Unit 3 Quiz - Quantitative Analysis of Consumer Data

Linear regression

so far, we have generally assumed a linear relationship between Y and X

• this is an assumption of OLS

• a non-linear relationship between Y and X violates this assumption

Non-linear relationship example

See image

OLS Failure Image

See image

Can we adjust our model to still use linear regression to estimate a non-linear relationship between X and Y?

Yes! (sometimes)

How Much Would You Buy?

• Extreme example:

  • How much more of X would you buy if Price goes $10 → $5?

    • ???

  • How much more of X would you buy if Price goes $5 → $0?

    • ???

      • ∞

  • How can we get an accurate, actionable estimate of consumers responses to changes in price?

Demand curve

• Change in Units Sold when Price increases by $1 is not constant

• A linear regression coefficient assumes it is

• Weird note: economics classes will flip these axes

Demand curve (continued)

• % change in Units Sold when Price increases by 1% is constant

  • This is a stable empirical finding in economics

• How can we capture that with regression?

Elasticities

• A crucial concept in most pricing strategies is elasticity

• Elasticity measures the how the percent change in one variable (e.g., price) affects the percent change in another variable (e.g., units sold)

Three common measures of elasticity in marketing/economics/business

1) own-price elasticity of demand

2) cross-price elasticity of demand

3) income elasticity of demand

own-price elasticity of demand

how does demand for a product changes when its price changes?

cross-price elasticity of demand

how does demand for one product change when the price of another product changes?

income elasticity of demand

how does demand for a product change when the customer’s income changes?

own-price elasticity of demand?

• OPE = %Δ in Units Sold / %Δ in Price

  • Almost always negative (we like lower prices)

• Inelastic Demand: -1 < OPE < 0

  • → |%Δ in Units Sold| < %Δ in Price

• Elastic Demand: OPE < -1

  • → |%Δ in Units Sold| > %Δ in Price

• Examples?

what drives OPE?

• what overarching aspects of purchase decisions drive whether demand for a good/service is elastic or inelastic?

  • want vs need

  • availability of substitutes

    • also includes brand loyalty, inertia, switching costs, path dependence

  • percent of budget

  • time horizon

own-price elasticity of demand (regular)

Non-linear relationships between Price and Units Sold!

Variable transformations

• it is common in real word data to transform variables for a variety of reasons

  • we have already seem examples of this → standardized regression coefficients scale variables in terms of standard deviations

• one useful reason to transform variables is to create a linear relationship

  • in most markets, the relationship between price and sales is approximately logarithmic

own-price elasticity of demand (ln)

Linear relationship between ln(Price) and ln(Units Sold)!

Log-Log Regressions

• When we run a linear regression with ln(Y) as our dependent variable and ln(X) as our independent variable…

  • A 1-unit increase in ln(X) predicts a β-unit change in ln(Y)

    • Is this helpful?

  • A 1% increase in X predicts a β% change in Y

    • This is our definition of elasticity!

      • → We can easily estimate elasticities from data by running “log-log” regressions

Cross-Price Elasticity of Demand

• CPE = %Δ in Units of X Sold / %Δ in Price of Y

• Substitutes: CPE > 0

  • → Y is an alternative to X

  • → If the price of Y goes down, I buy less X and more Y

• Compliments: CPE < 0

  • → Y and X are used together

  • → If the price of Y goes down, I buy more X and more Y

• Examples?

Income Elasticity of Demand

• IE = %Δ in Units Sold / %Δ in Customer Income

• Inferior Goods: IE < 0

  • As my income goes up, I buy less

  • Examples?

• Normal Goods: IE > 0

  • Necessities: 0 < IE < 1

    • |%Δ in Units Sold| < %Δ in Income

      • → As my income goes up, I buy more at a decreasing rate

• Luxuries: IE > 1

  • |%Δ in Units Sold| > %Δ in Income

    • → As my income goes up, I buy more at an increasing rate

So far…

• Basic question we’ve addressed thus far: How does X affect our prediction of Y?

  • Or, how does X1 affect our prediction of Y, holding other X's constant?

• New question: How does the effect of X1 on our predictions of Y change depending on the value of X2?

  • No longer linear

  • Need an interaction term!

Product Reviews Interaction (Example)

• You are studying the impact of a product’s average consumer rating on its sales

• Would you rather buy a product with…

  • A) an average rating of 4.9 on 10 total reviews

  • B) an average rating of 4.8 on 10,000 total reviews

• The effect of average rating likely depends on the total number of reviews

Product Reviews Interaction - R

• R syntax: regmod(Y ~ X1*X2, data)

  • R shorthand for regmod(Y ~ X1 + X2 + X1:X2 , data)

• regmod(Sales ~ Avg_rating * Num_Reviews, data = data_reviews)

The script for interpreting interactions

• General interaction interpretation: how does the effect of one predictor change depending on the value of another predictor?

