Interpreting Statistical Output — Interpret statistical output from software (estimates, SE, p-values, CIs)

Historical Context & Motivation

Before the age of electronic computation, statistical calculations were performed by hand — a laborious process that limited the scope and complexity of analyses that researchers could practically undertake. A single regression with a handful of predictors could consume days of arithmetic, and errors were common and difficult to detect. The development of statistical software revolutionized quantitative research by automating computation, but it introduced a new challenge: researchers now needed to correctly interpret dense tables of numerical output rather than derive each quantity themselves. This shift made interpretation skills — the ability to read and critically evaluate estimates, standard errors, p-values, and confidence intervals — among the most essential competencies in modern biostatistics.

Today, nearly every published biostatistical analysis relies on software-generated output. Whether you are using R, Stata, SAS, SPSS, or Python, the tables produced share a common vocabulary: coefficient estimates, standard errors, test statistics, p-values, and confidence intervals. The central question this lesson addresses is: What do these numbers actually mean, and how should a researcher interpret them responsibly?

Core Principles & Definitions

Statistical output is organized around four interconnected quantities, each answering a distinct question about the relationship under study. Understanding their definitions and how they relate to one another is the foundation for interpreting any regression table, ANOVA output, or test summary. The four pillars are the point estimate, the standard error, the p-value, and the confidence interval. Together, they communicate the magnitude, precision, and statistical evidence for an effect.

Visual Explanation — Anatomy of a Regression Output Table

The diagram below illustrates a typical regression output table as it might appear from R, Stata, or SPSS. Each column is annotated to show its role in the interpretive process. The key insight is that every column derives from, or relates to, the point estimate and the standard error: the test statistic is their ratio, the p-value is derived from the test statistic, and the confidence interval is constructed by adding and subtracting a critical value times the SE from the estimate.

Notice in the diagram that the Treatment variable has an estimate of −4.87 with a 95% CI of [−8.64, −1.10]. Because the entire interval lies below zero, the interval is consistent with a negative treatment effect, and the p-value of 0.013 confirms that this result would be unlikely under the null hypothesis of no treatment effect. In contrast, the Age variable has a CI of [−0.05, 0.67] that crosses zero, aligning with its p-value of 0.089, which exceeds the conventional α = 0.05 threshold. These two rows illustrate a fundamental principle: a 95% CI that excludes zero corresponds to a p-value below 0.05 for a two-sided test.

Mathematical Framework

The quantities reported in statistical output are connected by a small set of formulas. Understanding these relationships is essential for checking output, performing back-of-the-envelope calculations, and recognizing when software may be reporting unexpected results. We present the core equations below in the context of ordinary least squares (OLS) regression, though the logic generalizes to generalized linear models, survival analysis, and other frameworks.

Detailed Breakdown — Interpreting Each Output Component

Each component of statistical output conveys a distinct piece of information about the estimated relationship. Misinterpreting any one of them can lead to incorrect conclusions. In this section, we provide a detailed breakdown of how to read and report each quantity, accompanied by a visual depiction of the confidence interval and its relationship to hypothesis testing.

Point Estimate: Magnitude and Direction

The point estimate (often labeled Estimate, Coef, or B) represents the most likely value of the population parameter given the data. In a linear regression context, a coefficient of −4.87 for Treatment means that, on average, being in the treatment group is associated with a 4.87-unit decrease in the outcome relative to the reference group, holding all other predictors constant. Always note the sign (positive or negative) and the units of the outcome variable, as these determine the practical interpretation.

Standard Error: How Precise Is the Estimate?

The standard error is not the same as the standard deviation of the raw data. It estimates the variability of the coefficient estimate across hypothetical replications of the study. A large SE relative to the estimate signals that the data provide imprecise information about the parameter — this often occurs with small samples, high residual variability, or multicollinear predictors. A rule of thumb is that if the estimate is less than about twice its SE, the evidence for a non-zero effect is weak (since the test statistic will be less than ~2).

P-Value: What It Is and What It Is Not

The p-value is perhaps the most commonly misinterpreted statistic. It is not the probability that the null hypothesis is true, nor is it the probability that the results occurred by chance. Correctly stated, the p-value is the probability of observing data at least as extreme as what was observed, assuming the null hypothesis is true. A p-value of 0.013 means there is a 1.3% chance of obtaining a test statistic as extreme as −2.54 (or more extreme) if the true coefficient were exactly zero. The conventional threshold α = 0.05 is arbitrary; the ASA has emphasized that p-values should not be used in a mechanical pass/fail manner.

Confidence Interval: The Range of Plausible Values

The 95% confidence interval brackets the estimate with a range that, over repeated sampling, would contain the true parameter 95% of the time. Crucially, for a specific computed interval, we say we are 95% confident the parameter lies within it — we do not say there is a 95% probability the parameter is in that specific interval (the parameter is fixed; the interval is random). Narrower intervals indicate greater precision and are driven by larger sample sizes and lower variability. CIs are more informative than p-values alone because they communicate both the magnitude and the uncertainty of the effect.

Worked Example — Reading a Logistic Regression Output

Consider a study examining whether a new drug reduces the odds of hospital readmission within 30 days. A logistic regression is fit with readmission (yes/no) as the outcome and Drug (1 = new drug, 0 = standard care) and Age (years) as predictors. The software produces the following output for the Drug variable: Estimate = −0.82, SE = 0.31, z = −2.65, p = 0.008, 95% CI for the log-odds = [−1.43, −0.21]. We will interpret each quantity step by step.

Common Misinterpretations & Best Practices

Decades of research on statistical literacy have documented persistent misinterpretations of standard output quantities. Even experienced researchers sometimes fall into these traps. The table below contrasts common mistakes with correct interpretations, serving as a diagnostic checklist when you read or write results sections.

Connections to Advanced Topics

The core concepts of estimates, SE, p-values, and CIs extend well beyond simple linear and logistic regression. As you advance in biostatistics, you will encounter output from more complex models — mixed-effects models, Cox proportional hazards models, generalized estimating equations (GEE), and Bayesian frameworks — each with its own nuances in output interpretation. The table below connects the foundational concepts covered here to their advanced counterparts.

Another important extension involves effect size measures such as Cohen's d, R², and number needed to treat (NNT), which provide standardized or clinically anchored summaries of effect magnitude that complement the raw coefficient estimates. Many journals now require reporting of effect sizes and confidence intervals rather than relying on p-values alone, reflecting the evolving standards of statistical communication articulated in guidelines such as the STROBE, CONSORT, and ARRIVE reporting checklists.

Practice Problems

Summary — Interpreting Statistical Output