Product Management Fundamentals

Chapter 12
Product Metrics, Analytics and Experimentation

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Product Metrics, Analytics and Experimentation

Introduction

Product management is increasingly an evidence-based discipline. Product teams use behavioural data, customer research, financial information and experimentation to decide what to build, for whom, and in what sequence. Yet measurement is not synonymous with good decision-making. A team can monitor hundreds of dashboards and still misunderstand customer value, optimise a local funnel while damaging long-term retention, or interpret correlation as causation.

This chapter presents a rigorous approach to product measurement. It connects organisational goals to product outcomes through metric hierarchies, North Star metrics, OKRs, funnels, cohorts and retention analysis. It then examines quantitative and qualitative evidence, experimentation design, statistical interpretation, causal reasoning, governance and practical implementation.

A central principle is that metrics are models of reality. They are useful representations, not reality itself. A metric becomes valuable when its definition, data-generating process, decision use and limitations are understood.

Learning Objectives

By the end of this chapter, the reader should be able to:

  1. Construct a metric hierarchy connecting strategy, outcomes, drivers and guardrails.
  2. Evaluate North Star metrics and OKRs without confusing activity with value.
  3. Analyse funnels, cohorts, retention and segmentation.
  4. Combine qualitative evidence with quantitative evidence.
  5. Design and interpret A/B tests using appropriate statistical and causal logic.
  6. Identify common analytical errors, including selection bias, novelty effects, peeking and Simpson’s paradox.
  7. Establish trustworthy metric governance and responsible use of customer data.

1. The Logic of Product Measurement

A product metric is a defined measurement of a product-related phenomenon over a specified population and time period. Examples include weekly retained users, successful payment rate, average resolution time and percentage of new customers completing activation.

A useful metric definition specifies:

  • Object: what is being measured, such as users, orders or sessions.
  • Event or state: what happened or what condition exists.
  • Population: which users or entities are included.
  • Time window: daily, weekly, monthly or another interval.
  • Unit: users, sessions, dollars, minutes or percentages.
  • Aggregation: count, rate, mean, median, percentile or sum.
  • Exclusions: bots, internal accounts, duplicates, fraud or test traffic.

For example, “activation rate” is incomplete. A stronger definition is: “the percentage of newly registered, non-test accounts that complete the first successful project creation within seven days of registration.”

A basic rate is:

Rate = Numerator ÷ Denominator × 100%

The numerator must be a subset of the denominator, and both must refer to the same population and period. Rates are dimensionless, although their interpretation depends heavily on the denominator.

Leading, lagging and diagnostic measures

Lagging indicators measure outcomes after value has been delivered, such as retained revenue, renewal rate or customer lifetime value. Leading indicators occur earlier and may predict later outcomes, such as successful onboarding or repeated use of a core feature. Diagnostic metrics explain why an outcome changed, such as page-load time, error rate or abandonment at a particular funnel step.

A leading metric is not automatically a good metric. It should have conceptual relevance, empirical association with the desired outcome and sufficient sensitivity to product changes. A team should not promote a metric merely because it moves quickly.

2. Metric Hierarchies and Strategic Alignment

A metric hierarchy translates broad strategy into measurable product behaviour. A robust hierarchy usually includes four layers:

  1. Strategic objective: the business or mission-level result.
  2. Outcome metric: the customer or business value the product seeks to create.
  3. Driver metrics: behaviours or conditions that influence the outcome.
  4. Guardrail metrics: constraints that prevent harmful optimisation.

For a digital learning product, the hierarchy might be:

LayerExample
Strategic objectiveImprove sustainable learning outcomes
North Star outcomeWeekly learners completing meaningful learning sessions
DriversFirst lesson completion, relevant recommendations, reminder response
GuardrailsRefund rate, complaint rate, content quality, support contacts
Diagnostic metricsSearch latency, video failure rate, drop-off by lesson step

The hierarchy should express plausible relationships rather than imply that every driver causes the outcome. Causal claims require evidence.

North Star metrics

A North Star metric is a central measure of recurring customer value that helps align product, design, engineering, marketing and commercial teams. It typically combines:

  • a valuable customer action;
  • a defined frequency or time horizon;
  • an eligible population; and
  • a quality threshold where necessary.

