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In a normal linear model with an intercept, $n$ observations and $p$ independent mean coefficients, adjusted $R^2$ isIt compares the residual variance estimate with the unmodelled sample variance, adjusting the raw coefficient of determination for fitted dimension. It can be negative and is not guaranteed to increase when another regressor is added. In penalized regression a related convention uses the fit's effective degrees of freedom instead of $p$; the precise smoothing-software variance convention should be specified.

Analysis of covariance combines a categorical treatment and a quantitative covariate in a normal linear model. A common-slope model is $Y_i=\alpha +\beta x_i+{\sum }_{g=2}^G{\tau }_g{1}_{\{G_i=g\}}+{\epsilon }_i$, with independent $N(0,{\sigma }^2)$ errors. Treatment differences compare fitted means at the same covariate value. Adding treatment-by-covariate interactions permits separate slopes, and a partial F-test compares this extension with the parallel-lines model. The treatment test after adding the covariate differs from an unadjusted comparison, because treatment groups can have different covariate distributions.

Analysis of variance decomposes variation into contributions associated with model terms and residual error. For nested normal linear models, differences of residual sums of squares give sequential or partial F-tests.

In a complete two-factor design with equal cell replication, centred factor indicator columns for different factors have zero cross-products. Consequently the two main effects in an additive normal linear model are orthogonal, and their extra sums of squares do not depend on the order in which these main effects are entered. An omitted interaction term can still require a separate adequacy check.

In an ordinary least squares regression with an intercept and a nonconstant response, $R^2=1-\mathrm{RSS}/{\sum }_i(Y_i-\overline{Y}{)}^2$. It measures the fitted proportion of sample variation relative to an intercept-only baseline. It increases when regressors are added, even if those regressors are unhelpful for prediction. A high value does not establish the assumptions of a normal linear model or rule out a systematic regression residual pattern.

For the five-point example in linear regression, the total centered sum of squares is $10.8$ and the residual sum of squares is $0.8$, so $R^2=25/27\approx 0.926$. This summarizes the fitted reduction in squared error. It does not record the order or shape of the regression residuals, which are worth examining in a residual-versus-fitted plot.

Here is a fully constructed illustration. Use $x=(-2,-1,0,1,2)$ and the same fitted line $3+x$ for both datasets. Add respectively the residual vectorsBoth sum to zero, both have zero inner product with $x$, and both have squared norm $0.56$. Their ordinary least squares fit is therefore exactly the same line, with the same $R^2=10/(10+0.56)=125/132\approx 0.947$. But the first residual is $0.2(x^2-2)$, a systematic quadratic shape, while the second changes direction across the predictor values. These examples show directly which information the scalar $R^2$ discards; neither constructed pattern proves a random-error model.

In a normal linear model with an intercept and factor effects, set the effect of one factor level to zero. The intercept then denotes that reference level's mean, and the remaining effects are contrasts to it. The constraint removes redundant statistical parameters without altering the fitted mean space.

In a normal linear model with residual degrees of freedom $\nu \ge 2$, an internally studentized residual satisfies $|{\eta }_i|\le \sqrt{\nu }$ and ${\eta }_i^2/\nu \sim \mathrm{Beta}(1/2,(\nu -1)/2)$, provided its leverage is less than one. Choose an orthonormal basis of the residual space with its first direction proportional to the projection of the $i$th coordinate vector. The Gaussian coordinates in this basis are independent standard normal random variables $Z_1,\dots ,Z_{\nu }$, and ${\eta }_i=\sqrt{\nu }{Z}_1/({\sum }_j{Z}_j^2{)}^{1/2}$. The beta distribution follows from the independent chi-squared distributions of $Z_1^2$ and ${\sum }_{j=2}^{\nu }{Z}_j^2$. The bounded residual is not exactly distributed according to Student's t-distribution; it approaches the standard normal distribution as $\nu \to \infty$.

