pluto <- read_csv("https://raw.githubusercontent.com/samanthaseals/SDSII/refs/heads/main/files/data/lectures/W6_pluto.csv")
pluto %>% head()Visualizing the Model:
Poisson and Negative Binomial Regressions
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.
- Week 4: Gamma and beta regressions.
- Week 5: Binomial and multinomial logistic regressions.
In this lecture, we will focus on visualizing Poisson and negative binomial regression models.
Poisson & Negative Binomial Regressions
- Recall the Poisson and negative binomial models,
\ln(\hat{y}) = \hat{\beta}_0 + \hat{\beta}_1 x_1 + ... + \hat{\beta}_k x_k
The approach we will take will match the approach taken for gamma:
- Find the linear predictor.
- Exponentiate (run through
exp()).
Lecture Example Set Up
Pluto spends his days at Mickey’s park chasing squirrels and interacting with guests. Disney researchers are interested in understanding what factors influence the number of squirrels Pluto chases per hour.
For 300 observation periods, researchers recorded:
- squirrels_chased: Number of squirrels chased during the observation period
- temperature: Temperature (°F)
- crowd: Park crowd level (guests per 100 people)
- mickey_present: Whether Mickey is present (Yes/No)
- time_of_day: Time of day (Morning, Afternoon, Evening)
Lecture Example Set Up
- Pulling in the data,
Example 1: Negative Binomial Regression
- Recall the final model from the negative binomial lecture,
\ln\left( y \right) = 1.32 + 0.02 \text{ temp} + 0.02 \text{ crowd}
We want to construct a graph for this model. We will have:
- number of squirrels chased on the y-axis
- crowd level on the x-axis
- temperature will show different lines (let’s plug in 50, 75, and 100°F)
Creating Predicted Values
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}
- We saw this with gamma regression – remember that we exponentiate the linear predictor.
Example 1: Predicted Values
- Recall the final model,
\ln\left( y \right) = 1.32 + 0.02 \text{ temp} + 0.02 \text{ crowd}
- Finding our predicted values,
Example 1: Visualizing the Model
Coding our graph as we have before,
- Original values for the scatterplot (
geom_point()) - Predicted values for the lines (
geom_line())
- Original values for the scatterplot (
pluto %>% ggplot(aes(x = crowd)) +
geom_point(aes(y = squirrels_chased), alpha = 0.5) +
geom_line(aes(y = squirrels_50, color = "50°F"), linewidth = 1.5) +
geom_line(aes(y = squirrels_75, color = "75°F"), linewidth = 1.5) +
geom_line(aes(y = squirrels_100, color = "100°F"), linewidth = 1.5) +
scale_color_manual(values = c("50°F" = "#00BA38",
"75°F" = "#619CFF",
"100°F" = "#F8766D"),
breaks = c("50°F", "75°F", "100°F")) +
labs(x = "Guests per 100 people (Crowd Level)",
y = "Number of Squirrels Chased",
color = "Temperature") +
theme_bw()Example 1: Visualizing the Model
Example 2: Poisson Regression
Let’s now tackle the second example from the Poisson regression lecture.
- Remember, we saw a three-way interaction between temperature, Mickey’s presence, and time of day.
Example 2: Poisson Regression
We then stratified the analysis to “get rid” of the three way interaction.
When Mickey is present:
Example 2: Poisson Regression
We then stratified the analysis to “get rid” of the three way interaction.
When Mickey is not present:
Example 2: Poisson Regression
Let’s stop and think about our graph.
We want to graph the predicted number of squirrels chased on the y-axis and temperature on the x-axis.
We should have time of day define the lines.
We should have two graphs, one for when Mickey is present and one for when Mickey is not present.
