CMH MATH PATH — LESSON 5: LOGISTIC REGRESSION

Predicting Yes/No Events — The Revolution Probability Machine

PART 1: WHY WE NEED A DIFFERENT TOOL

The War Prediction Problem

Let’s say we want to predict: “Will Country X have a civil war next year?”

We try linear regression from Lesson 4:

  • X₁ = Poverty rate (%)
  • X₂ = Ethnic fractionalization (0-1)
  • Y = “War” (0 = No, 1 = Yes)

The Linear Regression Disaster:
We get: Ŷ = -0.8 + 0.05×(Poverty) + 0.6×(Ethnic Fractionalization)

Make a prediction for Country A:

  • Poverty = 30%, Ethnic Frac = 0.7
  • Ŷ = -0.8 + 0.05×30 + 0.6×0.7 = -0.8 + 1.5 + 0.42 = 1.12

The Problem: What does “1.12” mean for “War: Yes/No”?

  • Y=0 means “No war”
  • Y=1 means “Yes war”
  • But 1.12? “112% war”? Nonsense!

This is why we need logistic regression:

  • Predict probabilities (0% to 100%)
  • Not arbitrary numbers that can go outside 0-1

PART 2: THE S-CURVE — NATURE’S PROBABILITY MACHINE

The “Tipping Point” Phenomenon

Think about pushing a boulder up a hill:

  • At first: Pushing does little (boulder stays put)
  • Near top: Small push makes big difference (almost rolls!)
  • Past top: Tiny push sends it rolling (definitely goes!)

This is an S-curve (or sigmoid curve):

Probability of War
    ↑
100% |                   ┌───┐
     |                  /
 50% |               ──
     |             /
  0% |────────────┘
     +------------------------→ Poverty Rate
         (Tipping Point Here!)

Three Key Regions:

  1. Left flat: Low poverty → Very low war probability (<10%)
  2. Steep middle: Medium poverty → Small changes, BIG probability jumps!
  3. Right flat: High poverty → Very high war probability (>90%)

This matches reality!

  • Very stable countries rarely collapse
  • Very unstable countries often collapse
  • Middle countries: Small changes decide fate!

PART 3: FROM LINE TO S-CURVE — THE LOGIT TRANSFORM

Step 1: Predict “Log-Odds” (A Number That Can Be Anything)

Instead of predicting probability directly (0-1), we predict log-odds (-∞ to +∞):

Odds Definition:

Odds = P/(1-P)
(Where P = probability)

Examples:

  • P = 0.5 (50%) → Odds = 0.5/0.5 = 1 (“1 to 1”)
  • P = 0.75 (75%) → Odds = 0.75/0.25 = 3 (“3 to 1”)
  • P = 0.9 (90%) → Odds = 0.9/0.1 = 9 (“9 to 1”)

Log-Odds: Natural log of odds:

  • Odds = 1 → log(1) = 0
  • Odds = 3 → log(3) = 1.1
  • Odds = 0.33 → log(0.33) = -1.1

Key insight: Log-odds can be any number (-∞ to +∞), perfect for linear regression!

Step 2: The Logistic Regression Equation

We do linear regression on log-odds:

log(P/(1-P)) = b₀ + b₁X₁ + b₂X₂ + …

Then transform back to probability:

P = 1 / (1 + e^-(b₀ + b₁X₁ + b₂X₂ + …))

Don’t memorize — just know the computer does this automatically!

PART 4: INTERPRETING LOGISTIC COEFFICIENTS — THE “ODDS RATIO”

The Most Important Concept

In linear regression: b₁ = “For 1-unit increase in X, Y changes by b₁ units”

In logistic regression: e^b₁ = “For 1-unit increase in X, odds multiply by e^b₁”

Example: Civil War Model
Variables:

  • X₁ = Youth Unemployment Rate (%)

We get: b₁ = 0.2 for Youth Unemployment

Interpretation:

  • e^0.2 = 1.22
  • Meaning: Each 1% increase in youth unemployment MULTIPLIES odds of war by 1.22

Concrete example:

  • Baseline: YouthUnemp = 20% → Odds of war = 1:9 (10% probability)
  • Increase to 21%: New odds = 1:9 × 1.22 = 1.22:9 (≈ 1:7.4 → 11.9% probability)
  • Increase to 25%: New odds = 1:9 × (1.22)^5 = 1:9 × 2.7 = 2.7:9 (≈ 1:3.3 → 23% probability)

See the power? Small coefficient (0.2) → Big cumulative effect!

