Interaction Terms:
Continuous x Categorical

Introduction

Recall the general linear model,

y = \beta_0 + \beta_1 x_1 + ... + \beta_k x_k + \varepsilon

In the last lectures, we learned about both continuous \times continuous and categorical \times categorical interactions.
In this lecture, we will combine the two concepts to explore continuous \times categorical interactions.

Lecture Example Set Up

Clarabelle runs a popular milkshake booth in Toontown. On busy days, customers can pile up quickly, and long wait times may affect satisfaction. Clarabelle wants to understand which factors most strongly influence average customer wait time and whether some factors work differently together.
We have a dataset with 300 days of operation for:
- Average wait time (avg_wait_time): Average customer wait time (in minutes) on a given day.
- Orders per hour (orders_per_hour): Average number of milkshake orders per hour that day.
- Staff experience (staff_experience): Average years of experience among staff working that shift.
- Special flavor (flavor_special): Type of milkshake special offered that day (none, seasonal, or limited edition).
- Status of milkshake machine (machine_status): Whether the milkshake machine was fully operational or temperamental that day.

clarabelle <- read_csv("https://raw.githubusercontent.com/samanthaseals/SDSII/refs/heads/main/files/data/lectures/W3_clarabelle.csv")

Example 1: Describing the Data

We first want to consider average wait time as a function of avg orders per hour, status of milkshake machine, and their interaction.
Let’s examine summary statistics,

clarabelle %>% mean_median(avg_wait_time, orders_per_hour)

# A tibble: 2 × 3
  variable        mean_sd     median_iqr 
  <chr>           <chr>       <chr>      
1 avg_wait_time   8.1 (3.4)   7.4 (4.2)  
2 orders_per_hour 25.3 (14.2) 22.5 (18.6)

clarabelle %>% filter(machine_status == "Fully Operational") %>% mean_median(avg_wait_time, orders_per_hour)

# A tibble: 2 × 3
  variable        mean_sd     median_iqr 
  <chr>           <chr>       <chr>      
1 avg_wait_time   7.0 (2.7)   6.7 (3.2)  
2 orders_per_hour 26.0 (14.5) 23.5 (19.3)

clarabelle %>% filter(machine_status == "Temperamental") %>% mean_median(avg_wait_time, orders_per_hour)

# A tibble: 2 × 3
  variable        mean_sd     median_iqr 
  <chr>           <chr>       <chr>      
1 avg_wait_time   11.2 (3.3)  10.8 (4.3) 
2 orders_per_hour 23.4 (13.3) 21.6 (15.0)

Example 1: Fitting the Model

Estimating the coefficients,

m1 <- glm(avg_wait_time ~ orders_per_hour + machine_status + orders_per_hour:machine_status,
         family = "gaussian",
         data = clarabelle)
m1 %>% coefficients()

                                (Intercept) 
                                 4.87966729 
                            orders_per_hour 
                                 0.08167935 
                machine_statusTemperamental 
                                 2.87831326 
orders_per_hour:machine_statusTemperamental 
                                 0.06360992

Gives us the model,

\hat{\text{wait time}} = 4.88 + 0.08 \text{ orders} + 2.88 \text{ temp.} + 0.06 \text{ orders $\times$ temp.}

Example 1: Statistical Inference

Running our model results through tidy(),

m1 %>% tidy()

Because our interaction is represented with only one term in the model (continuous \times binary), we can use the results from tidy() to determine that yes, the interaction is significant (p = 0.011).

Example 1: Simplifying the Model

Because the interaction is significant, we are now justified in splitting our model into two.

\hat{\text{wait time}} = 4.88 + 0.08 \text{ orders} + 2.88 \text{ temp.} + 0.06 \text{ orders $\times$ temp.}

The model for when the machine is fully operational,

\begin{align*} \hat{\text{wait time}} &= 4.88 + 0.08 \text{ orders} + 2.88(0) + 0.06(0) \text{ orders } \\ &= 4.88 + 0.08 \text{ orders} \end{align*}

The model for when the machine is temperamental,

\begin{align*} \hat{\text{wait time}} &= 4.88 + 0.08 \text{ orders} + 2.88(1) + 0.06(1) \text{ orders } \\ &= 7.76 + 0.14 \text{ orders} \end{align*}

Example 1: Interpreting the Model

Let’s consider the two model results,

\begin{align*} \hat{\text{wait time}|\text{operational}} &= 4.88 + 0.08 \text{ orders} \\ \hat{\text{wait time}|\text{temperamental}} &= 7.76 + 0.14 \text{ orders} \end{align*}

When the machine is fully operational, for every additional order per hour, average wait time increases by 0.08 minutes.
When the machine is temperamental, for every additional order per hour, average wait time increases by 0.14 minutes.

