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ImpactMojo 101 Series · Free Forever

Multivariate
Analysis
101

Controlling for Several Things at Once — Multiple Regression and Its Cousins for Development Practitioners in South Asia

Research-BackedSouth Asia Focus100 SlidesFree Access

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Agenda

What We Cover

From Bivariate to Multivariate

Slides 3–11

Multiple Linear Regression

Slides 12–20

Interpreting Coefficients

Slides 21–29

Model Fit & Inference

Slides 30–39

Assumptions & Residual Diagnostics

Slides 40–48

Multicollinearity

Slides 49–57

Interactions & Non-Linearity

Slides 58–65

Logistic Regression

Slides 66–74

Data Reduction — PCA & Factor Analysis

Slides 75–83

Pitfalls

Slides 84–92

Reporting & Tools

Slides 93–99

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Section One

From Bivariate to Multivariate

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Starting Point

One outcome, one predictor

Bivariate analysis looks at two variables at a time: child stunting and household income; learning scores and class size. A simple regression or correlation summarises how they move together.

Bivariate

An analysis of the relationship between exactly two variables — one outcome and one predictor — with nothing else held constant.

The trouble: in the real world, almost nothing varies one at a time. Income, education, caste and location all move together.

Bivariate tells you	It cannot tell you
Whether X and Y move together	Whether X causes Y
The strength of that association	Whether a third variable drives both
A slope in the raw data	The slope among comparable units

A bivariate slope is not wrong; it answers a narrower question than people read it as. It describes the raw association, which mixes the effect of X with the effects of everything correlated with X.

It is still worth computing first. The gap between the bivariate and multivariate coefficient tells you how much of the raw association was confounding, which is informative in itself.

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The Real World

Outcomes have many causes at once

Whether a child is stunted depends on income, the mother's education, sanitation, diet, birth order and more — simultaneously. Look at any one in isolation and you mix up its effect with all the others.

Mother's education

→

Household income

→

Sanitation & water

→

→ Child stunting

Multivariate analysis is simply analysis that handles several predictors at the same time.

Determinant of stunting	Correlated with	So omitting it…
Mother's education	Income, sanitation, health-seeking	Inflates their coefficients
Household income	Diet, housing, everything	Inflates most coefficients
Sanitation	Income, village infrastructure	Inflates income
Birth order	Family size, mother’s age	Distorts household variables

The determinants are correlated with each other, which is the entire problem. They arrive as a bundle, and separating them is what multiple regression is for — imperfectly.

It is also why coefficients shrink as controls are added. Each new variable takes back some of the credit the earlier ones had absorbed, and the sequence of shrinkages is worth reporting.

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The Core Problem

Confounding: the lurking variable

Confounder

A variable that influences both the predictor and the outcome, creating a misleading association between them when it is left out of the analysis.

Districts with more private clinics may show worse average health — not because clinics harm anyone, but because clinics open where the population is older and sicker. Age confounds the clinic–health link.

For Z to be a confounder	Check
Z affects the outcome Y	Is there a plausible mechanism?
Z affects the predictor X	Does Z come earlier in the story?
Z is not caused by X	Otherwise it is a mediator, not a confounder

The third condition is the one that is skipped and it reverses the advice. A variable caused by X is a mediator: controlling for it removes part of the effect you were trying to measure.

The private-clinic example is a good one to hold onto: clinics locate where illness is, so the correlation runs from health to clinics, not the reverse. Confounding and reverse causality look identical in a table.

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See It

Why a raw comparison misleads

Suppose richer households both eat more diverse diets and have less stunting. A bivariate look at 'diet diversity vs stunting' credits diet with the whole gap — part of which is really just income.

Income (confounder)

→

raises diet diversity

→

AND lowers stunting

→

→ diet looks more powerful than it is

Without controlling for income, the diet coefficient is biased. This is the problem multivariate analysis exists to solve.

Estimate	What it captures
Bivariate diet coefficient	Diet’s effect plus income’s effect
Controlling for income	Diet’s effect among households with similar income
Plus education and sanitation	Net of those too
Still unmeasured	Prices, motivation, health environment

Each row narrows the claim and none reaches a causal one. The last row does not disappear as you add controls; it is defined as whatever remains outside the model.

This is why the honest phrasing is "associated with, among households similar on the measured characteristics". It sounds weaker because it is weaker, and it is what the method supports.

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The Fix

'Holding other variables constant'

Multiple regression estimates the effect of each predictor while statistically holding the others fixed — comparing households that differ in diet but have the same income, education and sanitation.

This is the single most important idea in the course. A multivariate coefficient is a partial effect: the contribution of one variable, net of the others in the model.

Comparison	Controls for
All households, diet versus stunting	Nothing
Within income bands	Income only
Within income, education and sanitation bands	Those three
A randomised diet intervention	Everything, observed and not

Each row narrows the comparison and each still leaves everything unmeasured. Regression is the third row done efficiently — it does not reach the fourth, which is the whole difference between adjustment and design.

"Holding constant" is doing arithmetic, not physically holding anything. The model compares units that differ in X and are statistically similar on the included variables — nothing more.

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Caveat

Control is not the same as causation

Holding observed variables constant removes those confounders — but only those. Anything you did not measure (motivation, local prices, unobserved health) can still bias the estimate.

Controlling for what you can measure is necessary but not sufficient for causal claims. Keep your conclusions honest about what remains unobserved.

Unmeasured variable	Why it is missing	Effect
Maternal motivation	No survey measures it	Biases schooling and diet coefficients
Local food prices	Rarely collected	Biases income coefficients
Unobserved health status	Hard to measure	Biases almost everything
Local governance quality	Not in household surveys	Biases service-access variables

Every one of these plausibly affects both a predictor and the outcome, which is the definition of a confounder — and none is in any standard household survey.

This is why credible causal claims come from designs (randomisation, thresholds, natural experiments) rather than from richer control sets. Control gets you a better description, not a counterfactual.

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Vocabulary

The words analysts use

Term	Also called	Meaning
Outcome	Dependent variable, Y	What you are trying to explain
Predictor	Independent variable, X, covariate	What you use to explain it
Coefficient	Slope, β	Effect of a predictor on the outcome
Control	Adjust for, condition on	Hold a variable constant
Residual	Error, e	What the model fails to predict

We will use 'predictor' and 'covariate' interchangeably; both just mean a right-hand-side variable.

Term	Also called	Watch out
Outcome	Dependent variable, Y	Must be measured after the predictors
Predictor	Independent variable, X, covariate	"Independent" implies nothing about causality
Coefficient	Slope, β	Glued to the units of X
Control	Adjust for, condition on	Only removes measured confounding
Residual	Error, e	Contains every omitted cause

"Independent variable" is the most misleading term in the vocabulary. It means independent in the model’s structure, not statistically independent of anything — predictors are usually heavily correlated.

The residual is where the trouble lives. Everything affecting the outcome that you did not include sits there, and exogeneity is precisely the claim that none of it correlates with your predictor.

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Roadmap

Where this course goes

The workhorse

Multiple linear regression
Reading coefficients correctly
Model fit, inference, diagnostics

Beyond the basics

Interactions and non-linearity
Logistic regression for yes/no outcomes
PCA & factor analysis; pitfalls; tools

Examples lean on NFHS, PLFS and the kind of data you actually meet at work.

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Section Two

Multiple Linear Regression

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The Model

Predicting Y from several Xs

Multiple linear regression models the outcome as a straight-line combination of predictors plus an error term:

Y = β0 + β1X1 + β2X2 + … + βkXk + e

It is the same machinery as simple regression — just with several Xs at once. Everything that follows builds on this one line.

The model assumes	Which fails when
Effects add up	One predictor’s effect depends on another — use an interaction
Each effect is a straight line	Returns diminish — use a log or squared term
The same effect for everyone	Effects vary by subgroup
Errors behave	Clustering or changing spread — fix the standard errors

Additivity and linearity are assumptions about the world, not about the maths, and both are fixable inside the same framework. That flexibility is why the linear model remains the workhorse.

The fourth row is different: it does not bias the coefficients, only the standard errors. Knowing which failures move the estimate and which move the uncertainty is half of diagnostics.

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The Pieces

What each symbol means

Symbol	Name	What it is
Y	Outcome	What you predict (e.g. test score)
β₀	Intercept	Predicted Y when every X = 0
β₁…β_k	Slope coefficients	Partial effect of each predictor
X₁…X_k	Predictors	Your covariates
e	Error / residual	Everything the model misses

The intercept is rarely interesting on its own — 'all Xs = 0' is often impossible (a household with zero adults). The slopes carry the story.

Symbol	Interpretable when
β₀ (intercept)	X = 0 is a real, observed value
β₁…βₖ	The units of each X are stated
e (residual)	It is uncorrelated with every X
Fitted values	Predictions stay inside the observed range

The intercept is often uninterpretable and is reported anyway. Predicted HAZ for a household with zero income, zero schooling and no toilet describes nobody in the sample.

