Residual Analysis

What Is Residual Analysis — And Why Your Regression Depends On It

Residual analysis is one of those topics that separates students who understand regression from those who just run it. You can fit a model in seconds. Knowing whether that model is actually telling you something true takes a lot more than a high R² and a significant F-statistic. That's precisely the gap that residual analysis closes — and it's where most statistical assignments either earn full marks or fall apart.

At its core, residual analysis is the systematic examination of the leftover information your model didn't explain. Every fitted regression model generates a prediction for each observation. The residual is the difference between what actually happened and what the model predicted. These leftover errors — if your model is well-specified — should look like noise: random, centered at zero, evenly spread, and normally distributed. When they don't, you have a problem the model hasn't accounted for. Understanding regression analysis as the backbone of predictive modeling means recognizing that residual diagnostics aren't optional — they're the validation step that makes prediction trustworthy.

What Is a Residual?

A residual is the observed value of the dependent variable minus the model's predicted (fitted) value for that same observation. In formal notation:

Where eᵢ is the residual for observation i, yᵢ is the observed response value, and ŷᵢ is the model's fitted value. A positive residual means the model underpredicted; a negative residual means it overpredicted. Residuals are the observable approximations of the theoretical error terms (ε) in the population regression model — but unlike errors, residuals can actually be computed and examined. Simple linear regression is typically where students first encounter residuals, but the concept extends to every model class in statistics.

The sum of residuals in an OLS model is always exactly zero — a mathematical property, not a coincidence. This is why we can't just look at the total; we need to look at the pattern. Anscombe's landmark 1973 paper in The American Statistician demonstrated this point with devastating clarity: four datasets with identical means, variances, correlations, and regression lines — but wildly different residual structures. Same summary statistics, completely different models. Residual plots caught what the numbers missed.

The Difference Between a Residual and an Error

This distinction trips up a lot of students — and it matters for both exams and written assignments. The error term (ε) is the theoretical, population-level deviation between an observation and the true, unknown regression line. It's unobservable — you can never compute it directly because you don't know the true parameters. The residual (e) is the estimated error computed from your fitted model. It's what you calculate from sample data. Residuals are estimates of errors. Good residual behavior gives you evidence that the errors are behaving the way OLS assumes they should. Understanding regression model assumptions is the prerequisite for interpreting what residual patterns actually tell you.

Why Residual Analysis Matters for Your Assignment

In statistics and econometrics courses at universities across the United States and United Kingdom, regression assignments routinely require residual diagnostics as part of the submission. Presenting a regression without residual analysis is like submitting a medical diagnosis without examining the patient. Professors at institutions ranging from Harvard to University College London (UCL) expect you to verify assumptions, not just report coefficients. Missing this step is consistently one of the top reasons students lose marks on quantitative assignments. Statistics assignment help for regression and residual analysis is among the most requested topics precisely because the gap between running a regression and interpreting it diagnostically is so significant.

Raw, Standardized, and Studentized Residuals — What Each One Tells You

Not all residuals are created equal. Residual analysis involves several different forms of residuals, each designed to answer a specific diagnostic question. Using the right type matters — raw residuals alone can mask problems that scaled versions reveal clearly. The distinction between descriptive and inferential approaches to residuals maps directly onto the choice between raw and scaled forms.

Raw (Ordinary) Residuals

Raw residuals are the simplest form: eᵢ = yᵢ − ŷᵢ. They're on the same scale as the response variable, which is useful for understanding the practical magnitude of prediction errors. But because they're not standardized, comparing them across observations with different leverage is misleading. A residual of 15 in a model predicting salaries in thousands means something very different from a residual of 15 in a model predicting temperature. Raw residuals are the starting point, not the endpoint, of rigorous residual analysis.

Standardized Residuals

Standardized residuals divide each raw residual by the estimated standard deviation of all residuals (the root mean square error, RMSE). This produces dimensionless residuals that can be compared to a standard normal distribution. Observations with standardized residuals beyond ±2 are typically flagged for closer examination; those beyond ±3 are strong outlier candidates. The formula is:

Where s is the root mean square error and hᵢᵢ is the leverage of observation i. Understanding normal distribution and its properties is essential for interpreting whether standardized residuals suggest a problem or are within expected range.

