Logistic regression and classification

Alex Tutor

Everything so far predicted a number: orders, revenue. But churn is yes or no. Can we still use a straight line?

You

Sure, just fit a line where y is 1 for churned and 0 for stayed?

Alex Tutor

Try it, and the line happily predicts things like 1.3 or -0.2. Those aren't probabilities, and the relationship isn't a straight line near the edges anyway.

You

So what do we use instead?

Alex Tutor

An S-shaped curve that's always squeezed between 0 and 1 — logistic regression.

Alex Tutor

Question 1

What's the main problem with fitting a plain linear regression to a yes/no outcome coded as 0 and 1?

Choose one answerIt can predict values below 0 or above 1, which aren't valid probabilitiesIt can't be computed at all for a 0/1 outcomeIt always gives exactly the same coefficient as logistic regressionIt requires more data than logistic regression

To build that curve, we first need a trick for turning a probability into something that can stretch from minus infinity to plus infinity.

Start with odds: the probability of the event divided by the probability that it doesn't happen.

Alex Tutor

\text{odds} = \frac{p}{1-p}

Odds turn a 0-to-1 probability into a 0-to-infinity number: how many times more likely the event is than its opposite.

You

So if 80% of similar customers churn, the odds are... 4?

Alex Tutor

Exactly — 0.8 / 0.2 = 4. Odds of 4 mean churning is four times as likely as staying.

Alex Tutor

Question 2

If the probability of a customer churning is 0.8, what are the odds of churning?

Your answer

Units: e.g. 2.0

Odds still only go from 0 to infinity. Take the natural log of the odds, and now it's centred at zero and unbounded both ways.

Alex Tutor

\ln(\text{odds}) = b_0 + b_1 x

Log-odds (the logit): this is the linear part logistic regression actually fits.

You

And that log-odds thing is what the regression actually predicts?

Alex Tutor

Right. Fit it just like linear regression, then run it back through a formula that squeezes it back into a 0-to-1 probability.

Alex Tutor

p = \frac{1}{1 + e^{-(b_0 + b_1 x)}}

The logistic (sigmoid) function: steep in the middle, flat near 0 and 1.

That's the whole trick: a straight line in log-odds space becomes a smooth S-curve in probability space.

You

Let's try it on an actual churn model.

Alex Tutor

Question 3

CatChow's churn model is $\ln(\text{odds}) = -2 + 0.8 \times \text{months\_since\_last\_purchase}$ . Predict the probability of churn for a customer with 3 months since their last purchase.

Your answer

Units: 0 to 1, e.g. 0.5

About 60% churn risk at three months of silence. Let's talk about what that coefficient, 0.8, actually means.

You

Each extra month adds 0.8 to something, right?

Alex Tutor

To the log-odds, yes. Undo the log, and it means each extra month multiplies the odds of churn by about

e^{0.8} \approx 2.23

— not the probability itself.

Alex Tutor

You

Isn't that the same as saying the probability goes up 2.23 times?

Alex Tutor

No — and that mix-up is common enough that it's your next question.

Alex Tutor

Question 4

An odds ratio of 2.23 for a predictor means that each extra unit of the predictor doubles (roughly) the predicted probability of the outcome.

TrueFalse

Alex Tutor

A probability is a number. To actually act — send a retention offer or not — you need a yes/no cutoff.

You

0.5, obviously? More likely than not means yes?

Alex Tutor

That's the default, but not always the smartest one. If a retention offer is cheap and a lost customer is expensive, you might flag anyone above 0.2.

Alex Tutor

You

So the threshold is a business call, not a statistics one?

Alex Tutor

Both — statistics gives you the probability, and the business decides how cautious to be with it.

Whatever threshold you pick, you'll want to score how good the resulting yes/no calls actually are.

Every prediction falls into one of four boxes, compared with what actually happened.

Alex Tutor

Question 5

Match each confusion-matrix cell to its meaning for a churn model.

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

Model predicted churn, and the customer really churned
Model predicted churn, but the customer stayed
Model predicted the customer would stay, but they churned
Model predicted the customer would stay, and they stayed

Answers left to place

Good. Two useful scores come straight out of that table: precision and recall.

Precision: of everyone you flagged as churning, what share actually churned. Recall: of everyone who actually churned, what share you caught.

You

Why not just use accuracy — the share of all predictions that were right?

Alex Tutor

Because churn is rare. A model that just says "nobody churns" can score 95% accuracy while catching zero actual churners.

You

Oh — so accuracy alone can lie to you.

Alex Tutor

Question 6

A churn model's confusion matrix: 40 true positives, 10 false positives, 20 false negatives, 130 true negatives. Compute precision.

Your answer

Units: 0 to 1, e.g. 0.5

Nicely computed. Now here's the part that trips people up: precision and recall pull in opposite directions.

You

Can't we just maximise both at once?

Alex Tutor

Rarely. Lower the threshold and you flag more people: recall climbs, but precision usually drops as false alarms creep in.

For a cheap retention email, that trade is often worth it — CatChow would rather email a few people who'd have stayed anyway than miss real churners.

Alex Tutor

Question 7

You lower the classification threshold for the churn model (flagging more customers as at risk). Which of these typically happen? Select all that apply.

Choose all that applyRecall increases — you catch a larger share of actual churnersPrecision usually decreases — more false alarms creep inAccuracy always increasesYou flag more people overall, catching more true churners but also more false positives

Alex Tutor

Time to put this into words.

You

What am I explaining this time?

Alex Tutor

A quick cost trade-off.

You

Ready.

Alex Tutor

Question 8Short answer · AI-checked task

CatChow's churn model flags customers above a threshold for a $5 retention email. A churned customer costs about $150 in lost lifetime value on average. Would you want a higher or lower classification threshold, and why? Answer in 2-4 sentences.

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Good instinct. Now the full case, one last time this module.

You

Last one?

Alex Tutor

Last one — the churn model, the confusion matrix, and a real decision to recommend.

Here's the scenario.

Alex Tutor

Question 9Case · AI-checked task

Recommend whether CatChow should lower the churn model's classification threshold.

CatChow's churn model ( $\ln(\text{odds}) = -2 + 0.8 \times \text{months\_since\_last\_purchase}$ ) is currently run at a 0.5 threshold, giving this confusion matrix on last month's customers: 40 true positives, 10 false positives, 20 false negatives, 130 true negatives. Retention emails cost about $5 each; a churned customer costs roughly $150 in lost lifetime value. The retention team wants to know whether to lower the threshold to flag more customers. Write a short recommendation covering the current precision and recall, the cost trade-off, and what you'd suggest next.

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Great work, Sofia. You've gone from a single scatter plot to a full churn model with a defensible threshold. Module 6 is next: turning models into forecasts and finding customer segments.

Alex Tutor

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

Next lesson: Models and forecasts