Centring the predictors makes the intercept the prediction at the sample mean, which is a real household and a meaningful number. It costs nothing and improves the table.

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Worked Setup

An example we will reuse

Suppose we model a child's height-for-age z-score (an anthropometric outcome) on three predictors:

HAZ = β0 + β1(income) + β2(mother's years of schooling) + β3(improved toilet) + e

All coefficients in this deck are illustrative — chosen to teach interpretation, not reported as real NFHS findings.

Model choice	Question it embeds
HAZ as the outcome	Chronic nutrition, not acute — stunting, not wasting
Income as a control	Income confounds; it is not the thing being studied
Mother’s schooling included	Assumed to affect HAZ directly
Improved toilet as a dummy	Any improved toilet treated as equivalent

Every modelling choice is a substantive claim. Coding toilets as a single 0/1 asserts that a shared improved latrine and a private flush toilet have the same effect, which may well be false.

Height-for-age captures cumulative deprivation over a child’s life; weight-for-height captures recent shortage. Choosing between them changes what your coefficients mean.

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The Geometry

Fitting a plane, not a line

With one predictor, regression fits a line through a cloud of points. With two predictors, it fits a plane; with many, a hyper-plane you cannot draw. The idea is unchanged: find the surface closest to the data.

'Closest' has a precise meaning — the surface that makes the prediction errors as small as possible, in a specific sense.

Predictors	Geometry	Can you plot it?
One	A line	Yes
Two	A plane	With effort
Three or more	A hyper-plane	No

You lose the ability to see the fit exactly when you most need it. That is why residual plots matter: they are the only view of the model’s shape that survives into higher dimensions.

Partial-regression (added-variable) plots restore some of it. They show one predictor’s relationship with the outcome after both have been netted of the other predictors — a two-dimensional view of a multivariate slope.

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How It's Fit

Ordinary least squares (OLS)

Ordinary least squares

The method that chooses the coefficients minimising the sum of the squared residuals — the squared vertical gaps between each observed Y and the value the model predicts.

Squaring the residuals penalises big misses heavily and treats over- and under-prediction symmetrically. The software solves it instantly; your job is to set up and read the model well.

Least squares	Consequence
Squares the residuals	Large misses are penalised heavily
Minimises the sum	The line is pulled toward outliers
Uses vertical distances	Assumes X is measured without error
Has a closed-form solution	Fast, deterministic, always returns an answer

"Always returns an answer" is a hazard as well as a convenience. OLS will fit a line through data that violates every assumption and report coefficients with confidence intervals as if nothing were wrong.

The vertical-distance convention matters: it treats X as exact and Y as noisy. Measurement error in X attenuates the coefficient toward zero, which is why noisy predictors understate real effects.

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Key Idea

Each slope is a partial effect

In our HAZ model, β₂ is the change in height-for-age associated with one more year of the mother's schooling, holding income and toilet access constant.

Compared with a bivariate estimate, the multivariate slope strips out the part of schooling's apparent effect that was really income or sanitation in disguise.

Coefficient type	What it includes
Bivariate slope on schooling	Schooling’s effect plus everything correlated with schooling
Multivariate slope on schooling	Schooling’s effect net of income and sanitation
Difference between them	How much of the raw association those controls explained

Report both. A schooling coefficient that falls from 0.15 to 0.06 when income enters tells the reader that most of the raw association was income — a substantive finding, not a technical detail.

If it barely moves, that is informative too, and worth saying: it means the included controls did not explain the association, though unmeasured ones still might.

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Fitted vs Residual

Prediction = signal + leftover

For each child the model gives a fitted value (its best guess of HAZ) and a residual (observed − fitted). The residual is what the predictors could not explain.

Observed Y

→

− Fitted Y (the model's guess)

→

= Residual (the leftover)

→

→ diagnostics live in the residuals

Quantity	Definition	Used for
Fitted value	The model’s prediction for that unit	Prediction; residual plots
Residual	Observed minus fitted	Diagnostics
Large positive residual	The model under-predicted	Investigate; possible outlier
Patterned residuals	The model is missing something	Respecify

Residuals should look like noise, and any pattern in them is unmodelled signal. A curve means the wrong functional form; clusters mean a missing grouping variable.

Look at the largest residuals individually. They are often data-entry errors, and finding one is usually more valuable than any refinement of the specification.

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Why Linear First

Start simple, then complicate

Linear regression is the default workhorse because it is transparent, fast, and surprisingly flexible — you can add interactions, squared terms and log transforms without leaving the framework.

Master the linear model and you have the scaffolding for almost every method that follows, including logistic regression.

Linear regression handles	Via
Curved relationships	Logs and squared terms
Effects that vary by group	Interaction terms
Categorical predictors	Dummy variables
Clustered data	Clustered standard errors
Binary outcomes	Not well — use logistic

Four of five extensions stay inside the framework, which is why mastering the linear model gets you most of applied quantitative work rather than a starting point you will soon abandon.

The remaining case — a 0/1 outcome — is the one genuinely different tool in this deck, and even there the interpretive discipline carries straight over.

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Section Three

Interpreting Coefficients

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The Golden Sentence

How to read any coefficient

Read every slope with one template: 'A one-unit increase in X is associated with a β-unit change in Y, holding the other variables constant.'

Get this sentence right and most interpretation errors vanish. The three load-bearing phrases are one-unit, associated with, and holding the others constant.

Phrase	Load it carries
"A one-unit increase in X"	Check the units — rupees or thousands?
"Is associated with"	Not "causes"
"Holding the other variables constant"	Only the ones in the model
"On average"	Individual effects may differ

Read the sentence out in full for every coefficient you report. Most interpretation errors are caught by hearing "associated with, holding the included variables constant" spoken aloud.

The phrase people drop first is the third one, and it is the most consequential. Without it the claim becomes causal by implication, which no observational regression supports.

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Units Matter

A coefficient is glued to its units

If income is measured in rupees, β_income is the effect of one extra rupee — a tiny number. Measure income in thousands of rupees and the coefficient is 1,000 times larger. The relationship is identical; only the scale changed.

Always state units. A coefficient of '0.002' is meaningless until you know 0.002 of what, per what.

Income measured in	Coefficient	Same relationship?
Rupees	0.00003	Yes
Thousands of rupees	0.03	Yes
Lakhs	3.0	Yes
Log rupees	Read as % change	Yes, differently expressed

A tiny coefficient can be a large effect and a large one can be trivial. Size means nothing until you know the units, which is why comparing raw coefficients across predictors is meaningless.

Rescale to a meaningful unit before reporting — per ₹1,000, per year of schooling, per standard deviation. The reader should be able to feel the magnitude without doing arithmetic.

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Continuous Predictors

Worked reading: a continuous X

Illustrative result: β_schooling = 0.06 in our HAZ model.

Read it as: 'Each additional year of the mother's schooling is associated with a 0.06 higher height-for-age z-score, holding income and toilet access constant.' Five extra years ≈ 0.30 z-score.

Illustrative figure — not a reported NFHS coefficient.

β(schooling) = 0.06	Translation
One extra year	0.06 higher HAZ
Five extra years	About 0.3 higher HAZ
Primary to secondary completion	Roughly 0.3–0.4
Compared with what?	Households alike on income and sanitation

Scaling to a policy-relevant change makes the coefficient legible. "0.06 per year" is abstract; "completing secondary rather than primary school" is a decision someone can act on.

The last row is the one to keep saying. The comparison is between otherwise-similar households, and that qualification is what distinguishes the estimate from a raw group difference.

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Dummy Variables

Encoding categories as 0/1

Dummy variable

A 0/1 indicator standing for a category — e.g. improved toilet = 1, else 0. Its coefficient is the gap between that category and the baseline.

For a dummy, 'a one-unit increase' simply means switching from 0 to 1 — from the baseline group to the indicated group.

Dummy coding choice	Effect on the coefficient
Improved toilet = 1, none = 0	Gap relative to no improved toilet
Reversed coding	Same magnitude, opposite sign
Three toilet categories	Two dummies; two gaps from the baseline
Continuous "sanitation score"	Effect per unit of the score

Collapsing categories into one dummy assumes they are equivalent. A shared improved latrine and a private flush toilet coded identically asserts they have the same effect on child height.

If the categories plausibly differ, use separate dummies and let the data show it. Combining them is a modelling assumption that can be tested cheaply.

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Baseline Categories

Every dummy needs something to compare to

A categorical variable with k categories becomes k−1 dummies; the omitted one is the baseline. For religion with Hindu/Muslim/Christian/Other, if 'Hindu' is omitted, each coefficient is that group's gap relative to Hindu.

Never interpret a dummy without naming the baseline. 'β = −0.4 for urban' means nothing until you know it is relative to rural.

Categorical variable	Dummies needed	Baseline
Toilet: improved / not	1	Not improved
Religion: 4 categories	3	Whichever you omit
Wealth quintile: 5	4	Usually the poorest
All k categories included	Model will not estimate	Perfect collinearity

Including all k dummies plus an intercept is the dummy-variable trap — they sum to one, so the model cannot separate them and the software will drop one for you, often silently.

