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 wait times at Magic Kingdom,

  • wait_time: posted standby wait time (minutes)
  • temp_f: outdoor temperature (°F)
  • crowd_index: a 1–10 index summarizing park busyness
  • ride_type: “family”, “dark”, or “thrill”

• We constructed the model,

\ln(y) = 1.28 + 0.015 \text{ temp} + 0.11 \text{ crowd} - 0.19 \text{ family} + 0.55 \text{ thrill}

Lecture Example Set Up

• Pulling in the data,

mk_wait <- read_csv("https://raw.githubusercontent.com/samanthaseals/SDSII/refs/heads/main/files/data/lectures/W4_wait_times.csv")
mk_wait %>% head()

• From here, we can create our predicted values.

Creating Predicted Values

• Recall the general gamma regression model,

\ln(\hat{y}) = \hat{\beta}_0 + \hat{\beta}_1 x_1 + \hat{\beta}_2 x_2 + ... \hat{\beta}_k x_k

• Now that the \ln() is present in the model, we need to be careful when creating predicted values.

  • We want to graph the predicted y, not \ln(y).

\hat{y} = e^{\hat{\beta}_0 + \hat{\beta}_1x_1 + \beta_2 x_2 + \cdot \cdot \cdot + \hat{\beta}_k x_k}

Creating Predicted Values: Example

• Let’s apply this to our example.

\ln(y) = 1.28 + 0.015 \text{ temp} + 0.11 \text{ crowd} - 0.19 \text{ family} + 0.55 \text{ thrill}

• Let’s have crowd index on the x-axis, wait time on the y-axis, and lines defined by ride type.

  • We will plug in the median temperature.

mk_wait <- mk_wait %>%
  mutate(ln_wait_dark = 1.28 + 0.015*median(temp_f) + 0.11*crowd_index - 0.19*0 + 0.55*0,
         ln_wait_family = 1.28 + 0.015*median(temp_f) + 0.11*crowd_index - 0.19*1 + 0.55*0,
         ln_wait_thrill = 1.28 + 0.015*median(temp_f) + 0.11*crowd_index - 0.19*0 + 0.55*1)

Creating Predicted Values: Example

• For demonstration purposes, let’s compare our current predicted values to the observed values,

mk_wait %>% select(wait_time, ln_wait_dark, ln_wait_family, ln_wait_thrill) %>% head()

• We can see that the scaling is not correct.

Creating Predicted Values: Example

• Trying again, but now exponentiating using exp(),

mk_wait <- mk_wait %>%
  mutate(wait_dark = exp(1.28 + 0.015*median(temp_f) + 0.11*crowd_index - 0.19*0 + 0.55*0),
         wait_family = exp(1.28 + 0.015*median(temp_f) + 0.11*crowd_index - 0.19*1 + 0.55*0),
         wait_thrill = exp(1.28 + 0.015*median(temp_f) + 0.11*crowd_index - 0.19*0 + 0.55*1))

Visualization of the Model: Example

• Now, we can build our visualization as normal,

mk_wait %>% ggplot(aes(x = crowd_index)) +
  geom_point(aes(y = wait_time, color = ride_type), alpha = 0.5) +
  geom_line(aes(y = wait_dark), color = "#F8766D", linewidth = 1) + #  
  geom_line(aes(y = wait_family), color = "#00BA38", linewidth = 1) + # color = "#00BFC4",
  geom_line(aes(y = wait_thrill), color = "#00BFC4", linewidth = 1) + # ,
  labs(x = "Crowd Index",
       y = "Wait Time (minutes)",
       color = "Ride Type") +
  theme_bw()

Visualization of the Model: Example

• The resulting graph,

Linking to Interpretations

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

• Temperature is a significant predictor (p < 0.001).

  • For a 1 degree F increase in temperature, we expect e^{0.015} = 1.015 times the wait time, or a 1.5% increase in expected wait time.
  • Note! We held this constant at the median temperature, so this does not need to be commented on.

• Crowd index is a significant predictor (p < 0.001).

  • For a 1 unit increase in crowd index, we expect e^{0.11} = 1.116 times the wait time, or an 11.6% increase in expected wait time.

Linking to Interpretations

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

• Ride type is a significant predictor (p < 0.001).

  • As compared to dark rides, we expect family rides to have e^{-0.19} = 0.827 times the wait time, or a 17.3% decrease in expected wait time.

  • As compared to dark rides, we expect thrill rides to have e^{0.55} = 1.733 times the wait time, or a 73.3% increase in expected wait time.

Linking to Interpretations

• As crowd index increases (i.e., Magic Kingdom is more crowded), wait times increase for all ride types (p < 0.001).

Linking to Interpretations

• There is a difference in wait time between the three ride types (p < 0.001). Family rides have lower times than dark rides (p < 0.001) and thrill rides have higher times than dark rides (p < 0.001).

Wrap Up

• In this lecture, we learned how to visualize gamma 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 exponentiate results to get the correct predicted values in gamma regression.

• Next lecture: Beta regression