Episode 90 — OLS Assumptions: What Violations Look Like in Real Problems
In Episode ninety, titled “O L S Assumptions: What Violations Look Like in Real Problems,” we focus on why understanding assumptions matters as much as understanding the regression equation. Ordinary Least Squares, abbreviated as O L S, can produce clean coefficients and impressive summaries even when the underlying conditions for reliable interpretation are not met. When that happens, the model output can look authoritative while quietly misleading anyone who treats it as policy grade evidence. Assumptions are not academic trivia, because in real problems they shape whether estimates are unbiased, whether uncertainty is trustworthy, and whether conclusions hold beyond the sample you happened to observe. The practical value is that assumption checks help you decide whether to trust the model as is, transform the problem, or choose a different approach altogether. This episode builds the habit of recognizing the symptoms of assumption violations in realistic settings so you can respond with discipline instead of false confidence.
Before we continue, a quick note: this audio course is a companion to the Data X books. The first book is about the exam and provides detailed information on how to pass it best. The second book is a Kindle-only eBook that contains 1,000 flashcards that can be used on your mobile device or Kindle. Check them both out at Cyber Author dot me, in the Bare Metal Study Guides Series.
The key assumptions you should keep in mind are linearity, independence, constant variance, and approximately normal errors, because these are the ones most often referenced and most often violated in practice. Linearity means the expected relationship between features and the target can be represented as a linear combination of predictors, at least after any intended transformations. Independence means the error terms are not correlated across observations, which is often challenged by time ordering or group structure. Constant variance, also called homoskedasticity, means the spread of errors is roughly the same across the range of predictions rather than expanding or shrinking systematically. Normal errors are often described as an assumption, but in many contexts it is primarily important for inference and confidence intervals rather than for point prediction. When these assumptions fail, the model may still produce a line, but the meaning of that line and the reliability of reported uncertainty can degrade in specific, recognizable ways.
Nonlinearity is one of the most visible violations, and it often shows up as curved patterns in residual plots or as systematic misses that are not evenly distributed. If residuals tend to be positive in one region of the predicted values and negative in another, the model is not capturing the shape of the relationship. You might see a U shaped pattern, an S shaped pattern, or residuals that drift upward as predictions increase, all of which indicate the linear form is forcing a straight line through a curved reality. In operational terms, this can manifest as consistent under prediction at high loads and over prediction at moderate loads, or vice versa, depending on the domain. These patterns matter because they indicate bias, meaning the model’s errors are not random noise but structured mistakes. When residuals carry structure, the model is telling you that the relationship is not being expressed in the right form.
Dependence is the violation that sneaks into many real datasets because observations often come from time series or from repeated measurements within groups. If you have multiple rows from the same user, device, account, or organization, the errors are likely to be correlated within those entities, breaking the independence assumption. Time dependence is another common form, where errors correlate across adjacent time points due to autocorrelation or drift, especially when you model a system that evolves. Dependence often leads to overconfident inference, because the model treats correlated observations as if they were independent pieces of evidence. The point estimates might look reasonable, but the uncertainty estimates can be too small, giving a false sense of precision. In narratives, dependence is hinted at by phrases like repeated readings, rolling windows, weekly cycles, or multiple transactions per customer, all of which signal that independence may be a fragile assumption.
Heteroskedasticity, meaning non constant variance, occurs when the spread of residuals changes as the predicted values or certain features change. A classic pattern is a funnel shape where residuals are tight at low predicted values and fan out at higher predicted values, indicating that errors grow with scale. In business terms, predicting demand or revenue often shows this pattern because variability is naturally larger when volumes are higher. In system performance modeling, latency errors can grow under heavy load because small perturbations create larger swings in response time. Heteroskedasticity matters because it affects the reliability of standard errors and confidence intervals, which can cause you to misjudge which predictors are statistically meaningful. Even if your point predictions seem acceptable, inference built on constant variance assumptions can be distorted. Recognizing the changing spread of residuals is therefore a key diagnostic habit.
Multicollinearity is not always obvious from residual plots, but it reveals itself through unstable coefficients and inflated uncertainty around those coefficients. When predictors are highly correlated, the model struggles to attribute effect cleanly to one feature versus another, so small changes in data can cause large swings in coefficient values while overall predictive performance stays similar. This can create confusion because the model’s fit looks fine but the interpretation becomes unreliable, with coefficients changing sign or magnitude in ways that do not match domain expectations. Multicollinearity often arises in telemetry, pricing, and operational datasets where multiple features measure similar underlying factors, such as different proxies for load or user activity. The practical danger is that stakeholders may interpret coefficients as stable marginal effects when they are actually artifacts of correlated inputs. Inference suffers because standard errors inflate, making it hard to assert which individual predictor matters even when the combined set predicts well.
Outliers and high leverage points can distort an O L S fit because least squares places disproportionate weight on points with large residuals, and leverage points can pull the fitted line toward themselves. An outlier is an observation with an unusual target value, but a leverage point is an observation with unusual feature values, and that combination can be especially influential. In real problems, outliers can come from data entry issues, rare operational incidents, or genuine extreme events that you might care about, so you need to distinguish error from signal. When a few influential points dominate, the fitted line can shift in a way that harms performance for the bulk of observations. This also makes inference fragile because the coefficients reflect the needs of those points rather than the typical case. A practical symptom is that removing or correcting a small number of cases causes large changes in coefficients and fitted values, which suggests the model is being held hostage by leverage.
