Numerical Narratives — Research Toolkit

Section 1

Purpose of a numerical narrative

Readers arrive with priors formed by the work they have already read. A contribution is a change in what an informed reader believes after seeing your evidence. The numerical narrative is the part of a paper that manages that change with the simplest evidence available: counts, rates, conditional probabilities, distributions, and well-chosen figures, presented before the multivariate analysis.

The narrative has three jobs: (1) establish the phenomenon of interest, (2) motivate the research question, and (3) justify the intellectual burden of the more complex analyses to come. A reader who has seen the pattern and where the obvious explanations fall short arrives at your regression tables wanting their answer.

Descriptives move beliefs; regressions defend them against objections. Earn the right to run the model.From the Session 3 slides, “Adjudicating Mechanisms with Numerical Narratives”

Seminar audiences often ask to see the regression. It belongs several tables in, because a reader who believes the argument before seeing the best evidence for it is more likely to update. A paper whose only figure is a post-estimation plot forgoes the chance to affirm the reader’s priors and to show what generates the estimate.

The narrative serves the differences-in-inferences argument: affirm some of the reader’s priors (“yeah, yeah, yeah…”), deny others or establish new ones (“…but…”), and reconcile the evidence with a mechanism (“…and therefore…”). In quantitative work, the argument takes the form of three models that replicate, diverge, and reconcile; in qualitative work, observations consistent with priors, observations inconsistent with them, and the belief the reader should update or replace.

The approach draws on Merton’s (1987) first fragment (establishing the phenomenon) and on Davis’s (1971) observation that an interesting claim denies part of what the audience takes for granted while affirming the rest. Damodaran (2017) makes a parallel case in valuation, where an analyst persuades investors by pairing a story with the numbers that scale it from one customer to many.

Section 2

The arc: possible, plausible, probable

Readers seldom move from indifference to conviction in one step. An effective narrative walks them through three belief states about your preferred explanation, each with its own kind of evidence.

Step 1

Possible

The argument is consistent with multiple, reasonable prior beliefs.

Evidence: the phenomenon, established statistically, plus suggestive evidence for more than one account.

Step 2

Plausible

Prima facie evidence is consistent with the preferred argument.

Evidence: a figure at the point where the accounts make different predictions.

Step 3

Probable

The best available evidence justifies updating posterior beliefs.

Evidence: multivariate models, research designs, and simulations.

The three thousand-word pictures carry the reader up the first two rungs. Picture 1 establishes the phenomenon and Picture 2 makes more than one mechanism look defensible; together they make your argument possible. Picture 3, the prima facie evidence at the point where the mechanisms diverge, makes it plausible. The multivariate models, the research design, and any simulations make it probable.

Each rung is a Bayesian update. The reader starts with a prior about how much of the phenomenon your mechanism explains. Each updated belief is a weighted average of that prior and the new evidence, weighted by the precision of each, so your evidence has to sit beyond the belief you want the reader to hold and carry enough weight to outweigh the evidence behind their prior. Two smaller updates ask less of the reader than one large jump, and shifting the mean and narrowing the interval of their beliefs toward yours is enough.

From prior to posterior Schematic

A reader’s belief about the share of the phenomenon attributable to your preferred mechanism, drawn as normal curves on the 0–1 scale. Shaded curves show the new evidence; a curve’s width reflects how much evidence stands behind it, and vertical lines mark each curve’s center. Each updated center is a precision-weighted average of the prior center and the evidence, so it lands between them. The plausible belief still places about 11% of its weight below 0.1; the probable belief places almost none there.

The intermediate steps

The table lists each move, the reaction it aims for, and the evidence it needs.

Move	Reader’s reaction	Evidence	What to draw
1. Establish the phenomenon (possible)	“That pattern is real.”	Counts, rates, means and standard deviations, conditional probabilities, distributions	Picture 1: the simplest chart that makes the pattern visible, leaving explanation for later
2. Affirm priors with suggestive evidence for several accounts (possible)	“Yeah, yeah, both of those could be true.”	Descriptive patterns consistent with each mechanism, including the obvious one	Picture 2: a figure in which at least two mechanisms from your Elster diagram (Elster, 2007) look defensible
3. Pivot with prima facie evidence (plausible)	“…but the obvious account leaves this unexplained.”	The point of empirical divergence between accounts	Picture 3: a figure that tilts the reader toward the preferred mechanism before any regression
4. State the “therefore”	“So the other mechanism deserves a closer look.”	The reconciled inference and the specified ignorance it leaves	One or two sentences that set up the design
5. Make it probable (probable)	“That probably happens.”	Models ordered as replicate, diverge, reconcile; credible designs; simulations	The three-model table (see Section 3)

Section 3

Guidance for building the narrative

1. Start from the belief you want to change

Write down what your audience believes before reading your paper and what you want them to believe afterward. That update is your “probable,” and the narrative is the path you build backward from it to the phenomenon. Learn their standard of evidence, too; if they find only field experiments convincing, learn that before you design the study.