For a full-rank $n\times p$ normal linear model with error variance ${\sigma }^2$, the maximum-likelihood residual variance satisfiesConsequently

Suppose a normal linear model gives an unrestricted mean to each cell and has $m$ independent observations per cell, with common error variance ${\sigma }^2$. Each fitted cell mean is its sample mean, with variance ${\sigma }^2/m$; different fitted cell means are independent. Estimate ${\sigma }^2$ by the pooled within-cell residual sum of squares divided by $n-q$, where $q$ is the number of cells. Thus a cell mean has standard error $s/\sqrt{m}$, a difference of two cell means has standard error $s\sqrt{2/m}$, and a two-by-two difference-of-differences has standard error $2s/\sqrt{m}$.

// Target: probability-and-statistics.bigb

For nested normal linear models, the decrease in residual sum of squares obtained by adding coefficients is their extra sum of squares. Dividing by the number of added coefficients and by the full-model residual mean square gives the partial F-test. A sequential analysis of variance attributes this decrease according to the specified order of entry; it need not equal the partial comparison when other terms are retained.

To pool factor levels in a normal linear model, impose equality of their fitted means while retaining any needed interaction terms. Compare the resulting nested model with the full model using a nested-model F-test. For two cells each of size $m$, pooling their fitted means adds $(m/2)({\overline{Y}}_1-{\overline{Y}}_2{)}^2$ to the residual sum of squares. If pooling two pairs separately within two other-factor levels, there are four restrictions.

For a full-rank ordinary least squares design matrix, the hat matrix $H$ and residual projection $G=I-H$ satisfy $HG=GH=0$. Hence fitted values and regression residuals are perpendicular as observed vectors. Under isotropic errors their cross-covariance matrix is zero, and under a normal linear model the two vectors are independent.

In the normal linear model, assume $\epsilon \sim N_n(0,{\sigma }^{2}I)$ is independent of $X$. Independent rows of $X$ with multivariate normal distribution $N_p(0,I_p)$ give a simple random design for Debiased Lasso. Let $p\to \infty$, $\log p=o(n)$ and $s\log p=o(\sqrt{n})$, with fixed positive noise scale. Choose the Lasso penalty and all Nodewise Lasso penalties as sufficiently large constant multiples of $\sqrt{\log p/n}$, including the noise scale in the former. Uniform sample cross-correlations are bounded by this scale, so zero is the nodewise solution with probability tending to one. Column squared norms divided by $n$ converge uniformly to one. To check the Compatibility condition for the Lasso directly, put $\hat{\Sigma }=X^{T}X/n$ and $\eta ={\max }_{j,k}|{\hat{\Sigma }}_{jk}-1_{\{j=k\}}|$. On the Lasso cone condition, $\Vert \delta {\Vert }_1\le 4\Vert {\delta }_S{\Vert }_1$, and the Cauchy-Schwarz inequality givesFor $s\ge 1$, entrywise concentration of a Gaussian Gram matrix gives $\eta \le D_0\sqrt{\log p/n}$ with probability tending to one for sufficiently large fixed $D_0$. The stated sparsity condition implies $s\eta \to 0$ on this event, so one can take ${\varphi }^2\ge 1/2$ eventually. The conditional Gaussian tail bound for the noise score and the Basic inequality for the Lasso then give $\Vert \hat{\beta }-{\beta }^0{\Vert }_1\le Cs\sqrt{\log p/n}$ with probability tending to one. Consequently the debiased-Lasso remainder bound is at most $As\log p/\sqrt{n}$ with probability tending to one, for a fixed constant $A$. These assumptions allow both $p$ and $s$ to grow, for example $p=n^2$, $s=\lfloor n^{1/4}\rfloor$.

When $L$ has full column rank, its Gram matrix $L^{T}L$ is positive-definite and $BL{L}^{T}L(L^{T}L{)}^{-1}=BL=0$. The stacked normal vector $(BY,L^{T}Y)$ therefore has zero cross covariance, so its blocks are independent. Since $Y^{T}AY=\Vert L^{T}Y{\Vert }^2$, the linear and quadratic statistics are independent. This is the projection argument behind independence of regression coefficient and variance estimators in a normal linear model.

