An independent study reference written by Dr Phuc V. Nguyen. It is not official subject material — for assessment requirements always follow your subject outline and vUWS.

What Business Analytics Isadvanced

Correlation and causation

Q: An online retailer finds that customers who write a product review spend 40 per cent more over the following year, and proposes paying customers to write reviews. What is the strongest objection?

Customers who choose to write reviews may already differ in ways that predict spending, so the gap mixes any effect of reviewing with selection. Writing a review is self-selected, so reviewers and non-reviewers can differ in ways that themselves predict spending. The observed gap is then any effect of reviewing plus a selection term, and these data do not say how large either part is. Paying a different population to write reviews need not transfer whatever made the original reviewers valuable. A randomised holdout would separate the two.

Q: A team wants to know whether a new onboarding flow increases activation. Which piece of evidence best supports a causal claim?

New users were randomly assigned to the old or the new flow and activation was compared. Before-and-after comparisons absorb everything else that changed at the same time, such as a campaign or seasonality. Comparing completers with abandoners compares people who differ in motivation. Adding controls helps only if every confounder happens to be measured, which is rarely knowable. Random assignment removes the selection term by construction.

Two variables moving together is evidence of association and nothing more. Common explanations include the following. One variable really does cause the other. The causation runs the other way. A third variable drives both, which is confounding. The sample was selected in a way that manufactured the pattern. Or the pattern is chance, which turns up easily once many variables are compared. The distinction matters because business decisions are interventions. Knowing that app users churn less predicts nothing about what happens when you push app installs onto people who never chose one. A prediction can live on association. An intervention needs a causal claim.

Try it yourself

Confounding and the pooled gap

A subscription business has 10,000 members, and the allocation it opens on shows app users churning far less than everyone else. Engagement is recorded as two categories, high and low, so there is no binning choice to argue about. Move members between the cells and watch two numbers at once. The pooled gap swings and can change sign. The adjusted difference within engagement is only ever what the third slider says, and it disappears once a group holds no comparison members. Neither number is a measured effect of the app.

Pooled gap (app minus no app)

−6.8 pp

Adjusted difference within engagement

−1.0 pp

Bar height is the number of members, on a fixed axis. The bold figure above each bar is that cell's annual churn rate, and an empty cell shows an em dash because nothing is observed in it.

Pooled app churn 5.5%Pooled no-app churn 12.3%Expected app churn (members a year) 220

High-engagement members with the app3,000 of 4,000 (75.0%)

Low-engagement members with the app1,000 of 6,000 (16.7%)

Assumed within-group difference1.0 pp lower

Why engagement is the confounder

High-engagement members hold the app more often than low-engagement members, 75.0% against 16.7%. Engagement feeds uptake and it feeds churn, which is what makes it a confounder rather than a detail.

Observed difference after holding engagement fixed: −1.0 pp. That figure sits outside the diagram on purpose. It is an adjusted association, so it belongs to the dashed link as a question and not as an answer.

Annual churn by engagement group. Gap is app churn minus no-app churn in percentage points, so a negative gap means app users churn less. Every figure here is an observed difference, never a measured effect, and a group holding members in only one cell shows no gap at all.
Group	App churn	No-app churn	Gap (pp)	Group size
High engagement	3.0% n = 3,000	4.0% n = 1,000	−1.0 pp	4,000
Low engagement	13.0% n = 1,000	14.0% n = 5,000	−1.0 pp	6,000
Pooled	5.5% n = 4,000	12.3% n = 6,000	−6.8 pp	10,000

Pooled, app users churn at 5.5% against 12.3% for everyone else, a gap of −6.8 pp. Holding engagement fixed, the observed difference in each group is −1.0 pp. The pooled figure splits exactly into that adjusted difference of −1.0 pp plus a composition term of −5.8 pp. The app group is loaded with high-engagement members, who churn at 4.0% before the app is considered at all, so the pooled comparison is partly reporting who holds the app rather than anything the app did. The −1.0 pp that survives the split is an association within engagement, not a measured causal effect. That is why the app to churn link in the diagram is drawn dashed. A randomised holdout or another defensible causal design is needed before anyone forecasts what a campaign delivers.

Stylised model. The churn rate of each cell is held fixed while members are moved between cells, which is an assumption of the teaching case rather than a fitted result. Churn counts are rate-implied expectations rather than a tally of people, so they can land on a fraction. The default allocation reproduces the worked figures on this page.

Why it matters

Association answers a spotting question. If I see this, what else should I expect to see. Causation answers a doing question. If I change this, what changes as a result. The two come apart whenever the people who already have the thing differ from the people who do not. Loyal customers install the app and loyal customers stay. Handing the app to everyone else does not hand them the loyalty that came with it.

Before you read on — recall

An online retailer finds that customers who write a product review spend 40 per cent more over the following year, and proposes paying customers to write reviews. What is the strongest objection?

