Week 11 Exercise

Nonlinear Effects, Nonlinear-by-Linear Terms, and 3-Way Interactions

Overview

Up to this point, most of our models have assumed that a predictor changes \(Y\) in a straight-line way:

One-unit increase in \(X\) → same expected change in \(Y\), regardless of where on the scale you start.

But real customer behavior is often more complicated than that.

Sometimes more of something helps — but only up to a point. Sometimes very low and very high values of a predictor are both associated with lower outcomes. Sometimes the bend itself depends on a second variable.

This week we explore these kinds of models in the context of CLV: customer lifetime value. Here, CLV is a 12-month dollar-value outcome reflecting how much revenue a given customer generates.

Our goal is not just to fit a model that reduces error — it is to fit a model we can actually explain to another human being. Your boss, client, or coworker is much more likely to ask:

  • “Are we getting diminishing returns here?”
  • “Does this pattern depend on how engaged customers are?”
  • “Should we treat all customers the same, or not?”

That means your job is to understand the pattern well enough to answer reasonable follow-up questions.

We are extending multiple regression in two related directions:

  1. Allow a predictor’s relationship with \(Y\) to bend (interact with itself) rather than stay a straight line: \[Y = b_0 + b_1 X + b_2 X^2 + \varepsilon\]

  2. Allow that bent relationship to depend on another predictor: \[Y = b_0 + b_1 X + b_2 X^2 + b_3 Z + b_4 (X \cdot Z) + b_5 (X^2 \cdot Z) + \varepsilon\]

This means:

  • The slope of \(X\) can change across values of \(X\) (curvature)
  • And the shape of that curvature can itself vary with \(Z\)

We also preview 3-way interactions among continuous predictors: \[Y = b_0 + b_1 X + b_2 Z + b_3 W + \ldots + b_7 (X \cdot Z \cdot W) + \varepsilon\]

Questions we will be asking today:

  • Is adding curvature to our model worth it? (quadratic vs. linear)
  • What does centering do to our interpretations in a curved model?
  • Does a nonlinear (curved) term vary with another predictor?
  • How do we think about a 3-way (linear) interaction in this framework?

Part 0 — Load and Refamiliarize with the Data

clv <- read.csv("Datasets/clv_nonlinear_exercise_full.csv")

head(clv)
str(clv)
summary(clv)

n <- length(clv$CLV)

Data Dictionary

Outcome:

Variable Description
CLV 12-month customer lifetime value in dollars

Continuous predictors:

Variable Description
tenure_months How long the customer has been with the firm
avg_monthly_spend Average monthly spend in dollars
visit_freq_month Average number of visits per month
email_open_rate Proportion of marketing emails opened (0 to 1)
engagement_score Internal engagement index
satisfaction Customer satisfaction rating (1 to 10)
age Customer age in years
income Annual income in dollars
day_of_year_joined Day of the year the customer relationship began

Categorical predictors:

Variable Levels
campaign_type Organic, Referral, BlackFriday, HolidayGift, NewYearsPromo
segment Value, Growth, At_Risk
primary_channel Online, Store, Omnichannel

For now, do not worry about using every variable immediately. Some are central to this week’s nonlinear work; others give us something to come back to later. This is a fake dataset built to have some interesting patterns, so treat it accordingly.

Before you start modeling, take a minute to actually look at the variables like a person.

Ask yourself:

  • Which predictors seem like they could reasonably have diminishing returns?
  • Which ones do you expect might have “sweet spots” rather than straight linear effects?
  • Which ones feel like they might interact/depend on another variable?

You do not need to know the answer yet. But you should start thinking through what might make sense before R starts throwing coefficients at you.

In other words: do not let the software have the first interesting thought.