When assumptions fail, the disciplined response is to use transformations or different models rather than forcing the original O L S form to work through wishful thinking. Transformations can address nonlinearity by expressing relationships in a more linear form, such as using logarithms for multiplicative effects or scaling features to reduce curvature. If dependence is the issue, you may need models that account for grouped structure or time correlation rather than treating each row as independent. If heteroskedasticity is present, a variance stabilizing transformation can sometimes help, or you might move to methods that model variance explicitly. The core idea is that assumptions are not rules you must obey, but conditions you must check, and failures are signals to adjust the modeling strategy. A model choice that respects the data generating process is more valuable than a familiar technique applied mechanically.
Robust standard errors are a specific mitigation when variance assumptions are violated, because they adjust how uncertainty is estimated without changing the coefficient estimates themselves. When heteroskedasticity is present, ordinary standard errors can be biased, leading to misleading confidence intervals and p values. Robust standard errors provide a way to make inference more reliable under certain forms of variance misspecification, though they do not fix nonlinearity, dependence, or leakage. This distinction matters because teams sometimes treat robust standard errors as a universal patch, when they are targeted to a specific problem. In applied work, robust standard errors can be a reasonable step when your primary concern is inference and you have evidence that heteroskedasticity is present. The important exam level takeaway is understanding what robust standard errors correct and what they do not.
Diagnosing violations from narrative scenario symptoms is a skill you can practice by translating story details into assumptions that are likely broken. If a scenario mentions time trends, seasonality, or repeated measures, independence is a suspect assumption. If it mentions that errors grow for larger customers, higher volumes, or heavier loads, heteroskedasticity is a likely issue. If the model consistently under predicts at extremes and over predicts in the middle, nonlinearity is implied through systematic residual patterns. If features are described as overlapping measurements of the same thing, multicollinearity becomes likely, especially if coefficient interpretations seem unstable. If a few rare events dominate outcomes or drive large changes in results, outliers and leverage should be considered. Turning narrative cues into diagnostic hypotheses is exactly how you recognize assumption issues before you waste time trusting misleading summaries.
Communicating assumption limits to stakeholders matters because regression outputs often get used to justify policies, budgets, or operational decisions. If assumptions are violated, you can still use the model for certain purposes, but you must be explicit about what is trustworthy and what is not. For example, you might trust predictions in a common operating range while warning that uncertainty grows at high volumes due to heteroskedasticity. You might explain that coefficients are unstable due to multicollinearity, so individual feature effects should not be interpreted as firm drivers. This communication should happen before decisions are framed as final, because it prevents stakeholders from treating statistical summaries as certainty. Assumptions are governance information, and sharing them is part of responsible analytics.
Avoiding blind reliance on p values is particularly important when assumptions are clearly broken, because p values depend on the validity of the underlying inference framework. When heteroskedasticity, dependence, or severe non normality is present, standard p values can be misleading, often appearing smaller than they should be due to underestimated variance. Even when p values are technically computed, they may not reflect the uncertainty you think they do, leading to overconfident claims about significance. This problem is magnified when stakeholders use p values as a binary decision rule, treating features as important or unimportant based on an arbitrary cutoff. A mature approach uses p values as one piece of evidence, conditioned on assumption checks, rather than as a substitute for understanding the data and the model. In exam terms, the correct stance is skepticism when assumptions are violated, not mechanistic interpretation.
The anchor memory for Episode ninety is that residuals reveal violations, fix process before inference. Residual patterns are the visible footprint of nonlinearity, heteroskedasticity, and sometimes dependence, and they often tell you more than the headline metrics. Fixing process means addressing the root cause through better splitting, appropriate modeling choices, transformations, or robust methods rather than adjusting your story to match a convenient summary table. Before you use a regression output to justify policy, you should have confidence that the assumptions supporting your inference are at least approximately reasonable or that you have applied mitigations that make inference more reliable. This anchor also reminds you that interpretation is not a right you earn by running a model, it is a privilege you earn by validating conditions. When you treat residuals as a diagnostic tool, you avoid many of the most common ways regression results mislead.
To conclude Episode ninety, titled “O L S Assumptions: What Violations Look Like in Real Problems,” choose one violation and one mitigation you prefer so you can respond clearly under exam pressure. If you observe heteroskedasticity, indicated by residuals that fan out as predictions increase, a mitigation you might prefer is using robust standard errors for inference or applying a transformation that stabilizes variance if prediction quality is the primary concern. If you observe dependence in time series data, indicated by correlated errors over time, a mitigation you might prefer is adopting a time aware modeling approach or restructuring splits to respect time order and reduce leakage of future patterns. If you observe nonlinearity through curved residual patterns, a mitigation you might prefer is applying a transformation or moving to a model family that captures nonlinear effects. The important habit is to match the mitigation to the violation rather than applying generic fixes. When you can name a violation and a targeted response, you demonstrate the core competency: recognizing when O L S summaries might mislead and adjusting the approach before making high stakes interpretations.