So, this will create variables for
Mickey present: morning, afternoon, and evening
Mickey not present: morning, afternoon, and evening
Example 2: Predicted Values
| Term | Mickey Present | Mickey Not Present |
|---|---|---|
| Intercept | 2.37 | 3.09 |
| Temperature | 0.02 | 0.009 |
| Time of Day (Evening) | -0.41 | -1.44 |
| Time of Day (Morning) | 0.41 | -1.69 |
| Temp x Evening | -0.001 | 0.01 |
| Temp x Morning | - 0.01 | 0.02 |
- The base model for when Mickey is present,
- The base model for when Mickey is not present,
Example 2: Predicted Values
- Creating predicted values for when Mickey is present,
pluto <- pluto %>%
mutate(squirrels_morning_mickey = exp(2.37 + 0.02*temperature - 0.41*0 + 0.41*1 - 0.001*temperature*0 - 0.01*temperature*1),
squirrels_afternoon_mickey = exp(2.37 + 0.02*temperature - 0.41*0 + 0.41*0 - 0.001*temperature*0 - 0.01*temperature*0),
squirrels_evening_mickey = exp(2.37 + 0.02*temperature - 0.41*1 + 0.41*0 - 0.001*temperature*1 - 0.01*temperature*0))- Creating predicted values for when Mickey is not present,
pluto <- pluto %>%
mutate(squirrels_morning_no_mickey = exp(3.09 + 0.009009*temperature - 1.44*0 - 1.69*1 + 0.01*temperature*0 + 0.02*temperature*1),
squirrels_afternoon_no_mickey = exp(3.09 + 0.009*temperature - 1.44*0 - 1.69*0 + 0.01*temperature*0 + 0.02*temperature*0),
squirrels_evening_no_mickey = exp(3.09 + 0.009*temperature - 1.44*1 - 1.69*0 + 0.01*temperature*1 + 0.02*temperature*0))Example 2: Predicted Values
- Checking the predicted values,
Example 2: Visualizing the Model
Coding our graph as we have before,
- Original values for the scatterplot (
geom_point()) - Predicted values for the lines (
geom_line()) - Separate graphs for Mickey present vs. not present
- Original values for the scatterplot (
pluto %>% ggplot(aes(x = temperature)) +
geom_point(aes(y = squirrels_chased), alpha = 0.5) +
geom_line(aes(y = squirrels_morning_mickey, color = "Morning"), linewidth = 1.5) +
geom_line(aes(y = squirrels_afternoon_mickey, color = "Afternoon"), linewidth = 1.5) +
geom_line(aes(y = squirrels_evening_mickey, color = "Evening"), linewidth = 1.5) +
scale_color_manual(values = c("Morning" = "#00BA38",
"Afternoon" = "#619CFF",
"Evening" = "#F8766D"),
breaks = c("Morning", "Afternoon", "Evening")) +
labs(x = "Temperature (°F)",
y = "Number of Squirrels Chased",
color = "Time of Day") +
theme_bw() Example 2: Visualizing the Model
- Running the code,
Example 2: Visualizing the Model
- The second graph (when Mickey is not present),
pluto %>% ggplot(aes(x = temperature)) +
geom_point(aes(y = squirrels_chased), alpha = 0.5) +
geom_line(aes(y = squirrels_morning_no_mickey, color = "Morning"), linewidth = 1.5) +
geom_line(aes(y = squirrels_afternoon_no_mickey, color = "Afternoon"), linewidth = 1.5) +
geom_line(aes(y = squirrels_evening_no_mickey, color = "Evening"), linewidth = 1.5) +
scale_color_manual(values = c("Morning" = "#00BA38",
"Afternoon" = "#619CFF",
"Evening" = "#F8766D"),
breaks = c("Morning", "Afternoon", "Evening")) +
labs(x = "Temperature (°F)",
y = "Number of Squirrels Chased",
color = "Time of Day") +
theme_bw() Example 2: Visualizing the Model
- Running the code,
Example 2: Visualizing the Model
- Putting the two graphs side by side (see .qmd file for details),
Wrap Up
In this lecture, we learned how to visualize Poisson and negative binomimal regression models.
- Remember that we have to “undo” the ln() to get the correct predicted values for either Poisson or negative binomial regression models.
This lecture completes the new material component of this course!
Always remember that visualization is a key component of data analysis!
We use data visualization to check model assumptions, interpret results, and communicate findings.
When constructing graphs, always think about your audience and the story you want to tell with your data.
Wrap Up
Now, we will focus on putting everything together with a final assignment.
- You will be presented research questions with corresponding datasets (similar to previous assignments).
Your main focus for the assignment will be to build appropriate models and visualize them effectively.
Your main focus for the (non-computational) final exam will be choosing the appropriate modeling strategy, interpreting model output, and performing statistical inference.