Choose the baseline deliberately. Every coefficient is a comparison with it, so an odd or tiny baseline group makes the whole table hard to read.

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Dummy Example

Reading a 0/1 coefficient

Illustrative result: β_{improved toilet} = 0.18 (baseline = no improved toilet).

Read it as: 'Children in households with an improved toilet have, on average, a 0.18 higher height-for-age z-score than otherwise-similar children without one.' The phrase 'otherwise-similar' is the controls doing their work.

Illustrative figure.

β(toilet) = 0.18	What it does and does not say
Children with improved toilets	0.18 higher HAZ on average
Compared with	Otherwise-similar children without one
Does it mean building toilets raises HAZ?	Not on its own
What would establish that?	Randomised or quasi-experimental variation

Toilet ownership is chosen, not assigned. Households that build improved toilets differ in ways no survey records — health knowledge, future orientation, local norms — and those differences also affect child height.

Sanitation is a well-studied case where the observational association substantially exceeds what trials of sanitation programmes have found. The gap is the confounding.

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Standardised

Unstandardised vs standardised coefficients

	Unstandardised (b)	Standardised (β)
Units	Original (rupees, years)	Standard deviations
Reads as	Effect of 1 real unit	Effect of a 1-SD change
Comparable across Xs?	No — different scales	Yes — common scale
Best for	Real-world meaning	Ranking relative importance

Standardised coefficients let you ask 'which predictor matters most?' — but lose the plain-language 'per rupee' meaning. Report unstandardised for interpretation, standardised for comparison.

Use	Unstandardised (b)	Standardised (β)
Reporting to a policymaker	Yes — real units	No
Comparing importance across predictors	No	Yes
Comparing across studies	Only if units match	More portable
Dummy variables	Natural	Awkward — an SD of a 0/1 has no meaning

Standardised coefficients on dummy variables are close to meaningless, since a standard deviation of a binary variable has no interpretation anyone can picture. Report those unstandardised.

Standardisation also makes coefficients depend on your sample’s variance, so two studies of the same effect in different populations produce different standardised values. Portability is partly an illusion.

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Sign & Size

Direction first, then magnitude

Sign: + means Y rises with X; − means Y falls as X rises
Size: how much Y moves per unit of X — in context
Always ask 'is this big?' — a 0.02 z-score gain may be real but trivial; a 0.4 gain may be programme-changing

A coefficient is not 'important' just because it is non-zero. Judge magnitude against what would matter for your decision.

Effect on HAZ	Interpretation
0.02 z-score	Real perhaps, but negligible in practice
0.18 z-score	Meaningful — comparable to real programme effects
0.40 z-score	Large; check for confounding before believing it
1.0 z-score	Implausible from a single covariate — investigate

Suspiciously large is as much a warning as suspiciously small. An implausible coefficient usually indicates confounding, a coding error or a collinearity artefact rather than a discovery.

Build a mental scale for your outcome. For HAZ, knowing what a typical intervention achieves lets you judge whether a coefficient is exciting or impossible — and that judgement is domain knowledge, not statistics.

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Section Four

Model Fit & Inference

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How Good Is the Fit?

R²: variance explained

R² (R-squared)

The share of the variation in the outcome that the model's predictors explain — ranging from 0 (explains nothing) to 1 (explains everything).

An R² of 0.34 means the predictors account for 34% of the variation in the outcome; the other 66% is residual — unmeasured causes and noise.

R² of	Means	Common in
0.05–0.15	Predictors explain a little	Individual behaviour
0.15–0.40	A substantial share	Household survey outcomes
Above 0.8	Excellent fit — or a problem	Physical systems; or a mediator in the model

A very high R² in social data usually means something is wrong. Most often a predictor is a near-restatement of the outcome, or a post-treatment variable has crept in.

R² measures prediction, not correctness. A model with an R² of 0.08 can carry an unbiased, policy-relevant coefficient; a model with 0.6 can be entirely confounded.

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See It

R² as a slice of total variation

Variation in the outcome: explained vs unexplained (illustrative)

Illustrative

In social data, R² values of 0.1–0.4 are common and not shameful — human behaviour is genuinely noisy. A high R² is not the goal; an honest, well-specified model is.

Field	Typical R²	Why
Individual behaviour	0.05–0.20	Behaviour is genuinely variable
Household outcomes	0.15–0.40	Structural factors explain more
Aggregated district data	Often higher	Averaging removes noise

Low R² in social data is normal, not shameful. Human outcomes have thousands of causes, and a model capturing a fifth of the variation may still contain a precisely estimated, policy-relevant coefficient.

Aggregation inflates R² without adding knowledge: a district-level regression looks better fitted than the household one underneath it and supports weaker conclusions.

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The Catch

R² only ever goes up

Add any predictor — even random noise — and R² will rise or stay flat; it can never fall. So a bigger R² does not prove the new variable belongs in the model.

Chasing R² by piling in predictors is a recipe for overfitting. You need a measure that penalises needless complexity.

Adding a predictor	R²	Adjusted R²
A genuinely useful one	Rises	Rises
A useless one	Rises slightly	Falls
Pure random noise	Rises slightly	Falls
Many predictors, small sample	Approaches 1	Falls sharply

R² can never fall when you add a variable, which makes it useless for model selection. With as many predictors as observations it reaches 1 exactly, fitting the noise perfectly.

Adjusted R² is the minimum correction and a weak one. For genuine model comparison, out-of-sample performance on held-out data is the honest test.

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The Fix

Adjusted R²

A version of R² that penalises each extra predictor. It rises only when a new variable improves the model by more than chance would predict — and can fall when you add a useless one.

Compare models on adjusted R², not raw R². If adjusted R² drops when you add a variable, that variable is earning its place by less than it costs.

Comparing models by	Sound?
Raw R²	No — it only rises
Adjusted R²	Better — penalises extra predictors
AIC / BIC	Yes, for non-nested models
Out-of-sample error	The strongest test
Which gives the result you wanted	No

Model comparison by fit statistics is prediction-oriented, and for causal work the control set should come from the causal story rather than from whichever specification fits best.

A falling adjusted R² when you add a variable is a signal it is not earning its place. It is not a reason to drop a genuine confounder, which belongs in the model whatever it does to fit.

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Is the Model Worth Anything?

The F-test

The F-test asks a whole-model question: do the predictors together explain more than nothing? Its null hypothesis is that all slope coefficients are zero at once.

A small F-test p-value says 'this model beats a model with no predictors'. It does not tell you which predictor matters — that is the t-tests' job.

Test	Null hypothesis	Answers
F-test	All slopes are zero	Does this model beat nothing?
t-test	This slope is zero	Does this predictor add anything?
Both together	—	A significant F with no significant t suggests collinearity

The third row is the diagnostic pattern worth memorising. A model that is jointly significant with no individually significant predictor is the classic signature of multicollinearity.

The F-test is a low bar in large samples. With NFHS-scale data almost any set of predictors will beat nothing, so passing it says very little about model quality.

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Is This Predictor Real?

t-tests on each coefficient

Each coefficient gets its own t-test: is this particular slope distinguishable from zero, given its uncertainty? The output is a t-statistic and a p-value per predictor.

Standard error

The estimated uncertainty in a coefficient. t = coefficient ÷ standard error; a large t (small p) means the slope is unlikely to be zero by chance.

Coefficient and its standard error	Reading
0.06 (0.02)	t ≈ 3 — clearly distinguishable from zero
0.06 (0.05)	t ≈ 1.2 — not distinguishable
0.06 (0.30)	Uninformative — the study could not tell
0.60 (0.02)	Precise and large — check for error

The standard error is the more informative of the two numbers and the one that vanishes when results are reported with asterisks. Always report it, or the interval derived from it.

Roughly, a t-statistic near 2 corresponds to the conventional 5% threshold. Being able to compute it mentally from a coefficient and its standard error is a useful reading habit.

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The Honest Range

Confidence intervals beat stars

A coefficient of 0.06 with a 95% confidence interval of 0.02 to 0.10 says the plausible effect lies in that band. The interval shows both direction and precision — far more than a lone p-value or a row of asterisks.

If a 95% interval comfortably excludes zero, the effect is 'significant' at the 5% level — but always read the width. A wide interval means you really do not know much.

Reported as	Reader learns
β = 0.06***	It differs from zero
β = 0.06 (se 0.02)	Precision — they can compute the rest
β = 0.06, CI [0.02, 0.10]	Direction, size and precision
β = 0.06, CI [−0.14, 0.26]	The study could not tell

The last two are indistinguishable if you only report stars. One is a precise small effect, the other is no information at all, and the asterisks look the same.

An interval containing zero is not proof of no effect. It means the data are compatible with zero — and often with a substantial effect in either direction.

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See It

Coefficients with confidence intervals

Illustrative coefficients on height-for-age, with 95% CIs

Illustrative — not reported NFHS estimates

Bars are the point estimates; the listed intervals (e.g. toilet: 0.07–0.29) are the 95% CIs. None crosses zero here, so each is 'significant' — but the toilet effect is the least precise.