“Monthly active users” is often too broad because any low-value visit may count. “Monthly users who complete at least one meaningful analysis used in a decision” is more value-oriented, provided “meaningful” is operationally defined.

A North Star metric should not be treated as a single executive target that replaces all other measures. It is supported by drivers and constrained by guardrails. Teams should also monitor whether the metric can be gamed. For example, increasing notifications may raise weekly sessions while reducing customer satisfaction.

OKRs and metrics

Objectives and Key Results (OKRs) connect qualitative ambition with measurable results. An objective states a directional aspiration; key results specify observable outcomes. “Make onboarding feel effortless” is an objective. “Increase the seven-day activation rate from 42% to 55% while keeping support contacts per new account below 0.18” is a stronger set of key results.

OKRs differ from a dashboard. A dashboard describes performance; an OKR expresses a time-bounded change the organisation intends to achieve. Key results should generally measure outcomes rather than the completion of internal tasks. “Release a tutorial” is an initiative or output, not evidence that customers benefited.

3. Funnels, Cohorts and Retention

Funnel analysis

A funnel represents a sequence of stages through which users may progress. A commerce funnel could include product view, add to cart, checkout initiation and purchase. For each stage, analysts calculate conversion and abandonment.

If Ni is the number of eligible users reaching stage i, the stage-to-stage conversion rate is:

ci = Ni+1 ÷ Ni

The overall conversion from the first to the final stage is:

C = Nk ÷ N1

where N1 is the number entering the funnel and Nk is the number completing the final stage. These formulas assume consistent user identity, a clearly defined ordering and no inappropriate mixing of cohorts.

Worked example. Suppose 10,000 users view a product, 2,000 add it to a cart, 1,200 begin checkout and 900 purchase. Stage conversions are 20%, 60% and 75%. Overall conversion is 900 ÷ 10,000 = 9%. The largest numerical loss occurs at the first transition, but that does not prove it is the highest-value intervention. The team should examine traffic quality, user intent, alternatives and the cost of improving each stage.

Funnels are often misused when naturally non-linear behaviour is forced into a fixed sequence. Some products require repeated discovery, comparison and return visits. In such cases, path analysis, state-transition analysis or cohort-based journeys may be more appropriate.

Cohorts

A cohort is a group sharing a defining characteristic, usually a common start date or acquisition period. Signup cohorts allow analysts to compare users who began under similar product conditions. Other cohorts may be defined by plan, geography, acquisition source, device or use case.

Cohort analysis reduces a major source of confusion: mixing old and new users. A rising monthly active-user count may result from acquisition growth even while each cohort retains less well. Cohort tables reveal whether behaviour improves for users who start after a product change.

Retention

Retention asks whether users continue to be active after an initial event. For a signup cohort, exact-day retention on day t can be defined as:

Rt = Users active on day t after signup ÷ Users in the signup cohort × 100%

This definition requires a specified qualifying activity and a consistent observation window. “Active” should mean meaningful product use, not merely opening an email.

Common forms include:

  • Classic retention: percentage active on a specific day or date.
  • Unbounded retention: percentage active on day t or any later time.
  • Rolling retention: percentage active on day t or within a subsequent interval.
  • Revenue retention: change in recurring revenue from an existing customer base, including expansion and contraction.

For recurring revenue, net revenue retention is commonly expressed as:

NRR = (Beginning recurring revenue + Expansion − Contraction − Churn) ÷ Beginning recurring revenue × 100%

The terms must refer to the same customer cohort and measurement period. New-customer revenue is excluded from the numerator when NRR is being calculated.

Retention is not identical to loyalty or satisfaction. A user may remain active because of switching costs, contractual obligation or lack of alternatives. Interpretation requires qualitative context.

4. Choosing and Evaluating Metrics

A useful metric is not simply one that is easy to collect. Product teams should assess metrics using the following properties:

  • Validity: does it represent the concept of interest?
  • Reliability: would the measurement be reasonably consistent?
  • Sensitivity: can it detect meaningful change?
  • Timeliness: is it available when decisions must be made?
  • Actionability: can the team identify interventions?
  • Interpretability: can stakeholders understand its meaning?
  • Resistance to gaming: can it be increased without creating value?
  • Comparability: are definitions stable across time and segments?

A metric can be statistically precise but strategically invalid. For example, clicks may be measured reliably while failing to represent customer success. Conversely, a valuable construct such as trust may require a carefully designed survey, interviews and behavioural proxies rather than one perfect number.

