Correlation & Regression

This page covers the Correlate, Regression, General Linear Model and Linear Mixed Model menu groups — everything for measuring association between variables and modelling an outcome from predictors.

Correlate

Analyze → Correlate.

Bivariate

When. Measure the strength and direction of association between scale (or ordinal) variables. Pearson's r summarises a linear relationship — when in doubt, look first with a scatterplot and its LOESS smoother: a curved trend says a single linear coefficient will under-describe the association.

Dialog. Move two or more variables into Variables; choose the coefficient(s): Pearson (linear), Spearman or Kendall's tau-b (rank). The Options section sets one- vs two-tailed significance, the missing-data policy (pairwise default, or listwise) and whether to Plot heatmap.

Output. A correlation matrix of coefficients with significance and the N behind each cell.

With Plot heatmap ticked, the coefficient matrix is also drawn as coloured cells on a diverging scale — one hue for negative, one for positive, neutral at zero — which is how you see the shape of a matrix too big to read cell by cell: a block of variables that move together, one that relates to nothing, a sign that flips. The scale is fixed at −1…+1, not stretched to the strongest correlation present, so a matrix of weak associations looks weak and two matrices can be compared. See Graphs.

Example. BMI, SystolicBP, FastingGlucose_mmolL — do they move together?

Permutation test

With exactly two variables selected (Pearson or Spearman), tick Permutation test in the Options section to also test the coefficient by randomization: one variable's values are reshuffled against the other's, and the permutation p is the share of arrangements with a coefficient at least as extreme as the observed one — no normality assumption behind the p. Small samples (n ≤ 9, where every arrangement can be enumerated) report an Exact p; larger ones report a Monte Carlo estimate at the dialog's replicate count with a 99% interval for its simulation uncertainty, made exactly reproducible by the Seed field. The asymptotic matrix above it is unchanged — the permutation row is an addition, not a replacement — and the option follows the dialog's tails setting. It is not available under Weight Cases, for Kendall's tau-b, or for a matrix of three or more variables (those stay asymptotic).

Partial

When. Measure the association between two variables controlling for one or more others.

Dialog. The two variables plus the controlling variables. (Partial correlation is inherently listwise — it needs cases complete on all variables.)

Output. The partial correlation coefficient(s) with significance.

Regression

Analyze → Regression. Model one outcome from one or more predictors. Pick the member that matches your outcome type:

Linear

Dialog. A Dependent scale variable and one or more Independent variables. Options include:

• Robust standard errors (HC0/HC1/HC3 sandwich estimators);
• Forward / backward stepwise variable selection;
• Include the intercept (constant) — on by default. Off fits regression through the origin (y = b·x, no constant), for calibration and physical-science models where a zero input must give a zero response. R² and the ANOVA total are then uncentred (about zero, with n total degrees of freedom) — the form NIST, R and statsmodels report for a no-intercept fit, and not comparable with an intercept model's R² — and the standardized Beta, Partial and Part columns are left blank;
• casewise influence diagnostics — Cook's distance and leverage;
• diagnostic plots — the classic three-plot assumption panel: a Residuals vs Fitted scatter (a random band around zero supports linearity; a curve says the linear form is wrong), a Scale-Location plot (√|standardized residual| against fitted — a flat band supports constant error variance, a rising wedge is the variance-grows-with-the-mean signature), and a Normal Q-Q of the standardized residuals (points on the reference line support the normality behind the significance tests);
• save predicted values and residuals back as new variables.

Output. The model summary (R², adjusted R², standard error), the ANOVA of the regression, and the coefficients table (B, standard error, beta, t, significance, and confidence intervals). Through the origin the coefficients table has no constant row and the model summary notes that R² is uncentred. If the predictors are severely ill-conditioned (raw polynomial powers of one variable, say), the model summary carries an accuracy note — the fit is still solved by an orthogonal (QR) decomposition, but centring or rescaling the predictors will give you more reliable digits.

Example. Predict SystolicBP from Age, BMI and Weight_kg.

Regularized (Ridge / Lasso / Elastic net)

When. Many or collinear predictors, and the goal is prediction rather than inference — or you want the model to choose its predictors. The penalty shrinks coefficients toward zero (ridge), sets some exactly to zero (lasso — automatic variable selection), or both (elastic net).