Example 2: Describing the Data

Clarabelle now wants to consider average wait time as a function of average orders per hour, if there are special flavors, and their interaction.
Let’s examine summary statistics,

clarabelle %>% filter(flavor_special == "None") %>% mean_median(avg_wait_time, orders_per_hour)

# A tibble: 2 × 3
  variable        mean_sd     median_iqr 
  <chr>           <chr>       <chr>      
1 avg_wait_time   6.8 (2.9)   6.7 (3.3)  
2 orders_per_hour 25.3 (14.6) 22.4 (18.9)

clarabelle %>% filter(flavor_special == "Limited Edition") %>% mean_median(avg_wait_time, orders_per_hour)

# A tibble: 2 × 3
  variable        mean_sd     median_iqr 
  <chr>           <chr>       <chr>      
1 avg_wait_time   11.4 (3.4)  11.4 (4.0) 
2 orders_per_hour 27.6 (15.9) 25.4 (21.0)

clarabelle %>% filter(flavor_special == "Seasonal") %>% mean_median(avg_wait_time, orders_per_hour)

# A tibble: 2 × 3
  variable        mean_sd     median_iqr 
  <chr>           <chr>       <chr>      
1 avg_wait_time   7.7 (2.7)   7.2 (3.3)  
2 orders_per_hour 24.0 (12.7) 22.2 (14.7)

Example 2: Fitting the Model

Estimating the coefficients,

m2a <- glm(avg_wait_time ~ orders_per_hour + flavor_special + orders_per_hour:flavor_special,
         family = "gaussian",
         data = clarabelle)
m2a %>% coefficients()

                           (Intercept)                        orders_per_hour 
                            8.84927244                             0.09105015 
                    flavor_specialNone                 flavor_specialSeasonal 
                           -3.94666767                            -2.78508254 
    orders_per_hour:flavor_specialNone orders_per_hour:flavor_specialSeasonal 
                           -0.01495659                            -0.02459592

Whoops! R used “Limited Edition” as the reference group. I think it makes more sense to use “None” – let’s pause to do some data management.

Example 2: Data Management

Checking original ordering (alphabetical),

levels(as.factor(clarabelle$flavor_special))

[1] "Limited Edition" "None"            "Seasonal"

Setting “None” as our reference group,

clarabelle <- clarabelle %>% mutate(flavor_special = factor(flavor_special,
                                                            levels = c("None", "Seasonal", "Limited Edition")))
levels(clarabelle$flavor_special)

[1] "None"            "Seasonal"        "Limited Edition"

Example 2: Re-fitting the Model

Re-estimating the coefficients,

m2b <- glm(avg_wait_time ~ orders_per_hour + flavor_special + orders_per_hour:flavor_special,
         family = "gaussian",
         data = clarabelle)
m2b %>% coefficients()

                                  (Intercept) 
                                  4.902604769 
                              orders_per_hour 
                                  0.076093559 
                       flavor_specialSeasonal 
                                  1.161585135 
                flavor_specialLimited Edition 
                                  3.946667673 
       orders_per_hour:flavor_specialSeasonal 
                                 -0.009639335 
orders_per_hour:flavor_specialLimited Edition 
                                  0.014956586

Gives us the model,

\hat{\text{wait time}} = 4.88 + 0.08 \text{ orders} + 2.88 \text{ temp.} + 0.06 \text{ orders $\times$ temp.}

Example 2: Statistical Inference

Running our model results through tidy(),

m2b %>% tidy()

Because our interaction is represented with two terms in the model, we must use car::Anova() here.

Example 2: Statistical Inference

Checking the omnibus test,

m2b %>% car::Anova(type = 3)

The interaction is not significant (p = 0.721). We do not have justification in stratifying the model by special flavor.
- Note: I would use a visualization to help illustrate the non-significance.