Interval	Conclusion
[0.02, 0.10]	Positive effect, reasonably precise
[−0.01, 0.13]	Probably positive; cannot exclude zero
[−0.20, 0.32]	Uninformative in both directions

The last row is the case large samples generate constantly: an effect that is unambiguously real, unambiguously not zero, and far too small to justify any programme.

Judge the interval against a threshold that matters, not against zero. "Does this exclude an effect large enough to be worth funding?" is the question a decision-maker actually has.

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Significance ≠ Importance

Statistically real, practically tiny

With a large sample — and NFHS has hundreds of thousands of records — even a microscopic effect can be 'statistically significant'. Significance is about certainty, not size.

Always report the effect size and its units alongside the p-value. 'Significant' answers 'is it real?', never 'does it matter?'

Sample size	Effect of 0.01 z-score
n = 500	Not significant
n = 5,000	Possibly significant
n = 200,000 (NFHS scale)	Almost certainly significant

With NFHS-scale data, significance stops carrying information. Nearly everything is significant, so the useful question becomes how large the effect is and whether it is large enough to matter.

Report effect sizes in units a decision-maker feels — children moved out of stunting, percentage points, rupees — and let significance be a footnote rather than the headline.

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Section Five

Assumptions & Residual Diagnostics

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The Fine Print

OLS rests on assumptions

Least-squares estimates are trustworthy only if a few assumptions roughly hold. Violations do not always bias the coefficients, but they can wreck the standard errors — and so the p-values and intervals.

Most assumptions are checked by looking at the residuals — the model's leftovers. Plotting them is non-negotiable.

Concern	Assumption at stake
Is the coefficient biased?	Exogeneity — not on the standard list
Is the functional form right?	Linearity
Are the standard errors right?	Independence and homoscedasticity
Are the intervals right in a small sample?	Normality of residuals

The first row is the one diagnostics cannot check. Every residual plot can look perfect while the coefficient is confounded, because omitted confounders leave no signature in the residuals.

So diagnostics tell you whether the model is internally well specified, never whether it is causally identified. Those are separate questions and only the second needs a research design.

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The Four

What linear regression assumes

Assumption	Plain meaning	Check with
Linearity	The true relationship is a straight line	Residual vs fitted plot
Independence	Observations don't lean on each other	Study design; clustering
Homoscedasticity	Error spread is constant across X	Residual vs fitted plot
Normal residuals	Errors are roughly bell-shaped	Q–Q plot, histogram

Note what is not required: the predictors themselves need not be normal, and Y need not be normal — only the residuals.

Assumption fails	Biases coefficients?	Wrecks inference?
Linearity	Yes	Yes
Independence	No	Yes — standard errors too small
Homoscedasticity	No	Yes
Normal residuals	No	Only in small samples

Only linearity threatens the estimate itself. The other three affect uncertainty, which is why robust and clustered standard errors solve most of what diagnostics find.

Exogeneity is the assumption that is not on this list and matters most. Every assumption here can hold perfectly while the coefficient remains confounded.

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Linearity

Is a straight line the right shape?

If the real relationship curves — income's effect on nutrition flattening at high incomes — a straight-line model will systematically over- and under-predict in patterns.

Diagnosis: plot residuals against fitted values. A curve or a smile in that plot says the linearity assumption is failing — consider a transform or a squared term (Section 7).

Residual plot shows	Means
A random cloud around zero	Linearity is fine
A curve or U-shape	The relationship bends — add a squared or log term
A widening fan	Heteroscedasticity
Distinct clusters	A grouping variable is missing

The residual-versus-fitted plot is the single most informative diagnostic, and it takes one line of code. Two of the four assumptions are visible in it at once.

A curve in the residuals is a specification error, not noise. Ignoring it means the model systematically over-predicts in one range and under-predicts in another — which will show up as a spurious effect somewhere.

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Independence

When observations cluster

Children in the same village share water, clinics and shocks, so their outcomes are correlated. Treating them as fully independent makes standard errors look smaller than they are — falsely confident results.

Survey data like NFHS is clustered by design. Use clustered or survey-adjusted standard errors, or you will overstate significance.

Clustering in survey data	Consequence if ignored
Children within a household	Standard errors too small
Households within a village	Same, usually worse
Villages within a PSU	The level NFHS sampling actually uses
Repeated observations of a unit	Same

NFHS and DHS use stratified multi-stage cluster designs, and ignoring that inflates significance substantially. Use survey weights and the survey commands — svy: in Stata, survey in R.

Clustering does not bias the coefficients; it makes them look more precise than they are. A result that only survives with naive standard errors is not a result.

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Homoscedasticity

Constant error spread

Homoscedasticity

The residuals have roughly the same spread across all fitted values. Its opposite, heteroscedasticity, is a fan or funnel shape in the residual plot.

Income data is a classic offender: the rich vary far more in spending than the poor, so residuals widen as predicted spending rises. Coefficients stay unbiased, but the standard errors are wrong.

Heteroscedasticity	Response
Funnel widening with fitted values	Robust standard errors
Caused by skew in Y	Log transform
Caused by group differences	Cluster, or model the variance
Detected but ignored	Overconfident intervals

Robust standard errors are cheap and should be close to a default in applied work. They cost a little precision when errors happen to be homoscedastic and save you from overconfidence when they are not.

Income and expenditure are the classic offenders: variation among rich households is far larger than among poor ones, so any model with them on either side is likely heteroscedastic.

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See It

Good residuals vs a heteroscedastic fan

Residual vs fitted: even band (good) vs widening fan (bad)

Illustrative

The green band stays flat; the red points fan out as the fitted value grows. The fan is the warning sign — reach for robust standard errors or a log transform.

Residual pattern	Diagnosis	Fix
Even band around zero	Fine	Nothing
Widening fan	Heteroscedasticity	Robust errors, or log Y
Curve	Wrong functional form	Add a squared or log term

The fan is the most common pattern in development data, because income, expenditure and most monetary outcomes have variance that grows with their level.

Log-transforming Y often fixes the fan and the curve together, since both usually arise from the same right-skewed distribution. It also changes how every coefficient is read.

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Normal Residuals

The Q–Q plot

Inference (t-tests, CIs) assumes the residuals are roughly normal. Check with a Q–Q plot: if the residuals are normal, the points hug the diagonal line; fat tails or skew show as departures at the ends.

Good news: with a large sample, mild non-normality barely matters thanks to the central limit theorem. Worry most about it in small samples.

Q–Q plot shape	Means	Serious?
Points on the diagonal	Residuals roughly normal	No issue
S-shape	Skew	Mild in large samples
Ends peeling away	Fat tails — outliers	Worth investigating
Severe departure, small n	Inference unreliable	Yes

Normality matters least of the four assumptions. With a large sample the central limit theorem makes inference robust to mild non-normality, so a wobbly Q–Q plot on n = 50,000 is not a crisis.

The tails are still worth reading, because fat tails usually mean influential outliers. Those can move a coefficient substantially even when the overall inference is sound.

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The Habit

Plot first, conclude later

Residual vs fitted: checks linearity and equal spread at once
Q–Q plot: checks normality of residuals
Leverage / influence plot: finds the points bending the fit
Scale–location plot: a sharper look at spread

Anscombe's lesson applies here too: numbers alone hide trouble. Run the diagnostic plots on every model before you trust a single coefficient.

Plot	Reveals
Residual vs fitted	Linearity and equal spread
Q–Q	Normality of residuals
Scale–location	Spread, more sensitively
Leverage / Cook’s distance	Observations bending the fit

The leverage plot is the one that catches data errors. A single household with income entered in the wrong units can drive a coefficient, and no summary statistic reveals it.

Anscombe’s quartet is the standing lesson: four datasets with identical means, variances, correlations and regression lines look completely different when plotted. Fit statistics do not describe shape.

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Section Six

Multicollinearity

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The Problem

When predictors overlap

Multicollinearity

When two or more predictors are highly correlated with each other, so they carry overlapping information about the outcome.

Household income, monthly expenditure and asset ownership all measure roughly the same thing — living standards. Put all three in a model and the regression struggles to separate their individual effects.

Predictor pair	Overlap	Same construct?
Income and expenditure	Very high	Essentially yes
Income and assets	High	Related but distinct
Education and income	Moderate	No — keep both
Age and age-squared	Very high by construction	Centre, do not drop

High correlation between predictors is not itself a problem. It becomes one when it is so high that the model cannot attribute variation between them, and where the threshold lies depends on your sample size.

The construct question is what decides the remedy. Two measures of the same thing should be combined or one dropped; two distinct causes that happen to correlate should both stay.

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Why It Hurts

Unstable, imprecise coefficients

When predictors overlap, the model cannot tell which one deserves the credit. The result: coefficients with huge standard errors, wild swings if you add or drop a variable, and sometimes nonsensical signs.

Crucially, multicollinearity does not bias the coefficients — it makes them imprecise. The overall prediction can still be fine; the individual effects become untrustworthy.