Dialog. A Dependent and at least two Predictors (each is standardized before penalization, so the penalty treats them alike). Penalty: Lasso (L1, variable selection) — the default — Ridge (L2, shrinkage) or Elastic net (½ L1 + ½ L2). The penalty strength λ is chosen by 10-fold cross-validation (the syntax/AI argument lambda fixes it instead; folds and seed control the CV).

Output. The Coefficients at the selected λ, on the original and the standardized scale (a lasso zero means "dropped"); the Regularization Path — for each λ on the path, how many coefficients are nonzero and the cross-validated error with its standard error (the selected λ is the error minimum); and a Model Summary. Penalized estimates carry no standard errors or p-values by design — use Linear regression on the selected predictors if you need inference, knowing the selection was data-driven.

Example. SystolicBP on Age, BMI, Weight_kg, TotalCholesterol_mmolL, FastingGlucose_mmolL, PhysicalActivity_minWk and HeartRate with the lasso: the path shows which predictors survive as the penalty relaxes.

Robust (Huber / Tukey)

When. A diagnostic plot of the Linear fit shows a few influential outliers dragging the line. M-estimation down-weights large residuals in the fit itself, so the coefficients describe the bulk of the data.

Dialog. Dependent, Predictors, and the Estimator: Huber (clip) — the default — clips the influence of far residuals; Tukey bisquare (redescending) rejects gross outliers entirely. The syntax/AI argument tuning sets the constant c (defaults 1.345 and 4.685, each ~95% efficient under normality; smaller is more resistant, larger is closer to OLS).

Output. The robust Coefficients with sandwich (H1) standard errors, t tests and confidence intervals; a Model Summary with the robust scale estimate and iterations; and Down-Weighted Cases — the cases the fit trusted least, with their weights, which is where to look first.

Example. SystolicBP on Age and BMI when a handful of implausible blood-pressure entries pull the OLS slope; compare the robust slope with the Linear one.

Nonlinear

When. The relationship has a known functional form that is not linear in its parameters — a growth or decay curve, a dose–response, a Michaelis–Menten saturation, a sum of exponentials. (If the curve is a polynomial or a transform makes it linear, Linear regression on computed variables is simpler and exact.)

Dialog. A Dependent, the Predictors the model may refer to by name, the Model expression (for example b0 + b1*exp(b2*x); operators + - * / ^, functions exp, ln, log10, sqrt, sin, cos, abs, pow, …; a name that matches a predictor is that case's value, every other name is a parameter) and the Start values as name=value pairs (b0=1, b1=1, b2=0.1). Every parameter in the model must have a start value; their order is the order of the output. Good starts matter — a least-squares surface can have several valleys, and the fit finds the one your starts lead to. When a fit fails to converge, try starts nearer plausible values, or a rescaled predictor.

Output. Parameter Estimates (estimate, asymptotic Gauss–Newton standard error, t, significance, confidence interval) and a Model Summary (residual sum of squares, residual standard error, a descriptive 1 − SSE/SST pseudo-R² that need not lie in [0, 1], iterations, and Converged — No means the 1000-iteration cap was reached before the fit settled; the Levenberg–Marquardt fit is certified against all 27 NIST StRD nonlinear datasets, so a No points at the model or the starts, not the optimizer).

Artificial zero-residual data. If the data lie exactly on the curve (generated data, or a model that reproduces every point to rounding), the fit reaches the floor of double precision: the parameter estimates are right, but the residual sum of squares and hence the standard errors are not meaningful beyond a few digits — there is no noise left to estimate. Real measurements never do this; textbook or simulated data sometimes do.

Example. FastingGlucose_mmolL as an exponential in Age, model b0 + b1*exp(b2*Age), start b0=4, b1=0.1, b2=0.03.

Instrumental Variables (2SLS)

When. A regressor is endogenous — correlated with the error through simultaneity, an omitted confounder or measurement error — so its OLS coefficient is biased, and you have instruments: variables that move the endogenous regressor but affect the outcome only through it.

Dialog. A Dependent; the Endogenous regressor(s) being instrumented; the excluded Instruments (at least as many as endogenous regressors); and any included Exogenous regressors, which enter both the equation and the instrument set. Two-stage least squares projects the endogenous regressors onto the instruments and regresses the outcome on the projection.

