The BRSR / ESG Data Story

How 99 companies' ESG reports become numbers, correlations, and a regression model — told as one continuous story, start to finish

How to use this page: Read it top to bottom, like a story — each chapter builds on the one before it. You don't need a statistics background. Every technical term is explained the moment it appears, with a plain-English box, a formula, and a worked example. By the end, you'll be able to read the workbook's results and explain them in your own words.

SIMPLE TERMS

0. The whole story in simple terms (no jargon at all)

Read this first if you want the entire idea in two minutes, with zero technical words. Every chapter after this explains the same story again, slower and with the real numbers.

Step 1 — Look at the companies. We have 99 companies. For each one, we checked: "does this company do X ESG thing? yes or no." Every "yes" becomes a 1, every "no" becomes a 0. That's all the raw data is — a big table of yeses and noes.

Step 2 — Add up the yeses. For each company, we count how many "yes" answers it has in different buckets — risk-related things, strategy-related things, day-to-day operations, and the actual benefits (value) it got from doing ESG. More "yeses" in a bucket = more "breadth" in that bucket. It's just counting.

Step 3 — Ask a simple question. Do companies that score higher on risk/strategy/operations also score higher on value (the benefits)? In everyday words: "Do companies that try harder at ESG actually get more out of it?"

Step 4 — Check if the pattern is real or just luck. We compare each pair of scores across all 99 companies and get a "relatedness score" between −1 and +1 (close to 0 means no pattern, close to +1 means a strong matching pattern). Then we ask: "could this pattern have shown up just by random chance?" If it's very unlikely to be chance, we call it a real pattern.

Step 5 — Don't get fooled by testing too many things. We checked three things at once (risk, operations, strategy). If you check enough things, one of them will look "special" purely by luck — like flipping three coins and being impressed one landed on heads five times in a row. So we apply a stricter, fairer bar before believing any one result. One result (strategy) that looked promising didn't survive this fairer check.

Step 6 — Look at everything together, not one at a time. Instead of checking risk, operations, and strategy separately, we can also throw them all into one calculation together and ask "how much does each one matter, accounting for the others at the same time?" Because the thing we're predicting is a count (like a score from 0 to 19), we use a counting-friendly version of this calculation, called Poisson regression.

What that calculation actually found, in plain words: An average company has about 12.5 out of 19 ESG capabilities. Compared to that average company, one that puts in a bit more risk-related effort tends to have about 11% more capabilities; a bit more day-to-day operational effort tends to have about 8% more; and a bit more strategic effort tends to have about 12% more. All three of these patterns are strong enough that they almost certainly aren't due to luck.

Step 7 — Translate the math back into English. The end result is a sentence like: "risk-related effort is the strongest connected to getting value out of ESG. Day-to-day operations barely matters. Strategy effort has a weak link that isn't strong enough to trust once we're careful about multiple testing."

The one-sentence version of the whole workbook: We turned ESG report checklists into numbers, counted how many boxes each company ticked in different areas, checked whether ticking more risk/strategy/operations boxes lines up with getting more real-world value, and made sure any "yes it lines up" claim survives a fair, strict double-check before we trust it.

What this is NOT saying: None of this proves that doing more ESG work causes a company to get more value. It only shows whether the two tend to go up and down together across the 99 companies we looked at.

SYMBOLS DECODER

0b. Symbols & notation — decode every symbol before you meet it

Assume zero math background. Every symbol used anywhere in this document is explained here, with a plain meaning and a small worked example, so nothing later ever feels like an unexplained code.

e — "Euler's number" (≈ 2.71828)

What it actually is: Just a fixed number, a bit like π (pi ≈ 3.14159). It's not a variable, not a mystery — it's always the same value, approximately 2.71828. Mathematicians discovered that this exact number shows up naturally whenever something grows continuously (like compound interest, population growth, or — here — the way probabilities/counts scale). We didn't choose it; it just happens to be the "correct" number for this kind of conversion.

Concrete example: e¹ = e ≈ 2.718. e² ≈ 7.389. e⁰ = 1 (always — anything to the power 0 is 1).
In our formulas you'll see e^0.10 — this means "e raised to the power 0.10" ≈ 1.105. You never need to calculate this by hand; Excel's EXP() function or any calculator does it instantly.

log — "logarithm"

What it actually is: The reverse/undo operation of raising a number to a power. If 10² = 100, then log₁₀(100) = 2 — the logarithm just answers "what power do I need?" In our statistics, we specifically use the natural logarithm (sometimes written ln), which is the "undo" button for e specifically.

Concrete example: Since e² ≈ 7.389, that means log(7.389) = 2 — the log "undoes" the e-power and hands you back the original exponent.

Why we bother with log at all: A count (like Capability Total) can never be negative — you can't have "−3 capabilities." But an ordinary regression model can produce negative predictions. Putting a log on the outcome side of the equation is a mathematical safety trick that guarantees the model's predictions, once converted back, can never dip below zero. You (the reader) never have to do log math yourself — you just need to know that Beta lives on this "logged" scale, and that's exactly why we convert it back using e (next entry).

E[Y] — "expected value of Y"

What it actually is: A fancy way of saying "the number the model predicts for the outcome, on average, given certain predictor values." It is a prediction, not a guarantee for any one specific company. E stands for "Expected," Y is just the name given to the outcome variable (here, Capability Total).

Everyday analogy: If a weather model says "the expected rainfall this month is 80mm," that doesn't mean every single day gets exactly that — some days will be wetter, some drier. E[Y] = 12.53 for the Intercept means: "for an average company, we expect around 12.53 capabilities" — not that every average company has exactly 12.53 (which isn't even a whole number — it's an average, not a single company's actual score).

log(E[Y]) = β₀ + β₁X₁ + β₂X₂ + β₃X₃ — putting it together

Now that you know all three pieces: this equation says "take the logged version of our expected count prediction, and build it up by adding a starting point (β₀, the intercept) plus a contribution from each predictor (β₁X₁ etc.)." To get back to an actual, readable expected count, you exponentiate (raise e to the power of) both sides — that's exactly the IRR calculation explained in Chapter 20.

Σ — "sigma" / summation

What it actually is: An instruction meaning "add up everything that follows." It's not a mystery symbol — it's shorthand so we don't have to write out long addition by hand.