Studentized Residuals

Studentized residuals (also called internally studentized residuals) are an improvement on standardized residuals because they account for the fact that high-leverage observations have smaller residuals by construction — the regression line gets pulled toward them. By adjusting for each observation's specific leverage, studentized residuals give a fairer comparison across all data points. Externally studentized residuals go one step further: they refit the model without the observation in question, then compute the residual. This makes them the most sensitive tool for detecting individual outliers.

The distinction matters practically in R (where rstandard() gives internally studentized and rstudent() gives externally studentized residuals), Python's statsmodels library (get_influence() object), and SPSS (where both are available through the regression save dialog). Finding appropriate datasets to practice residual diagnostics is itself a useful skill for statistical assignments.

Pearson and Deviance Residuals (Generalized Linear Models)

When you move beyond ordinary linear regression — into logistic regression, Poisson regression, or other generalized linear models (GLMs) — raw residuals are no longer directly interpretable because the response isn't continuous. GLMs use Pearson residuals (raw residual divided by the square root of the estimated variance function) and deviance residuals (signed square roots of the contribution to the model's deviance). Both are used in residual analysis for GLMs the same way raw and standardized residuals are used in OLS. Logistic regression diagnostics rely almost entirely on these scaled residual forms rather than OLS-style raw residuals.

How to Read Residual Plots — The Core of Residual Analysis

Residual analysis lives and dies by its plots. Formal tests are useful, but the human eye is remarkably good at detecting structure in scatter — which is why every major regression textbook from Montgomery, Peck, and Vining to Gelman and Hill at Columbia University leads with graphical diagnostics. The four plots you need to know are described below, along with exactly what to look for in each one.

Residuals vs. Fitted Values Plot

This is the workhorse of residual analysis. Plot the raw or standardized residuals on the y-axis against the fitted values (ŷ) on the x-axis. What you want to see: a horizontal band of points randomly scattered around zero with no obvious pattern. What different patterns mean:

• Curved or U-shaped pattern: Non-linearity. Your model is missing a polynomial term or the relationship between X and Y isn't linear. The fix is adding a quadratic or higher-order term, or applying a data transformation. Polynomial regression is the direct solution for this pattern.
• Fan or funnel shape: Heteroscedasticity — the variance of residuals changes with the level of fitted values. This is one of the most common violations of OLS assumptions in practice.
• Random scatter around zero: Linearity and homoscedasticity assumptions are satisfied. This is the outcome you want.
• Systematic wave or S-shape: Possible autocorrelation or a missing periodic variable in time series data.

Normal Q-Q Plot (Quantile-Quantile Plot)

The Normal Q-Q plot plots the quantiles of your standardized residuals against the quantiles of a theoretical normal distribution. If residuals are normally distributed, the points fall approximately along a straight 45-degree line. Deviations from this line tell a specific story depending on where they appear:

• S-shaped curve: Heavy tails (leptokurtosis) — residuals have more extreme values than a normal distribution predicts. This is common in financial data.
• Banana-shaped curve: Skewness — residuals are consistently pulled in one direction. A log transformation of Y often helps.
• Points diverging at ends: Outliers. Individual observations pulling away from the normal line are candidates for further investigation.
• Points on the line throughout: Normality assumption satisfied.

Formal normality tests supplement the Q-Q plot. The Shapiro-Wilk test, developed by Samuel Shapiro and Martin Wilk in 1965, is generally considered the most powerful normality test for samples under 50. The Anderson-Darling test is preferred for larger samples. Hypothesis testing principles apply directly to these formal normality tests — understanding p-values and test statistics is essential for interpreting their output correctly.

Scale-Location Plot (Spread-Location Plot)

Also called the spread-location or √|residuals| vs. fitted values plot, this graph shows the square root of the absolute values of standardized residuals against fitted values. It's designed specifically to detect heteroscedasticity. A roughly horizontal line with evenly spread points indicates constant variance. An upward slope indicates that variance increases with fitted values — which is the most common form of heteroscedasticity in practice, particularly in income, expenditure, or biological data. Understanding variance and expected values is the theoretical backbone for interpreting why constant variance matters for OLS.

Residuals vs. Leverage (Cook's Distance Plot)

The leverage of an observation measures how far its predictor values are from the center of the predictor space — high leverage means the observation occupies an unusual position on the X-axis and can exert disproportionate influence on the regression coefficients. The residuals vs. leverage plot, with Cook's Distance contour lines overlaid, combines both pieces of information: is this observation far from the fit AND from the center of the data? Points in the upper-right or lower-right corners (high residual AND high leverage) are the most concerning. They are both outliers and influential — the worst combination for model stability.