2. Match the statistic to what you want to explain

In late 2016, a widely shared New York Times graphic (Taub, 2016) drew on survey data analyzed by Foa and Mounk (2017) to show a steep, cohort-by-cohort decline in the share of respondents who rated living in a democracy as “essential” (10 on a 10-point scale). Voeten (2016) replotted the same data as mean scores on a full vertical axis, and the decline looked far less dramatic. Neither graph is untruthful; they answer different questions. Decide whether you are explaining a mean, an extreme, or a whole distribution, and choose the display that fits the explanandum. When your audience is anchored on one summary, show both, so they see that the mean barely moves while the share at the top of the scale falls. Anticipate the checks each display invites: other survey items, other surveys, reweighted samples.

3. Show distributions, magnitude, and variation

The distribution often carries more of the argument than a mean difference does, as the Syverson (2004) example in Section 4 shows. Plot the raw data before modeling it to see how sensitive a relationship is to outliers, where observations fall in a joint distribution, and how the main constructs correlate (Healy & Moody, 2014). Show how the pattern varies across contexts.

4. Make the rival accounts live

Picture 2 makes the obvious explanation as strong as the data honestly allow, so that the reader credits you later for showing where it falls short; using it to rule the alternative out forfeits that credit. Name at least two rival mechanisms before claiming your own. If you are stuck, start with omitted variables and reverse causation, then replace those with the specific competing mechanisms your audience cares about. State each rival in a form its strongest proponent would accept.

5. Give each mechanism a signature

Several mechanisms can produce the same main effect (equifinality). Data can adjudicate among them only when each mechanism implies a distinct secondary pattern: a different slope, intercept, subgroup, or outcome. Before collecting or analyzing data, sketch what each mechanism predicts, map what each possible result would mean (supported, null, reversed), and prefer a design under which each outcome is informative.

6. Let readers raise the alternatives before you control for them

When a descriptive figure invites readers to propose their own explanations, the controls in your specification answer concerns they have already voiced. Skip to the regression and you have to justify each control from scratch.

7. Build the priors into the theory up front

Alternative explanations handled only as statistical controls, or introduced late in response to reviewers, leave readers unprepared to update. Present the rival accounts in the introduction, build them into the theory, and let the evidence adjudicate.

8. Keep the evidence as simple as the argument allows

Use descriptive statistics as far as they carry the argument. The more complex the evidence, the harder it is to guide a reader through each update. Report effects in the reader’s units, too: a multiplier or a percentage-point difference reads faster than a coefficient with stars.

9. Write the results table as the narrative

The columns of a well-built results table are the beats of the narrative, in order. Model 1 replicates the prior, Model 2 introduces the difference, and Model 3 reconciles with a design that isolates the surviving mechanism. Decide which models in which tables play these three roles before you run them; other models are supplementary. The distance between the naïve estimate and the design-based estimate is the part of the result a reader repeats to a colleague, so display both.

	Model 1 Naïve	Model 2 + ability proxy	Model 3 Lottery design
Effect of mentorship on promotion	+18 pp	+12 pp	+9 pp
95% confidence interval	[16, 20]	[9, 14]	[6, 12]
Ability accounted for?	No	Partly (noisy proxy)	Yes, by randomization
Narrative move	Replicate the prior	Diverge	Reconcile
Arc beat	“Yeah, yeah…”	“…but…”	“…and therefore.”

Illustrative estimates from a simulation in the book draft. Ability raises both the chance of being mentored and the chance of promotion. The planted effect of mentorship is +8 percentage points; mentored employees are promoted at 43% and unmentored employees at 25%. Section 5, step 9 shows code for a simulation of this kind.

Section 4

Worked examples

The figures below are original sketches of each published figure’s logic; the reference list links to the originals.