Part 0.5 — Create Centered Variables

clv$tenure_c <- clv$tenure_months     - mean(clv$tenure_months,     na.rm = TRUE)
clv$spend_c  <- clv$avg_monthly_spend - mean(clv$avg_monthly_spend, na.rm = TRUE)
clv$visit_c  <- clv$visit_freq_month  - mean(clv$visit_freq_month,  na.rm = TRUE)
clv$open_c   <- clv$email_open_rate   - mean(clv$email_open_rate,   na.rm = TRUE)
clv$eng_c    <- clv$engagement_score  - mean(clv$engagement_score,  na.rm = TRUE)
clv$sat_c    <- clv$satisfaction      - mean(clv$satisfaction,      na.rm = TRUE)

Reminder: Centering changes the “magic zero.” It changes what the intercept means and what lower-order terms mean in the presence of interactions and quadratics — but it does not change overall model fit and does not change the actual underlying pattern in the data.

Part 1 — Linear vs. Quadratic: CLV ~ Tenure

Research question: Does the relationship between customer tenure and CLV appear curved?

In a purely linear model, every additional month of tenure changes CLV by the same amount, regardless of whether a customer is brand new or has been around for years.

In a quadratic model, that effect can change. For example:

  • Tenure may help a lot at first and then level off
  • Or it may rise, peak, and then taper
  • Or, less commonly, it could accelerate

This is one of those situations where a manager might reasonably ask: “Are longer-tenure customers always more valuable in the same way, or are we hitting diminishing returns?”

1A — Linear model (compact)

m_lin_tenure <- lm(CLV ~ tenure_months, data = clv)
summary(m_lin_tenure)

1B — Quadratic model (augmented)

m_quad_tenure <- lm(CLV ~ tenure_months + I(tenure_months^2), data = clv)
summary(m_quad_tenure)

OK, take a second and try to think about what this means before we move on.

At minimum, resist the urge to immediately declare victory based on a p-value.

We can start by examining how much additional error we’re explaining in this model compared to the compact, linear model.

1C — PRE for adding curvature

SSE_lin_tenure  <- sum(residuals(m_lin_tenure)^2)
SSE_quad_tenure <- sum(residuals(m_quad_tenure)^2)

PRE_quad_over_lin_tenure <- (SSE_lin_tenure - SSE_quad_tenure) / SSE_lin_tenure
PRE_quad_over_lin_tenure

Interpretation: How much additional error in the linear model do we explain by allowing the relationship to bend?

And more importantly: if your boss asked whether tenure has a basically straight-line relationship with CLV or whether it levels off or changes shape, what would you say?

TipNote this for the exercise

Record your value of PRE_quad_over_lin_tenure. You will be asked to identify it in the exercise you’ll submit this week.

1D — Partial-F test for the quadratic term

df_small_tenure <- df.residual(m_lin_tenure)
df_big_tenure   <- df.residual(m_quad_tenure)

df_num_tenure <- df_small_tenure - df_big_tenure   # should be 1
df_den_tenure <- df_big_tenure

F_quad_tenure <- ((SSE_lin_tenure - SSE_quad_tenure) / df_num_tenure) /
  (SSE_quad_tenure / df_den_tenure)

p_quad_tenure <- pf(F_quad_tenure,
                    df1 = df_num_tenure,
                    df2 = df_den_tenure,
                    lower.tail = FALSE)

F_quad_tenure
p_quad_tenure
TipNote this for the exercise

Record your value of F_quad_tenure. You will be asked to identify it in the exercise you’ll submit this week.

1E — Connect to summary() output

summary(m_quad_tenure)

Reminder: The \(t\)-test on I(tenure_months^2) is testing the same hypothesis as the 1-df \(F\) test above. For a single-parameter test: \[F = t^2\]

You can verify this by squaring the \(t\) value reported in the summary output.

Is the added curvature statistically worth it? In this class, that essentially translates to: did the more complicated model reduce error enough to justify its complicated self? And can you describe what kind of curvature you think you are seeing?

TipNote this for the exercise

Look at the sign of the coefficient on I(tenure_months^2) in the summary output. Is it positive or negative? You will be asked what that sign tells you about the shape of the curve.

Part 2 — Centering Tenure: What Changed and What Didn’t?

