What Factors Drive the Results You See?
Moving from describing what happened and when — to understanding which variables are responsible.
Sets 01–03 described data, diagnosed failures, and projected trends. Set 04 answers the next logical question: which variables actually drive the outcomes we care about? If revenue is growing, what is driving it — price, volume, mix, or something else? If customer satisfaction is declining, which factors explain the most variation?
The techniques in this set quantify relationships — attaching numbers to the question "how much does Y change when X changes?" This shifts analysis from observation to explanation.
variance explained
coefficient
Pearson Correlation
Measures the strength and direction of the linear relationship between two continuous variables. Produces r between −1 and +1.
Spearman Correlation
Non-parametric alternative to Pearson. Uses ranks instead of raw values. Appropriate when data is ordinal or non-normally distributed.
Simple Linear Regression
Models the relationship between one predictor (X) and one outcome (Y). Produces a slope (β) and intercept (α): Y = α + βX.
Multiple Linear Regression
Extends simple regression to multiple predictors simultaneously. Estimates the unique contribution of each predictor while controlling for the others.
Partial Correlation
Measures the relationship between two variables after removing the effect of one or more control variables. Isolates the "pure" relationship.
Elasticity Analysis
Measures the % change in Y for a 1% change in X. Used extensively in pricing, demand analysis, and economic modelling.
Driver Analysis (Relative Weight)
Ranks independent variables by their proportional contribution to R². Answers: "Which driver explains the most variation in the outcome?"
Path Analysis (IBM AMOS)
Tests direct and indirect relationships between variables. Goes beyond regression by modelling causal chains — X affects Y both directly and through mediating variable M.
All regression and correlation techniques identify statistical associations — even when they are strong, significant, and robust. Causal interpretation requires the disciplined reasoning covered in Sets 05 (Multivariate Analysis) and the analytical thinking programme's Modules 04–05. Use these techniques to quantify relationships; use Sets 05+ to defend causal claims.
Pearson & Spearman Correlation
The first question in any driver analysis: do these two variables move together — and how strongly?
Pearson Correlation Coefficient (r)
Pearson r measures the strength and direction of the linear relationship between two continuous variables. It ranges from −1 (perfect negative) to +1 (perfect positive), with 0 indicating no linear relationship.
| r value | Strength | Business interpretation | Typical example |
|---|---|---|---|
| +0.9 to +1.0 | Very strong positive | X almost always rises when Y rises | Ad spend and revenue in a growing market |
| +0.6 to +0.9 | Strong positive | Clear positive pattern — X is likely a meaningful driver | Training hours and performance scores |
| +0.3 to +0.6 | Moderate positive | Some positive relationship; other factors also matter | Customer age and product spend |
| −0.3 to +0.3 | Weak or no relationship | Little evidence of linear association | Shoe size and job performance |
| −0.6 to −0.3 | Moderate negative | X tends to rise as Y falls | Wait time and satisfaction score |
| −0.9 to −0.6 | Strong negative | Strong inverse relationship | Machine age and output quality |
Pearson assumes both variables are continuous and normally distributed, and measures only linear relationships. Use Spearman instead when: data is ordinal (satisfaction ratings, rankings) · data is not normally distributed · the relationship is monotonic but non-linear (always increasing or always decreasing, but not at a constant rate).
Spearman Rank Correlation (ρ)
Spearman's ρ (rho) works the same way as Pearson — also ranges from −1 to +1 — but instead of using raw values it converts values to ranks first, then calculates the correlation of the ranks. This makes it robust to outliers and non-normal distributions.
An HR analyst wants to understand whether employee satisfaction rank correlates with productivity rank. Because satisfaction is measured on a 1–5 Likert scale (ordinal data) and productivity has a few extreme outliers, Spearman is the correct choice. The ranks are stable and the outliers do not distort the correlation. Pearson would give a misleadingly weak result because of the outliers.
What is R² (Coefficient of Determination)?