The Three Types of Effects:

1. b > 0 (e^b > 1): Risk factor

  • e^b = 1.5 → 1-unit increase in X multiplies odds by 1.5
  • Example: Poverty → War risk

2. b < 0 (e^b < 1): Protective factor

  • e^b = 0.7 → 1-unit increase in X multiplies odds by 0.7 (reduces!)
  • Example: Trust in government → Less war risk

3. b = 0 (e^b = 1): No effect

  • e^b = 1 → Odds don’t change
  • Example: Maybe “mountainous terrain” has no effect on civil war?

PART 5: HANDS-ON LOGISTIC REGRESSION LAB

Exercise 1: The “Revolution Risk” Calculator (Simplified)

Model:
log(P/(1-P)) = -5 + 0.1×(YouthUnemp) + 0.05×(Poverty) + 0.8×(RecentProtest)

Where:

  • YouthUnemp = Youth unemployment rate (%)
  • Poverty = Population below poverty line (%)
  • RecentProtest = 1 if major protest last year, 0 if not

Calculate probability for Country A:

  • YouthUnemp = 25%
  • Poverty = 30%
  • RecentProtest = 1 (Yes)

Step 1: Calculate log-odds
log-odds = -5 + 0.1×25 + 0.05×30 + 0.8×1
= -5 + 2.5 + 1.5 + 0.8
= -0.2

Step 2: Convert to probability
P = 1 / (1 + e^-(-0.2)) = 1 / (1 + e^0.2) = 1 / (1 + 1.221) = 1/2.221 = 0.45

Interpretation: Country A has 45% chance of revolution next year!

Exercise 2: Spreadsheet Simulation

Step 1: Make up data for 20 “countries”:

Col A: Country ID (1-20)
Col B: YouthUnemp = RANDBETWEEN(10, 50)
Col C: Poverty = RANDBETWEEN(10, 60)
Col D: RecentProtest = RANDBETWEEN(0, 1)
Col E: WarNextYear (we'll make this up logically)

Step 2: Make “WarNextYear” somewhat predictable:

=IF((B2>30)+(C2>40)+(D2=1)>=2, 1, 0)

(If at least 2 risk factors present, war=1, else war=0)

Step 3: Run logistic regression (Google Sheets add-on or Excel Data Analysis)

  • Or use online calculator
  • Get coefficients b₀, b₁, b₂, b₃

Step 4: Make predictions for new countries

  • Use formula: P = 1/(1+exp(-(b₀+b₁×Unemp+b₂×Poverty+b₃×Protest)))

Exercise 3: The “Threshold” Game

Most real decisions use a threshold:

  • P > 50% → Predict “War”
  • P > 70% → Sound alarm
  • P > 90% → Evacuate embassy

Play with thresholds:

  1. Calculate P for 5 hypothetical countries
  2. Try threshold = 40%, 50%, 60%
  3. For each threshold:
  • How many predicted wars?
  • Check “accuracy” if you knew truth
  • Trade-off: Higher threshold = fewer false alarms, but miss more real wars

This is the CMH policymaker’s dilemma!

PART 6: CMH CASE STUDY — PREDICTING REVOLUTIONS

The Goldstone Model (Simplified)

Political scientist Jack Goldstone found key revolution predictors:

Variables:

  1. Elite Overproduction: Too many educated elites for available jobs
  2. State Fiscal Crisis: Government running out of money
  3. Popular Grievances: Mass discontent
  4. International Pressure: Foreign threats/interference

Logistic model from historical data:
log(P/(1-P)) = -4.3 + 1.2×(EliteOverprod) + 0.8×(FiscalCrisis) + 0.6×(Grievances) + 0.4×(IntlPressure)

Interpret coefficients:

  • e^1.2 = 3.32 → Elite overproduction TRIPLES odds of revolution!
  • e^0.8 = 2.23 → Fiscal crisis more than DOUBLES odds
  • Combined: 3.32 × 2.23 = 7.4 → Both together multiply odds by 7.4×!