Testing OLS Assumptions Through Residual Analysis

Every ordinary least squares regression rests on a set of assumptions. Residual analysis is fundamentally the process of testing whether those assumptions hold in your data. The Gauss-Markov Theorem — named after German mathematician Carl Friedrich Gauss and Russian mathematician Andrei Markov — guarantees that OLS estimators are Best Linear Unbiased Estimators (BLUE) only when these assumptions are satisfied. Violate them, and your estimates may still be computable, but they won't have the properties you're relying on for inference. The full breakdown of regression model assumptions provides the theoretical foundation for everything this section covers.

Linearity

OLS assumes a linear relationship between the predictors and the response variable. A residuals-vs-fitted plot that shows a curved pattern is your primary diagnostic. Remedies include polynomial terms (quadratic, cubic), interaction terms, or transformations of X or Y. Non-linearity doesn't make regression useless — it means the linear model is misspecified. Polynomial regression and other flexible approaches address non-linearity while remaining interpretable.

Independence (No Autocorrelation)

OLS assumes residuals are independent — no observation's error is related to another's. In cross-sectional data this is usually satisfied. In time series data or panel data, serial autocorrelation is common and dangerous: it inflates the apparent precision of estimates, producing artificially narrow confidence intervals and overstated significance. Time series analysis and ARIMA models are the appropriate framework when residuals show serial structure.

The Durbin-Watson statistic, developed by James Durbin and Geoffrey Watson at the London School of Economics in 1950–51, is the standard test for first-order autocorrelation. Its value ranges from 0 to 4:

• ≈ 2: No autocorrelation — the desired outcome
• < 1.5: Positive autocorrelation — residuals in sequence tend to be similar
• > 2.5: Negative autocorrelation — residuals alternate in sign

The original Durbin-Watson paper in Biometrika (1950) remains the primary scholarly reference for this test. When autocorrelation is detected, remedies include adding lagged variables, using Newey-West standard errors, or switching to a GLS (Generalized Least Squares) estimator.

Homoscedasticity (Constant Variance)

Heteroscedasticity — when residual variance changes across fitted values — is one of the most common problems in real-world regression. It doesn't bias OLS coefficient estimates but makes standard errors incorrect, which invalidates all inference (hypothesis tests and confidence intervals). The Breusch-Pagan test (developed by Trevor Breusch and Adrian Pagan at the Australian National University) and the White test (developed by Halbert White at UC San Diego) are the standard formal tests. White's 1980 paper in Econometrica introduced the heteroscedasticity-consistent covariance estimator that remains standard in applied economics today.

Practical remedies for heteroscedasticity include:

• Log transformation of Y: Works when variance grows proportionally with the mean — common in income, financial, and biological data
• Square root transformation: Appropriate for count data (Poisson-distributed responses)
• Weighted Least Squares (WLS): Explicitly down-weights high-variance observations
• Robust standard errors (Huber-White sandwich estimator): Corrects standard errors without changing coefficients — the most practical solution in most academic and professional contexts

Normality of Residuals

OLS doesn't require normality for coefficient estimates to be unbiased (a common misconception). But for valid t-tests and F-tests in small samples, normality of residuals matters. In large samples, the Central Limit Theorem typically saves you — test statistics converge to their target distributions even with non-normal residuals. The Q-Q plot and Shapiro-Wilk test address this assumption directly. When normality is violated: log or Box-Cox transformations of Y often resolve the issue; alternatively, bootstrapped confidence intervals bypass the normality assumption entirely. Bootstrapping methods are increasingly standard in academic and professional statistical practice for exactly this reason.

No Multicollinearity (Predictor Independence)

While multicollinearity isn't strictly a residual-analysis issue (residual plots don't directly reveal it), it's a key assumption in multiple regression that deserves mention in this context. The Variance Inflation Factor (VIF) is the standard diagnostic — VIF values above 5 or 10 suggest problematic collinearity. Multicollinearity inflates standard errors and makes individual coefficient estimates unstable, even if the overall model fit is good. Ridge and Lasso regularization were specifically developed as responses to collinearity in high-dimensional data.