R² = r × r. If r = 0.80, then R² = 0.64. This means 64% of the variation in Y is explained by the linear relationship with X. The remaining 36% is explained by other factors not in the model. R² is the single most important number to understand in regression output.
| r value | R² value | Plain interpretation |
|---|---|---|
| 0.90 | 0.81 | X explains 81% of the variation in Y. Very high — most of Y's movement is accounted for by X. |
| 0.70 | 0.49 | X explains 49% of variation. Substantial — but half of Y's variation comes from other factors. |
| 0.50 | 0.25 | X explains only 25%. Real relationship, but most of Y is explained by things not in this model. |
| 0.30 | 0.09 | X explains only 9%. Weak — X matters but barely. Probably not the key driver. |
Simple Linear Regression
One predictor. One outcome. How much does Y change for each unit increase in X?
Correlation tells you that two variables are related. Regression quantifies the relationship. Simple linear regression fits the best possible straight line through a scatter of points — the line that minimises the total prediction error across all observations.
Outcome variable
(what we want to predict)
Intercept
(value of Y when X=0)
Slope coefficient
(change in Y per unit of X)
Error term
(unexplained variation)
Regression of monthly revenue per sales rep (Y) on training hours completed (X) produces: Revenue = $42,000 + $1,850 × Training Hours. R² = 0.67.
Interpretation: Each additional training hour is associated with $1,850 more monthly revenue per rep. Training hours explain 67% of the variation in revenue across reps. A rep with zero training is predicted to generate $42,000 (the base intercept). A rep with 20 training hours: $42,000 + (20 × $1,850) = $79,000.
Caveat: this is an association, not a proven causal effect. High performers may both train more AND generate more revenue for independent reasons.
Reading regression output — the key numbers
| Output element | What it means | Rule of thumb |
|---|---|---|
| R² (R-squared) | % of outcome variance explained by the model | Higher is better. Context-dependent: R²=0.30 may be excellent for behaviour prediction; R²=0.30 is poor for cost modelling. |
| Adjusted R² | R² penalised for adding extra predictors that don't help | Always use Adjusted R² when comparing models with different numbers of predictors. |
| β coefficient | Change in Y per 1-unit increase in X | Must be interpreted in the units of both X and Y. "$1,850 per training hour" is clear. "0.023" is not — always add units. |
| p-value of β | Probability that this coefficient is due to chance if the true relationship is zero | p < 0.05 is the conventional threshold (5% chance of false positive). p < 0.01 is stronger. p > 0.05 = not statistically significant. |
| Standard Error (SE) | Uncertainty in the β estimate | 95% confidence interval ≈ β ± 2×SE. A wide SE means the coefficient is imprecisely estimated. |
| F-statistic (overall model) | Tests whether the model explains significantly more than chance | p-value of F should be < 0.05 for the model to be considered statistically useful. |
| Residual plot | The unexplained variation — should look random | If residuals show a pattern (curved, fan-shaped), the model is misspecified — try a transformation or a different functional form. |
Multiple Linear Regression
Multiple predictors, one outcome — estimating each driver's unique contribution while controlling for the others.
Multiple regression extends simple regression to include several predictors simultaneously. This is more realistic — most business outcomes are driven by more than one factor. The critical advantage is control: each coefficient represents the change in Y per unit of that predictor, holding all other predictors constant.
A subscription company runs multiple regression with CLV ($) as Y and three predictors: Onboarding Score (X₁), Number of Features Used (X₂), Support Tickets Raised (X₃).
Output: CLV = $800 + $42×Onboarding + $95×Features − $180×Support Tickets. Adjusted R² = 0.71.
Interpretation: Holding other factors constant, each additional feature used is associated with $95 more CLV. Each support ticket is associated with $180 less CLV. Features used has the largest positive driver effect; support tickets have the largest negative effect. This tells the product team where to invest: feature adoption is worth more than fixing onboarding.