Concrete example: In the Spearman formula, Σd² means: for every company, square its rank-difference (d²), then add up all 99 of those squared numbers into one total.

β (beta)

The Greek letter used for a regression coefficient — the raw, model-estimated number showing a predictor's effect, on the log scale. Subscripts like β₀, β₁, β₂, β₃ just number them: β₀ = intercept, β₁ = the first predictor's effect, and so on.

ρ (rho)

The Greek letter used specifically for Spearman's correlation value. Ranges from −1 to +1 (see Chapter 9).

α (alpha)

The significance threshold you decide on before running any test — almost always 0.05 (5%) in social-science and business research. It's the line in the sand: if your p-value is smaller than α, you call the result "statistically significant."

z (z-score)

A standardised value: how many standard deviations a number is from the average. See Chapter 21 for the full explanation with examples.

p (p-value)

The probability of seeing a result this extreme (or more extreme) purely by random chance, assuming there's actually no real effect. Small p = "this probably isn't just luck."

n

Sample size — how many observations (companies, in our case) the calculation is based on. Here, n = 99.

SD (standard deviation) and mean

Mean = the plain average (add everything up, divide by how many there are). SD = how spread out the values are around that average — a small SD means most companies cluster close to the average; a large SD means they're scattered widely.

CI (confidence interval)

A range, not a single number — the band of plausible values the true effect probably sits within. "95% CI" is the most common version used here.

H₀ and H₁ (null and alternative hypothesis)

H₀ = the boring "nothing's going on" starting assumption (e.g. "these two things are not related" / "these group means are equal"). H₁ = the more interesting claim we're checking evidence for (e.g. "these two things ARE related"). Statistics never proves H₁ — it only tells you whether there's enough evidence to doubt H₀.

% (percent) and % change

A way of expressing "out of 100" — used throughout to translate awkward decimal numbers (like IRR = 1.1077) into an everyday phrase ("about 11% more").

Quick reference — you'll now recognise every one of these on sight: e, log, E[Y], Σ, β, ρ, α, z, p, n, SD, mean, CI, H₀/H₁, %.

CHAPTER 1

Meet the data

Imagine 99 companies, each one having published a BRSR (Business Responsibility and Sustainability Report). Someone has gone through all 99 reports and asked the same set of yes/no questions about each company: "Does this company have board-level ESG oversight?" "Does it manage ESG risk?" "Does it invest in sustainable innovation?"

Each answer becomes a number: 1 if yes, the capability is present. 0 if no, it's absent.

In simple words: One company = one row = one observation. With 99 companies, we say n = 99. Every ESG feature becomes a column of 1s and 0s.

Company	RISK1	RISK2	RISK3
A	1	0	1
B	1	1	1
C	0	0	1

That's it — that's the whole starting point. Everything from here on is just doing arithmetic on these 1s and 0s, then asking whether the patterns we see mean anything.

CHAPTER 2

Turning ESG into numbers (operationalisation)

You can't measure something abstract like "ESG capability" with a ruler. So the researcher breaks it into small, answerable questions — each becomes a short code like GOV1, RISK1, STR1.

ESG Capability ↓ Different dimensions (governance, risk, strategy...) ↓ Specific indicators (GOV1, RISK1, STR1...) ↓ 0 / 1 coding

Code	Meaning
GOV1	Board-level ESG oversight
GOV2	Dedicated ESG/sustainability committee
SUS1	Time-bound ESG target with baseline
SUS2	Formal management system/certification
PEO1	ESG training and awareness
BEN2	External assessment/assurance
GOV3	Executive accountability/ESG-linked incentives
RISK1	ESG integrated into enterprise risk management
SUS3	ESG data monitoring/digital system
SUS4	Operational improvement with measured outcome
STR1	ESG integrated into business strategy
STR2	ESG-related capital allocation
STR3	Sustainable product/service/process innovation
RISK2	Climate-risk assessment/scenario analysis
RISK3	Supply-chain ESG risk management
STR4	Business-model/market adaptation

Simple definition: Operationalisation = turning a big fuzzy idea into small countable variables.

CHAPTER 3

What is "breadth"?

Now that every company has a bunch of 1s and 0s, the next natural question is: how many different capabilities does each company have, in each area? That count is called breadth.

In simple words: Breadth = how many different capabilities/domains a company covers in one category. Not depth, not quality — just how many boxes are ticked.

✏️ Correction — Operational breadth: Operational breadth is not just the four SUS indicators. The correct rule is: Operational breadth = every indicator that is NOT part of Risk, Strategic, Value, or COM. That means it pulls in all the governance, sustainability-system, people, benefit, and supply-chain indicators together — twelve indicators in total, not four.

Breadth measure	Built from	Formula	Max
Risk breadth	RISK1, RISK2, RISK3	RISK1 + RISK2 + RISK3	3
Operational breadth (corrected)	GOV1, GOV2, GOV3, SUS1, SUS2, SUS3, SUS4, PEO1, BEN1, BEN2, BEN3, SUP1	GOV1+GOV2+GOV3+SUS1+SUS2+SUS3+SUS4+PEO1+BEN1+BEN2+BEN3+SUP1	12
Strategic breadth	STR1, STR2, STR3, STR4	STR1 + STR2 + STR3 + STR4	4
Value breadth	VAL1, VAL2, VAL3, VAL4	VAL1 + VAL2 + VAL3 + VAL4	4

The easy way to remember it: Every one of the 24 total indicators belongs to exactly one bucket. Four buckets have their own dedicated letter-code family (Risk = RISK1–3, Strategic = STR1–4, Value = VAL1–4, and the standalone COM1). Everything left over — every indicator that isn't in one of those four buckets — is, by definition, Operational. That "leftover" bucket happens to include the GOV (governance), SUS (sustainability systems), PEO (people), BEN (benefit/external assurance), and SUP (supply chain) codes.

Value breadth specifically captures four kinds of ESG-related payoff: VAL1 = cost saving/productivity, VAL2 = resilience/risk reduction, VAL3 = revenue/commercial benefit, VAL4 = stakeholder trust/reputation.

CHAPTER 4

Calculating breadth — a worked example

Let's actually do the arithmetic once, so the formula stops feeling abstract.