Let’s refit the same curved model, but centered.

m_quad_tenure_c <- lm(CLV ~ tenure_c + I(tenure_c^2), data = clv)
summary(m_quad_tenure_c)

Compare:

summary(m_quad_tenure)
summary(m_quad_tenure_c)

Let’s think through what we’re seeing here:

  • How did the intercept change? What does it now mean?
    • Predicted CLV for a customer at mean tenure
  • What does the tenure_c coefficient now mean?
    • The local slope of tenure at mean tenure
  • What happened to the quadratic coefficient?
    • Same curvature — just a different coding of zero
TipNote this for the exercise

Record the intercept from m_quad_tenure_c. You will be asked to identify its value and what it represents. Also confirm that the SSE from the centered model and the uncentered model.

Quick visual:

plot(clv$tenure_months, clv$CLV,
     pch = 16, cex = .5,
     xlab = "Tenure (months)",
     ylab = "CLV",
     main = "CLV by Tenure with Quadratic Fit")

ten_seq  <- seq(min(clv$tenure_months, na.rm = TRUE),
                max(clv$tenure_months, na.rm = TRUE),
                length.out = 150)
pred_ten <- data.frame(tenure_months = ten_seq)
lines(ten_seq,
      predict(m_quad_tenure, newdata = pred_ten),
      lwd = 2)

Part 3 — Another Quadratic: CLV ~ Spend

We have now seen one example with tenure. Let’s examine another predictor where theory might also suggest diminishing returns: average monthly spend.

Fit and compare

m_lin_spend  <- lm(CLV ~ spend_c, data = clv)
m_quad_spend <- lm(CLV ~ spend_c + I(spend_c^2), data = clv)

summary(m_lin_spend)
summary(m_quad_spend)

Compute PRE and partial-F

SSE_lin_spend  <- sum(residuals(m_lin_spend)^2)
SSE_quad_spend <- sum(residuals(m_quad_spend)^2)

PRE_quad_over_lin_spend <- (SSE_lin_spend - SSE_quad_spend) / SSE_lin_spend
PRE_quad_over_lin_spend

df_small_spend <- df.residual(m_lin_spend)
df_big_spend   <- df.residual(m_quad_spend)

df_num_spend <- df_small_spend - df_big_spend
df_den_spend <- df_big_spend

F_quad_spend <- ((SSE_lin_spend - SSE_quad_spend) / df_num_spend) /
  (SSE_quad_spend / df_den_spend)

p_quad_spend <- pf(F_quad_spend,
                   df1 = df_num_spend,
                   df2 = df_den_spend,
                   lower.tail = FALSE)

F_quad_spend
p_quad_spend

Interpretation

Before you read the coefficients, look at the sign on the quadratic term. A negative quadratic coefficient means the curve opens downward — the effect of spend on CLV gets weaker as spend increases. A positive quadratic coefficient would mean the effect accelerates. Which is it here?

Now look at the quadratic coefficient’s magnitude. It is probably very small, which makes sense: spend is measured in dollars, so \(b_2\) is operating on dollars squared. That does not make it unimportant; it just means you need to calculate predicted values at realistic spend levels to understand the practical size of the bend. Try asking: what is the predicted CLV for a customer at mean spend vs. at mean + 1 SD spend?

Work through these questions:

  • Is there evidence of curvature in spend? (What’s your PRE? What’s your F and p?)
  • What is the direction of the curve — does the effect of spend increase or decrease as spend gets larger?
  • Is the direction of curvature the same as with tenure?
  • Which predictor shows stronger nonlinearity in this dataset?
  • What would you tell a manager who asked: “At what point does getting customers to spend more not really move the needle as much?”
TipNote this for the exercise

Record your PRE_quad_over_lin_spend value. You will be asked to identify it and compare it to the tenure PRE from Part 1. Also note the sign of the quadratic coefficient on I(spend_c^2) — you will be asked what it tells you about the direction of the curve.

Part 4 — Nonlinear-by-Linear: Does the Tenure Curve Depend on Engagement?

