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Stratified Analysis and Adjustment for Confounding

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Confounding: Definition, Identification, and Causal Criteria Effect Modification and Statistical Interaction→→Inverse Probability Weighting Multivariable Regression in Epidemiology

Core Idea

Stratified analysis divides data by levels of a confounder to examine the exposure-outcome association within homogeneous strata, then combines stratum-specific estimates (e.g., via Mantel-Haenszel). This approach allows simultaneous control of confounding and detection of effect modification. Stratification is transparent and interpretable but becomes cumbersome with multiple confounders.

Explainer

You already understand confounding: a variable associated with both the exposure and the outcome distorts the apparent exposure–outcome relationship. The intuitive fix is simple — look at the exposure–outcome association *separately* within groups that are homogeneous with respect to the confounder. Within each stratum, the confounder cannot confound because it does not vary. If the crude (unstratified) association differs from the stratum-specific associations, confounding was present and the stratum-specific estimates are the valid ones.

To make this concrete: suppose you observe a crude odds ratio of 2.0 for the association between coffee drinking and myocardial infarction (MI). But coffee drinkers also smoke more than non-drinkers. When you stratify by smoking status, you find an odds ratio of 1.0 in smokers and 1.0 in non-smokers. The stratum-specific estimates are homogeneous (both null) and differ from the crude estimate — classic positive confounding. Smoking was making coffee look harmful because it was associated with both the exposure (coffee) and the outcome (MI). Within strata of smoking status, that distortion disappears.

If the stratum-specific estimates are similar across strata, you can combine them into a single summary estimate that is adjusted for the stratifying variable. The Mantel-Haenszel (MH) estimator is the standard approach for 2×2 tables: it computes a weighted average of stratum-specific odds ratios (or risk ratios), where the weights reflect the amount of information in each stratum. The MH estimator is computationally simple, statistically efficient when homogeneity holds, and interpretable as the confounder-adjusted association. You can also use the Woolf method (variance-based weighting) as an alternative.

The same analysis that controls for confounding also detects effect modification: if stratum-specific estimates *differ* substantially across strata, the variable you stratified on is not merely a confounder — it is a modifier of the exposure effect. For example, if the coffee–MI odds ratio is 2.0 among smokers but 0.8 among non-smokers, the effect is genuinely different in the two groups. In this case, combining the estimates into a single adjusted summary would be misleading — the correct report is stratum-specific. The decision to report separate estimates versus a combined estimate hinges on whether heterogeneity is present. This is why stratified analysis is not merely a confounding-control tool but also the foundational method for identifying who benefits or is harmed differently — a question central to clinical and public health decision-making.

Stratification's limitation is the sparse data problem: each additional stratifying variable divides the dataset into more cells, many of which may have too few observations to produce stable estimates. Two binary confounders require four strata; three binary confounders require eight. This is why stratification gives way to multivariable regression — logistic, Poisson, Cox — when there are several confounders to control simultaneously. Regression can be understood as a generalization of stratification that handles sparse data through model-based smoothing, trading the transparency of stratification for the capacity to adjust for many variables at once.