Example — Risk breadth: A company has RISK1 = 1, RISK2 = 0, RISK3 = 1.
Risk breadth = 1 + 0 + 1 = 2.
Meaning: the company demonstrates 2 out of 3 risk-related capabilities.

Example — Operational breadth (corrected, 12 indicators): Suppose a company scores:
GOV1=1, GOV2=1, GOV3=0, SUS1=1, SUS2=1, SUS3=0, SUS4=1, PEO1=1, BEN1=1, BEN2=0, BEN3=1, SUP1=1.
Operational breadth = 1+1+0+1+1+0+1+1+1+0+1+1 = 9 (out of a possible 12).
Meaning: the company demonstrates 9 out of 12 operational-category capabilities.

Example — Value breadth: VAL1 = 1, VAL2 = 1, VAL3 = 0, VAL4 = 1.
Value breadth = 1 + 1 + 0 + 1 = 3.
Meaning: the company demonstrates 3 out of 4 value-related outcomes.

Do this for all 99 companies, for all four breadth measures, and now every company has four new numbers describing it — instead of a wall of 1s and 0s.

CHAPTER 5

Maturity levels & Capability Total (the bigger picture)

Separately from breadth, the workbook also groups all 19 indicators into four "maturity levels," from basic to advanced.

Level	What it means	Indicators	Count
Level 1 — Foundational	Basic ESG infrastructure in place?	GOV1, GOV2, SUS1, SUS2, PEO1, BEN2	6
Level 2 — Managed	Is ESG a repeatable process?	GOV3, RISK1, SUS3, SUS4, BEN1, BEN3	6
Level 3 — Integrated	Is ESG built into strategy/value chain?	RISK3, SUP1, STR1, STR2, STR3	5
Level 4 — Adaptive	Is the company planning ahead?	RISK2, STR4	2

Capability Total = L1 Count + L2 Count + L3 Count + L4 Count (max = 19)

Example: L1=6, L2=4, L3=3, L4=1 → Capability Total = 14 out of 19.

Note the difference: Breadth (Chapter 3) measures depth within one category (e.g. risk). Capability Total measures overall progress across all 19 indicators. Both are useful, for different questions.

CHAPTER 6

Look at the data first (descriptive statistics)

Before asking whether things are "related" (correlation) or "different" (ANOVA), a careful researcher always looks at the plain shape of the data first.

Mean — the average: Mean = ΣX / n
Median — the middle value once sorted
Standard deviation (SD) — how spread out the values are. Large SD = companies vary a lot. Small SD = companies are similar.
Min / Max — the range of observed values

Why bother? If almost every company scores the same, there's nothing to "explain" statistically. Understanding the shape of your data first prevents you from misreading later results.

CHAPTER 7

The research question

Now we're ready for the actual question this part of the dissertation is asking:

Research question: Are companies with greater ESG breadth (risk, operational, strategic) also showing greater value breadth?

This gets tested three separate times — once for each predictor:

Risk breadth → Value breadth
Operational breadth → Value breadth
Strategic breadth → Value breadth

⚠️ Note: the Operational breadth correlation result in the next few chapters (ρ = 0.0362) was computed using the earlier, narrower definition (SUS1–4 only). Since Chapter 3 corrected Operational breadth to the full 12-indicator definition, that one number would need to be recalculated on the raw data before being trusted in the final dissertation — the Risk breadth and Strategic breadth results are unaffected.

CHAPTER 8

Correlation — are two things related?

Correlation is the simplest tool for "are X and Y related?" It is not asking "does X cause Y?" — that's a much harder, separate question.

Correlation = relationship. Causation = cause → effect. Don't mix them up — this is one of the most common mistakes in research writing.

CHAPTER 9

Spearman's rho explained

The workbook uses Spearman's rho (ρ) rather than the more famous Pearson correlation. Here's why, and what it means.

Pearson cares about the exact numeric distance between values (assumes a straight-line relationship). Spearman only cares about the order — does company A rank higher than company B on both variables? This makes it more forgiving of lopsided or non-linear data.

ρ value	Simple meaning
+1	Perfect positive relationship
+0.7	Strong positive relationship
+0.5	Moderate positive relationship
0	Almost no relationship
−0.5	Moderate negative relationship
−1	Perfect negative relationship

ρ = 1 − [6Σd² / n(n²−1)]

d = difference between a company's two ranks (its rank on X minus its rank on Y), n = number of companies.

Important: ρ tells you direction + strength. It does not by itself tell you whether the relationship is statistically significant — that's the p-value's job (next chapter).

CHAPTER 10

The real results from this workbook

Here is what actually came out of running this analysis on the 99 companies:

Predictor	Spearman ρ	Raw p-value
Risk breadth	0.5122	0.0000000599
Operational breadth	0.0362	0.7223
Strategic breadth	0.2149	0.03264

Three numbers, three stories. Let's unpack each one.

CHAPTER 11

Reading each result in plain English

Risk breadth: ρ = 0.5122, p = 0.0000000599
Positive and moderately strong. Companies with more risk-related ESG capabilities tend to also show more value-related outcomes. The p-value is far smaller than 0.05 — this looks like a real, meaningful pattern, not random noise.

Operational breadth: ρ = 0.0362, p = 0.7223
Practically zero relationship. p = 0.7223 is way above 0.05, so there's no statistical evidence connecting operational breadth to value breadth in this sample.

Strategic breadth: ρ = 0.2149, p = 0.03264
Weak-to-moderate positive relationship. p = 0.03264 is technically below 0.05 — but remember, we ran three tests, not one. Before we celebrate this "significant" result, we need to check if it survives correction for multiple testing. (Spoiler: see Chapter 15.)

CHAPTER 12

What exactly is a "raw" p-value?

Simple definition: the raw p-value is the original significance result straight out of the test, before any adjustment for the fact that multiple tests were run.

Example: ρ = 0.2149, Raw p = 0.03264. Since 0.03264 < 0.05, our first impression is "significant." But this is only a first impression — it hasn't been corrected yet.

Why "two-sided"? Because we test for a relationship in either direction (positive or negative) — we don't assume in advance which way it will go.