Syverson (2004): an abstract that tells the narrative

Chad Syverson’s study of U.S. ready-mixed concrete plants carries the arc in its abstract, and two early figures show the phenomenon in the form the abstract predicts. Each sentence of the abstract performs a distinct move.

PhenomenonMany studies have documented large and persistent productivity differences across producers, even within narrowly defined industries.

Affirm, then departThis paper both extends and departs from the past literature, which focused on technological explanations for these differences, by proposing that demand-side features also play a role in creating the observed productivity variation.

MechanismThe specific mechanism investigated here is the effect of spatial substitutability in the product market.

ImplicationIncreases in substitutability truncate the productivity distribution from below, resulting in higher minimum and average productivity levels as well as less productivity dispersion.

Research site…taking advantage of geographic variation in substitutability created by the industry’s high transport costs.

The implication sentence predicts the shape of a distribution, and the figures deliver that shape. Syverson splits plants by whether their market sits above or below the median demand density in the sample, then plots the two productivity distributions.

Figure 1 logic Schematic

Productivity distributions by market density. In the paper: “At low productivity levels, the distribution of the high-density-market plants consistently has less weight than the low-density-market distribution. At high productivity levels, this pattern is reversed.”

Figure 2 logic Schematic

Plant-size distributions by market density. In the paper: “The average-sized high-density-market plant is 0.454 log points (s.e. = 0.019) larger than the average producer in a low-density market.”

Figure 1 establishes the phenomenon as a distribution: in denser markets, productivity sits further right, is more tightly clustered, and has a thinner lower tail. The prior (plants differ in technology) could still produce that picture, so the narrative stays at “possible.” Figure 2 adds a pattern that the substitutability account implies: plants are larger in dense markets. In a high-fixed-cost industry, larger plants spread fixed costs over more output, which is consistent with high-cost producers exiting where customers can switch suppliers easily. The reader now has a reason to expect the multivariate analysis that accounts for technology differences, and a sense of what it should show.

Rider, Wade, Swaminathan & Schwab (2023): racial disparity in NFL coaching promotions

The pipeline of coaches of color in the National Football League has diversified, while representation among head coaches has lagged. Prior research offers two accounts of such disparities in leadership. Under allocative bias, individuals of color are sorted (at hire) or stacked (after hire) into positions with lesser promotion prospects. Under valuative bias, individuals of color receive lesser rewards for equivalent performance within the same position. The narrative affirms the first account before showing that the disparity persists after accounting for allocation.

A · Establish the phenomenon Schematic

Representation of coaches of color over time, assistant versus head coaches. The gap between the lines is the explanandum.

B · Conditional probabilities, no controls Schematic

Positions ordered by the probability of ever becoming head coach given one’s initial position. Bars show each group’s share of initial years in each position; dots show each group’s probability of reaching head coach, P(HC), from that position. Weighting the dots by the bars gives each group’s overall probability: 0.168 for white coaches and 0.040 for coaches of color at the initial position (4.2×), and 3.1× for current positions, per the course slides.

C · Regressions, all else equal

Regression coefficients expressed as white-coach promotion multipliers, from the course slides. Across specifications that progressively add controls (playing position, initial and current position, performance), the multiplier for all promotions stays between 1.60 and 1.75; the bar shows the fully controlled estimate. The disparity is not significant for promotion to head coach given current position and is largest for promotion to coordinator.

D · The thousand-word picture

Simulated white-coach advantage in reaching head coach under two interventions staged at the initial position, from the course slides. Only the valuative intervention brings the ratio near parity.

Mapped onto the arc, the column comparisons carry the reader through possible and plausible with no controls. The raw conditional probabilities make the argument possible: white coaches are about four times as likely as coaches of color to reach head coach, and the bars show the allocation the allocative account predicts. The conditional probabilities by position (the dots) make the valuative account plausible: at most positions, white coaches are more likely to reach head coach, so accounting for sorting by position leaves the disparity in place. The regression multipliers in Picture C make it probable: the disparity holds as controls accumulate, and it concentrates in promotions to coordinator rather than in the final step to head coach. The figure shows the three specifications that carry this argument and leaves the intermediate columns to the paper’s table. Picture D translates the estimates into a comparison of interventions. Section 5, step 7 condenses this narrative into four sentences.

Rider & Tan (2015): a dyadic picture that invites the regression

The study follows lateral partner hiring among the 200 largest U.S. law firms. Every pair of firms is at risk of a partner move, and each pair can be placed by the hiring firm’s status and profitability relative to the source firm.