Outliers, Leverage, and Cook's Distance in Residual Analysis

One of the most important things residual analysis does is distinguish between observations that merely don't fit the model well and observations that are actively distorting it. These are very different problems with very different solutions, and conflating them leads to bad decisions about which data points to investigate or remove. Factor analysis and data reduction methods encounter similar issues with influential observations — the logic of leverage and influence applies broadly across multivariate statistical methods.

Outliers: Large Residuals

An outlier in regression terms is an observation with an unusually large residual — the model predicts poorly for this observation. Standardized residuals beyond ±2 are commonly flagged; beyond ±3 are strong candidates for investigation. But — and this is critical — an outlier isn't automatically a problem. It may reflect a genuine, meaningful observation that the model doesn't account for (in which case, you need a better model or an additional predictor), or it may reflect a data entry error (in which case, it should be corrected). Never delete an outlier simply because it doesn't fit your model. The question is always: why doesn't it fit? Missing data and imputation principles apply when outliers turn out to be data quality issues that require correction.

Leverage: Unusual Predictor Values

Leverage (denoted hᵢᵢ, the i-th diagonal of the hat matrix H) measures how far observation i's predictor values are from the center of the predictor space. High-leverage observations occupy unusual positions in X-space and have the potential to strongly influence the regression coefficients — even if their residual happens to be small (because the regression line got pulled toward them). The conventional threshold for high leverage is 2p/n, where p is the number of predictors and n is sample size. Hoaglin and Welsch's 1978 paper in The American Statistician formalized the hat matrix approach to leverage that remains standard today.

Cook's Distance: Combining Outlier and Leverage Information

Cook's Distance is the single most important influence statistic in residual analysis. Developed by R. Dennis Cook at the University of Minnesota in 1977, it measures how much the entire vector of fitted values would change if a single observation were removed from the analysis. It combines both the residual (how poorly the model fits this observation) and the leverage (how much potential this observation has to influence the coefficients):

Equivalently: Dᵢ = (eᵢ² / p · MSE) · (hᵢᵢ / (1 − hᵢᵢ)²)

The common thresholds: Dᵢ > 4/n is a commonly used rule of thumb; Dᵢ > 1 indicates more serious concern. But these are heuristics — context matters. In small samples, even moderate Cook's Distances may deserve attention. In large samples, they may be negligible even numerically. Cook's original 1977 paper in Technometrics provides the theoretical justification for this influential measure.

DFFITS and DFBETAS

Two additional influence diagnostics complement Cook's Distance in thorough residual analysis. DFFITS measures the change in the fitted value for observation i when i is deleted (scaled by its standard error). DFBETAS measure the change in each individual regression coefficient when an observation is removed — useful when you want to know which specific coefficients an influential observation is distorting. For standardized DFBETAS, values beyond ±2/√n are flagged. These measures are available in R's influence.measures() function, SPSS's casewise diagnostics, and Stata's dfbeta command.

How to Perform Residual Analysis: A Step-by-Step Guide

Understanding the theory of residual analysis is one thing. Executing it systematically on your own regression output is another. The following steps give you a complete process — from model fitting through formal testing and remediation — that covers what university-level statistics assignments and professional statistical practice both expect.

Key Figures, Organizations, and Frameworks in Residual Analysis

The development of residual analysis as a formal discipline spans two centuries of statistical innovation. Understanding who the key figures are — and why their specific contributions changed practice — elevates a university assignment from textbook recitation to genuine disciplinary awareness.

Carl Friedrich Gauss — The Origin of Least Squares

Carl Friedrich Gauss (1777–1855), the German mathematician and physicist who spent most of his career at the University of Göttingen, developed the method of least squares — the foundation of OLS regression and, therefore, of residual analysis. What makes Gauss uniquely significant is that he formalized the mathematical framework that defines residuals: minimize the sum of squared differences between observed and predicted values. He published this method in 1809 in Theoria Motus Corporum Coelestium, using it to predict the orbit of Ceres. Without least squares, there are no residuals; without residuals, there is no residual analysis. Every regression diagnostic traces its origin to his insight that the squared sum of errors is the natural objective to minimize.