CHAPTER 13

Why testing three things at once is risky

We ran three separate correlation tests (risk, operational, strategic breadth). Here's the catch:

If you run enough tests, some will look "significant" purely by chance — even if nothing real is going on. Using p < .05 for every one of three tests inflates your risk of a false alarm. So researchers apply a correction. The workbook uses the Holm correction.

CHAPTER 14

Holm correction, step by step

Step 1 — Rank the p-values, smallest to largest ("p-order"):

Predictor	Raw p	p-order
Risk breadth	0.0000000599	1 (smallest)
Strategic breadth	0.03264	2
Operational breadth	0.7223	3 (largest)

Step 2 — Calculate a stricter threshold for each rank (with α = 0.05, m = 3 tests):

Threshold at rank i = α / (m − i + 1)

Predictor	p-order	Holm threshold
Risk breadth	1	0.05 / 3 = 0.01667
Strategic breadth	2	0.05 / 2 = 0.02500
Operational breadth	3	0.05 / 1 = 0.05000

Step 3 — Compare and adjust. The smallest p-value must clear the strictest bar; if it does, move to the next.

CHAPTER 15

The corrected final table — the real answer

Predictor	Raw p	Holm-adjusted p
Risk breadth	0.0000000599	0.0000001797
Strategic breadth	0.03264	0.06529
Operational breadth	0.7223	0.7223

Predictor	Spearman ρ	Raw p	p-order	Holm threshold	Holm-adjusted p	Decision
Risk breadth	0.5122	<.000001	1	.01667	<.000001	Significant
Strategic breadth	0.2149	.03264	2	.02500	.06529	Not significant
Operational breadth	0.0362	.7223	3	.05000	.7223	Not significant

The key lesson: Strategic breadth looked significant before correction (p = .03264) but flipped to not significant after correction (p = .06529). This is exactly why multiple-testing correction matters — it stops us from over-claiming.

Bottom line: Only Risk breadth shows a statistically significant association with Value breadth after Holm correction.

CHAPTER 16

Doing it yourself in Excel

Suppose Risk breadth = column A, Operational breadth = B, Strategic breadth = C, Value breadth = D.

Excel's CORREL() computes Pearson correlation, not Spearman — so first convert values to ranks:
=RANK.AVG(A2,$A$2:$A$100,1)
Do the same for the Value breadth column.
Then correlate the two rank columns:
=CORREL(Risk_Ranks,Value_Ranks)
That result is Spearman's rho.

CHAPTER 17

Doing it yourself in SPSS

Go to Analyze → Correlate → Bivariate
Select Risk breadth, Operational breadth, Strategic breadth, and Value breadth
Tick Spearman
Tick Two-tailed
Click OK

SPSS gives you three numbers to read: Correlation coefficient (= ρ), Sig. (2-tailed) (= raw p-value), and N (= sample size).

What "two-tailed" means: we test for a relationship in either direction (positive or negative), without assuming beforehand which way it goes.

CHAPTER 18

From correlation to regression

Correlation looks at one relationship at a time. But what if Risk breadth, Operational breadth, and Strategic breadth are all a little related to each other too? Regression lets us ask a sharper question:

Regression question: How is Value breadth (or Capability Total) associated with each predictor, once we account for the other predictors at the same time?

Correlation: X ─────────→ Y (one relationship at a time) Regression: ┌→ Risk breadth │ Predictors ─────┼→ Operational breadth │ └→ Strategic breadth ↓ Count outcome (all together)

CHAPTER 19

Why Poisson regression?

The outcome we care about (e.g. Capability Total) is a count — a whole number like 0, 1, 2, ... up to 19. That's exactly the kind of outcome Poisson regression is built for.

Outcome type	Typical model
Continuous value	Linear regression
Yes/No	Logistic regression
Count (0,1,2,3...)	Poisson regression
Count with extra variability	Negative Binomial (often considered)

log(E[Y]) = β₀ + β₁X₁ + β₂X₂ + β₃X₃

Y = the count outcome, β₀ = intercept, β₁/β₂/β₃ = predictor effects. You don't need to memorise the equation — just know: predictors combine to explain how the expected count changes.

CHAPTER 20

Beta and IRR — from hard-to-read to easy-to-read

The raw regression coefficient (Beta, β) is on an awkward "log scale." Positive β = predictor pushes the outcome up; negative β = pushes it down. But the easier number to actually talk about is the IRR (Incidence Rate Ratio).

IRR = e^β

Example: β = 0.20 → IRR = e^0.20 ≈ 1.22. Meaning: a one-unit increase in the predictor is linked to roughly a 22% higher expected count, holding the other predictors constant.

IRR	Meaning
1.00	No change
1.20	20% higher expected count
1.50	50% higher
2.00	100% higher
0.80	20% lower
0.50	50% lower

If IRR > 1: % change = (IRR − 1) × 100 If IRR < 1: % change = (1 − IRR) × 100

What IRR actually means in practical, everyday terms

The formula tells you how to calculate IRR. This section tells you why anyone should care.

Analogy 1 — a recipe scaling multiplier. Imagine a recipe that currently makes 10 servings (that's your baseline, like the Intercept). IRR is a multiplier, not an "add-on." An IRR of 1.10 doesn't mean "add 1.10 servings" — it means "multiply your baseline by 1.10." 10 servings × 1.10 = 11 servings. That's exactly how it works here: an average company with ~12.53 expected capabilities, when you improve Risk breadth by one notch (IRR = 1.108), is expected to have about 12.53 × 1.108 ≈ 13.9 capabilities — not simply "12.53 + a fixed number."

Analogy 2 — a fitness tracker. Suppose your average daily step count is 8,000 steps (your baseline). If wearing better running shoes is associated with an IRR of 1.10, that means on days you wear those shoes, you tend to walk about 10% more — roughly 8,800 steps — compared to your normal baseline. It doesn't guarantee any single day will hit exactly 8,800; it describes the average pattern across many days (or here, many companies).

Why this matters for THIS dissertation, in plain business terms: IRR tells a sustainability manager which lever gives the most "bang for the buck." If Strategic breadth has the highest IRR (1.116, i.e. +11.6%) compared to Operational breadth (1.076, i.e. +7.6%), that's a practical signal: pushing the same amount of extra effort into strategy-related ESG work is associated with a bigger jump in overall ESG capability than putting that same effort into day-to-day operations. This can help prioritise where limited time or budget goes — even though (important caveat) it doesn't prove that investing there will cause the improvement; it only shows which factor has moved together with more capabilities in this sample of 99 companies.

