Multiple regression

Alex Tutor

One predictor was a nice start, but real orders depend on ad spend, holiday weeks, maybe price too. What if we used all of them together?

You

So we just add up their separate effects?

Alex Tutor

Almost — multiple regression fits one equation with several predictors at once, each with its own coefficient.

Alex Tutor

\hat{y} = b_0 + b_1 x_1 + b_2 x_2 + \dots + b_k x_k

Multiple regression: one intercept plus one slope per predictor, still fit by minimising squared residuals.

Each coefficient is the effect of its own predictor, holding all the other predictors constant — as if nothing else in the equation changed.

You

"Holding constant"? We can't actually freeze the other variables in real life.

Alex Tutor

True — it's a mathematical statement about the fitted equation, not something you did during data collection. Still useful for comparing predictors fairly.

Alex Tutor

Question 1

In the equation orders = b0 + b1 x ad_spend + b2 x is_holiday_week, what does b1 represent?

Choose one answerThe predicted change in orders per extra dollar of ad spend, holding holiday-week status constantThe total effect of ad spend, including any overlap with holiday timingThe correlation between ad spend and ordersThe predicted orders when ad spend is zero

Let's make this concrete with a CatChow-style equation.

\text{orders} = 12 + 0.05 \times \text{ad\_spend} + 4 \times \text{is\_holiday\_week}

, where is_holiday_week is 1 in a holiday week and 0 otherwise.

You

So being a holiday week is worth a flat 4 extra orders, no matter the spend?

Alex Tutor

Exactly, in this equation — a fixed bump on top of whatever ad spend predicts.

Alex Tutor

Question 2

Using orders = 12 + 0.05 x ad_spend + 4 x is_holiday_week, predict orders for a holiday week (is_holiday_week = 1) with ad_spend = 400.

Your answer

Units: orders, e.g. 30

Nice. Now flip the holiday flag off.

You

Same spend formula, just drop the +4?

Alex Tutor

Exactly that.

You

Let's check.

Alex Tutor

Question 3

Same equation, a non-holiday week (is_holiday_week = 0) with ad_spend = 600. Predict orders.

Your answer

Units: orders, e.g. 30

Two coefficients down. But here's a trap: throwing in more and more predictors always makes R² look better on this data, even useless ones.

You

Wait, even random nonsense predictors help R²?

Alex Tutor

By a tiny amount, yes — pure chance gives every extra column a little credit. That's exactly why adjusted R² and holdout testing exist.

You

So R² alone can be misleading with lots of predictors?

Alex Tutor

Question 4

Adding another predictor to a regression can only increase or leave unchanged its R² on the data it was fit on, even if the new predictor is pure random noise.

TrueFalse

Right instinct. There's a second, sneakier problem when predictors are correlated with each other, not just with the outcome.

You

Like ad spend and... what else moves together?

Alex Tutor

Ad spend and discount depth, say — CatChow often deepens discounts during the same weeks it spends more on ads.

Alex Tutor

When two predictors move together that tightly, the model can't cleanly tell which one deserves the credit. Coefficients become unstable — add one data point and they can swing wildly.

Alex Tutor

Question 5

Match each situation to the term that describes it.

Tap an answer, then tap the row it belongs to. You can also drag.

Two predictors move together so tightly that their coefficients swing wildly when the sample changes slightly
A model fits its training data almost perfectly but predicts poorly on new data
Fitting a model on one part of the data and checking its performance on the other, unseen part
A score that rewards a good fit but penalises extra predictors, useful for comparing models

Answers left to place

Good matches. Both problems have fixes.

For multicollinearity: drop one correlated predictor, combine them, or fit a stepwise regression. For overfitting: penalise extra predictors with an information criterion, or test on holdout data.

You

So more predictors isn't automatically a better model?

Alex Tutor

Correct. More predictors is only better if each one earns its place.

Alex Tutor

Question 6

Which of these are true about multicollinearity? Select all that apply.

Choose all that applyIt happens when two or more predictors are strongly correlated with each otherIt makes individual coefficients unstable and hard to interpretIt means the model as a whole can no longer predict the outcome wellDropping one of the correlated predictors is one common fix

Alex Tutor

Time to put this into words.

You

Here we go again.

Alex Tutor

Quick one this time.

You

Fine, hit me.

Alex Tutor

Question 7Short answer · AI-checked task

A teammate proposes adding both ad_spend and discount_depth to the model, noting CatChow usually raises both together during sales. What risk should you flag, and what would you suggest? Answer in 2-4 sentences.

Write your answer and get a score with feedback from our AI reviewer.

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Good instinct — flag it, don't just ignore it. Now the full case.

You

Three predictors this time?

Alex Tutor

Three predictors, and one multicollinearity risk hiding among them. Find it.

Here's the scenario.

Alex Tutor

Question 8Case · AI-checked task

Review the fitted model and write your recommendation.

A colleague fit orders = 10 + 0.04 x ad_spend + 6 x discount_depth - 2 x price + 3 x is_holiday_week (R² = 0.88) and wants to tell leadership: "discount_depth has the biggest coefficient, so discounting is our best lever — let's cut ad spend and discount harder instead." You happen to know CatChow always deepens discounts in the same weeks it increases ad spend. Write a short recommendation: what you'd check before trusting the discount_depth coefficient, and how you'd advise leadership to proceed.

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Solid work, Sofia. You can now read a multi-predictor equation, and you know its two classic traps. Next: what if the outcome itself is yes/no, not a number — logistic regression.

Alex Tutor

That’s the lesson. You answered every task — nicely done.

Next lesson: Logistic regression and classification