The multicollinearity problem
Multicollinearity occurs when two or more predictors are highly correlated with each other. This makes it difficult for the model to isolate the unique contribution of each — coefficients become unstable and standard errors inflate. It is the most common problem in multiple regression.
| VIF value | Diagnosis | Action |
|---|---|---|
| 1.0 – 2.0 | No multicollinearity | No action needed |
| 2.0 – 5.0 | Moderate — acceptable in most cases | Monitor; coefficients are still interpretable |
| 5.0 – 10.0 | High — problematic | Consider removing one of the correlated predictors or combining them |
| > 10.0 | Severe — coefficients are unreliable | Remove predictors until VIF is below 10. Do not interpret individual coefficients. |
VIF measures how much the variance of a coefficient is inflated due to correlation with other predictors. In Excel: calculate manually as 1/(1−R²j) where R²j is the R² from regressing predictor j on all other predictors. In SPSS: . In Python: .
Standardised coefficients (Beta weights)
Raw regression coefficients (β) cannot be compared directly if predictors are in different units — "training hours" and "onboarding score" are on completely different scales. Standardised coefficients (Beta weights) convert all predictors to the same scale (standard deviations), allowing direct comparison: which predictor has the largest standardised effect on the outcome?
Partial Correlation & Elasticity Analysis
Two specialist techniques for more precise driver quantification.
Partial Correlation
Partial correlation measures the relationship between two variables after statistically removing the influence of one or more control variables. It answers: "If I could hold Z constant, what is the true relationship between X and Y?"
A company finds a strong Pearson r = 0.75 between advertising spend and revenue. But both tend to peak in Q4 (Christmas). Is the correlation real, or is it driven by the shared seasonality?
The partial correlation between advertising and revenue, controlling for the month of year, comes back at r = 0.52. Still meaningful, but lower — confirming that some of the simple correlation was seasonal artefact. The "pure" advertising effect is moderate, not strong.
Elasticity Analysis
Elasticity measures the % change in Y for a 1% change in X. Unlike regression coefficients (which are in raw units), elasticity is unit-free — making it easy to compare the sensitivity of different variables.
| Elasticity value | Interpretation | Business meaning |
|---|---|---|
| > 1.0 (elastic) | Y changes more than proportionally to X | A 10% price increase leads to >10% demand drop. Customers are price-sensitive. Revenue falls if you raise price. |
| = 1.0 (unit elastic) | Y changes proportionally to X | A 10% price rise leads to exactly 10% demand drop. Revenue is unchanged. |
| 0 to 1.0 (inelastic) | Y changes less than proportionally to X | A 10% price rise leads to <10% demand drop. Customers are price-insensitive. Revenue increases if you raise price. |
| < 0 (negative) | Y moves in the opposite direction to X | Standard demand: as price rises, demand falls. Negative price elasticity is normal. |
A FMCG company tests price changes across two product categories. Bottled water: price elasticity = −0.3 (inelastic). A 10% price increase reduces volume by only 3% — customers still buy it because it's perceived as essential. Revenue rises. Premium snack range: price elasticity = −1.8 (elastic). A 10% price increase reduces volume by 18% — customers switch to alternatives. Revenue falls. Same 10% price increase, very different strategic implications.
If you run a log-log regression: , the coefficient β is directly the elasticity. A 1% increase in X is associated with a β% change in Y. This is the standard approach in demand modelling and economic analysis.
Driver Analysis & Path Analysis
Which predictor matters most — and do variables work directly or through others?
Driver Analysis (Relative Weight Analysis)
Multiple regression tells you whether each predictor is significant. Driver analysis answers a different question: which predictor contributes the most to explaining the outcome — accounting for correlations between predictors?
Standard beta weights from regression can be misleading when predictors are correlated. Relative Weight Analysis (RWA) decomposes R² into the portion attributable to each predictor, even when predictors share variance. The result: a percentage contribution that sums to 100%.
A telecom company runs driver analysis on NPS (Y) with four predictors: Product Quality, Price Fairness, Customer Service, and Network Reliability. Relative weight results:
| Driver | Relative Weight (%) | Beta (standardised) | Priority |
|---|---|---|---|
| Network Reliability | 38% | 0.52 | 🔴 Top priority |
| Customer Service | 29% | 0.41 | 🟡 High priority |
| Price Fairness | 20% | 0.34 | 🟡 Monitor |
| Product Quality | 13% | 0.28 | 🟢 Maintain |
Invest in network reliability first — it accounts for 38% of NPS variance. Price is real but less central than reliability or service.