The one thing IRR does NOT tell you: it does not tell you whether improving that predictor will definitely cause more capabilities to appear if you go and change it. It's an association measured across companies that already existed with different natural levels of Risk/Operational/Strategic breadth — not a controlled experiment. Treat it as a strong clue for where to look further, not a guaranteed recipe for what to do.

CHAPTER 21

Z-scores — putting predictors on the same ruler

You'll notice the predictors are written as "Risk breadth (z)". That "(z)" means the variable has been standardised.

z = (X − Mean) / SD

z-score	Meaning
0	Exactly average
+1	1 SD above average
−1	1 SD below average
+2	2 SD above average

Why bother? Risk breadth (0–3) and Strategic breadth (0–4) are on different scales. Standardising both puts them on the same "how unusual is this company" ruler, so their effects become directly comparable.

CHAPTER 22

Robust standard errors, p-values, and confidence intervals

Every estimate comes with some uncertainty. Three tools quantify that uncertainty:

Robust standard error (HC3 SE)

SE = Standard Error = how uncertain the estimated coefficient is. "Robust" (HC3) means it stays reliable even when the data is uneven or has outliers that would confuse an ordinary SE calculation.

Robust p-value

Test statistic ≈ β / Robust SE

p < .05 → evidence the coefficient differs from zero. p ≥ .05 → not enough evidence.

Confidence interval (CI)

Estimate ± 1.96 × SE (approx. 95% CI)

CI position (for IRR)	Meaning
Entirely above 1	Positive association
Includes 1	Not statistically significant at 5%
Entirely below 1	Negative association

Example: IRR = 1.30, 95% CI = 1.10–1.52. Since the interval doesn't include 1, it's significant at roughly the 5% level.

FIELD GUIDE

22a. Every column of the Poisson table, explained

Before looking at the actual numbers, here's what every single column in the results table means, where it comes from, and exactly how it's calculated. Think of this as the "data dictionary" for the table in the next chapter.

Role

What it is: A label telling you what kind of term the row is — not a calculated value at all, just a category.

There are two roles in this model: "Model constant" (the Intercept — the baseline expected count when every predictor is at its average) and "Predictor" (a variable whose effect on the outcome we're estimating: Risk breadth, Operational breadth, Strategic breadth). One row is labelled "Primary predictor" simply to flag it as the main variable of research interest, ahead of the other two.

Beta (linked source input)

What it is: The raw regression coefficient that the Poisson model estimates directly from the data. It's called a "linked source input" because it's not something you calculate by hand from the other columns — it comes straight out of the statistical software (SPSS, R, Python, etc.) when you run the Poisson regression. Every other column in the table is then built from this number.

log(E[Y]) = β₀ + β₁X₁ + β₂X₂ + β₃X₃

Beta lives on the log scale — that's the awkward part. A beta of 0.10 does not mean "10% more." It has to be converted (exponentiated) before it means anything intuitive. That conversion is the next column.

HC3 robust SE (linked source input)

What it is: Like Beta, this is also a "linked source input" — the software calculates it directly, using the HC3 heteroskedasticity-consistent formula, rather than you computing it manually from a simple textbook formula.

In plain terms: it measures how much the estimated Beta might wobble if you drew a slightly different sample of companies. "HC3" is simply the specific robust recipe used to compute it — one that stays trustworthy even when some companies behave very differently from others (uneven variance / a few unusual companies), which is common in real business data.

Rule of thumb: smaller SE relative to Beta → a more precise, more trustworthy estimate.

Expected count / IRR (formula)

What it is: Unlike Beta and SE, this column is a formula — you calculate it yourself from Beta. "IRR" stands for Incidence Rate Ratio, and it's simply Beta converted from the log scale into an easy "how many times higher/lower" scale.

IRR = e^Beta

For the Intercept row specifically, exponentiating gives the expected count itself (not a ratio) — the model's baseline predicted Capability Total when all predictors sit at their average.

CI low (formula) and CI high (formula)

What they are: The lower and upper edges of the 95% confidence interval for the IRR — the plausible range the true effect probably falls within. Both are formulas built from Beta and SE, in two steps.

Step 1 (on the log/beta scale): Lower limit = Beta − 1.96 × SE Upper limit = Beta + 1.96 × SE Step 2 (convert to the IRR scale): CI low = e^(Beta − 1.96 × SE) CI high = e^(Beta + 1.96 × SE)

Why exponentiate at the end, not the start? Because Beta and SE both live on the log scale — you build the interval there first (where the ± 1.96 math is valid), then convert both ends to the easy-to-read IRR scale.

Robust p (formula)

What it is: The probability of seeing a Beta this large (or larger) purely by chance, if the predictor actually had no real effect. Also a formula, built from Beta and the robust SE.

Test statistic (z) = Beta / HC3 robust SE Robust p = 2 × (1 − Φ(|z|)) where Φ = the standard normal cumulative distribution function

In plain words: divide Beta by its SE to get a "how many SEs away from zero" score. The further that score is from zero, the smaller (more significant) the p-value. A z-score around 11 or higher — as seen for all three predictors here — corresponds to a p-value far below .001, essentially "as close to impossible-by-chance as this kind of test can show."

% change (predictors only)

What it is: The IRR translated into an everyday percentage — the single most quotable number in the whole table. Calculated only for predictor rows, not the Intercept, because "the intercept increased by X%" isn't a meaningful sentence — the intercept is a baseline, not a change.

If IRR > 1: % change = (IRR − 1) × 100 If IRR < 1: % change = (1 − IRR) × 100 (a decrease)

Manual interpretation

What it is: Not a number at all — this is the plain-English sentence a human writes after looking at every other column together. It's "manual" because software can hand you Beta, SE, IRR, CI, and p, but it can't write the research sentence for you; that's the researcher's job.

The template to build a manual interpretation:

State the predictor and its standardisation: "A one-standard-deviation increase in [predictor]..."
State the direction and size, from % change: "...is associated with an approximately [X]% [increase/decrease] in..."
State the outcome: "...the expected [outcome variable]..."
State the control condition: "...holding the other predictors constant."
State the confidence, from the p-value and CI: "This effect is statistically significant (p < .001; 95% CI excludes 1)."

