Visualizing the Model: Interactions

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.

• We have seen this week that there are three types of interactions:

  • Continuous \times continuous
  • Categorical \times categorical
  • Continuous \times categorical

• Graphing a model with only categorical variables and interactions will follow what we learned for models with only categorical variables in Week 2.

• In this lecture we will focus on models with at least one continuous predictor in them.

Lecture Example Set Up

• Recall the Clarabelle data,

• 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

• In the last lecture, we constructed the following model:

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

• We saw that the interaction was significant (), so we were justified in stratifying our models:

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

• We will use the statified models for visualization.

  • We will have one regression line for when the machine is fully operational and another for when it is temperamental.

Example 1

• First, we will find predicted values,

clarabelle <- clarabelle %>% 
  mutate(y_op = 4.88 + 0.08 * orders_per_hour,
         y_temp = 7.76 + 0.14 * orders_per_hour)

Example 1

• Then, our code to graph,

clarabelle %>% ggplot(aes(x = orders_per_hour)) +
  geom_point(aes(y = avg_wait_time, color = machine_status), alpha = 0.5) +
  geom_line(aes(y = y_op), color = "#F8766D", size = 1) +
  geom_line(aes(y = y_temp), color = "#00BFC4", size = 1) +
  labs(x = "Orders per Hour",
       y = "Average Wait Time (minutes)",
       color = "Machine Status") +
  theme_bw()

Example 1

• Looking at our graph,

Example 2

• We also constructed this 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*}

• Although the interaction was not significant (p = 0.7206), we can still provide a data visualization of that specific model.

  • I will do this to demonstrate to others what I mean by “there is not an interaction”.

Example 2

• Thus, our stratified models (see last lecture for derivation):,

\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*}

• We will have three lines on our graph: one for each type of special flavor offering.

Example 2

• Finding predicted values,

clarabelle <- clarabelle %>% 
  mutate(y_none = 4.90 + 0.08 * orders_per_hour,
         y_seasonal = 6.06 + 0.07 * orders_per_hour,
         y_limited = 8.85 + 0.10 * orders_per_hour)

Example 2

• Then, our code to graph,

clarabelle %>% ggplot(aes(x = orders_per_hour)) +
  geom_point(aes(y = avg_wait_time, color = flavor_special), alpha = 0.5) +
  geom_line(aes(y = y_none), color = "#00BA38", size = 1) +
  geom_line(aes(y = y_seasonal), color = "#619CFF", size = 1) +
  geom_line(aes(y = y_limited), color = "#F8766D", size = 1) +
  labs(x = "Orders per Hour",
       y = "Average Wait Time (minutes)",
       color = "Flavor") +
  theme_bw()

Example 2

• Looking at our graph,

Example 2

• Wait!!!

• The test for interaction was not significant, so why do the slopes look different?

  • Eh…

  • Remember that we see parallel lines when we include categorical predictors – the interaction should cause different rates of change. We are not seeing that here.

• Thus – we can visualize models with non-significant interactions, but we should be careful about how we interpret them.

  • Always be careful with the word “significant” – it has a very specific meaning in statistics.

Wrap Up

• This lecture has demonstrated how to visualize models with interaction terms.

• To keep it simple, we focused on models with at least one continuous predictor.

  • To deal with models with only categorical predictors (and interactions), please refer back to Week 2’s material.

• We now have the general building blocks for regression analysis.

  • After this week, we will leave the normal distribution, meant for continuous outcomes that are mound-shaped and symmetric, and begin exploring distributions for other types of outcomes.