Practice Questions 5 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → One-to-One Correspondence → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Making 10 as an Addition Strategy → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Dividing Integers → Unit Rates → Proportions → Percent Concept → Converting Between Fractions, Decimals, and Percents → Operations with Rational Numbers → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Sine, Cosine, and Tangent Ratios → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Iterated Integrals and Fubini's Theorem → Double Integrals in Cartesian Coordinates → Double Integrals in Polar Coordinates → Double Integrals in Polar Coordinates → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Triple Integrals in Cartesian Coordinates → Triple Integrals in Cylindrical and Spherical Coordinates → Change of Variables and the Jacobian Determinant → Applications of Triple Integrals: Volume and Mass → Vector Fields and Their Representations → Line Integrals of Vector Fields → Work and Circulation → Line Integrals of Scalar and Vector Functions → Fundamental Theorem for Line Integrals → Conservative Vector Fields → Conservative Vector Fields and Potential Functions → Curl and Divergence of Vector Fields → Curl and Divergence → Divergence Theorem → Electric Flux and Divergence Theorem → Gauss's Law: Integral Form and Meaning → Solving Problems with Gauss's Law → Conductors in Electrostatic Equilibrium → Capacitance and Capacitors → Dielectrics → Dielectric Constant and Relative Permittivity → Electric Field Inside Dielectric Materials → Dielectric Materials and Polarization → Dielectric Susceptibility and Permittivity → Energy Density in Electric Fields → Electric Current and Current Density → Electrical Resistance and Resistivity → Ohm's Law and Circuit Elements → Electromotive Force (EMF) and Batteries → Kirchhoff's Circuit Laws: Voltage and Current → DC Circuit Network Analysis Methods → Transient Response in RC Circuits → RC Circuits → LC and RLC Circuits → AC Circuits: Fundamentals → Impedance and Reactance → AC Power and Resonance → Electromagnetic Waves → Postulates of Special Relativity → Time Dilation → Length Contraction → Lorentz Transformation → Relativistic Velocity Addition → Relativistic Momentum and Energy → Mass-Energy Equivalence and E=mc² → Photons as Particles with Energy and Momentum → Planck-Einstein Relation: Energy and Frequency → Photoelectric Effect → The Photon: Light as Quanta → Compton Scattering → Wave-Particle Duality → de Broglie Wavelength → The Schrödinger Equation → State Vectors and Wavefunctions → Quantum Superposition → The Measurement Problem → Interpretations of Quantum Mechanics → Postulates of Quantum Mechanics → Observables and Quantum Operators → Commutators and Commutation Relations → Quantum Angular Momentum → Quantum Mechanical Treatment of Hydrogen → Solving the Schrödinger Equation for Hydrogen Atom → Quantum Numbers → Electron Configuration → Periodic Trends → Covalent Bonding → Electronegativity and Bond Polarity → Ionic Bonding → Lewis Structures → VSEPR Theory and Molecular Geometry → Molecular Geometry and Electron Pair Geometry → Molecular Polarity and Dipole Moments → Intermolecular Forces → States of Matter and Phase Changes: Melting, Boiling, and Sublimation → Gas Laws and the Ideal Gas Equation → Gas Stoichiometry and Volume-Volume Calculations → Thermochemistry and Enthalpy → Heat Capacity and Calorimetry → Entropy and Molecular Disorder → Spontaneity and ΔG → Entropy and Gibbs Free Energy → Chemical Equilibrium → Acid-Base Chemistry → Weak Acid Ionization → Weak Base Ionization → Acid and Base Strength: Ka, Kb, and Ionization → Leaving Groups and Nucleofugality → SN2 Substitution Reactions → SN1 Substitution Reactions → E1 Elimination Reactions → Alcohols and Ethers: Structure, Properties, and Nomenclature → Reactions of Alcohols → Aldehydes and Ketones: Structure and Reactivity → Oxidation Reactions in Organic Chemistry → Oxidation of Alcohols to Aldehydes and Ketones → Aldehyde and Ketone Structure and Nomenclature → Nucleophilic Addition to Aldehydes and Ketones → Carboxylic Acids and Their Derivatives → IUPAC Nomenclature of Carbonyls and Carboxylic Acids → IUPAC Nomenclature of Alkenes → Electrophilic Addition to Alkenes → Aromaticity and Benzene → Electrophilic Aromatic Substitution (EAS) → Nucleophilic Aromatic Substitution (SNAr) → Nucleophilic Acyl Substitution → Amines: Structure, Basicity, and Reactions → Amine Reactivity: Nucleophilicity and Basicity → Amino Acid Structure and Properties → Peptide Bonds and Polypeptide Formation → Protein Primary Structure → Protein Secondary Structure → Protein Tertiary Structure → Enzyme Structure and Function → Transcription: DNA to RNA → RNA Types and Structure → RNA Structure and Intramolecular Base Pairing → RNA Processing and Splicing → Translation: RNA to Protein → Ribosomes: Protein Synthesis Machines → Translation: Initiation and Elongation → Post-Translational Modifications → Proteasomal Degradation and Ubiquitin-Mediated Marking → Cell Cycle Regulation and Checkpoints → Cell Cycle Checkpoints: Ensuring Genome Integrity → Cell Cycle Checkpoints and Cancer Prevention → Mitotic Spindle Checkpoint and Chromosome Segregation → Kinetochore Structure and Function → Mitochondria: Structure and Function → Cellular Respiration Overview → Bacterial Metabolism Overview → Antibiotic Resistance Mechanisms → Infectious Disease Epidemiology → Foundations of Epidemiology → Measuring Disease Frequency: Incidence and Prevalence → Epidemiologic Study Designs → Confounding: Definition, Identification, and Causal Criteria → Effect Modification and Statistical Interaction → Stratified Analysis and Adjustment for Confounding

Longest path: 229 steps · 1314 total prerequisite topics

Prerequisites (2)

Confounding: Definition, Identification, and Causal Criteriahard Effect Modification and Statistical Interactionsoft

Leads To (2)

Inverse Probability Weightingsoft Multivariable Regression in Epidemiologyhard