Research question: Does the relationship between tenure and CLV vary depending on engagement score — and in particular, does the shape of the curvature change across engagement levels?

This is more than asking whether the lines are just shifted up or down for different customers. We are asking whether the bend itself looks different for highly engaged customers versus less engaged ones.

A note on using : vs. * in model formulas

In many statistics courses that use R, you may have used * to specify interactions and let R handle the expansion:

lm(Y ~ X * Z)   # R expands this to: Y ~ X + Z + X:Z

That shorthand is completely fine — and for simple two-variable interactions it is exactly what you want. However, when we are building nonlinear-by-linear models, we need more precise control over which terms are included. Here is the problem:

# What you might try:
lm(CLV ~ tenure_c * eng_c + I(tenure_c^2) * eng_c)

# What R actually expands this to:
# CLV ~ tenure_c + eng_c + tenure_c:eng_c +
#       I(tenure_c^2) + eng_c + I(tenure_c^2):eng_c

This can add terms you did not intend to add and make your step-by-step model comparisons less transparent.

When we are building models one piece at a time and comparing them step by step (which is exactly what we are doing), we want to specify each term explicitly using :. The : operator gives you only the interaction term, without automatically adding main effects. That means you are in control of the hierarchy, not R’s expansion logic.

So:

  • X:Z gives you the interaction term only
  • X * Z gives you X + Z + X:Z

In simple models, these lead to the same place. In the kind of incremental model-building we do in this class, using : explicitly keeps things unambiguous - particularly when you want to be extremely clear about what model C you are comparing against. This is one of those cases where the shortcut that works in many situations becomes a source of confusion in more complex ones. Use : here.

Hierarchy reminder

When we include higher-order terms, we keep the lower-order pieces they build on. If we include \(X^2\), we also keep \(X\). If we include \(X \cdot Z\) or \(X^2 \cdot Z\), we keep the relevant lower-order terms as well. This keeps the model interpretable and prevents higher-order terms from absorbing effects they were not meant to capture.

4A — Model 1: Curvature only (no interaction)

m_ten_curve <- lm(CLV ~ tenure_c + I(tenure_c^2) + eng_c, data = clv)
summary(m_ten_curve)

4B — Model 2: Add linear interaction (X:Z)

m_ten_curve_linint <- lm(CLV ~ tenure_c + I(tenure_c^2) + eng_c +
                           tenure_c:eng_c,
                         data = clv)
summary(m_ten_curve_linint)

4C — Model 3: Add nonlinear-by-linear term (X²:Z)

m_ten_curve_full <- lm(CLV ~ tenure_c + I(tenure_c^2) + eng_c +
                         tenure_c:eng_c + I(tenure_c^2):eng_c,
                       data = clv)
summary(m_ten_curve_full)

Compare Model 1 vs. Model 2 (adding X:Z)

SSE_curve_only   <- sum(residuals(m_ten_curve)^2)
SSE_curve_linint <- sum(residuals(m_ten_curve_linint)^2)

PRE_linint_over_curve <- (SSE_curve_only - SSE_curve_linint) / SSE_curve_only
PRE_linint_over_curve

df_small_linint <- df.residual(m_ten_curve)
df_big_linint   <- df.residual(m_ten_curve_linint)

F_linint <- ((SSE_curve_only - SSE_curve_linint) /
               (df_small_linint - df_big_linint)) /
  (SSE_curve_linint / df_big_linint)

p_linint <- pf(F_linint,
               df1 = df_small_linint - df_big_linint,
               df2 = df_big_linint,
               lower.tail = FALSE)

F_linint
p_linint
TipNote this for the exercise

Record PRE_linint_over_curve and F_linint. You will be asked to identify both values.