Francis Anscombe — The Case for Visual Diagnostics

Francis John Anscombe (1918–2001), a British statistician and professor at Yale University, transformed how statisticians and students think about residual analysis with a single paper. His 1973 paper "Graphs in Statistical Analysis" in The American Statistician introduced what became known as Anscombe's Quartet — four datasets that are statistically identical (same mean, variance, correlation, and regression line) but look completely different when plotted. Dataset I is a clean linear relationship; Dataset II is a perfect quadratic; Dataset III has a single outlier distorting an otherwise perfect linear relationship; Dataset IV has a leverage point pulling the line toward a single extreme value. What makes Anscombe uniquely significant: he proved, visually and persuasively, that summary statistics alone are insufficient and that graphical residual analysis is indispensable. No statistics assignment on regression should be submitted without acknowledging this principle.

James Durbin & Geoffrey Watson — Autocorrelation Testing

James Durbin (1923–2012) and Geoffrey Watson (1921–1998) were statisticians at the London School of Economics who co-authored the landmark paper introducing the Durbin-Watson statistic in 1950 in Biometrika. What makes this test uniquely significant is its combination of computational simplicity and diagnostic precision — it provides a single number that captures first-order serial autocorrelation, which became the standard check for time series and economic regression residuals globally. The test is now reported by default in virtually every regression software package. Durbin later developed additional tests for higher-order autocorrelation, and Watson went on to contribute foundational work in time series econometrics.

R. Dennis Cook — Influence Analysis

R. Dennis Cook (born 1944), a statistician at the University of Minnesota, changed regression diagnostics with his 1977 paper in Technometrics introducing Cook's Distance. Before Cook's contribution, statisticians had separate tools for identifying outliers (large residuals) and high-leverage points — but no unified measure of influence that combined both. Cook's Distance filled that gap, providing a single, interpretable statistic for each observation's overall impact on the model. He later co-authored (with Sanford Weisberg) the influential text Residuals and Influence in Regression (1982), which remains the scholarly standard reference for applied regression diagnostics. His work at Minnesota continues to influence statistical software implementations globally.

Halbert White — Robust Standard Errors

Halbert White (1950–2012), econometrician at the University of California, San Diego, addressed heteroscedasticity with a practical solution that didn't require specifying the form of the variance structure. His 1980 paper in Econometrica introduced the heteroscedasticity-consistent (HC) covariance estimator — now universally called White standard errors or sandwich errors. What makes White uniquely significant: his estimator made valid inference under heteroscedasticity accessible to any practitioner without requiring transformation or model respecification. He also developed the White test for heteroscedasticity, a formal test using the residuals of an auxiliary regression. Applied economists at institutions from MIT to the UK Treasury routinely use White standard errors as a default specification.

Residual Analysis in R, Python, SPSS, Stata, and Minitab

Knowing the theory of residual analysis only gets you halfway. You need to know how to execute it in the statistical software your course or employer uses. Here's a practical breakdown of how each major platform handles the core diagnostic tasks.

Residual Analysis in R

R, the open-source statistical computing environment maintained by the R Foundation for Statistical Computing (Vienna, Austria), has the richest ecosystem for residual analysis diagnostics. The base plot(model) function on any lm object produces all four standard diagnostic plots instantly. The car package (Companion to Applied Regression, developed at Princeton University by John Fox) extends this with the influencePlot(), avPlots(), and crPlots() functions. The lmtest package provides bptest() (Breusch-Pagan) and dwtest() (Durbin-Watson). The sandwich package computes White standard errors. For visual diagnostics with publication-quality graphics, ggplot2's residual plotting via fortify(model) is the preferred approach in applied research. Logistic regression in R uses the DHARMa package for simulation-based residual analysis specific to GLMs.

Residual Analysis in Python

Python's primary statistical computing libraries — statsmodels and scipy — handle residual analysis through the OLSInfluence class in statsmodels. Key commands: model.resid (raw residuals), influence.resid_studentized_internal (internally studentized), influence.cooks_distance (Cook's Distance), and influence.hat_matrix_diag (leverage values). The seaborn library provides clean residual plot functions via sns.residplot(). For heteroscedasticity-robust standard errors in statsmodels, the HC3 covariance type (White's estimator) is requested via model.fit(cov_type='HC3'). Pingouin, a newer Python statistics library, adds user-friendly normality tests. PCA and other multivariate methods in Python encounter related diagnostic challenges that the same libraries address.