Formulas

Pearson correlation

r_{XY} = \frac{\operatorname{cov}(X, Y)}{s_X \, s_Y}

Covariance divided by both standard deviations, which puts

r

between -1 and 1 and strips out the units. It captures straight-line co-movement only. A perfect U-shaped relationship can return

r

near zero, and a correlation of 0.8 says nothing about which variable moved first.

The effect you actually want

\text{ATE} = E\big[Y(1) - Y(0)\big]

Write

Y(1)

for the outcome if a customer gets the treatment and

Y(0)

for the outcome if the same customer does not. Only one of the two is ever observed for any individual, which is the whole difficulty. The average treatment effect is the average of that gap across everyone.

What a naive group comparison measures

E[Y \mid D = 1] - E[Y \mid D = 0] = \underbrace{E[Y(1) - Y(0) \mid D = 1]}_{\text{effect on the treated}} + \underbrace{E[Y(0) \mid D = 1] - E[Y(0) \mid D = 0]}_{\text{selection bias}}

Comparing the treated group against the untreated group returns the effect you want plus a selection term, which is how the two groups would have differed anyway with no treatment at all. Random assignment forces that second term to zero because the groups are then alike in expectation. Nothing else does so automatically.

Worked examples

Scenario

A subscription business finds that members using the mobile app churn at 5.5 per cent a year against 12.3 per cent for members who do not. Marketing proposes a campaign to push app installs.

Solution

Split by how engaged members already were. Among 4,000 high-engagement members, 3,000 have the app and churn at 3 per cent while 1,000 do not and churn at 4 per cent. Among 6,000 low-engagement members, 1,000 have the app and churn at 13 per cent while 5,000 do not and churn at 14 per cent. Those figures reproduce the headline gap of nearly seven points, yet inside each engagement group the difference is one point. Engagement drives both installing the app and staying. Even that one point is an association adjusted for engagement, not a measured effect, so a randomised holdout or another defensible causal design is needed before anyone forecasts what the campaign delivers.

Scenario

A hospital finds that patients receiving a particular intensive therapy have higher mortality than those who do not, and a manager proposes withdrawing it.

Solution

Consider the direction of the arrow. Clinicians assign the intensive therapy to the sickest patients, so severity causes both the treatment and the death rate. The comparison measures who was selected, not what the therapy did. The selection term is large here and works against the therapy, which can make a helpful treatment look harmful. The available repairs are to compare within severity bands, to use a design that mimics random assignment, or to state plainly that these data cannot answer the question.

Common mistakes

✗A strong correlation means a strong causal effect. Strength describes how tightly points sit around a line. It says nothing about direction, and nothing about whether a third variable is producing both series.
✗Controlling for more variables always moves you closer to the truth. Adding the wrong variable makes things worse. Conditioning on something that sits between the cause and the effect, or on a common consequence of both, can create bias where there was none.
✗Correlation is useless. Association is enough for prediction, which is most of what analytics does. A model that spots which accounts are about to lapse does not need to know why. The causal question becomes unavoidable only when you intend to intervene.
✗Only a randomised experiment can support a causal claim. Randomisation is the cleanest route, but comparisons exploiting a policy change, a threshold rule or a staggered rollout can also support causal claims, provided the assumptions are stated and checked.

Revision bullets

•Sources of correlation: cause, reverse cause, confounding, selection, chance
•Prediction can live on association; intervention needs causation
•A naive group gap = effect on the treated + a selection term
•Random assignment is what drives the selection term to zero
•Confounding example: engagement drives both app use and retention
•Controlling for the wrong variable adds bias rather than removing it

Quick check

An online retailer finds that customers who write a product review spend 40 per cent more over the following year, and proposes paying customers to write reviews. What is the strongest objection?

A team wants to know whether a new onboarding flow increases activation. Which piece of evidence best supports a causal claim?

Connected topics

Three Analytics Types Sampling Bias Models & Abstraction A/B Testing Purpose of Visuals Visual Mistakes Why Ethics Matters Algorithmic Bias Changing Analyst Role

More in What Business Analytics Is

What Analytics Is BI vs Analytics Analytics Workflow Analytics in Practice Why Analytics Stalls

Sources

Pearl & Mackenzie (2018)
Pearl, J., & Mackenzie, D. The Book of Why: The New Science of Cause and Effect. Basic Books, 2018.
A non-technical treatment of why questions about intervention cannot be answered from association alone.
Angrist & Pischke (2009)
Angrist, J. D., & Pischke, J.-S. Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press, 2009.
Source of the decomposition of a naive group comparison into a treatment effect plus selection bias.
Bickel, Hammel & O'Connell (1975)
Bickel, P. J., Hammel, E. A., & O'Connell, J. W. "Sex Bias in Graduate Admissions: Data from Berkeley." Science, 187(4175), 398-404, 1975.
The classic worked case in which an aggregate association changes once the confounding variable, here the department applied to, is taken into account.

How to cite this page

Dr. Phil's Quant Lab. (2026). Correlation and causation. Business Analytics Atlas. https://phucnguyenvan.com/analytics_atlas/concept/ba-correlation-causation