Fully worked example (Risk breadth): "A one-standard-deviation increase in Risk breadth is associated with an approximately 10.77% increase in the expected Capability Total, holding the other predictors constant. This effect is statistically significant (p < .001; 95% CI 1.088–1.127, which excludes 1)."

WORKED EXAMPLE

22b. The real Poisson numbers from this workbook

Chapters 19–22 explained the theory with a made-up β = 0.20. Now here are the actual calculated results from running the Poisson model on the 99 companies, with:

Outcome = Capability Total (0–19)
Risk breadth = RISK1 + RISK2 + RISK3
Operational breadth = every non-Risk, non-Strategic, non-Value, non-COM indicator (GOV1, GOV2, GOV3, SUS1–4, PEO1, BEN1–3, SUP1 — 12 indicators, see corrected Chapter 3)
Strategic breadth = STR1 + STR2 + STR3 + STR4
All three predictors standardised as z-scores
HC3 robust standard errors

⚠️ Important note on the numbers below: The table below was originally supplied using Operational breadth = SUS1+SUS2+SUS3+SUS4 (4 indicators). Chapter 3 has since been corrected to the proper 12-indicator definition of Operational breadth. That means the Risk breadth and Strategic breadth rows below are unaffected (their definitions never changed), but the Operational breadth row (β = 0.0736, IRR = 1.0764) was calculated under the old, narrower definition and would need to be re-run on the raw data using the corrected 12-indicator Operational breadth before it can be trusted in the final dissertation. Treat that one row as provisional until recalculated.

Term	Role	Beta	HC3 robust SE	IRR	CI low	CI high	Robust p	% change
Intercept	Model constant	2.5279	0.00979	12.5274	12.2893	12.7700	<0.001	—
Risk breadth (z)	Primary predictor	0.1023	0.00897	1.1077	1.0884	1.1274	<0.001	10.77%
Operational breadth (z) ⚠️	Predictor (old definition — needs re-run)	0.0736	0.00812	1.0764	1.0594	1.0937	<0.001	7.64%
Strategic breadth (z)	Predictor	0.1100	0.00994	1.1163	1.0948	1.1383	<0.001	11.63%

✓ Cross-checked: every number in this table is internally consistent. IRR = e^β for all four rows (e.g. e^0.1023 ≈ 1.1077; e^2.5279 ≈ 12.527). Each confidence interval matches Estimate ± 1.96×SE on the beta scale, then exponentiated (e.g. Risk breadth: 0.1023 ± 1.96×0.00897 = 0.0847–0.1199 → e^0.0847=1.088, e^0.1199=1.127 ✓). The robust test statistic for Risk breadth (0.1023 / 0.00897 ≈ 11.4) is large enough to produce a p-value far below .001, so "p < .001" for all three predictors checks out.

Manual interpretation — in the simplest possible words

Term	Manual interpretation (simple terms)
Intercept	An average company (average risk, average operations, average strategy effort) is expected to have about 12.5 out of 19 ESG capabilities.
Risk breadth	A company that scores one notch above average on risk-related ESG effort tends to have about 11% more capabilities overall than an average company — and this is a real, reliable pattern (not luck).
Operational breadth	A company that scores one notch above average on day-to-day ESG operations tends to have about 8% more capabilities overall — a real, reliable pattern, though the smallest effect of the three.
Strategic breadth	A company that scores one notch above average on ESG strategy tends to have about 12% more capabilities overall — the strongest effect of the three, and a real, reliable pattern.

Put together in one plain sentence: "Trying harder at any one of risk, operations, or strategy is linked to having more ESG capabilities overall — strategy effort seems to matter slightly more than risk effort, which matters slightly more than operational effort, and none of these three results look like they happened by chance."

WORKED EXAMPLE

22c. Reading each predictor's result

Risk breadth: β = 0.1023 → IRR = 1.1077 (+10.77%)
A one-standard-deviation increase in Risk breadth is associated with about a 10.77% increase in the expected Capability Total, holding the other predictors constant. (Because the predictor is standardised as "(z)", the correct phrase is "one standard-deviation increase" — not "one unit increase".)

Operational breadth: β = 0.0736 → IRR = 1.0764 (+7.64%)
A one-standard-deviation increase in Operational breadth is associated with about a 7.64% increase in expected Capability Total, holding the others constant.

Strategic breadth: β = 0.1100 → IRR = 1.1163 (+11.63%)
A one-standard-deviation increase in Strategic breadth is associated with about an 11.63% increase in expected Capability Total — the largest of the three effects in this model.

Intercept: β₀ = 2.5279 → e^2.5279 ≈ 12.53
Because all predictors are z-scored, zero means "average company." So when Risk, Operational, and Strategic breadth are all at their average levels, the model expects a baseline Capability Total of about 12.53.

Predictor	IRR	% change	Simple interpretation
Risk breadth	1.108	+10.77%	1 SD higher Risk breadth → ~10.8% higher expected Capability Total
Operational breadth	1.076	+7.64%	1 SD higher Operational breadth → ~7.6% higher expected Capability Total
Strategic breadth	1.116	+11.63%	1 SD higher Strategic breadth → ~11.6% higher expected Capability Total

Because all three CIs sit entirely above 1 and all three p-values are < .001 — all three predictors show statistically significant positive associations with Capability Total in this model. Unlike the correlation results in Chapter 15, nothing drops out here.

WORKED EXAMPLE

22d. Correlation vs. regression — don't mix them up

This is a subtle but important trap. The correlation analysis (Chapters 10–15) and this Poisson model are asking different questions with different outcomes — the results are not directly comparable.

	Predictors	Outcome	Result
Correlation (Ch. 10–15)	Risk / Operational / Strategic breadth, one at a time	Value breadth	Only Risk breadth survives Holm correction
Poisson regression (here)	Risk + Operational + Strategic breadth, together	Capability Total	All three are significant

It's tempting to say "wait, earlier only Risk breadth mattered, now suddenly all three matter — which is it?" The answer: they're not contradicting each other. One study asks whether breadth links to Value breadth; the other asks whether breadth links to Capability Total. Different outcome variable, different question, different (valid) answer.