Residual Analysis in SPSS

SPSS Statistics, developed by IBM (Armonk, New York), handles residual diagnostics through the Analyze → Regression → Linear dialog. The "Save" submenu allows saving raw residuals, standardized residuals, studentized residuals, Cook's Distance, leverage values, DFFITS, and DFBETAS as new variables in the dataset. The "Plots" submenu generates residuals vs. fitted plots and P-P/Q-Q plots. SPSS is widely used in UK university statistics courses and psychology departments — many university assignments in the United States and United Kingdom specify SPSS output format, which makes knowing where each diagnostic lives essential. MANOVA in SPSS also generates residual diagnostics relevant to multivariate assumption checking.

Residual Analysis in Stata

Stata, developed by StataCorp (College Station, Texas), is the dominant statistical software in econometrics courses at universities in both the United States and United Kingdom. After regress y x1 x2, the key residual commands are: predict e, residuals (raw), predict r, rstandard (standardized), rvfplot (residuals vs. fitted), avplot x1 (added variable plot), predict d, cooksd (Cook's Distance), predict lev, leverage (leverage), and actest for autocorrelation testing. Stata's xtserial tests for serial correlation in panel data models. Its vce(robust) option applies White standard errors to any regression command. For heteroscedasticity testing, hettest and whitetst run Breusch-Pagan and White tests respectively.

Essential Vocabulary and Related Concepts for Residual Analysis

Graduate-level residual analysis assignments and professional statistical reporting require command of precise vocabulary. The following terms appear frequently in rubrics, professor feedback, peer-reviewed journals, and the standard reference texts for applied regression diagnostics.

Core Statistical and Procedural Terms

Residual Sum of Squares (RSS) — the total of all squared residuals; the quantity OLS minimizes. Also called Sum of Squared Errors (SSE). Mean Square Error (MSE) — RSS divided by degrees of freedom; the estimate of the error variance σ². R-squared — the proportion of variance in Y explained by the model; does NOT indicate whether residual assumptions are met. High R² with violated assumptions is common and dangerous. Adjusted R-squared — R² adjusted for the number of predictors; penalizes unnecessary model complexity.

Hat matrix (H) — the projection matrix H = X(XᵀX)⁻¹Xᵀ; its diagonal elements hᵢᵢ are the leverage values. BLUE (Best Linear Unbiased Estimator) — the status of OLS estimators when all Gauss-Markov assumptions hold. GLS (Generalized Least Squares) — a generalization of OLS that accounts for non-spherical error structure (heteroscedasticity, autocorrelation). WLS (Weighted Least Squares) — a special case of GLS where each observation is weighted by the inverse of its error variance. FGLS (Feasible GLS) — GLS with the error covariance structure estimated from the data rather than known a priori.

Newey-West estimator — a heteroscedasticity and autocorrelation consistent (HAC) standard error estimator, developed by Whitney Newey and Kenneth West at Princeton University, particularly important in time series regression. Box-Cox transformation — a family of power transformations parameterized by λ that includes log (λ=0), square root (λ=0.5), and identity (λ=1); used to normalize residuals and stabilize variance. Partial regression plot (added-variable plot) — shows the relationship between Y and one predictor after controlling for all others; also visualizes the partial residuals for that predictor. Correlation vs. causation is a critical conceptual distinction that partial regression plots help clarify in multiple regression contexts.

Related Statistical Frameworks and NLP Concepts

Broader conceptual themes important for advanced residual analysis work include: model misspecification (the consequences of omitting relevant variables or including irrelevant ones — both show up in residual patterns); cross-validation and out-of-sample residuals (computing residuals on held-out test data to assess generalization); simulation-based residual analysis for non-Gaussian models (the DHARMa approach in R); partial residuals and component-plus-residual plots for checking functional form assumptions for individual predictors; and recursive residuals in time series for detecting structural breaks in regression relationships. Cross-validation and bootstrapping extend residual analysis logic into resampling frameworks that are increasingly standard in machine learning model validation.

For students writing assignments on Bayesian inference, posterior predictive checks — comparing observed data against data simulated from the fitted posterior — are the Bayesian equivalent of residual analysis, serving the same model-checking function within a different inferential framework. The principle is identical: does the model generate predictions consistent with what we observed? If not, where does it fail? MCMC methods used in Bayesian computation produce trace plots and convergence diagnostics that serve a similar purpose to residual plots in frequentist regression — both are diagnostic tools ensuring the model's output can be trusted.

Frequently Asked Questions: Residual Analysis