Multicollinearity	Effect
Biases the coefficients	No
Inflates standard errors	Yes
Makes signs unstable	Yes
Affects prediction	Barely
Affects joint significance	No — the F-test stays strong

Multicollinearity is a precision problem, not a bias problem. The coefficients remain unbiased; the model simply cannot separate the overlapping predictors, so each estimate is noisy.

That is why it barely matters if you only want prediction. It matters enormously if you want to interpret an individual coefficient, which in policy work you usually do.

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The Symptoms

How to spot it

A high overall R² and significant F-test, but no individual predictor is significant
Coefficients flip sign or size when you add/remove a variable
Implausibly large standard errors on variables you expected to matter
Predictors you know are related (income & expenditure) are both in the model

These symptoms point you toward a formal diagnostic — the variance inflation factor.

Symptom	Alternative explanation
High R², no significant predictor	Collinearity — the classic signature
Signs flip when a variable is dropped	Collinearity, or confounding
Huge standard errors	Collinearity, or a small sample
Known-related predictors both included	Collinearity by construction

The second row is ambiguous and important. A sign flip can mean the model cannot separate two overlapping predictors, or that the dropped variable was a genuine confounder — and those need opposite responses.

Which it is depends on the causal structure, not the data. Draw the diagram before deciding whether a sign flip is a symptom to fix or a finding to report.

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The Diagnostic

The Variance Inflation Factor (VIF)

VIF

For each predictor, how much its coefficient's variance is inflated by correlation with the other predictors. VIF = 1 means no overlap; higher means more.

It is computed by regressing each predictor on all the others. The more predictable a variable is from the rest, the higher its VIF — and the shakier its coefficient.

VIF	Interpretation
1	No correlation with other predictors
2–4	Ordinary; not a concern
5–10	Watch it
Above 10	Severe overlap — investigate

VIF thresholds are conventions, not tests. A VIF of 12 on a control variable you do not interpret is harmless; a VIF of 6 on your variable of interest may already be a problem.

Interactions and squared terms produce mechanically high VIFs because they are built from the same variable. Centring before creating them removes most of that artefact.

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Rules of Thumb

How high is too high?

Illustrative VIFs — expenditure & income overlap badly

Illustrative

Common rule of thumb: VIF above 5–10 signals a problem. Here income and expenditure both blow past it — they are measuring the same underlying wealth.

VIF above 10 on	Response
A variable you interpret	Serious — combine, drop, or accept wide intervals
A control you do not interpret	Usually harmless
A squared or interaction term	Centre the base variable first

VIF is a per-predictor diagnostic, and a high value only matters for a coefficient you intend to read. Applying a blanket threshold across every variable in the model over-corrects.

Income and expenditure both blowing past it is the signature of two measures of one construct — pick one or build an index.

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The Remedies

What to do about it

Remedy	How	When
Drop one	Remove a redundant predictor	Two variables measure the same thing
Combine	Build one index (e.g. PCA)	Several proxies for one concept
Centre / rescale	Subtract the mean before squaring/interacting	Collinearity from interaction terms
Get more data	Larger / more varied sample	Overlap is mild, not structural
Accept it	Keep the model, widen the CIs	You only care about prediction

If two predictors are basically the same concept, do not agonise — keep one, or fold them into a single index. The wealth index in Section 9 is exactly this move.

Remedy	Use when	Cost
Drop one predictor	They measure the same thing	Confounding if it was distinct
Combine into an index	Several proxies for one concept	The index needs interpreting
Centre before interacting	Collinearity from a product term	None
Accept wider intervals	You need both coefficients	Less precision, honestly reported

The fourth remedy is the most under-used and often the right one. Collinearity does not bias anything, so reporting a wide interval is an honest description rather than a failure.

Combining is what PCA does — and why the wealth index exists. Twenty asset variables collapse into one component precisely because they overlap almost completely.

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A Subtlety

Don't 'fix' collinearity you need

If two predictors are genuinely distinct concepts that happen to correlate — education and income — dropping one can reintroduce confounding. The cure may be worse than the disease.

Decide based on your question. If you need the partial effect of education net of income, keep both and accept wider intervals rather than deleting a real confounder.

Two correlated predictors	Drop one?
Income and expenditure	Yes — nearly the same construct
Education and income	No — distinct, and each confounds
Age and age-squared	No — centre instead
Two asset indicators	Combine into an index

Dropping a genuine confounder to lower a VIF trades a precision problem for a bias problem, which is a bad exchange: wide intervals are visible and honest, while omitted-variable bias is invisible.

Let the question decide. For a partial effect of education net of income, you must keep income and accept the imprecision that comes with it.

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Recap

Multicollinearity in one breath

It is about predictors overlapping with each other, not with Y
It inflates standard errors; it does not bias coefficients
Diagnose with VIF (watch for > 5–10) and unstable signs
Remedy by dropping, combining, or simply accepting wider uncertainty

Multicollinearity	True or false
It biases coefficients	False
It inflates standard errors	True
It is about predictors overlapping with each other	True
It matters for prediction	Barely
Dropping a variable always fixes it	True — and may cause a worse problem

The last row is the trap. Dropping a collinear predictor always lowers the VIF, and if that predictor was a genuine confounder you have exchanged imprecision for bias.

Precision problems are visible in the interval; bias problems are invisible. Given a choice, take the one the reader can see.

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Section Seven

Interactions & Non-Linearity

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Beyond Straight Lines

When one slope isn't enough

The plain model assumes each predictor's effect is the same for everyone and constant at every level. Reality is richer: effects can depend on other variables, or change as a variable grows.

Interaction

X's effect depends on another variable (e.g. schooling helps more where toilets exist).

Non-linearity

X's effect changes with its own level (e.g. income matters more at the bottom).

The plain model assumes	Reality
The same effect for everyone	Effects often differ by group
A constant effect at every level of X	Returns usually diminish
Effects that simply add up	They can depend on each other

All three assumptions are testable inside the same framework — interactions for the first, transformations for the second, and both together for the third. None requires abandoning linear regression.

Testing them costs statistical power, which is why they are often assumed rather than checked. Say which you assumed rather than letting the reader believe you tested it.

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Interactions

An effect that depends on context

Interaction term

A predictor formed by multiplying two variables (X1 × X2). Its coefficient captures how the effect of one variable changes across levels of the other.

Model: HAZ = … + β₁(schooling) + β₂(toilet) + β₃(schooling × toilet). β₃ tells you whether schooling's payoff differs for households with and without a toilet.

Model	Says
β₁(schooling) + β₂(toilet)	Schooling has the same effect everywhere
… + β₃(schooling × toilet)	Schooling’s effect differs by sanitation
β₃ positive	Schooling helps more where sanitation exists
β₃ negative	Schooling compensates where sanitation is absent

The last two rows are substantively opposite policy conclusions, and both are plausible stories. The interaction term is what lets the data distinguish them.

Interactions need statistical power. Detecting a difference between two slopes takes considerably more data than detecting a single slope, so null interactions are often uninformative.

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See It

Two slopes from one interaction

Schooling's effect on HAZ, by toilet access (illustrative)

Illustrative

The steeper green line means schooling helps more where sanitation is in place — an interaction. Two different slopes, one model. Without the interaction term you would force them to be parallel.

Interaction result	Policy reading
Schooling helps more with sanitation	Sequence sanitation first, or bundle them
Schooling helps more without it	Schooling substitutes where infrastructure is absent

A null interaction in a modest sample is usually uninformative rather than evidence of a uniform effect — interactions need far more data than main effects.

Show interactions as a plot of predicted values. Two fitted lines communicate the finding; the product term does not.

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Reading Interactions

The main effect changes meaning

Once you include X1×X2, the coefficient on X1 alone is no longer 'the effect of X1'. It is the effect of X1 when X2 = 0. The full effect of X1 is β₁ + β₃·X2.

This trips people up constantly. In a model with interactions, never read a main-effect coefficient in isolation — always say 'at what level of the other variable?'

In a model with X1 × X2	The coefficient on X1 is
Without the interaction	The average effect of X1
With the interaction	The effect of X1 when X2 = 0
If X2 is never 0	A meaningless extrapolation
After centring X2	The effect of X1 at the mean of X2

Centring makes the main effect interpretable again, which is why it is standard practice with interactions — and why an uncentred interaction model can report a main effect for a value nobody has.

Never read a main effect in an interaction model as "the effect of X1". The full effect is β₁ + β₃X2, so it differs at every value of X2 and is best shown as a plot rather than a number.

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Non-Linearity

Quadratic and polynomial terms

Add a squared term — β₁X + β₂X² — to bend the line into a curve. This captures diminishing returns (income's effect flattening) or U-shapes (age and earnings).

Centre X before squaring to tame the collinearity between X and X², and resist going past a quadratic — high-order polynomials wiggle wildly and overfit.