WORKED EXAMPLE

22e. A caution before finalising the dissertation

Before these exact numbers go into a final dissertation, two things are worth double-checking with the researcher/supervisor:

1. Confirm the specification. Is Capability Total definitely the intended dependent (count) variable? Are the three breadth measures exactly RISK1–3, SUS1–4, and STR1–4, matching the workbook's Code Map? These were the definitions used to produce the table above — if the researcher intended something slightly different, the numbers would change.

2. Check the model fit. For a doctoral-level analysis, it's worth checking Poisson dispersion/fit (i.e. whether the data is "overdispersed" and might need a Negative Binomial model instead) before treating these p-values as final. Using HC3 robust standard errors helps guard against some problems, but it doesn't by itself prove Poisson was the right choice of model.

CHAPTER 23

ANOVA — comparing groups (a different kind of question)

Everything so far compares numbers to numbers. But sometimes the question is about groups: does average ESG capability differ across industries?

Industry	Mean Capability
Banking	15.2
Manufacturing	13.1
Healthcare	12.8
Power	16.0

H₀ (null hypothesis): all group means are equal. H₁ (alternative): at least one group mean is different.

F = Between-group variation / Within-group variation

Example: F = 4.82, p = .006. Since .006 < .05, we reject H₀ — there's significant evidence that ESG capability differs across at least some industries. ANOVA doesn't say which industries differ though; for that you need a post-hoc test (e.g. Tukey or Games-Howell).

CHAPTER 24

The whole journey, in one picture

Research Question ↓ Identify variables ↓ Operationalise (turn ideas into 0/1 indicators) ↓ Calculate breadth / capability measures ↓ Descriptive statistics (look at the data) ↓ Correlation (Spearman ρ, raw p) ↓ Multiple-testing correction (Holm) ↓ Regression (Poisson, Beta → IRR) ↓ Check robustness (HC3 SE, robust p, CI) ↓ Interpret in plain language ↓ Write the research conclusion

CHAPTER 25

The 7-step rule for reading ANY statistical result

Once you know this checklist, you can read almost any SPSS or Excel output — even outside this dissertation.

What is the research question? — What are we actually trying to find out?
What are X and Y? — Identify the predictor and the outcome.
What test/model was used? — Spearman? ANOVA? Regression?
What is the effect? — Look at ρ, β, IRR, or mean difference.
What is the p-value? — Is it below .05?
What is the confidence interval? — Does it include the "no effect" value (0 for correlation/coefficients, 1 for IRR)?
What does it mean in the real world? — Translate the numbers into a plain business/ESG sentence.

CHAPTER 26

Significance vs. importance vs. causality — don't confuse them

Statistical significance: "This evidence is unlikely to have happened by chance alone."

Practical significance: "The size of the effect actually matters in the real world."

Causality: "X causes Y."

These are three different claims. A result can be statistically significant but practically tiny (e.g. ρ = .05 with p < .001 in a huge sample). And a moderate correlation (ρ = .60) never by itself proves that one thing causes another.

CHAPTER 27

The final story, told in one paragraph

This is the payoff — after all those steps, here's what the numbers actually say, written the way it would appear in the dissertation itself:

"Risk breadth has the clearest relationship with Value breadth. The association is moderate and positive (ρ = 0.512) and remains statistically significant after Holm correction. Operational breadth shows virtually no association with Value breadth (ρ = 0.036, p = .722). Strategic breadth has a weak positive association (ρ = 0.215), but this does not remain statistically significant after Holm correction (adjusted p = .065)."

Notice: every sentence in that paragraph is traceable back to a number in the tables above. That's the whole goal of statistics writing — nothing stated that isn't backed by a number you calculated.

CHAPTER 28

Five things worth memorising

1. Spearman ρ — strength + direction of a relationship.

2. Raw p — first impression of statistical significance, before correction.

3. Holm-adjusted p — the final, trustworthy significance decision after correcting for multiple tests.

4. IRR — how much the expected count changes for a one-unit increase in a predictor.

5. p < .05 — evidence against the null hypothesis at the 5% level. It is not proof that the research hypothesis is true.

FORMULA SHEET

29. All formulas, updated & in one place

Every formula used anywhere in this document, gathered here, double-checked, and written with plain variable definitions. If you forget how something was calculated, come back to this page.

1. Level percentage

Level % = Level Count ÷ Number of indicators in that level

Level Count = how many indicators a company has in that maturity level. Denominators: L1÷6, L2÷6, L3÷5, L4÷2.

2. Capability Total

Capability Total = L1 Count + L2 Count + L3 Count + L4 Count (maximum = 19)

3. Breadth measures

Risk breadth = RISK1 + RISK2 + RISK3 (max 3) Operational breadth = GOV1+GOV2+GOV3+SUS1+SUS2+SUS3+SUS4+PEO1+BEN1+BEN2+BEN3+SUP1 (max 12) → i.e. every indicator that is NOT Risk, NOT Strategic, NOT Value, and NOT COM Strategic breadth = STR1 + STR2 + STR3 + STR4 (max 4) Value breadth = VAL1 + VAL2 + VAL3 + VAL4 (max 4)

4. Z-score (standardisation)

z = (X − Mean) / SD

X = a company's raw value; Mean and SD = the average and standard deviation of that variable across all 99 companies.

5. Spearman's rho (correlation)

ρ = 1 − [6 × Σd² / (n × (n² − 1))]

d = difference between a company's rank on X and its rank on Y; n = number of companies (99); Σd² = sum of all the squared rank differences.

6. Holm correction threshold

Threshold at rank i = α / (m − i + 1)

α = significance level (usually 0.05); m = total number of tests being run together; i = this p-value's rank (1 = smallest).

7. Poisson regression model

log(E[Y]) = β₀ + β₁X₁ + β₂X₂ + β₃X₃

Y = the count outcome (e.g. Capability Total); E[Y] = the expected/predicted count; β₀ = intercept; β₁, β₂, β₃ = each predictor's coefficient; X₁, X₂, X₃ = the (standardised) predictor values.

8. IRR (Incidence Rate Ratio)

IRR = e^β

Converts the log-scale Beta into an easy-to-read multiplier. e ≈ 2.71828 (a fixed constant — see the Symbols Decoder chapter).