• Let β1 = the effect of X1 on predictions of Y (aka Ŷ)

• Let ΔX2 = what it means to change the value of X2

• Interaction effect = When ΔX2 , β1 would change by β3

• Note: multiplication is symmetric. Deciding which predictor is X1 and which is X2 depends on our hypothesis

  • X1 = "focal" variable that we are primarily interested in its impact on the outcome

  • X2 = "moderator" that can alter the effect of the "focal" variable

Product Reviews Interaction - R (Continued)

Table breakdown

Multicollinearity and Variable Selection

• Tolerate VIFs > 10 (we are explicitly adding a correlated predictor)

  • regmod() suppresses the VIF for the interaction term

  • Multicollinearity problem if VIF for main effects > 10

• Never omit the "main effects" from an interaction model

  • The main effects (coefficients, effect size) are no longer interpretable on their own because they now have a multiplicative effect on the outcome

  • The question of interest is about the interaction

• Interaction R2i is generally small, even when effect size is large

  • The variables were already there for the model to use for predictions, the interaction term clarifies the best way to use them

You try - Income “Elasticity” (Example)

• You work for a company that is trying to understand how sensitive its customers are to changes in price

• You know low-income customers will be more price sensitive than high-income customers

• Your goal is to see how large of an effect income has on price sensitivity

• You measure total units sold to test this

You try - Income “Elasticity” (Example Continued)

If income were $1k higher, the effect of a $1 increase in price on Units Sold would be 1.06 units larger

→ note: smaller impact

Types of Interactions

• Continuous X Continuous

• Continuous X Categorical

• Categorical X Categorical

Is Advertising More Effective for Startups? (Example)

• You want to know if advertising is more effective for startups or established brands

• People already aware of established brands, more ads reinforce awareness

• People may be unaware of startups, more ads generate more awareness

• You observe this pattern in your data but you need to formally test it

Is Advertising More Effective for Startups? (Example - R)

See image for output

The Script for Interpreting Interactions

• Y ~ β0 + β1*X1 + β2*X2 + β3* X1 *X2

• General interaction interpretation: how does the effect of one predictor change depending on the value of another predictor?

• Let β1 = the effect of X1 on predictions of Y (aka Ŷ)

• Let ΔX2 = what it means to change the value of X2

• Interaction effect = When ΔX2 , β1 would change by β3

Startup Advertising Interaction (Continued)

See image for table breakdown

Is Advertising More Effective for Startups? (Conclusion)

• Is ad spending more effective for startups?

• Yes, the change in sales from a one-unit increase in Ad_Spend is $7.92 larger for startups than established brands

• Slope for Startups – Slope for Established = 7.92

• Slope for Established = 5.38

• Slope for Startups = 13.3

You Try – Discounts for Loyalists

• You work for a company that is trying to understand how loyalty membership affects discount sensitivity

• You observe the data on the right and need to determine whether that interaction exists

You Try – Discounts for Loyalists (Continued)

Effect for non-members = 30.18 + 105.47 = 135.65

Effect of discount for members = 30.18

Effect of discount for nonmembers – Effect of discount for members = 105.47

Regression with Binary Outcome Variables

• OLS assumes a linear relationship between the predictors and the outcome

• If Y is binary (buy/didn’t buy), this is not true

• Using OLS in this case violates assumptions and makes suboptimal predictions

Did a Customer Buy a Subscription? (Example)

• We're doing market research studying how price impacts whether people subscribe to a streaming service

• For each customer, we observe the price they saw and whether they subscribed

• OLS?

• Running OLS with this data reveals a glaring problem…

• Predicted probabilities can fall outside [0,1]

  • Impossible

Logistic Regression

• take the OLS line and “bend” it into a sigmoid (S-shaped) curve

• this keeps predicted probabilities within [0,1]

• it also improves model fit (sometimes dramatically)

Logistic Regression explained

• Thus far, everything we've done has been housed under OLS

• Why introduce logistic regression?

  • Binary outcomes are very common in marketing

  • Logistic regressions predict those outcomes more accurately than OLS

  • ML/AI

  • Demystifying the difference between linear and logistic regression

    • The way it's typically presented is odd

Logistic Regression – The Key Change

• OLS assumes the effect of a predictor on the outcome is constant

• When the outcome is binary, the effect is smaller near the bounds

Marginal Effect (ME):

Change in Ŷ for a one-unit change in a X, starting from a specific value of X

→ Not constant in logistic regression

How do we Capture this Nonlinearity? (marginal effect)

• OLS → constant linear effect

• log-log → nonlinearity captured by changing units

• Interactions → nonlinearity captured by multiplicative interaction

• Logistic regression → nonlinearity captured by a decay factor

Logistic Regression as Decaying ME's

• In logistic regression, predictors have their largest impact when P(Y=1) = 50%

• Predictors' ME's approach zero as P(Y=1) approaches 0 or 1

• Why? …

Logistic Regression as Decaying ME's (Continued)

The effect of a predictor on a binary outcome combines two values

1) Maximum effect the predictor can have (i.e., ME at p̂ = 50%)

2) How that effect decays as p̂ moves away from 50%

Logistic Regression as Decaying ME's (About the Decay Factor)

• Same for every predictor in every logistic regression

• The decay accelerates as you approach the bounds

So How Do We Do This?

Logistic regression is simply OLS + an "activation function"

Logistic Regression = OLS + Sigmoid Activation

• The GIF is real!

• Mathematically, we are just applying a function to bend the OLS line

→ This lets us keep almost everything the same, just add the decay

Note on "OLS + Activation Function"

Framing logistic regression in this way has two key advantages

1) Most of our previous machinery and interpretations remain

2) This is a broader concept that generalizes outside our class

→ There are many activation functions used in other fields or for other purposes, especially in ML/AI

→ Knowing this creates an easy bridge for you

Running Logistic Regression in R

• Don't ask AI to give you code for this

• Default logistic regression output/interpretation is more complicated

• Default R functions for it make it even more complicated

• So how do we do it?

• regmod() ☺

Logistic Fit > OLS Fit

Logistic regression is the winner due to a higher R²

Multiple Predictors? No Problem

• Let's say some of the customers received a free trial of the streaming service before deciding whether to subscribe

• As before, the presence of a free trial has the largest ME for customers on the fence

• The ME of price remains the same, holding trial status constant