🎲 Regression Coefficient

Regression Coefficient#

The estimated effect of a predictor on the response in a regression model.

Important

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What it is#

A regression coefficient \(\beta\) quantifies the relationship between a predictor \(X\) and the outcome \(Y\) in a regression model: how much \(Y\) changes when \(X\) rises by one unit, holding all other predictors fixed.

Simple and multiple regression#

In simple linear regression, \(Y = \beta_0 + \beta_1 X + \varepsilon\), where \(\beta_0\) is the intercept (the value of \(Y\) at \(X = 0\)) and \(\beta_1\) the slope. In multiple regression, \(Y = \beta_0 + \beta_1 X_1 + \dots + \beta_k X_k + \varepsilon\), each \(\beta_i\) is a partial coefficient — the effect of \(X_i\) controlling for the other predictors.

Reading a coefficient#

Its sign gives direction (positive: \(Y\) rises with \(X\); negative: it falls), its magnitude the strength of the effect, and its p-value whether it differs significantly from 0 (a non-significant coefficient may not contribute). For example, Salary = 30000 + 2000 × years says each extra year adds about 2,000 in salary; Price = 50000 + 100 × sqft + 20000 × garage says each square foot adds 100 holding garage fixed, and a garage adds 20,000 holding size fixed.

Standardised and logistic#

Unstandardised coefficients are in original units (dollars, cm); standardised ones (\(\beta^*\)) are in standard-deviation units, so effects on different scales can be compared. In logistic regression the coefficients are in log-odds; exponentiating gives an odds ratio — e.g. \(\beta = 0.7\) gives \(e^{0.7} \approx 2.0\), so a one-unit increase roughly doubles the odds of the event.

Why it matters#

Regression coefficients are the parameters of interest in regression — read in context, alongside p-values, confidence intervals and effect sizes, and (in multiple models) always as partial effects.

Theme: Probability & Statistics Foundations · All terminology

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Mind map — connected ideas

Parameter(s) of Interest · P-Value (probability value) · Logistic Regression · Sample Mean · Frequentist · Standard Error (SE)

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More in Probability & Statistics Foundations

Beta Distribution · Confidence Level · Correlation · Critical Value · Cumulative Distribution Function (CDF) · Frequentist · IID (Independent and Identically Distributed) · Likelihood · Margin of Error (MoE) · Mean · Median · Normal Distribution · Outlier · Population Proportion