Historical test:

  • French Revolution (1789): All 4 factors present → P ≈ 85%
  • Stable Britain (same era): Only 1 factor → P ≈ 15%

PART 7: BEYOND YES/NO — MULTINOMIAL LOGISTIC REGRESSION

When There Are Multiple Possible Outcomes

Not just “War/Peace” but:

  • Regime change type: Democracy, Military rule, Same regime, Civil war
  • Election outcome: Party A win, Party B win, Coalition, Crisis
  • Crisis resolution: Reform, Repression, Revolution, Collapse

The Idea: Instead of one S-curve, we have multiple comparisons:

  • Probability of Democracy vs. Status Quo
  • Probability of Military rule vs. Status Quo
  • Probability of Civil war vs. Status Quo

CMH Example: Arab Spring Outcomes (2011)

  • Tunisia → Democracy (P=60%)
  • Egypt → Military rule (P=70%)
  • Libya → Civil war (P=80%)
  • Saudi Arabia → Status quo (P=90%)

Different factors mattered for each outcome!

PART 8: YOUR CMH LOGISTIC REGRESSION FIELD KIT

Quick Decision Guide

Use Logistic Regression When:

  • Predicting categories (Yes/No, Win/Lose, War/Peace)
  • Outcome is binary or few categories
  • You want probability estimates, not just predictions
  • Real-world decision needs risk assessment (40% chance vs. 80% chance)

Use Linear Regression When:

  • Predicting continuous numbers (GDP, population, temperature)
  • Relationship is roughly linear
  • You want to know “how much” not just “if”

The Logistic Regression Report Card

Every logistic model should report:

  1. Coefficients (b): For interpretation as log-odds
  2. Odds Ratios (e^b): More intuitive!
  3. Predicted Probabilities: For example cases
  4. Classification Table: How good is it at predicting?
  5. AUC-ROC: Overall predictive power (0.5=random, 0.7=decent, 0.9=excellent)

PART 9: REAL-WORLD APPLICATIONS BEYOND CMH

Everywhere You See Probability Estimates:

  1. Medicine:
  • “Based on your age, weight, symptoms: 30% chance of disease X”
  • Treatment decisions based on probability thresholds
  1. Finance:
  • “5% probability of default on this loan”
  • Credit scores = logistic regression predictions!
  1. Marketing:
  • “40% chance this customer will buy if shown this ad”
  • Target advertising to high-probability customers
  1. Criminal Justice (Carefully!):
  • “Risk scores” for bail decisions
  • HUGE ethical concerns! Models can encode bias

The Common Pattern: Never certain, always probabilistic. Better information → better probabilities → better decisions.

PART 10: ETHICAL DANGERS — THE “PREDICTIVE POLICING” TRAP

When Probability Prediction Becomes Oppression

Scenario: Police use logistic regression:

  • Predict “probability of violent crime by neighborhood”
  • Deploy more officers to high-probability areas
  • More officers → More arrests (confirmation!)
  • More arrests → Model sees “proof” neighborhood is high-risk
  • Feedback loop: Prediction creates reality

The CMH Ethical Rule:

Prediction should inform service, not surveillance. Probability should guide help, not punishment.

Better approach: High predicted unrest probability → Send social workers, job programs, community mediators, not just police.