Counter-metrics and guardrails

Every target creates incentives. A team targeting delivery speed may reduce testing; a team targeting conversion may use aggressive defaults; a team targeting engagement may increase low-value notifications. Guardrails make trade-offs visible.

Typical guardrails include:

  • complaint and refund rates;
  • cancellation and opt-out rates;
  • error and latency measures;
  • accessibility failures;
  • fraud and abuse indicators;
  • customer support burden;
  • employee or partner impact.

Guardrails are not merely risk controls. They help define the quality dimension of growth.

5. Quantitative and Qualitative Evidence

Quantitative evidence describes magnitude, frequency and variation. Qualitative evidence helps explain meaning, motivation, confusion and unmet needs. Neither is inherently superior.

A decline in checkout conversion may be detected quantitatively. Interviews or usability sessions may reveal that customers distrust a new payment screen. Session recordings may show repeated attempts to find delivery information. Customer-support text may expose a vocabulary mismatch that event data cannot capture.

Triangulation means comparing multiple evidence sources. A practical sequence is:

  1. Detect an unusual pattern in behavioural data.
  2. Segment it by customer type, device, geography and acquisition source.
  3. Form competing explanations.
  4. Investigate using interviews, usability research, support data or qualitative coding.
  5. Test the most credible intervention.
  6. Re-measure both the target outcome and guardrails.

Qualitative evidence also has biases: participants may be unrepresentative, interviews may be leading, and researchers may selectively interpret responses. Analysts should document recruitment, prompts, coding procedures and disconfirming evidence.

6. Experimentation and A/B Testing

An A/B test randomly assigns eligible units to a control condition or a treatment condition. Randomisation aims to make the groups comparable in expectation, so differences in outcomes can be attributed to the treatment, subject to implementation quality and assumptions.

Designing an experiment

A sound experiment specifies:

  • the decision to be made;
  • the target population and unit of randomisation;
  • the treatment and control conditions;
  • a primary outcome;
  • guardrail outcomes;
  • the minimum practically important effect;
  • the intended duration and sample size rationale;
  • exclusion rules and stopping criteria;
  • the analysis plan before examining results.

The unit of randomisation may be a user, account, household, store or geographic region. It should match the level at which treatment is assigned and interference is plausible. If users influence one another, individual randomisation may violate the no-interference assumption.

Difference in conversion rates

Let pT be the treatment conversion rate and pC the control conversion rate. The absolute treatment effect is:

Δ = pT − pC

The relative lift, when pC is non-zero, is:

Lift = (pT − pC) ÷ pC × 100%

For example, if treatment conversion is 10.8% and control conversion is 10.0%, the absolute effect is 0.8 percentage points and relative lift is 8%. These are different statements and both should be reported.

Under independent Bernoulli observations, an approximate standard error for the difference in proportions is:

SE(Δ) = √[pT(1 − pT) ÷ nT + pC(1 − pC) ÷ nC]

where nT and nC are the numbers of observations in treatment and control. The approximation assumes independent observations and sufficiently appropriate sample sizes. Clustered, repeated or heavily imbalanced data require methods that account for dependence and design.

A confidence interval communicates sampling uncertainty. A 95% confidence interval is constructed by a method whose repeated-use procedure would contain the true parameter approximately 95% of the time under its assumptions. It does not mean there is a 95% probability that the fixed parameter lies inside this particular interval.

Statistical significance and practical significance

A small p-value does not establish that an effect is important, causal in every context, or replicable. With a very large sample, a trivial effect may be statistically detectable. With a small sample, a meaningful effect may be inconclusive.

Decision-makers should consider:

  • effect size and confidence interval;
  • practical and financial value;
  • risk and reversibility;
  • duration of the effect;
  • impact on segments and guardrails;
  • implementation and measurement quality.

Power and sample size

Statistical power is the probability of detecting an effect of a specified size under a chosen significance threshold, assuming the effect exists and the design assumptions hold. Sample-size planning depends on baseline rate, minimum detectable effect, significance level, desired power, allocation ratio and variance structure. It should be performed with an appropriate statistical method or validated software, not by an arbitrary rule such as “run it for one week.”

Common experimentation errors

Peeking: repeatedly checking results and stopping when a favourable p-value appears increases false-positive risk unless sequential methods are planned.