Dyadic hiring by status and profit differences Stylized

Each gray point is a pair of firms at risk of a partner move from firm j to hiring firm i; the solid points clustered near the center are actual hires. Quadrant labels give the hire rate and the number of at-risk dyads. Based on 5,000+ hires within the AmLaw 200, 2002–2009. Point positions are simulated to show the pattern; rates and counts are from the paper’s figure.

The figure establishes where moves happen relative to the opportunity structure. Status and profit differences correlate positively, so most at-risk dyads fall in the upper-right and lower-left quadrants, and most hires cluster near the center. Hire rates are highest where the hiring firm is lower status but more profitable (3.2%) and next highest where it is higher status but less profitable (2.3%), against 2.1% and 1.8% on the diagonal. That pattern is consistent with partners treating status and pay as substitutes. Readers will propose other explanations: practice area, geography, firm age, client base. Those proposals are the controls in the dyadic regressions that follow, and the reader who raised them arrives at the table ready to see them addressed.

Barach & Rider (2023): draw your theory

Freelancers on an online labor platform observe the jobs that employers post and the bids that win them. The gap between a winning bid and the next-best bid from a comparably rated freelancer implies an opportunity’s profit potential, and the number of distinct employers a freelancer has worked for, holding experience constant, proxies for access to market information. Three mechanisms could link information access to founding a business, and each leaves a different signature.

Schematic signatures. Solid: high access to information; dashed: low access. Discovery shifts which opportunities an individual observes; discernment changes the slope of founding in profitability; exploitation raises the founding rate at every profitability level above the threshold.

This is the “draw your theory” technique for developing and articulating an argument. Sketching the three panels before analyzing any data forces the precision that prose hides: each mechanism has to commit to a shape, and if two mechanisms produced the same panel, no dataset could separate them. Step 4 in Section 5 asks you to do the same for your own mechanisms.

Section 5

Build your own, step by step

Work through the steps in order. Pencil and paper suit steps 1 through 7; open a laptop at step 8.

Write the belief update
State what your audience believes now and what they should believe after reading your paper. Name the audience: a literature, a method community, a policy audience. If you cannot find someone who would bet against your hypothesis, the update is too small to motivate a paper.
Output: “Before: … After: …” in two sentences.
Establish the phenomenon statistically
Name the phenomenon in a short phrase; a phenomenon that takes a paragraph to name is too broad to review. Then quantify it with counts and rates, means and standard deviations, and conditional probabilities. Plot the raw data and check outliers and joint distributions.
Output: Picture 1, plus one sentence stating the fact and the question it raises.
Name the priors and the rival mechanisms
List the obvious explanation your audience already holds and at least two rival mechanisms besides your preferred one. Draw them on an Elster diagram with your preferred mechanism at the center. A language model can help here: give it your phenomenon and preferred mechanism, and ask for the three most credible rival explanations a fair but skeptical referee would raise, each stated as its strongest proponent would state it. Treat the output as a checklist of priors to affirm; the adjudication comes later.
Output: an Elster diagram with one preferred and two or more rival mechanisms.
Draw each mechanism’s signature
For each mechanism, sketch the figure you would expect if it operated alone. Mark where the sketches differ. Then write what a supported, a null, and a reversed result would each mean.
Output: one small panel per mechanism, as in the Barach & Rider example.
Hand-sketch three thousand-word pictures
Give yourself 20 minutes. Picture 1 establishes the phenomenon: the chart that would convince a skeptical reader the pattern exists. Picture 2 seeds at least two mechanisms from your diagram. Picture 3 tilts the reader toward your preferred mechanism before any regression. Label the axes and say what each line or bar shows.
Output: three sketches on one page.
Find the point of divergence
Identify the evidence that would hold if your mechanism operates and fail if only the alternatives do. If allocation were the only story, the promotion gap would close once position is held constant; in the NFL data it does not. Check that the proxy carrying this comparison is one of your strongest measures.
Output: a caption for Picture 3 that says why it adjudicates.
Write the narrative in four sentences
Synthesize steps 1 through 6 in the differences-in-inferences order.

(1) The racial gap among NFL head coaches persists despite a diversifying pipeline of assistant coaches.