YOUR MATH HOMEWORK

Exercise 1: Build a “College Admission” Predictor

Hypothetical data on 100 students:

  • X₁ = GPA (0-4.0)
  • X₂ = SAT score (400-1600)
  • X₃ = Extracurriculars (0-10 scale)
  • Y = Admitted (1) or Not (0)

Your tasks:

  1. Make up reasonable data (or find real dataset online)
  2. Run logistic regression
  3. Interpret: “Each 0.1 GPA increase multiplies admission odds by _“
  4. Calculate: Student with GPA=3.5, SAT=1200, EC=7 has what admission probability?
  5. If college wants to admit ~30%, what probability threshold should they use?

Exercise 2: The “Tipping Point” Analysis

Using the revolution model:
log(P/(1-P)) = -5 + 0.1×(YouthUnemp)

Calculate P for:

  • YouthUnemp = 20% → P = ?
  • YouthUnemp = 30% → P = ?
  • YouthUnemp = 40% → P = ?
  • YouthUnemp = 50% → P = ?

Graph it! Notice the S-curve:

  • Where is the “tipping point” (steepest part)?
  • Why might a government ignore 20→30% rise but panic at 40→50% rise?
  • Policy insight: Small interventions before tipping point = huge risk reduction!

Exercise 3: Design a CMH Early Warning System

You’re building a system to predict “Mass Protest Next 6 Months” for 50 countries.

Available monthly data:

  • Social media unrest mentions
  • Inflation rate
  • Youth unemployment
  • Previous protest (Y/N)
  • Government approval rating

Design questions:

  1. What would Y variable be? (How define “mass protest”?)
  2. Which variables as X? Why?
  3. What probability threshold for “warning”?
  • 30%? (Many warnings, some false alarms)
  • 60%? (Fewer warnings, miss some events)
  1. How would you test if system works?
  2. What ethical safeguards would you build in?

Exercise 4: The “Base Rate” Problem

Scenario: Disease affects 1% of population. Test is 95% accurate.

  • If test says “positive”, what’s true probability of disease?

Apply Bayes’ Theorem (related to logistic regression!):

  • Prior odds = 1:99 (1% disease)
  • Likelihood ratio = 0.95/0.05 = 19 (for positive test)
  • Posterior odds = 1:99 × 19 = 19:99 ≈ 1:5.2
  • Probability = 1/(1+5.2) = 16%

Even with 95% accurate test, positive result only means 16% chance!

CMH analogy: Even with good model, rare events (revolutions) are hard to predict accurately!

KEY TAKEAWAYS:

  1. Logistic regression predicts probabilities (0% to 100%) for yes/no events
  2. Uses S-curve (sigmoid) — flat at ends, steep in middle (tipping points!)
  3. Coefficients as odds ratios: e^b = “multiplies odds by this much”
  4. Essential for CMH: Revolutions, wars, elections are yes/no events
  5. Threshold decisions: Choose probability cutoff based on costs of errors
  6. Ethical dangers: Predictive systems can create self-fulfilling prophecies

The CMH Mantra:

“Linear regression predicts how much rain will fall.
Logistic regression predicts the chance it will rain at all.
In human affairs, the yes/no questions matter most:
War or peace? Revolution or stability? Collapse or resilience?
But remember: a 30% chance is not a prediction of what WILL happen,
it’s a warning of what COULD happen — and therefore what we should prepare for.”

NEXT LESSON PREVIEW:

Lesson 6: Time Series Analysis — Predicting the Future from the Past

Where we’ll learn:

  • How to find patterns over time (trends, cycles, seasons)
  • ARIMA models: The workhorse of time prediction
  • How to separate signal from noise in historical data
  • Why “momentum” exists in social systems (good times tend to continue, crises tend to cascade)

Thought to Ponder: Logistic regression tells us risk at one moment. But societies have memory — today’s risk depends on yesterday’s events. Time series analysis adds this crucial dimension: history matters. The past doesn’t just suggest the future; in social systems, it actively creates it through momentum, trauma, institutional memory, and cycles…

Remember: In CMH, we’re not fortune tellers. We’re risk assessors. Logistic regression gives us the language of risk — not “there will be a war” but “the risk of war has doubled from 20% to 40%.” This changes everything about how leaders should respond. It’s the difference between panic and preparedness, between overreaction and prudent caution.

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