Shape	Specification	Example
Diminishing returns	Add X² with a negative coefficient	Income and nutrition
U-shape	X² positive	Age and earnings, inverted
Proportional change	Log X	Income effects generally
Threshold	A dummy for above/below	Poverty line effects

Centre X before squaring it. Otherwise X and X² are almost perfectly correlated, the VIFs explode, and the individual coefficients become impossible to interpret.

Do not read the coefficient on X alone in a quadratic model. The marginal effect is β₁ + 2β₂X, which changes at every value — plot it rather than reporting a number.

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Log Transforms

Taming skew, reading elasticities

Logging a right-skewed variable like income compresses its long tail and often straightens a curved relationship. It also changes how you read the coefficient — in percentage terms.

Elasticity

In a log–log model, the coefficient is an elasticity: the percentage change in Y for a 1% change in X. A coefficient of 0.4 means a 1% rise in X is linked to a 0.4% rise in Y.

Why log income	Effect
Compresses the right tail	Reduces outlier influence
Often straightens the relationship	Fixes a curve without adding terms
Stabilises variance	Frequently cures the residual fan
Changes interpretation	Coefficients become proportional

Logging often fixes three problems at once — skew, curvature and heteroscedasticity — because all three come from the same underlying multiplicative structure in income data.

It also drops zero and negative values, which matters in poor samples where some households report zero income or expenditure. Decide explicitly how those are handled rather than letting them vanish.

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Log Cheat-Sheet

Four ways logs change the reading

Model	Coefficient β reads as…
Y on X (level–level)	β-unit change in Y per 1-unit X
Y on log X (level–log)	β/100 change in Y per 1% rise in X
log Y on X (log–level)	~100·β % change in Y per 1-unit X
log Y on log X (log–log)	β % change in Y per 1% change in X (elasticity)

Logs are the practitioner's friend for money variables: they fix skew, ease heteroscedasticity, and give intuitive percentage interpretations.

Model	β reads as	Use for
Y on X	β units of Y per 1 unit of X	Most variables
Y on log X	β/100 change in Y per 1% rise in X	Skewed predictors
log Y on X	About 100β% change in Y per unit X	Wages, expenditure
log Y on log X	Elasticity: β% per 1%	Demand, output

The log-Y approximation degrades above about 0.1. For β = 0.5 the true change is roughly 65%, not 50% — use exp(β) − 1 whenever the coefficient is large.

Logs drop zeros, which matters when the variable is income or expenditure in a poor sample. Inverse hyperbolic sine is the usual substitute and carries its own interpretation difficulties.

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Section Eight

Logistic Regression

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A Different Outcome

When Y is yes/no

Many development outcomes are binary: a child is fully immunised or not; a woman delivered in a facility or not; a household is below the poverty line or not. Linear regression is the wrong tool for these.

Fit a straight line to a 0/1 outcome and it will happily predict probabilities below 0 or above 1 — nonsense. We need a model that stays inside 0–1.

Linear model on a 0/1 outcome	Problem
Predicted probabilities	Can fall below 0 or above 1
Error structure	Necessarily heteroscedastic
Functional form	Assumes a constant effect at every level
Still used as an LPM?	Yes — with robust errors, for interpretability

The linear probability model has real defenders. Its coefficients are marginal effects on the probability directly, which is much easier to communicate than an odds ratio.

The practical compromise many applied economists use: report the LPM for interpretation, run logistic as a robustness check, and note that the two agree.

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The Solution

Model the probability with an S-curve

Logistic regression models the probability of 'yes' using an S-shaped (logistic) curve that flattens near 0 and 1, so predictions always stay in the valid range.

The trick: instead of modelling the probability directly, it models the log-odds of the outcome as a linear function of the predictors. That keeps the familiar 'linear in the Xs' structure.

Logistic regression	Because
Models the log-odds linearly	Log-odds are unbounded, so a line is safe
Produces an S-curve in probability	The transformation bends it
Never predicts outside 0–1	By construction
Is fitted by maximum likelihood	No closed-form solution

The S-curve is a consequence of the link function, not an assumption about behaviour. Modelling log-odds linearly is what forces predicted probabilities to flatten near 0 and 1.

Maximum likelihood is iterative, so it can fail to converge — usually because of perfect separation, where some predictor perfectly predicts the outcome. That is a data structure problem, not a computing one.

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Odds & Log-Odds

From probability to log-odds

Quantity	Definition	Range
Probability p	Chance of 'yes'	0 to 1
Odds	p ÷ (1 − p)	0 to ∞
Log-odds (logit)	ln(odds)	−∞ to +∞

A probability of 0.8 is odds of 4 (4-to-1 on) and log-odds of about 1.39. Logistic regression's coefficients live on this log-odds scale — which is why they are not directly readable.

Probability	Odds	Log-odds
0.5	1	0
0.8	4	About 1.39
0.9	9	About 2.20
0.95	19	About 2.94

Odds grow explosively as probability approaches 1, which is the source of most misreading: a doubling of the odds means very different things at p = 0.1 and at p = 0.8.

Log-odds are unbounded in both directions, which is exactly why the model works on them — a linear function of predictors can range freely without ever producing an impossible probability.

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The Key Move

Coefficients are in log-odds — exponentiate them

A logistic coefficient β is the change in log-odds per one-unit increase in X. That is hard to feel. So we take exp(β) to get an odds ratio — a multiplicative effect on the odds.

This is the most-mangled idea in applied statistics: the raw coefficient is log-odds; exp(coefficient) is the odds ratio. Never read the raw logit coefficient as a probability.

Quantity	Scale	Report to
β	Log-odds	Nobody — uninterpretable directly
exp(β)	Odds ratio	Technical audiences
Marginal effect	Probability change	Everyone else
Predicted probabilities	Probability	The clearest for policy

Predicted probabilities at meaningful covariate values are the clearest output logistic regression produces, and they are the least often reported. "62% versus 41%" needs no explanation.

Average marginal effects are the usual compromise: one number per predictor, expressed as a change in probability, available in every statistical package.

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Reading Odds Ratios

Above 1, below 1, equal to 1

OR > 1: the predictor raises the odds of 'yes' (OR 1.5 → odds 50% higher)
OR < 1: the predictor lowers the odds (OR 0.7 → odds 30% lower)
OR = 1: no association — this is the null value

Because odds ratios are multiplicative, the null is 1, not 0. A 95% CI for an OR that includes 1 is 'not significant'.

Odds ratio	Means	Null value
1.5	Odds 50% higher	—
0.7	Odds 30% lower	—
1.0	No association	This is the null, not zero
CI [0.9, 2.1]	Not significant — it contains 1	—

For odds ratios the null value is 1, not 0. A confidence interval is significant when it excludes 1, and reading it against zero is a common and consequential error.

Odds ratios are multiplicative, so their intervals are asymmetric on the natural scale. Plot them on a log axis or the visual comparison misleads.

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See It

Odds ratios for facility delivery

Illustrative odds ratios for facility delivery (vs home)

Illustrative — patterned on NFHS-style predictors

Read across the line at OR = 1: urban residence and wealth raise the odds of a facility delivery; high birth order (OR 0.62) lowers them by about 38%. All illustrative.

Predictor	Illustrative OR	Reading
Urban residence	Above 1	Raises the odds of facility delivery
Higher wealth quintile	Above 1	Raises the odds
High birth order	0.62	Lowers the odds by 38%

Read across the line at OR = 1, not at zero. Anything above 1 raises the odds and anything below lowers them, and an interval spanning 1 is the non-significant case.

Plot odds ratios on a log scale. On a linear axis, an OR of 2 and an OR of 0.5 — equal and opposite effects — look wildly different in magnitude.

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Careful Reading

Odds ratios are not risk ratios

An odds ratio of 2 does not mean the probability doubles. Odds and probability diverge sharply when outcomes are common. For a rare outcome the two are close; for a common one they are not.

Say 'the odds are 85% higher', not '85% more likely'. If your audience needs probabilities, report predicted probabilities at chosen covariate values instead.

Baseline probability	OR of 2 gives	Risk ratio
0.05 (rare)	About 0.095	About 1.9 — close to the OR
0.20	About 0.33	About 1.7
0.50 (common)	0.67	1.33 — far from the OR

The rarer the outcome, the closer the odds ratio is to a risk ratio. For common outcomes — facility delivery, immunisation — they diverge sharply and the OR substantially overstates the change in probability.

Say "the odds are 85% higher", never "85% more likely". The second phrasing is a risk-ratio claim and is usually wrong for the common outcomes development work deals with.

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Same Discipline

Everything else carries over

Coefficients are still partial effects — net of the other predictors
Each odds ratio still comes with a confidence interval — report it
Multicollinearity, interactions and dummies all behave as before
Fit is judged differently (pseudo-R², classification, AUC), not by OLS R²

Logistic regression is the same way of thinking on a new scale — learn the log-odds/odds-ratio translation and you are most of the way there.

Carries over from linear regression	Differs
Coefficients are partial effects	They are on the log-odds scale
Confidence intervals matter	Null value is 1, not 0
Interactions and dummies work the same	Interactions are harder to interpret
Multicollinearity behaves the same	Fit uses pseudo-R², not R²

Pseudo-R² measures are not shares of explained variance and are not comparable across models the way R² is. McFadden values around 0.2–0.4 already indicate a good fit.

