begin{align*} \text{logit}(\mu) = \ & 2.38 + 0.83 \text{ Minnie} - 0.04 \text{ task load} -0.08 \text{ shift hours } + \\ & 0.02 \text{ Main Street} - 0.31 \text{ Toontown} - \\ & 0.05 \text{ Minnie} \times \text{task} \end{align*}

Lecture Example Set Up

  • Pulling in the data,
operations <- read_csv("https://raw.githubusercontent.com/samanthaseals/SDSII/refs/heads/main/files/data/lectures/W4_daisy.csv")
operations %>% head()
  • From here, we can create our predicted values.

Creating Predicted Values

  • In order to create predicted values, we must isolate \mu.

\begin{align*} \text{logit}(\mu) &= \beta_0 + \beta_1 x_1 + ... + \beta_k x_k \\ \ln\left(\frac{\mu}{1 - \mu}\right) &= \beta_0 + \beta_1 x_1 + ... + \beta_k x_k \\ \frac{\mu}{1 - \mu} &= e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k} \\ \mu &= (1-\mu) e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k} \\ \mu &= e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k} - \mu(e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k}) \\ \mu + \mu(e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k}) &= e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k} \\ \mu(1+e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k}) &= e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k} \end{align*}

Creating Predicted Values

  • In order to create predicted values, we must isolate \mu.

\begin{align*} \text{logit}(\mu) &= \beta_0 + \beta_1 x_1 + ... + \beta_k x_k \\ & \ \ \vdots \\ \mu(1+e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k}) &= e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k} \\ \mu &= \frac{e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k}}{1 + e^{\beta_0 + \beta_1 x_1 + ... + \beta_k x_k}} \end{align*}

Creating Predicted Values: Example

  • Recall our stratified models.

  • For Minnie,

\begin{align*} \text{logit}(\mu) = \ & 3.21 - 0.09 \text{ task load} -0.08 \text{ shift hours } + \\ & 0.02 \text{ Main Street} - 0.31 \text{ Toontown} \end{align*}

  • For Daisy,

\begin{align*} \text{logit}(\mu) = \ & 2.38 - 0.04 \text{ task load} -0.08 \text{ shift hours } + \\ & 0.02 \text{ Main Street} - 0.31 \text{ Toontown} \end{align*}

Creating Predicted Values: Example

  • Let’s have the shift hours on the x-axis, task completion on the y-axis, and lines defined by character and location.

    • We will plug in the median task load.
operations <- operations %>%
  mutate(logit_minnie_ms = 3.21 - 0.09*median(task_load) - 0.08*shift_hours + 0.02*1 - 0.31*0,
         logit_minnie_tt = 3.21 - 0.09*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*1,
         logit_minnie_ws = 3.21 - 0.09*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*0,
         logit_daisy_ms = 2.38 - 0.04*median(task_load) - 0.08*shift_hours + 0.02*1 - 0.31*0,
         logit_daisy_tt = 2.38 - 0.04*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*1,
         logit_daisy_ws = 2.38 - 0.04*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*0)

Creating Predicted Values: Example

  • For demonstration purposes, let’s compare our current predicted values to the observed values,
operations %>% 
  select(completion_rate, 
         logit_minnie_ms, logit_minnie_tt, logit_minnie_ws, 
         logit_daisy_ms, logit_daisy_tt, logit_daisy_ws) %>% 
  head(n=4)

  • Like in gamma regression, we can see that the scaling is not correct.

    • There are predicted values greater than 1, which is impossible for a proportion.

Creating Predicted Values: Example

  • Trying again, but now transforming back to \mu using exp(),
operations <- operations %>%
  mutate(minnie_ms = exp(3.21 - 0.09*median(task_load) - 0.08*shift_hours + 0.02*1 - 0.31*0)/(1+exp(3.21 - 0.09*median(task_load) - 0.08*shift_hours + 0.02*1 - 0.31*0)),
         minnie_tt = exp(3.21 - 0.09*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*1)/(1+exp(3.21 - 0.09*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*1)),
         minnie_ws = exp(3.21 - 0.09*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*0)/(1+exp(3.21 - 0.09*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*0)),
         daisy_ms = exp(2.38 - 0.04*median(task_load) - 0.08*shift_hours + 0.02*1 - 0.31*0)/(1+exp(2.38 - 0.04*median(task_load) - 0.08*shift_hours + 0.02*1 - 0.31*0)),
         daisy_tt = exp(2.38 - 0.04*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*1)/(1+exp(2.38 - 0.04*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*1)),
         daisy_ws = exp(2.38 - 0.04*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*0)/(1+exp(2.38 - 0.04*median(task_load) - 0.08*shift_hours + 0.02*0 - 0.31*0)))