Path Analysis (IBM AMOS)
Path analysis tests whether a variable affects the outcome both directly and through other mediating variables. Standard regression only captures direct effects. Path analysis maps the full causal chain.
A simple regression: Training → Customer Satisfaction (β = 0.45, significant). But there may be a mediating path: Training → Service Quality → Customer Satisfaction. Path analysis (AMOS) tests both paths simultaneously:
Results: Training → Customer Satisfaction directly: β = 0.12 (weak, now non-significant). Training → Service Quality → Customer Satisfaction: indirect effect = 0.38 (significant).
Conclusion: Training improves satisfaction primarily through service quality, not directly. Investment in training must be paired with service quality measurement — otherwise it won't move the satisfaction needle.
After running a path model in AMOS, assess model fit with these indices: CFI > 0.95 (Comparative Fit index — how well model fits vs a null model) · RMSEA < 0.06 (Root Mean Square Error of Approximation — lower is better) · SRMR < 0.08 (Standardised Root Mean Residual) · χ² p-value > 0.05 (non-significant means model fits the data). All four should be checked — no single index is sufficient.
Microsoft Excel
Correlation, simple regression, and multiple regression — all via the Data Analysis ToolPak.
Table 2 — ANOVA: F-statistic and its p-value. p<0.05 = model is statistically significant overall.
Table 3 — Coefficients: Intercept and each X coefficient (β) · Standard Error · t Stat · P-value · 95% Confidence Interval. Look for p-values <0.05.
| Formula | Purpose |
|---|---|
| Pearson r between two columns | |
| R² for simple regression | |
| β coefficient (slope) | |
| α (intercept) | |
| Predicted Y for a new X value | |
| Standard error of the regression |
SPSS & IBM AMOS
Professional regression with full diagnostics in SPSS. Path analysis and structural models in AMOS.
IBM AMOS (Analysis of Moment Structures) is a separate application that connects to SPSS. It uses a visual drag-and-drop interface to build path diagrams — no code required.
Python — statsmodels, scipy & pingouin
Full regression pipeline — from correlation to multiple regression, VIF, and standardised coefficients.
Three Simulators
Build intuition for correlation, regression, and multicollinearity through live interaction.
Simulator 1 · Correlation Strength Explorer
Adjust the slider to control the true correlation between two variables. Watch how the scatter plot and r value change — and feel the difference between a "strong" and "weak" relationship.
Simulator 2 · Regression Coefficient Reader
Three business scenarios are presented. Read the regression output and answer what the coefficient means in plain language. Tests your ability to translate numbers into business insight.
Simulator 3 · Multicollinearity Impact
Increase the correlation between two predictors and watch what happens to their regression coefficients and VIF. This demonstrates why multicollinearity makes coefficients unreliable.
Common Mistakes & Knowledge Check
Six mistakes that turn good data into misleading conclusions — followed by five assessment questions.
Claiming causation from regression
A significant β coefficient shows a statistical association — not a cause. Omitted variables, reverse causality, and confounding are always possible. Never say "X causes Y" from regression alone.
Ignoring multicollinearity
When predictors are highly correlated, individual coefficients become unstable and uninterpretable. Always check VIF. If VIF >10, do not interpret the individual coefficients — they are unreliable.
Comparing raw β coefficients across variables in different units
"Training hours: β=1,850" vs "Satisfaction score: β=420" cannot be compared. $1,850 per hour vs $420 per score point are completely different scales. Always use standardised Beta weights for driver ranking.
Not checking residuals
A significant model with good R² can still be badly misspecified. If the residuals show a pattern (curve, funnel, clusters), the model is wrong. Always plot residuals before reporting results.
Overfitting with too many predictors
Adding more predictors always increases R² — even if they are irrelevant. Use Adjusted R² (penalises for extra predictors) and always check out-of-sample performance before acting on a model with many variables.
Extrapolating beyond the data range
If your training data ranged from 0–20 hours, predicting for 50 hours is unreliable. The regression line is valid only within the observed range. Extrapolation assumes the linear relationship continues indefinitely — which it rarely does.