Example 2: Statistical Inference

Driving home that the reference group does not affect the omnibus test,

m2a %>% car::Anova(type = 3)

m2b %>% car::Anova(type = 3)

Example 2: Simplifying the Model

Let’s simplify the model for illustrative purposes.
The overall model,

\begin{align*} \hat{\text{wait time}} = 4.&90 \\ & + 0.08 \text{ orders} \\ & + 1.16 \text{ seasonal} + 3.95 \text{ limited edition} \\ & - 0.01 \text{ orders $\times$ seasonal} + 0.02 \text{ orders $\times$ limited edition} \end{align*}

No specials: seasonal = 0, limited edition = 0.
Seasonal flavors: seasonal = 1, limited edition = 0.
Limited edition flavors: seasonal = 0, limited edition = 1.

Example 2: Simplifying the Model

Let’s simplify the model for illustrative purposes.
No special flavor,

\begin{align*} \hat{\text{wait time}} = 4.&90 \\ & + 0.08 \text{ orders} \\ & + 1.16 (0) + 3.95 (0) \\ & - 0.01 (0) \text{ orders} + 0.02 (0) \text{ orders} \\ = 4.&90 + 0.08 \text{ orders} \end{align*}

Example 2: Simplifying the Model

Let’s simplify the model for illustrative purposes.
Seasonal,

\begin{align*} \hat{\text{wait time}} = 4.&90 \\ & + 0.08 \text{ orders} \\ & + 1.16 (1) + 3.95 (0) \\ & - 0.01 (1) \text{ orders} + 0.02 (0) \text{ orders} \\ = 6.&06 + 0.07 \text{ orders} \end{align*}

Example 2: Simplifying the Model

Let’s simplify the model for illustrative purposes.
Limited edition,

\begin{align*} \hat{\text{wait time}} = 4.&90 \\ & + 0.08 \text{ orders} \\ & + 1.16 (0) + 3.95 (1) \\ & - 0.01 (0) \text{ orders} + 0.02 (1) \text{ orders} \\ = 8.&85 + 0.10 \text{ orders} \end{align*}

Example 2: Intepreting the Model

Considering the three model results,

\begin{align*} \hat{\text{wait time}|\text{no special}} &= 4.90 + 0.08 \text{ orders} \\ \hat{\text{wait time}|\text{seasonal}} &= 6.06 + 0.07 \text{ orders} \\ \hat{\text{wait time}|\text{limited edition}} &= 8.85 + 0.10 \text{ orders} \end{align*}

When there are no special flavors, for every additional order per hour, average wait time increases by 0.08 minutes.
When there are seasonal flavors, for every additional order per hour, average wait time increases by 0.07 minutes.
When there are limited edition flavors, for every additional order per hour, average wait time increases by 0.10 minutes.

Example 2: Intepreting the Model

Thinking about the practical implications,
- When there are no special flavors, for every additional order per hour, average wait time increases by 0.08 minutes (4.8 sec).
- When there are seasonal flavors, for every additional order per hour, average wait time increases by 0.07 minutes (4.2 sec).
- When there are limited edition flavors, for every additional order per hour, average wait time increases by 0.10 minutes (6 seconds)
These interpretations help us drive home the message that the flavor situation really doesn’t affect the relationship with wait time and number of orders.
- For every 10 additional orders per hour, average wait time only changes by about 1 minute regardless of flavor special.
- Given what we see here, we would need a substantial change in average number of orders per hour in order to see a meaningful change in average wait time.

Wrap Up

This lecture is brief because continuous \times categorical interactions are conceptually similar to both continuous \times continuous and categorical \times categorical interactions.
Remember that everything in this course, especially to this point, are different building blocks.
- This lecture in particular begins to combine several concepts.
- As we move forward, we will continue to combine concepts to build more complex models.
It is impossible to prepare you for all of the “what if” situations you will encounter in your career.
- That’s why we focus on building a strong foundation of understanding the core concepts.
In the next lecture, we will review how to visualize interactions.

Interaction Terms: Continuous x Categorical

Introduction

Lecture Example Set Up

Example 1: Describing the Data

Example 1: Fitting the Model

Example 1: Statistical Inference

Example 1: Simplifying the Model

Example 1: Interpreting the Model

Example 2: Describing the Data

Example 2: Fitting the Model

Example 2: Data Management

Example 2: Re-fitting the Model

Example 2: Statistical Inference

Example 2: Statistical Inference

Example 2: Statistical Inference

Example 2: Simplifying the Model

Example 2: Simplifying the Model

Example 2: Simplifying the Model

Example 2: Simplifying the Model

Example 2: Intepreting the Model

Example 2: Intepreting the Model

Wrap Up

Interaction Terms:
Continuous x Categorical