Multiple comparisons: testing many outcomes, segments or variants creates more opportunities for chance findings. Primary outcomes and correction or hierarchical procedures should be planned.

Novelty and primacy effects: users may react temporarily to a new interface or initially resist it. Duration should reflect the adoption cycle.

Sample-ratio mismatch: if observed allocation differs materially from the planned allocation, investigate instrumentation, eligibility, randomisation or exposure problems before interpreting outcomes.

Interference: one user’s treatment may affect another user, violating independence assumptions.

Survivorship bias: analysing only users who remain active can make a product appear more successful than it is.

Regression to the mean: extreme observations often move closer to average on subsequent measurement, even without intervention.

Simpson’s paradox: an aggregate trend may reverse within important subgroups because segment composition differs. Overall conversion should therefore be examined alongside relevant stratifications.

7. Causal Inference Beyond A/B Tests

Randomised experiments are powerful because assignment breaks the usual relationship between treatment and pre-existing characteristics, in expectation. Observational data generally cannot support the same causal claim without stronger assumptions or designs.

A causal question should be stated explicitly: “What would happen to these eligible users if exposed to the treatment instead of the current experience?” This is a counterfactual question. The individual treatment effect cannot normally be observed directly because the same unit cannot simultaneously receive both conditions.

Correlation may arise from:

  • confounding, where a third variable affects both exposure and outcome;
  • reverse causality, where the outcome influences the apparent cause;
  • selection effects, where inclusion depends on variables related to outcome;
  • measurement artefacts;
  • common trends or seasonality.

When randomisation is impossible, product analysts may use difference-in-differences, interrupted time series, matching, instrumental variables or regression discontinuity. Each requires substantive assumptions. For example, difference-in-differences relies on a credible parallel-trends assumption: absent treatment, treatment and comparison groups would have followed similar trends. Statistical adjustment does not automatically create causal identification.

8. Metric Infrastructure and Governance

Trustworthy analytics depends on more than dashboards. A mature measurement system includes:

  • a documented event taxonomy;
  • a single definition for important metrics;
  • stable user and account identifiers;
  • data-quality checks for volume, freshness, uniqueness and validity;
  • version-controlled transformations;
  • a metric dictionary and ownership model;
  • lineage from source event to reported number;
  • access controls and audit logs;
  • incident procedures for broken tracking.

A metric dictionary should record name, definition, formula, owner, source, refresh frequency, known limitations and change history. “Active user” should not have different meanings in product, marketing and finance reports without explicit qualification.

Ethics, privacy and responsible measurement

Product analytics involves personal and sometimes sensitive data. Responsible practice includes data minimisation, purpose limitation, appropriate consent or legal basis, retention limits, access control, encryption and transparent communication. Teams should distinguish aggregated insight from re-identification risk.

Metrics can produce discriminatory outcomes when data reflects historical exclusion or when model performance differs across groups. Segmentation should be used to detect unequal impact, not to exploit vulnerability. Automated personalisation should be reviewed for unfair treatment, manipulation and explainability. AI-generated analyses require human verification because language fluency does not guarantee valid causal interpretation.

9. Illustrative Case: Improving Activation in a Collaboration Product

Background. A fictional project-management platform acquires many trial accounts, but paid conversion is stagnant. Leadership proposes a new onboarding tour.

Problem. The headline seven-day activation rate is 38%. However, its definition counts any account that opens a workspace. Product research suggests that customers experience value only after creating a project, inviting a colleague and completing one task.

Analysis. The team revises the activation definition to completion of the three meaningful actions within seven days. It examines signup cohorts and finds that activation is 46% for desktop users and 24% for mobile users. Funnel analysis shows a major mobile drop at colleague invitation. Interviews reveal that the invitation screen requests permissions before explaining why collaboration is useful. Support tickets also show confusion about whether invited colleagues need paid accounts.

Approach. The team designs an experiment among eligible mobile trial accounts. Control receives the existing flow; treatment receives clearer permission language, an explanation of collaboration value and an option to invite later. The primary outcome is seven-day meaningful activation. Guardrails are opt-out rate, support contacts and privacy complaints. Randomisation occurs at the account level, and the analysis plan specifies one primary outcome and a fixed observation window.