Creating Predicted Values: Example

  • For demonstration purposes, let’s compare our current predicted values to the observed values,
operations %>% 
  select(completion_rate, 
         minnie_ms, minnie_tt, minnie_ws, 
         daisy_ms, daisy_tt, daisy_ws) %>% 
  head(n=4)
  • All is right with our predicted values.

Visualization of the Model: Example

  • Now, we can build our visualizations as normal.

  • For Minnie,

operations %>% ggplot(aes(x = shift_hours)) +
  geom_point(aes(y = completion_rate, color = location), alpha = 0.5) +
  geom_line(aes(y = minnie_ms), color = "#00BA38", linewidth = 1) + 
  geom_line(aes(y = minnie_tt), color = "#00BFC4", linewidth = 1) + 
  geom_line(aes(y = minnie_ws), color = "#F8766D", linewidth = 1) + 
  labs(x = "Hours on Shift",
       y = "Task Completion",
       color = "Location") +
  theme_bw())

Visualization of the Model: Example

  • Now, we can build our visualizations as normal.

  • For Minnie,

Minnie's scatterplot of task completion rate versus shift hours with three regression lines, one for each location

Visualization of the Model: Example

  • Now, we can build our visualizations as normal.

  • For Daisy,

operations %>% ggplot(aes(x = shift_hours)) +
  geom_point(aes(y = completion_rate, color = location), alpha = 0.5) +
  geom_line(aes(y = daisy_ms), color = "#00BA38", linewidth = 1) + 
  geom_line(aes(y = daisy_tt), color = "#00BFC4", linewidth = 1) +
  geom_line(aes(y = daisy_ws), color = "#F8766D", linewidth = 1) + 
  labs(x = "Hours on Shift",
       y = "Task Completion",
       color = "Location") +
  theme_bw()

Visualization of the Model: Example

  • The resulting graph,
Daisy's scatterplot of task completion rate versus shift hours with three regression lines, one for each location

Visualization of the Model: Example

  • Putting these graphs side-by-side for comparison,
Side-by-side scatterplots of task completion rate versus shift hours with three regression lines, one for each location and character

Linking to Interpretations: Example

  • Now that we have the visualization, we can link it back to interpretations and inference. Recall,

  • The interaction between character and task load is significant (p < 0.001).

    • We cannot discuss the individual main effects of character or task load.

  • Shift length is a significant predictor of completion rate (p < 0.001).

  • Location is a significant predictor of completion rate (p = 0.008).

    • There is not a difference in the mean completion rate between Epcot’s World Showcase and Main Street (p = 0.867).

    • There is a difference in the mean completion rate between Epcot’s World Showcase and Toontown (p = 0.010).

Linking to Interpretations: Example

  • As the number of shift hours increases by 1 hour, the odds of the mean completion rate are multiplied by 0.92 – this is a decrease of approximately 8%.
Side-by-side scatterplots of task completion rate versus shift hours with three regression lines, one for each location and character

Linking to Interpretations: Example

  • While there is a difference in locations (p = 0.008), when comparing against World Showcase, there is no detectable difference with completion rates on Main Street (p = 0.867), but an increased completion rate in Toontown (p = 0.010).
Side-by-side scatterplots of task completion rate versus shift hours with three regression lines, one for each location and character

Linking to Interpretations: Example

  • Finally, we can see that on average, Daisy has a higher task completion rate than Minnie.
Side-by-side scatterplots of task completion rate versus shift hours with three regression lines, one for each location and character

Wrap Up

  • In this lecture, we learned how to visualize beta regression models.

  • Again, we are building upon what we have already learned in this course.

  • Now that we are venturing outside of the normal distribution, we need to think carefully about how to create predicted values.

    • Remember that we have to “undo” the logit() function to get the correct predicted values in beta regression.

  • Next week: Logistic regression