For $n$ independent normal responses of common variance at $g$ distinct predictor settings, a rank-$p$ normal linear model has residual degrees $n-p$. The saturated setting-mean model has $g$ parameters and pure error degrees $n-g$. Splitting orthogonal residual spaces gives lack of fit degrees $g-p$. Under the fitted mean model, the two squared lengths divided by the common variance are independent chi-squared variables; their ratio of mean squares has the F-distribution shown. This test is unavailable without repeated settings or positive lack-of-fit degrees.

Linear regression models a response's conditional expectation as a linear combination of specified predictors. In matrix form,where the design matrix $X$ describes the observations, $\beta$ contains the regression coefficients, and $\epsilon$ contains the errors. The fitted values $X\hat{\beta }$ estimate the response means. The model is linear in its regression coefficients; its columns may include powers, interactions, or other transformed predictors, so polynomial regression can fit curved relationships using the same machinery.

For a concrete example, take five constructed observationsAn ordinary least squares fit with an intercept gives the line $\hat{y}=0.8+x$. Its regression residuals are $(0.2,0.2,-0.8,0.2,0.2)$ and its residual sum of squares is $0.8$. A one-unit increase in the predictor corresponds to a one-unit increase in the fitted response. This is a description of this fitted association; interpreting a change as a causal effect requires the assumptions of a causal inference problem.

The geometric reason for the fit is simple. The possible mean vectors $X\beta$ form the column space of $X$. Ordinary least squares chooses the orthogonal projection of the data vector onto that vector subspace, so the remaining regression residual is orthogonal to every permitted fitted direction. The least-squares normal equations express this orthogonality as $X^T(Y-X\hat{\beta })=0$. The data's explained component and unexplained component are therefore separated by the same geometry used in a Hilbert space.

The optimization definition makes sense for any observed data. Quantifying the uncertainty of the regression coefficients needs additional assumptions about the sampling and errors. With fixed $X$ of full column rank and mean-zero errors with covariance matrix ${\sigma }^{2}I$, ordinary least squares is unbiased and has covariance ${\sigma }^2(X^{T}X{)}^{-1}$; the Gauss-Markov theorem compares it with other linear unbiased estimators. A normal linear model additionally makes Student t-distribution intervals and tests exact in finite samples. Model adequacy and predictive assessment remain relevant even when these calculations are available.

Two historical settings help explain the subject's appeal. Adrien-Marie Legendre's Nouvelles méthodes pour la détermination des orbites des comètes (1805) includes an appendix on the method of least squares: observations give more equations than unknowns, and squared errors provide a criterion for reconciling them. Its astronomical setting shows the motivation for combining imperfect measurements. ^[ref] Francis Galton's Regression Towards Mediocrity in Hereditary Stature (1886) examines the association between parental and offspring heights and the tendency of group averages toward the population average. This is a historical use of the regression name; the modern linear model is a general statistical formulation. ^[ref]

If a design matrix $X$ has a nonzero null vector $z$, the normal linear model has identical observation distributions at $\beta$ and $\beta +z$. Any estimator has the same expectation at those two parameters, and therefore cannot be an unbiased estimator of both coefficient vectors. Full column rank is required for unbiased estimation of the entire unconstrained coefficient vector; identifiable linear combinations can still be estimated without it.

For a normal linear model with coefficient vector $\eta$ and residual variance ${\sigma }^2$, a normal-inverse-gamma prior has $\eta \mid {\sigma }^2\sim N(m,{\sigma }^{2}V)$ and ${\sigma }^2\sim \mathrm{IG}(a,b)$, with $V$ a positive-definite matrix and $a,b>0$. The inverse-gamma density is proportional to $({\sigma }^2{)}^{-a-1}{e}^{-b/{\sigma }^2}$. Multiplying by the normal linear model likelihood function and completing the square preserves the family, giving an analytic posterior and evidence integral.