Compare Model 2 vs. Model 3 (adding X²:Z)

SSE_curve_full <- sum(residuals(m_ten_curve_full)^2)

PRE_nl_over_linint <- (SSE_curve_linint - SSE_curve_full) / SSE_curve_linint
PRE_nl_over_linint

df_small_nl <- df.residual(m_ten_curve_linint)
df_big_nl   <- df.residual(m_ten_curve_full)

F_nl <- ((SSE_curve_linint - SSE_curve_full) /
           (df_small_nl - df_big_nl)) /
  (SSE_curve_full / df_big_nl)

p_nl <- pf(F_nl,
           df1 = df_small_nl - df_big_nl,
           df2 = df_big_nl,
           lower.tail = FALSE)

F_nl
p_nl
TipNote this for the exercise

Record PRE_nl_over_linint and F_nl. You will be asked to identify both values.

What the coefficients mean

  • tenure_c: The local slope of tenure at mean engagement and mean tenure (both centered to zero). This is the expected change in CLV per additional month of tenure for a customer at the average engagement level.
  • I(tenure_c^2): The curvature in tenure at mean engagement. Negative means diminishing returns; positive means acceleration.
  • tenure_c:eng_c: Whether the linear component of the tenure slope differs by engagement. Positive means higher-engagement customers have a steeper tenure slope at mean tenure.
  • I(tenure_c^2):eng_c: Whether the curvature itself changes with engagement. This is the true nonlinear-by-linear term — it tells us whether the shape of the tenure→CLV curve depends on engagement, not just whether it shifts up or down.

A useful test of whether you understand this: imagine someone asks, “So what is different about high-engagement customers here?” If your answer is only “there is an interaction,” you are not done yet.

TipNote this for the exercise

From your summary(m_ten_curve_full) output, record the coefficient on eng_c. You will be asked to interpret it. Also think about what it means that this coefficient is evaluated at tenure_c = 0 — i.e., at mean tenure. One question will ask you to use the tenure_c and tenure_c:eng_c coefficients together to compute the simple slope of tenure at +1 SD engagement (SD of eng_c ≈ 2.00).

Part 5 — Quick Visual of the Nonlinear-by-Linear Pattern

When the model starts getting more complicated, plots become less optional. It’s really difficult to hold 3 nonlinear effects or more than 2 dimensions in your head. Let’s visualize predicted tenure curves for relatively low, average, and high engagement.

eng_sd <- sd(clv$eng_c, na.rm = TRUE)
eng_lo <- -eng_sd
eng_md <- 0
eng_hi <-  eng_sd

# Note: because eng_c is already centered, its mean is approximately 0.
# So eng_lo and eng_hi are simply ±1 SD from the center.

x <- seq(min(clv$tenure_c, na.rm = TRUE),
         max(clv$tenure_c, na.rm = TRUE),
         length.out = 150)

plot(clv$tenure_c, clv$CLV,
     pch = 16, cex = .4,
     xlab = "Centered Tenure",
     ylab = "CLV",
     main = "Tenure Curves at Low / Mean / High Engagement")

lines(x,
      predict(m_ten_curve_full,
              newdata = data.frame(tenure_c = x, eng_c = eng_lo)),
      lty = 2)

lines(x,
      predict(m_ten_curve_full,
              newdata = data.frame(tenure_c = x, eng_c = eng_md)),
      lty = 1)

lines(x,
      predict(m_ten_curve_full,
              newdata = data.frame(tenure_c = x, eng_c = eng_hi)),
      lty = 3)

legend("topleft", bty = "n",
       legend = c("Low engagement (-1 SD)",
                  "Mean engagement",
                  "High engagement (+1 SD)"),
       lty = c(2, 1, 3))

Are these just vertically shifted curves? Or does the shape itself appear to change?

The models are getting more complex, which is precisely why we need to slow down and think carefully rather than stare at the output until meaning arrives.

Part 6 — Three-Way Interaction: CLV ~ Spend × Visit × Open

Research question: Does the relationship between monthly spend and CLV depend jointly on visit frequency and email open rate?

In other words: is the spend→CLV relationship different for customers who visit often vs. rarely, and does that itself depend on how engaged they are with email?