(2) Many attribute the gap to sorting and stacking, and the data show that coaches of color are more often allocated to less favorable positions than white coaches are. [Yeah, yeah, yeah…]

(3) But the gap persists after we account for these positional differences. […but…]

(4) Therefore, an allocative account is insufficient on its own, and valuative bias (lesser rewards for equivalent performance) warrants investigation. […and therefore…]

Output: your own four sentences, one per beat.
Name the three model cells before running them
Build an empty table shell: Model 1 replicates the prior on standard covariates, Model 2 introduces your construct and shows the divergence, Model 3 reconciles (the prior effect attenuates, depends on a condition, or changes sign). Resist the kitchen-sink specification; each added control should answer a rival the reader already knows about.
Output: a table shell with the three columns labeled by role.

Simulate the narrative (optional, recommended)

Before trusting a design on real data, watch it recover a truth you planted. State the planted value beside every estimate. The code below simulates a version of the mentorship example in Section 3; with this seed it returns +19.5, +14.4, and +6.8 percentage points against the planted +8. To reconstruct a published paper instead, simulate data that match its descriptive statistics and correlation matrix (Table 1), then walk the simulated data through possible, plausible, and probable.

import numpy as np
import statsmodels.api as sm

rng = np.random.default_rng(20260527)
N, b_true = 6000, 0.08                      # planted effect: +8 pp

def world(randomize):
    A = rng.normal(0, 1, N)                 # ability (unobserved)
    if randomize:                           # lottery assigns mentors
        M = (rng.uniform(0, 1, N) < 0.5).astype(int)
    else:                                   # able employees get mentored
        M = (rng.uniform(0, 1, N) < 1/(1 + np.exp(-0.95*A))).astype(int)
    p = np.clip(0.30 + b_true*M + 0.15*A, .02, .98)
    Y = (rng.uniform(0, 1, N) < p).astype(int)
    A_proxy = A + rng.normal(0, 1, N)       # noisy ability measure
    return M, Y, A_proxy

def est(X, Y):
    r = sm.OLS(Y, sm.add_constant(X)).fit(cov_type="HC1")
    return 100*r.params[1], 100*r.conf_int()[1]

M, Y, Ap = world(False)
print("Model 1 (naive):   ", est(M, Y))
print("Model 2 (+ proxy): ", est(np.column_stack([M, Ap]), Y))
M3, Y3, _ = world(True)
print("Model 3 (lottery): ", est(M3, Y3), " b_true = 8.0")

Stata users can run the data-generation block inside python: … end and estimate the three models with regress.

Output: a script whose printed estimates sit beside the planted truth.

Write the aspirational abstract
Suspend disbelief and write 150 words in six sentences as if each claim were fully supported: hook, gap, contribution, test, finding, implication. Make the finding specific enough to be falsifiable; if you cannot picture the evidence that would force you to abandon it, sharpen it. If the abstract does not excite you, reconsider the project before investing in it.
Output: a 150-word abstract to revisit every 30 days.
Test the narrative on a reader
Pitch your question and Elster diagram in 60 seconds, then stay silent. Ask the reader to name the prior the paper affirms, where the divergence sits, and which mechanism supplies the reconciliation. Answers that differ from yours show where the narrative needs work.
Output: a revised diagram, pictures, and abstract.

Section 6

Common failure modes

These patterns recur in doctoral assignments and submitted manuscripts.

Failure mode	What it looks like	Repair
Straight to the “but”	The introduction overturns a consensus before stating it accurately, so the reader has no prior loaded to update.	Affirm the prior with evidence before pivoting.
Only the “yeahs”	The paper confirms what readers believed and calls the confirmation a contribution.	Find the evidence the obvious account cannot explain.
Pseudo-divergence	“Prior work found a large effect; we find a smaller one,” with no theoretical reason for the gap. An enthusiastic supporter of the prior would not update.	Tie the divergence to a mechanism the prior omits.
Reconciliation collapse	The takeaway is “context matters,” “both,” or “more research is needed.”	Name the single process that produces both the prior’s evidence and yours.
A straw-person middle picture	Picture 2 rules the rival out instead of showing that it has support.	Show the rival’s best descriptive case first.
Picture 3 argues for the rival	The adjudicating figure shows a pattern the rival account also predicts, such as a moderator the rival implies.	Condition on the rival’s signature (e.g., show the effect within the rival’s subgroups).
Rivals at the wrong level	A randomized design switches off distal rivals by construction, yet the pictures still adjudicate among those distal rivals.	Adjudicate among the mechanisms that remain live under your design.
The weakest proxy does the adjudicating	The construct that separates your mechanism from its rival is the one measured least well.	Invest in that measure, or choose a different point of divergence.
Pictures that disagree	Picture 1 shows a decline where Picture 3 and the abstract claim an increase.	Commit to one prediction and let the earlier picture foreshadow it.
A table with no story	Columns are ordered by software default; Model 3 does not answer the question Model 1 raised.	Order columns as replicate, diverge, reconcile.
A story with no table	The prose says “we address this concern” where a column should be doing the addressing.	Point to the model cell that answers each concern.