Marginal effects at the mean, or average marginal effects, convert logistic output back into probability changes. Most software reports them and they are far easier for a non-technical reader.

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Section Nine

Data Reduction — PCA & Factor Analysis

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The Problem

Too many indicators, one concept

You may have twenty asset and housing variables — fridge, TV, motorcycle, pucca walls, electricity, toilet type — all proxying one idea: household living standards. Putting all twenty in a regression is messy and collinear.

Data reduction compresses many correlated indicators into a few summary scores that capture most of their shared information.

Twenty asset variables	Options
All twenty in the regression	Collinear, uninterpretable, degrees of freedom lost
Pick one or two	Arbitrary; discards information
Count them	Treats a fridge and a fan as equivalent
PCA into one index	The standard approach

A simple count is more defensible than it looks and is sometimes used deliberately, because it is transparent and does not depend on the sample used to construct the weights.

PCA weights are estimated from your data, so an index built on one survey is not directly comparable to one built on another. That is a real limitation of the standard approach.

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PCA

Principal Component Analysis

A technique that re-expresses many correlated variables as a smaller set of uncorrelated 'components', ordered so the first captures the most variance, the second the next-most, and so on.

The first principal component is the single weighted combination of the variables that explains the largest share of their joint variation — usually the 'size' or 'level' they share.

PCA does	It does not
Re-express correlated variables as components	Explain anything causally
Order components by variance captured	Name them for you
Produce uncorrelated components	Remove information about the outcome
Reduce dimensions	Handle categorical variables gracefully

PCA is blind to your outcome. It finds the directions of greatest variance among the predictors, which may or may not be the directions that predict Y — a real limitation when the goal is regression.

Standardise before running it. PCA on unstandardised variables lets whichever variable has the largest units dominate the first component entirely.

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How Much to Keep

The scree plot

Variance explained by each component (illustrative scree)

Illustrative

The first component dwarfs the rest — the classic signal that one dimension (wealth) underlies the asset variables. Keep components up to the 'elbow' where the bars level off.

Scree plot shows	Means
One dominant component	A single dimension underlies the variables
A gradual decline	No clear structure — reduction may not help
A clear elbow	Keep components up to it

The elbow rule is a convention with no statistical basis, and different readers see the elbow in different places. Report the variance shares and let the reader judge.

For an asset index, a dominant first component is the result you want — it is the evidence that the variables really are proxies for one underlying dimension.

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The Canonical Use

The DHS / NFHS wealth index

The standard DHS/NFHS wealth index is built with PCA. Asset, housing and utility variables go in; the first principal component becomes each household's wealth score — the textbook example of PCA in development.

Households are then ranked and split into five wealth quintiles. This is the 'wealth quintile' you see throughout NFHS tables — a PCA score in disguise.

DHS / NFHS wealth index	Detail
Inputs	Assets, housing materials, utilities, water and sanitation
Method	First principal component of those variables
Output	A continuous score per household
Reported as	Quintiles — five equal fifths of households
Known limitation	Urban-skewed asset baskets

The index measures relative standing within the sample, not absolute welfare. The poorest quintile in one country is not comparable to the poorest quintile in another, and neither is a poverty line.

Because the asset basket favours urban goods, rural households cluster in the lower quintiles partly by construction. DHS produces separate urban and rural components partly to address this.

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See It

The wealth-index distribution

Distribution of a PCA wealth score, split into quintiles (illustrative)

Illustrative

By construction the quintiles each hold ~20% of households — the index is a continuous score cut into five equal bands. The underlying score itself is continuous and right-skewed in level.

Wealth quintile	What it means	What it does not
Poorest 20%	Bottom fifth of this sample	Below a poverty line

Quintiles are relative by construction: 20% of households are always in the poorest one, even where absolute poverty has fallen sharply.

Cutting a continuous score into bands also discards information and creates artificial cliffs. Use the underlying score where it is available.

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PCA vs Factor Analysis

Cousins, not twins

	PCA	Factor analysis
Goal	Summarise / compress variance	Find latent underlying factors
Direction	Variables → components	Factors → cause the variables
Model of error	None — pure re-expression	Separates shared vs unique variance
Typical use	Indices (wealth index)	Psychometrics, attitude scales

In practice they often give similar results for index-building. PCA is the default for wealth indices; factor analysis suits questionnaire scales where a true latent trait is assumed.

	PCA	Factor analysis
Goal	Compress variance	Find latent causes
Direction of the model	Variables → components	Factors → variables
Error	None — pure re-expression	Separates shared from unique variance
Typical use	Wealth indices, dimension reduction	Attitude and psychometric scales

They answer different questions and are routinely conflated. PCA summarises what you measured; factor analysis posits something unmeasured that produced what you measured.

If the construct is theoretical — empowerment, wellbeing, an attitude — factor analysis is the appropriate framework. For a wealth index built from observable assets, PCA is the right choice.

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Watch-Outs

Reduction has costs

Components can be hard to name — what does PC2 mean?
PCA is scale-sensitive — standardise variables (or use a suitable method) first
An asset index reflects what is in the basket — urban-skewed assets bias it
Compressing always discards some information by design

Used well, data reduction tames multicollinearity and yields one clean predictor. Used blindly, it buries the very structure you wanted to study.

Watch-out	Practical response
Components hard to name	Interpret only the first, or use factor analysis
Scale sensitivity	Standardise before running PCA
Urban-skewed asset basket	Consider separate urban/rural components
Information discarded	Report how much variance the retained components hold

The first principal component is usually interpretable and the rest usually are not. PC2 in an asset index often separates urban from rural goods rather than measuring any coherent concept.

Report the share of variance the first component captures. If it is 30%, calling it "wealth" is a stretch; if it is 60%, the single-dimension story is defensible.

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Recap

Data reduction in one breath

Many correlated indicators → a few uncorrelated summary scores
PCA's first component = the dominant shared dimension (often 'level' or 'wealth')
The NFHS/DHS wealth index is the first principal component of assets
Quintiles are that continuous score cut into five equal groups

Statement	True?
PCA finds latent causes	No — that is factor analysis
The first component is often interpretable	Yes
The wealth index is the first component of assets	Yes
Quintiles are the score cut into fifths	Yes
Reduction is lossless	No

The first row is the most common misstatement about PCA, and it matters because it implies the components measure something real underlying the data rather than summarising the data itself.

Always report how much variance you kept. "The first component explains 48% of the variance" tells a reader exactly how much was discarded to get one number.

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Section Ten

Pitfalls

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Pitfall 1

Overfitting

Throw enough predictors at a small sample and the model fits the noise, not the signal. It looks brilliant in your data and fails on the next dataset — it memorised rather than learned.

Symptoms: many predictors relative to observations, dazzling in-sample R², coefficients that change wildly across samples. Cure: simpler models, adjusted R², out-of-sample testing.

Overfitting signal	Guard
Many predictors relative to observations	Rule of thumb: at least 10–20 obs per predictor
R² far above the field norm	Compare adjusted R²
Coefficients change wildly across samples	Cross-validate
Predictors chosen by significance	Pre-specify the model

Stepwise selection is the most common route to overfitting and still appears in software menus. Choosing predictors by their p-values invalidates every p-value the final model reports.

Hold out a random 20% of the data before modelling. If performance collapses there, the model memorised rather than learned — and no in-sample statistic would have told you.

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Pitfall 2

Omitted-variable bias

Leave out a real confounder and its effect contaminates the variables you did include. This is the mirror image of controlling: omit income and the diet coefficient absorbs income's effect.

The bias's direction depends on how the omitted variable relates to both X and Y. Unlike multicollinearity, this one genuinely biases coefficients — it is the more dangerous problem.

Omitted variable Z	Bias on the coefficient of X
Z raises X and raises Y	Upward — too big
Z lowers X and lowers Y	Upward
Z raises X and lowers Y	Downward
Z unrelated to X	None — not a confounder

You can sign the bias without measuring Z. If income raises both diet diversity and nutrition, omitting it biases the diet coefficient upward — so the true effect is smaller than reported.

A signed bound is often enough for a decision. If the upward-biased estimate is already too small to justify a programme, the direction of the bias settles the question.

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A False Trade-Off

Multicollinearity vs causality

These two pitfalls pull in opposite directions. Dropping a collinear predictor cures the imprecision — but if that predictor was a real confounder, dropping it creates omitted-variable bias.

The resolution is your question. For an unbiased causal estimate, keep the confounder and tolerate wide intervals. For pure prediction, collinearity may not matter at all.

Situation	Keep the collinear predictor?
It is a genuine confounder	Yes — accept the imprecision
It duplicates another variable	No — drop or combine
It is a mediator	No — keeping it blocks the effect
You only want prediction	Either — collinearity barely matters

The two pitfalls pull in opposite directions and the causal diagram resolves them. Whether a variable should stay depends on its role in the causal story, not on its VIF.