A 3-way interaction is not a personal attack on your sanity even if it feels like it is. Sometimes the world is actually structured that way. But because breaking one down can feel that way, we want to make sure that we are extremely programmatic in the way that we interpret them. Our job is still the same: compare models, reduce error, and explain the pattern clearly enough that a non-statistician could follow the basic logic.

6A — Additive model

m_add_3way <- lm(CLV ~ spend_c + visit_c + open_c, data = clv)
summary(m_add_3way)

6B — Full 2-way interaction model

m_2way_3way <- lm(CLV ~ spend_c + visit_c + open_c +
                    spend_c:visit_c + spend_c:open_c + visit_c:open_c,
                  data = clv)
summary(m_2way_3way)

6C — Full 3-way model

m_full_3way <- lm(CLV ~ spend_c + visit_c + open_c +
                    spend_c:visit_c + spend_c:open_c + visit_c:open_c +
                    spend_c:visit_c:open_c,
                  data = clv)
summary(m_full_3way)

Compare 2-way vs. 3-way

SSE_2way_3way <- sum(residuals(m_2way_3way)^2)
SSE_full_3way <- sum(residuals(m_full_3way)^2)

PRE_3way_over_2way <- (SSE_2way_3way - SSE_full_3way) / SSE_2way_3way
PRE_3way_over_2way

df_small_3way <- df.residual(m_2way_3way)
df_big_3way   <- df.residual(m_full_3way)

F_3way <- ((SSE_2way_3way - SSE_full_3way) /
             (df_small_3way - df_big_3way)) /
  (SSE_full_3way / df_big_3way)

p_3way <- pf(F_3way,
             df1 = df_small_3way - df_big_3way,
             df2 = df_big_3way,
             lower.tail = FALSE)

F_3way
p_3way

Interpretation: A significant 3-way means the 2-way interaction between any two of these variables is itself changing across levels of the third. So maybe spend and visit frequency work together one way when email open rate is low but that combined pattern looks different when email open rate is high.

This is more complicated than just “the lines are not parallel,” which means if someone asks you what the 3-way means, your best move is usually not to wave vaguely at the coefficient table. Your best move is to look at predicted values or plots and describe the pattern in slices.

TipNote this for the exercise

Record PRE_3way_over_2way and F_3way. You will be asked to identify both values. Also note the coefficient on spend_c in m_full_3way — you will be asked for the simple slope of spend when both visit_c and open_c are at their means (hint: when both moderators equal zero, all interaction terms drop out).

Part 7 — Probe the 3-Way Interaction with Slices

Let’s hold email open rate at low and high values and examine the predicted spend × visit relationship.

open_sd <- sd(clv$open_c, na.rm = TRUE)
open_lo <- -open_sd
open_hi <-  open_sd

visit_sd <- sd(clv$visit_c, na.rm = TRUE)
visit_lo <- -visit_sd
visit_hi <-  visit_sd

spend_seq <- seq(min(clv$spend_c, na.rm = TRUE),
                 max(clv$spend_c, na.rm = TRUE),
                 length.out = 40)

plot(spend_seq,
     predict(m_full_3way,
             newdata = data.frame(spend_c = spend_seq,
                                  visit_c = visit_lo,
                                  open_c  = open_lo)),
     type = "l", lty = 2,
     ylim = c(0, 700),
     xlab = "Centered Monthly Spend",
     ylab = "Predicted CLV",
     main = "3-Way Interaction: Spend × Visit × Open Rate")

lines(spend_seq,
      predict(m_full_3way,
              newdata = data.frame(spend_c = spend_seq,
                                   visit_c = visit_hi,
                                   open_c  = open_lo)),
      lty = 1)

lines(spend_seq,
      predict(m_full_3way,
              newdata = data.frame(spend_c = spend_seq,
                                   visit_c = visit_lo,
                                   open_c  = open_hi)),
      lty = 3)

lines(spend_seq,
      predict(m_full_3way,
              newdata = data.frame(spend_c = spend_seq,
                                   visit_c = visit_hi,
                                   open_c  = open_hi)),
      lty = 4)

legend("topleft", bty = "n",
       legend = c("Low visit / Low open",
                  "High visit / Low open",
                  "Low visit / High open",
                  "High visit / High open"),
       lty = c(2, 1, 3, 4))

Prompt: Do the “high visit” vs. “low visit” differences look the same at low and high open rate? If not, that is exactly the kind of pattern a 3-way interaction is capturing.