Outcome. Treatment increases activation from 24.0% to 27.0%, an absolute improvement of 3 percentage points and relative lift of 12.5%. The confidence interval is sufficiently above zero under the pre-specified analysis. Support contacts do not increase, but invitation completion rises less for users acquired through a partner channel. The team therefore rolls out gradually and investigates the partner segment rather than assuming the aggregate effect applies equally everywhere.

Learning points. The original metric measured low-value activity. Cohort and segment analysis exposed heterogeneity. Qualitative research explained the funnel loss. The experiment tested a specific causal intervention, while guardrails prevented conversion optimisation from being treated as the only objective. The result supports the tested experience for the studied population; it does not prove that every onboarding change will improve paid conversion.

10. Best-Practice Operating Model

A product organisation can institutionalise disciplined measurement through a recurring cycle:

  1. State the customer and business decision.
  2. Define the outcome and guardrails.
  3. Map drivers and possible mechanisms.
  4. Audit data quality and metric definitions.
  5. Segment and establish a baseline.
  6. Gather qualitative evidence and competing explanations.
  7. Select an intervention and, where feasible, randomised test.
  8. Pre-specify outcomes, duration and analysis.
  9. Interpret effect size, uncertainty, heterogeneity and costs.
  10. Decide, document the result and monitor long-term effects.

The most mature teams also ask what evidence would falsify their interpretation. This practice reduces confirmation bias and encourages learning rather than ceremonial experimentation.

Challenges and Limitations

Product measurement remains difficult because user value is multidimensional, behaviour changes over time, data is incomplete and experimentation can alter the system being measured. Short-term metrics may conflict with long-term trust. Privacy restrictions can reduce observability. Small products may lack sufficient sample sizes for frequent A/B testing. Network effects and marketplace dynamics can invalidate simple user-level experiments. Organisational incentives may encourage metric gaming.

Consequently, metrics should support managerial judgement rather than replace it. A decision is stronger when it combines validated definitions, appropriate quantitative analysis, customer understanding, ethical review and explicit uncertainty.

Chapter Summary

Product metrics are formal representations of customer or business phenomena. Effective measurement begins with precise definitions of object, event, population, time window, unit and aggregation. Metric hierarchies connect strategy to outcomes, drivers, diagnostics and guardrails. North Star metrics should represent recurring customer value, while OKRs translate strategic intent into time-bounded outcomes.

Funnels identify progression and loss, but require consistent populations and may oversimplify non-linear journeys. Cohorts separate user groups by shared starting conditions, and retention measures whether meaningful activity persists. Quantitative evidence establishes magnitude; qualitative evidence explains motivations and mechanisms. Their combination supports stronger product decisions.

A/B tests use randomisation to estimate causal effects under explicit assumptions. Teams should report absolute effects, relative lift, uncertainty and practical significance. Peeking, multiple comparisons, novelty, interference, regression to the mean, survivorship bias and Simpson’s paradox can produce misleading conclusions. Observational analyses require careful causal assumptions and cannot automatically substitute for randomisation.

Finally, trustworthy product analytics requires metric governance, data-quality controls, privacy protection, fairness review and responsible interpretation. The aim is not to maximise a number, but to improve customer and organisational outcomes through credible evidence.

Key Terms

A/B test: A randomised comparison between a control condition and one or more treatment conditions.

Activation: A defined early behaviour indicating that a user has experienced initial product value.

Cohort: A group sharing a defining characteristic, often a common start date.

Confidence interval: An interval produced by a procedure designed to contain a parameter at a stated long-run frequency.

Conversion rate: The proportion of an eligible population completing a specified action.

Driver metric: A measure believed to influence or explain an outcome metric.

Experiment unit: The entity assigned to treatment or control.

Funnel: A sequence of stages through which users may progress.

Guardrail metric: A constraint monitored to prevent harmful optimisation.

Key Result: A measurable outcome used within an OKR.

Leading indicator: An earlier measure that may predict a later outcome.

Metric hierarchy: A structured relationship among strategic objectives, outcomes, drivers, diagnostics and guardrails.

North Star metric: A central measure of recurring customer value used for cross-functional alignment.

Null hypothesis: A specified reference claim, often that an effect or difference is zero.

OKR: Objectives and Key Results, a framework linking aspiration to measurable outcomes.