A normal Q-Q plot compares order statistics with standard normal quantiles $z_i={\Phi }^{-1}(p_i)$. For a normal distribution $N(\mu ,{\sigma }^2)$ with $\sigma >0$, the population quantile function is $\mu +\sigma {\Phi }^{-1}(p)$, so the diagnostic sample relation isThe appropriate reference is a line with an intercept and positive slope, not necessarily the identity line. A line can be fitted through two central sample quantiles; R's qqline uses the quantile levels $p=0.25$ and $p=0.75$ by default. Neither that line nor its fitting rule is part of the definition of a normal Q-Q plot. ^[ref]

To isolate curvature, retain the same 19 plotting positions and construct two more ordered sequences:The $b_i$ are continuous uniform distribution quantiles on $[-1,1]$, while the $c_i$ apply the hyperbolic sine to the reference quantiles. Both examples are symmetric about zero. At ranks $1,10,19$, $b_i$ is approximately $-0.947368,0,0.947368$ and $c_i$ is approximately $-1.812618,0,1.812618$. Again, these are constructed values rather than random draws or real data.

Read these shapes relative to their central slope. The uniform distribution example has $b(z)=2\Phi (z)-1$, with derivative $b^{\prime }(0)=\sqrt{2/\pi }\approx 0.797885$; at the upper endpoint the tangent predicts $1.546246$, while $b_{19}=0.947368$. Its lower end lies above the tangent and its upper end below it: the outer quantiles are compressed. The hyperbolic sine example has $c^{\prime }(0)=0.7$; its tangent predicts $1.356552$ at the upper endpoint, while $c_{19}=1.812618$. Its lower end lies below the tangent and its upper end above it: the outer quantiles are stretched. Unequal departures at the two ends can suggest skewness. The direction of these descriptions assumes that reference quantiles are horizontal and observed values vertical. The graphs illustrate finite-range shape differences; a sample pattern cannot establish an asymptotic tail probability law or membership in a heavy-tailed distribution class.

A nearly straight normal Q-Q plot is a diagnostic, not a proof of a normal distribution or a formal hypothesis test. Random statistical samples fluctuate, particularly at extreme order statistics, and apparent curvature depends on sample size, plotting positions and the reference line. A calibrated hypothesis test or uncertainty band requires its own assumptions and must account for parameters estimated from the same observations. Sorting also discards the original observation order: rearranging a time series leaves its Q-Q plot unchanged, so the plot cannot establish independence.

For a normal linear model, R's diagnostic normal Q-Q plot uses standardized regression residuals, also called internally studentized residuals. ^[ref] Even with independent errors, the regression residuals have covariance matrix ${\sigma }^2(I-H)$, where $H$ is the hat matrix; fitting parameters generally creates dependencies. Dividing by a shared estimated standard deviation creates further dependence, and the distribution of an internally studentized Gaussian residual is not exactly the standard normal distribution. The normal Q-Q plot therefore helps assess error shape alongside other diagnostics rather than certifying independent normal random variables.

In a normal linear model $Y\mid b,s\sim N(Xb,sI)$, assign the coefficient vector $b$ a proper normal distribution with fixed positive-definite covariance matrix $D$, independently of the variance $s$, whose prior distribution is inverse-gamma distribution. Completing the square givesMultiplying the residual likelihood function by the inverse-gamma distribution density also givesThese full conditional distributions permit a Gibbs sampler. The marginal coefficient posterior distribution is obtained by marginalization over $s$. Its conditional mean and covariance matrix both depend on $s$, so it is generally not a Student's t-distribution. Even if the columns of $X$ and the prior coordinates make the conditional coefficient components independent, averaging over a shared unknown $s$ need not preserve that independence.

A normal-inverse-gamma prior instead scales the conditional coefficient prior covariance matrix by $s$, producing a multivariate Student's t-distribution on marginalization. It is a different joint prior distribution.