9. Percentage change from IRR

If IRR > 1: % change = (IRR − 1) × 100 → an increase If IRR < 1: % change = (1 − IRR) × 100 → a decrease

10. Confidence interval (95%, approximate)

On the log/beta scale: Lower limit = β − 1.96 × SE Upper limit = β + 1.96 × SE Converted to the IRR scale: CI low = e^(β − 1.96 × SE) CI high = e^(β + 1.96 × SE)

11. Robust test statistic and robust p-value

z = β / HC3 robust SE Robust p = 2 × (1 − Φ(|z|)) where Φ = the standard normal cumulative distribution function

12. ANOVA

F = Between-group variation / Within-group variation

13. Basic descriptive statistics

Mean = ΣX / n SD = how spread out the values are around the Mean Median = the middle value once all values are sorted

14. Correlation table — every column, one by one

Matching exactly the header row: Spearman ρ · Raw p · p-order · Holm threshold · Holm-adjusted p

Column	Formula / rule	Type
Spearman ρ	`ρ = 1 − [6 × Σd² / (n × (n² − 1))]`	Calculated (formula above, or software)
Raw p	Not hand-calculated from a simple formula — it comes from converting ρ into a test statistic (`t = ρ × √[(n−2)/(1−ρ²)]`) and looking up that value's probability on a t-distribution with (n−2) degrees of freedom. In practice, always read straight from software output.	Software output (linked source input)
p-order	Not a formula — just sort the Raw p column smallest → largest, and number them 1, 2, 3…	Ranking rule
Holm threshold	`Threshold at rank i = α / (m − i + 1)`	Calculated
Holm-adjusted p	For the p-value at rank i: `Adjusted pᵢ = max( pⱼ × (m − j + 1) )` for all j ≤ i — i.e. multiply each raw p by its own Holm multiplier, then carry forward whichever adjusted value so far is largest, so the adjusted p-values never decrease as rank increases.	Calculated

Worked check — Risk breadth: rank 1, multiplier = m/(m−1+1) = 3/3 = 1×. Raw p (≈0.00000006) × 1 = still ≈0.00000006 → matches the Holm-adjusted p shown in Chapter 15.

15. Poisson regression table — every column, one by one

Matching exactly the header row: Role · Beta · HC3 robust SE · Expected count/IRR · CI low · CI high · Robust p · % change

Column	Formula / rule	Type
Role	Not a formula — a text label: "Model constant" for the Intercept row, "Predictor" (or "Primary predictor") for every X variable.	Label only
Beta	Not hand-calculated — the Poisson model estimates it directly from the raw data by finding the value that best explains the observed counts (technically: maximum-likelihood estimation). Every other numeric column below is built from this one.	Software output (linked source input)
HC3 robust SE	Not hand-calculated — computed by the software using the HC3 heteroskedasticity-consistent formula, which adjusts for uneven variance across companies.	Software output (linked source input)
Expected count / IRR	`IRR = e^Beta` (for the Intercept row, this gives the baseline expected count itself, not a ratio)	Calculated
CI low	`CI low = e^(Beta − 1.96 × SE)`	Calculated
CI high	`CI high = e^(Beta + 1.96 × SE)`	Calculated
Robust p	`z = Beta / HC3 robust SE`, then `Robust p = 2 × (1 − Φ(\|z\|))` where Φ = standard normal cumulative distribution function	Calculated
% change	If IRR > 1: `(IRR − 1) × 100`. If IRR < 1: `(1 − IRR) × 100`. (Predictor rows only — not calculated for the Intercept.)	Calculated

Worked check — Risk breadth: Beta = 0.1023, SE = 0.00897 → IRR = e^0.1023 = 1.1077 ✓, CI low = e^(0.1023−0.01758) = e^0.0847 = 1.088 ✓, CI high = e^(0.1023+0.01758) = e^0.1199 = 1.127 ✓, z = 0.1023/0.00897 ≈ 11.4 → Robust p ≪ .001 ✓, % change = (1.1077−1)×100 = 10.77% ✓. Every column in Chapter 22b's table traces back to exactly these formulas.

Reminder — the 99-company, 3-breadth, one-outcome numbers actually used in this workbook: n = 99 companies · Risk breadth (0–3) · Operational breadth (0–12, corrected) · Strategic breadth (0–4) · Value breadth (0–4) · Capability Total (0–19).

GLOSSARY

📖 Every term, in plain words

Term	Simple meaning
Operationalisation	Turning a vague idea (e.g. "ESG maturity") into things you can count (1/0 indicators)
Indicator	A single yes/no question measuring one small piece of ESG capability
Breadth	How many different capabilities a company has ticked off in one category (risk, strategy, etc.)
Maturity level	A "stage" grouping of indicators, from basic (Level 1) to advanced (Level 4)
Capability Total	Total count of ESG capabilities a company has, out of 19
Domain Count	How many different ESG areas a company touches at all, regardless of depth
Correlation	Checking if two things move together — not proof one causes the other
Spearman's rho	A rank-based correlation measure, good for lopsided or non-linear data
p-value	Probability of seeing a result this extreme by pure chance, if there's really no effect
Statistically significant	p-value small enough (usually < .05) that the result is unlikely to be random
Multiple testing	Running several tests at once, which raises the risk of a false alarm
Holm correction	A method that tightens the significance bar when multiple tests are run together
Regression	A method to see how an outcome relates to several predictors at the same time
Poisson regression	Regression built for count outcomes (0, 1, 2, 3...)
Beta (β)	The raw regression coefficient — direction/size of effect on a log scale
IRR (Incidence Rate Ratio)	An easier-to-read version of beta — how many times higher/lower the expected count is
Z-score / standardised	How many standard deviations a value is from the average — makes different variables comparable
Standard error (SE)	How much uncertainty surrounds an estimated number
Robust SE (HC3)	A sturdier SE calculation that still works with uneven data or outliers
Confidence interval (CI)	A range of plausible values for the true effect, usually at 95% confidence
Intercept	The model's baseline value when all predictors are at their average
ANOVA	A test comparing the average of a number across 3+ groups (e.g. industries)
Descriptive statistics	Basic summary numbers (mean, median, SD, min, max) describing raw data
Two-tailed / two-sided	Testing for a relationship in either direction, without assuming which way it goes