Retention: Continued meaningful activity or economic relationship after an initial event.

Statistical power: The probability of detecting a specified effect under stated assumptions.

Treatment effect: The difference in an outcome attributable to exposure to an intervention under a causal design.

Validity: The extent to which a measure represents the concept it is intended to measure.

Review Questions

Short Answer Questions

  1. What elements should be included in a precise metric definition?
  2. How does a North Star metric differ from a dashboard metric?
  3. Distinguish absolute treatment effect from relative lift.
  4. Why are cohorts useful in retention analysis?
  5. Name two risks associated with repeatedly checking experiment results.

Long Answer Questions

  1. Explain how a metric hierarchy can connect an organisational objective to drivers and guardrails. Illustrate with a product of your choice.
  2. Compare classic, unbounded and rolling retention, including situations in which each may be useful.
  3. Explain why qualitative and quantitative evidence should be triangulated in product discovery and optimisation.
  4. Describe the assumptions and practical limitations of an A/B test.
  5. Discuss how privacy, fairness and data governance affect product analytics.

Analytical / Application Questions

  1. A checkout funnel has 50,000 product views, 8,000 cart additions, 4,000 checkout starts and 3,000 purchases. Calculate each stage conversion and overall conversion. Identify what additional evidence is needed before selecting an intervention.
  2. A treatment produces 12% conversion compared with 10% in control. Calculate absolute improvement and relative lift, then explain why a decision-maker should also request a confidence interval and guardrail results.
  3. A product’s overall retention rises after a redesign, but retention falls in every major acquisition channel. Use cohort composition and Simpson’s paradox to explain how this could occur and propose an investigation.

Multiple Choice Questions with Answers

Question 1

Which metric is most clearly an outcome-oriented North Star metric?

A. Number of push notifications sent

B. Number of design tickets completed

C. Weekly customers completing a meaningful analysis

D. Number of sessions opened regardless of activity

Correct Answer: C. Weekly customers completing a meaningful analysis

Question 2

A funnel has 1,000 users at stage one and 250 at stage two. What is the stage conversion rate?

A. 4%

B. 25%

C. 40%

D. 75%

Correct Answer: B. 25%

Question 3

Which statement best describes a cohort?

A. All users currently logged in

B. A group sharing a defining characteristic such as signup month

C. All users who clicked an advertisement

D. A randomly selected treatment group only

Correct Answer: B. A group sharing a defining characteristic such as signup month

Question 4

Control conversion is 10% and treatment conversion is 12%. What is the relative lift?

A. 2%

B. 12%

C. 20%

D. 22%

Correct Answer: C. 20%

Question 5

What is the principal purpose of randomisation in an A/B test?

A. To guarantee a large effect

B. To eliminate the need for a control group

C. To make treatment groups comparable in expectation

D. To ensure every user receives treatment

Correct Answer: C. To make treatment groups comparable in expectation

Question 6

Which practice can increase false-positive risk in experimentation?

A. Pre-specifying a primary outcome

B. Repeatedly checking results and stopping when favourable

C. Monitoring guardrail metrics

D. Using a fixed observation window

Correct Answer: B. Repeatedly checking results and stopping when favourable

Question 7

What does a guardrail metric do?

A. Replaces the primary outcome

B. Prevents or reveals harmful side effects of optimisation

C. Measures only engineering effort

D. Guarantees statistical significance

Correct Answer: B. Prevents or reveals harmful side effects of optimisation

Question 8

Which is an example of survivorship bias?

A. Including all eligible users in the denominator

B. Analysing only users who remained active and ignoring those who churned

C. Randomly assigning accounts to variants

D. Comparing signup cohorts

Correct Answer: B. Analysing only users who remained active and ignoring those who churned

Question 9

Why can statistical significance be insufficient for a product decision?

A. It never involves uncertainty

B. A statistically detectable effect may be too small to matter commercially

C. It can only be used for qualitative data

D. It proves that the metric definition is valid

Correct Answer: B. A statistically detectable effect may be too small to matter commercially

Question 10

Which statement about observational correlation is correct?

A. Correlation always proves causation

B. Correlation can result from confounding or reverse causality

C. Correlation removes the need for segmentation

D. Correlation is impossible in product analytics

Correct Answer: B. Correlation can result from confounding or reverse causality

Product Management Fundamentals
Chapter 12 of 15