Part 8 — Be Smarter Than R: day_of_year_joined

This part is mostly conceptual and exploratory.

We have a variable that is numeric (day 1 to 365), but it is really cyclical. Day 365 and day 1 are not actually “far apart” in any meaningful seasonal sense.

If we treat this as an ordinary numeric predictor, R will happily go along with that plan — because R is extremely accommodating in that way. But we should still stop and ask whether that is substantively smart.

A model can run (and even produce significant results) and still be a bad idea.

8A — Naive linear model

m_day_lin <- lm(CLV ~ day_of_year_joined, data = clv)
summary(m_day_lin)

8B — Naive quadratic model

m_day_quad <- lm(CLV ~ day_of_year_joined + I(day_of_year_joined^2), data = clv)
summary(m_day_quad)

SSE_day_lin  <- sum(residuals(m_day_lin)^2)
SSE_day_quad <- sum(residuals(m_day_quad)^2)

PRE_day_quad <- (SSE_day_lin - SSE_day_quad) / SSE_day_lin
PRE_day_quad
plot(clv$day_of_year_joined, clv$CLV,
     pch = 16, cex = .4,
     xlab = "Day of Year Joined",
     ylab = "CLV",
     main = "CLV by Day Joined")

day_seq <- seq(1, 365, length.out = 200)
lines(day_seq,
      predict(m_day_quad,
              newdata = data.frame(day_of_year_joined = day_seq)),
      lwd = 2)

Is a quadratic capturing something here? Probably. Is it necessarily the smartest way to represent this pattern? Probably not.

Why not? Because seasonality can create patterns that look nonlinear without being well described by one simple parabola. Day 365 and day 1 represent consecutive days in the real world but in a quadratic model they are treated as the endpoints of an arch, which imposes a shape the data may not actually follow.

We’ll talk about some solutions to this next week.

For now, this is a good reminder: do not let the software do all the thinking.

Part 9 — Optional Preview: Categorical Predictors

Not central for this week, but worth a look to see where we are headed:

table(clv$campaign_type)
table(clv$segment)
table(clv$primary_channel)

m_campaign <- lm(CLV ~ campaign_type, data = clv)
summary(m_campaign)

We will come back to categorical predictors and factor coding more explicitly. For now, just notice that meaningful group differences are already present in this data.

Part 10 — Practice: Additional Nonlinear Models

NoteThis section is for practice only — it is not part of the graded exercise submission.

The questions below cover aspects of the material some students find trickiest: building a nonlinear-by-linear model from scratch, running a second 3-way interaction, and constructing your own analysis from the ground up. Work through whichever option is most useful to you. Answer callouts are provided so you can check your work.

Option A: Nonlinear-by-linear with spend and email open rate

Fit these three models:

  • Model 1: CLV ~ spend_c + I(spend_c^2) + open_c
  • Model 2: CLV ~ spend_c + I(spend_c^2) + open_c + spend_c:open_c
  • Model 3: CLV ~ spend_c + I(spend_c^2) + open_c + spend_c:open_c + I(spend_c^2):open_c

Questions to answer:

  1. Is there evidence of curvature in spend? (You already fit this in Part 3 — use what you found.)
  2. Does the linear effect of spend on CLV depend on email open rate?
  3. Does the curvature in spend depend on email open rate?
  4. In plain English, what pattern do you think this model is capturing?