Section 7

Reading and tools

Differences-in-Inferences

Stage replicate, diverge, and reconcile for your project.

Contributing to a Literature

Build and refine an Elster diagram.

Research Toolkit: Framing Arguments

Companion tip sheet on framing.

Research Toolkit: Delivering Insights

Companion tip sheet on presenting results.

Barach, M. A., & Rider, C. I. (2023). Discovery, discernment, and exploitation: Entrepreneurial mechanisms at the nexus of individual and opportunity. Strategic Management Journal, 44(12), 2858–2887. https://doi.org/10.1002/smj.3528

Damodaran, A. (2017). Narrative and numbers: The value of stories in business. Columbia Business School Publishing. https://doi.org/10.7312/damo18048

Davis, M. S. (1971). That’s interesting! Towards a phenomenology of sociology and a sociology of phenomenology. Philosophy of the Social Sciences, 1(2), 309–344. https://doi.org/10.1177/004839317100100211

Elster, J. (2007). Explaining social behavior: More nuts and bolts for the social sciences. Cambridge University Press. https://doi.org/10.1017/CBO9780511806421

Foa, R. S., & Mounk, Y. (2017). The signs of deconsolidation. Journal of Democracy, 28(1), 5–15. https://doi.org/10.1353/jod.2017.0000

Healy, K., & Moody, J. (2014). Data visualization in sociology. Annual Review of Sociology, 40, 105–128. https://doi.org/10.1146/annurev-soc-071312-145551

Merton, R. K. (1987). Three fragments from a sociologist’s notebooks: Establishing the phenomenon, specified ignorance, and strategic research materials. Annual Review of Sociology, 13, 1–29. https://doi.org/10.1146/annurev.so.13.080187.000245

Rider, C. I., & Tan, D. (2015). Labor market advantages of organizational status: A study of lateral partner hiring by large U.S. law firms. Organization Science, 26(2), 356–372. https://doi.org/10.1287/orsc.2014.0907

Rider, C. I., Wade, J. B., Swaminathan, A., & Schwab, A. (2023). Racial disparity in leadership: Evidence of valuative bias in the promotions of National Football League coaches. American Journal of Sociology, 129(1), 227–275. https://doi.org/10.1086/725389

Syverson, C. (2004). Market structure and productivity: A concrete example. Journal of Political Economy, 112(6), 1181–1222. https://doi.org/10.1086/424743

Taub, A. (2016, November 29). How stable are democracies? ‘Warning signs are flashing red.’ The New York Times.

Voeten, E. (2016, December 5). That viral graph about millennials’ declining support for democracy? It’s very misleading. The Washington Post (Monkey Cage). Link

Purpose of a numerical narrative

The arc: possible, plausible, probable

The intermediate steps

Guidance for building the narrative

1. Start from the belief you want to change

2. Match the statistic to what you want to explain

3. Show distributions, magnitude, and variation

4. Make the rival accounts live

5. Give each mechanism a signature

6. Let readers raise the alternatives before you control for them

7. Build the priors into the theory up front

8. Keep the evidence as simple as the argument allows

9. Write the results table as the narrative

Worked examples

Syverson (2004): an abstract that tells the narrative

Rider, Wade, Swaminathan & Schwab (2023): racial disparity in NFL coaching promotions

Rider & Tan (2015): a dyadic picture that invites the regression

Barach & Rider (2023): draw your theory

Build your own, step by step

Write the belief update

Establish the phenomenon statistically

Name the priors and the rival mechanisms

Draw each mechanism’s signature

Hand-sketch three thousand-word pictures

Find the point of divergence

Write the narrative in four sentences

Name the three model cells before running them

Simulate the narrative (optional, recommended)

Write the aspirational abstract

Test the narrative on a reader

Common failure modes

Reading and tools