Draw the diagram before fitting anything. Ten minutes with a pen settles most control-set decisions that would otherwise be made by looking at which specification gives a nicer result.

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Pitfall 3

Controlling for a mediator

Mediator

A variable on the causal path between X and Y — X causes the mediator, which in turn causes Y. It transmits the effect rather than confounding it.

Education raises income, and income improves child nutrition. Income is a mediator of education's effect — not a confounder to be controlled away.

Variable	Confounder or mediator?	Test
Income, for education → nutrition	Mediator	Education comes first and causes income
Income, for diet → nutrition	Confounder	Income causes both
Sanitation, for income → nutrition	Could be either	Depends on the causal story

The same variable is a confounder in one model and a mediator in another. Its role depends on which effect you are estimating, not on the variable, which is why the diagram has to be drawn per question.

Ask which comes first in time and which causes which. If your predictor causes the candidate control, it is a mediator and controlling for it removes part of the effect you are measuring.

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Why It's a Trap

'Controlling for' can bias the answer

If you 'control for' income while estimating education's effect on nutrition, you block the very pathway through which education works — and understate its total effect. More controls is not always better.

Rule: control for confounders (common causes), never for mediators (intermediate effects) or things caused by the outcome. Draw the causal story before choosing controls.

Variable role	Control for it?	Effect if you do
Confounder	Yes	Removes bias
Mediator	No	Blocks the pathway; understates the effect
Collider	No	Creates bias where there was none
Predictor of Y only	Optional	Improves precision

Two of the four make things worse, which is why "control for everything available" is not a strategy. The role of the variable in the causal structure decides, and the data cannot tell you the role.

The mediator error is easy to make with survey data assembled after the fact. If education raises income and income improves nutrition, controlling for income removes most of education’s effect.

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Pitfall 4

Extrapolation

A model fitted on households earning ₹5,000–₹40,000 a month says nothing reliable about a household earning ₹5 lakh. Predicting outside the range of your data assumes the line keeps going — it rarely does.

Stay within the support of your data. The neat straight line is an artefact of the range you observed, not a law of nature beyond it.

Prediction	Data range	Reliable?
Household at ₹20,000/month	Inside the observed range	Yes
Household at ₹45,000	Just outside	Cautiously
Household at ₹5 lakh	Far outside	No
A district not in the sample	Different population	Depends on similarity

Extrapolation assumes the line continues, and relationships in social data almost always bend at the extremes. The straight line is an approximation over the range you observed and nowhere else.

Report the range of your predictors alongside the results. It tells the reader where the model applies, and it is the information most often missing from a regression table.

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Pitfall 5

p-hacking & the forking paths

Try enough specifications — add this control, drop that one, split by region, redefine the outcome — and something will cross p < 0.05 by chance. Reporting only the winning run manufactures false findings.

Defend against it: pre-specify your model, report what you tried, show robustness across specifications, and trust replication over a single surprising result.

Forking path	Guard
Many specifications, one reported	Pre-specify the main model
Subgroups explored after seeing results	Declare subgroups in advance
Outcome redefined mid-analysis	Fix the outcome before analysis
Controls chosen for the result	Choose from the causal diagram

This does not require bad faith. A researcher making each choice reasonably, informed by what the data showed, produces the same inflation of false findings as deliberate fishing.

Report the specification curve if you ran many models — show all the estimates, not the one you liked. It is more work and far more convincing than a single result.

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Pitfalls Recap

Six traps, one habit

Pitfall	What goes wrong	Guard
Overfitting	Fits noise, fails out of sample	Simplicity, adjusted R², hold-out
Omitted-variable bias	Confounder contaminates coefficients	Include real confounders
Collinearity confusion	Drop a confounder to 'fix' VIF	Let the question decide
Controlling a mediator	Blocks the causal pathway	Don't control intermediates
Extrapolation	Predicts beyond the data	Stay within range
p-hacking	Chance result sold as finding	Pre-specify, replicate

Pitfall	Guard
Overfitting	Simplicity; adjusted R²; a hold-out sample
Omitted-variable bias	Include real confounders
Collinearity confusion	Do not drop a confounder to lower a VIF
Controlling for a mediator	Draw the causal story first
Extrapolation	Report the range of your data
p-hacking	Pre-specify; show all specifications

Four of the six are prevented by one habit: deciding the model from the causal diagram before opening the data, rather than from what the output looks like afterwards.

The other two — overfitting and extrapolation — are prevented by honesty about what the sample contains. Neither requires any additional technique.

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Section Eleven

Reporting & Tools

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A Workflow

How to build a model, in order

ASK: a precise question & outcome

→

DRAW: the causal story — confounders vs mediators

→

FIT: choose linear or logistic; add controls

→

CHECK: residuals, VIF, fit

→

REPORT: effects, CIs, caveats

Notice the thinking happens before the software runs. Drawing the causal story first is what tells you which variables to control — and which to leave alone.

Step	What goes wrong if skipped
Ask a precise question	You fit a model with no interpretation
Draw the causal story	Mediators and colliders enter as controls
Choose linear or logistic	A 0/1 outcome fitted with a line
Check residuals and VIF	Overconfident intervals; unstable coefficients
Report effects with intervals	A table of stars nobody can act on

The first two steps involve no software at all, and they determine whether anything that follows means anything. Most modelling errors are made before the data is opened.

Write down the model you intend to fit before fitting it. Every deviation afterwards is then a visible decision rather than an invisible one.

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The Regression Table

What a results table must show

Predictor	Coefficient	95% CI	p
Income (per ₹1,000)	0.03	0.01 – 0.05	<0.01
Mother's schooling (yr)	0.06	0.02 – 0.10	<0.01
Improved toilet (vs none)	0.18	0.07 – 0.29	<0.01
Intercept	−1.42	—	—

Always report coefficients with confidence intervals, the sample size, R²/adjusted R², and the units. Illustrative figures shown.

A results table must show	Because
Coefficients with units	A number without units cannot be judged
Confidence intervals	Precision and direction together
Sample size	Determines what significance means
Which controls are included	The coefficient is conditional on them
Weighting and clustering	Survey data requires both

The last row is routinely omitted in applied reports using NFHS data, and without it a reader cannot tell whether the standard errors are meaningful.

State the reference category for every dummy. A coefficient of 0.18 on "improved toilet" is uninterpretable unless the reader knows what it is being compared with.

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Good Reporting

Habits that earn trust

State your question and model before the numbers
Give effect sizes with units, not just stars
Show confidence intervals; note the sample size and any weights
Report the diagnostics you ran and what you found
Be explicit about what is association vs causal claim

Habit	What it signals to a reader
Question and model stated first	The result was not chosen afterwards
Effect sizes with units	You know what the number means
Intervals, n, weights reported	The uncertainty is honestly described
Diagnostics reported	You looked, and said what you found
"Associated with", not "causes"	You know what the design supports

Reporting a diagnostic that failed, and what you did about it, builds more credibility than reporting none. A clean results section with no mention of checks reads as checks not performed.

State the limitations before a reader finds them. It is the cheapest form of credibility available and costs one paragraph.

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The Toolbox

R, Python and Stata

Tool	Strengths	Note
R	Stats-first, superb diagnostics & graphics	Free; lm(), glm(), broom
Python (statsmodels)	Cleaning + modelling in one place	Free; pandas, scikit-learn
Stata	Survey data, clustered SEs, ubiquitous in econ	Paid; svy: prefix
SPSS	Menu-driven, common in academia	Paid; gentle on-ramp

For NFHS/PLFS work, whatever you choose must handle survey weights and clustered standard errors — R's survey package, Python's statsmodels, or Stata's svy commands.

Tool	Strength	Note
R	Diagnostics and graphics; free	`lm()`, `glm()`, `survey`, `broom`
Python	Cleaning and modelling in one place	statsmodels, pandas
Stata	Survey data, clustered errors; ubiquitous in economics	Paid; `svy:` prefix
SPSS	Menu-driven	Harder to reproduce

For NFHS or DHS work, use tools with real survey support. Stata’s svy: and R’s survey package handle weights, strata and clusters correctly; naive commands do not.

Keep the analysis as a script whatever you choose. A menu-driven sequence cannot be re-run, checked or corrected, and every claim in this deck depends on being able to show your working.

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Text	Best for
Wooldridge, Introductory Econometrics	The standard undergraduate reference
Angrist & Pischke, Mostly Harmless	Causal designs beyond regression
James et al., Introduction to Statistical Learning	Prediction, overfitting, cross-validation — free PDF
Gelman, Hill & Vehtari, Regression and Other Stories	Modelling judgement and interpretation

Takeaway	The error it prevents
A coefficient is a partial effect	Reading it as a total effect
Adjusted R² and intervals, not stars	Mistaking significance for importance
Plot the residuals	Missing a curve or a fan you could have fixed
Logistic coefficients are log-odds	Reporting an odds ratio as a probability change
Control for confounders, not mediators	Blocking the pathway you were measuring