Starter skeleton:

m1 <- lm(CLV ~ spend_c + I(spend_c^2) + open_c, data = clv)
m2 <- lm(CLV ~ spend_c + I(spend_c^2) + open_c +
            spend_c:open_c, data = clv)
m3 <- lm(CLV ~ spend_c + I(spend_c^2) + open_c +
            spend_c:open_c + I(spend_c^2):open_c,
          data = clv)

summary(m1)
summary(m2)
summary(m3)

SSE_m1 <- sum(residuals(m1)^2)
SSE_m2 <- sum(residuals(m2)^2)
SSE_m3 <- sum(residuals(m3)^2)

PRE_m2_vs_m1 <- (SSE_m1 - SSE_m2) / SSE_m1
PRE_m3_vs_m2 <- (SSE_m2 - SSE_m3) / SSE_m2

PRE_m2_vs_m1
PRE_m3_vs_m2

# Use the same df.residual() and pf() pattern from Parts 1 and 3
# to compute F and p for each comparison.
# You will need:
#   df_num = df.residual(smaller model) - df.residual(larger model)
#   df_den = df.residual(larger model)

PRE, Model 2 vs. Model 1 (adding spend_c:open_c): approximately 0.044

F for Model 2 vs. Model 1: approximately 184.1, p < .001 — the linear interaction between spend and email open rate is statistically significant.

PRE, Model 3 vs. Model 2 (adding I(spend_c^2):open_c): approximately 0.006

F for Model 3 vs. Model 2: approximately 24.2, p < .001 — the nonlinear-by-linear term is also significant, though it accounts for considerably less additional variance than the linear interaction did.

Plain-English interpretation: Customers who open more emails get more CLV out of higher monthly spend — the spend→CLV relationship is steeper for high-open-rate customers. Model 3 adds the further finding that the curvature in that relationship also differs: at high open rates, the diminishing returns on spend are somewhat less pronounced than at low open rates.

Option B: Another 3-way interaction

Fit: CLV ~ tenure_c * sat_c * eng_c

Then:

  1. Compare additive vs. 2-way vs. 3-way models
  2. Compute PRE and F for the 3-way term
  3. Create at least one useful slice plot
  4. Write 2–4 sentences explaining what the 3-way means — without just saying “there is a 3-way interaction”

To check your model comparison, the key test is the 3-way term: tenure_c:sat_c:eng_c. Your F statistic for that single term (3-way model vs. 2-way model) should be significant at conventional levels. If you are getting an F near zero or a PRE near zero, double-check that you built your 2-way model with all three pairwise interactions (tenure_c:sat_c, tenure_c:eng_c, and sat_c:eng_c) before adding the 3-way.

For your slice plot, hold one variable at ±1 SD and plot the relationship between the other two. A clean approach: hold eng_c at low and high, then show how the tenure_c × sat_c relationship changes across those panels.

Plain-English framing to aim for: The 3-way means that the way satisfaction and tenure work together to predict CLV is itself different for high- vs. low-engagement customers. It is not enough to say “tenure and satisfaction both matter” or even “they interact” — the shape of that interaction depends on where a customer falls on engagement.

Option C: Build your own nonlinear story

Use one continuous \(X\) with clear theoretical motivation, and optionally add one \(Z\) and/or \(W\). Your analysis must include:

  1. A compact linear model
  2. A quadratic model
  3. At least one model comparison using PRE
  4. Correct interpretation of centering
  5. A plain-English explanation of the pattern

There is no single right answer here, but your write-up should be able to answer yes to all of the following:

  • Does your PRE comparison have a clear numerator (SSE of the simpler model minus SSE of the more complex model) and denominator (SSE of the simpler model)?
  • Is your F statistic computed using the correct degrees of freedom — df_num from the difference in residual df, df_den from the larger model?
  • After centering, does your intercept now represent a predicted value at a meaningful reference point (the mean of X) rather than at X = 0?
  • Can you state in one or two plain sentences what the curvature means — not just that the quadratic term is significant, but what the shape tells you about the relationship?

If you used a second variable and included interaction terms, also ask: did you keep the lower-order terms? And did you use : rather than * for the interaction terms so that your model comparisons are unambiguous?