Output. The Coefficients with 2SLS-corrected standard errors, t tests and confidence intervals; a Model Summary; and First-Stage Diagnostics — the weak-instrument F for each endogenous regressor (the rule of thumb: F below 10 is weak, and weak instruments make 2SLS unreliable).

Example. SystolicBP on BMI treated as endogenous, instrumented by PhysicalActivity_minWk, with Age exogenous — illustratively; whether an instrument is valid is a subject-matter argument, not a statistical test.

Binary Logistic

For a yes/no outcome (e.g. Hypertensive). A Link option chooses how the probability relates to the predictors: Logit (the default — coefficients are log-odds and report Exp(B) odds ratios with confidence intervals), Probit (a normal latent variable) or Complementary log-log (an asymmetric link suited to rare events). Logit reports odds ratios; probit and cloglog report coefficients on the link scale with a Wald interval. Optional robust standard errors and forward/backward selection. Diagnostic plots are available and work differently from the Linear ones — see Diagnostic plots for the other families.

Poisson / Negative Binomial

For counts (e.g. a count of events). Negative Binomial adds an estimated dispersion parameter for over-dispersed counts. Both report rate ratios. Both offer diagnostic plots, as do the zero-inflated and hurdle count models.

Gamma

For a strictly positive, right-skewed continuous outcome — costs, durations, concentrations — where the spread grows with the mean. A Link option offers the Log link (the default — a multiplicative model whose Exp(B) is the factor change in the mean per unit predictor) or the canonical Inverse link. Reports the coefficients, an estimated dispersion φ (so the standard errors account for the scale), and a deviance / Pearson goodness-of-fit table. Diagnostic plots are available; as the outcome is continuous, Gamma's are exact rather than randomised.

Diagnostic plots for the other families

The Diagnostic plots checkbox is offered on Binary Logistic, Poisson, Negative Binomial, Gamma and the zero-inflated / hurdle count models too — but what it draws is not the Linear panel above, and the difference is worth understanding before you compare a plot against one from another package.

Why not the same three plots. A residual is "observed minus expected", and for a yes/no outcome the observed value is only ever 0 or 1. So at any given fitted probability there are just two possible residuals, and a Residuals-vs-Fitted scatter of a logistic model collapses into two curved bands — one for the yeses, one for the noes. Those bands are a property of the outcome being binary; they say nothing about whether the model fits. Read as though they were the OLS panel, they invite you to see structure that is not there. Count models band the same way when the counts are small.

What is drawn instead: quantile residuals. Each case is put through its own fitted distribution and back through the normal curve. The result is a residual that, if the model is right, behaves like a standard normal value — whatever the family. That gives you two plots that read the same way for every model on this page:

Two details that matter if you are checking someone else's numbers:

• The x axis is the linear predictor, not the fitted mean — the same convention R's plot.glm uses. On the response scale the fitted values of a logistic or count model bunch up against 0 or 1 and the plot loses its resolution.
• The Q-Q reference is the 45° identity line, not a fitted line. Quantile residuals are standard normal by construction when the model is right, so the line you are comparing against is known in advance. (The Linear panel's Q-Q fits its line through the quartiles, because standardized residuals have no such guarantee.) A fitted line here would quietly absorb the very misfit the plot exists to show.

Randomised, but reproducible. For a discrete outcome — yes/no, or a count — there is a range of residual values consistent with each observation rather than a single one, so ChakataStat draws one at random from that range. This is the standard method (Dunn & Smyth) and it is what removes the banding. It means two things for you:

• The plot is reproducible: the draw is seeded, so the same command always produces the same picture, and a re-run of a recorded analysis reproduces the plot in your report exactly.
• You can vary the draw to sanity-check what you are seeing. Set the seed argument (see Reproducibility) to a different number and re-run: a pattern that survives several seeds is the model, and one that comes and goes is the randomisation. Inspecting a few realisations is the recommended practice.

Gamma is the exception: its outcome is continuous, so there is no range to draw from and the residual is exact. Its plots carry no seed and never change between runs.

Multinomial / Ordinal

For categorical outcomes with three or more levels — unordered (Multinomial, baseline-category logit) or ordered (Ordinal, proportional-odds). Both report the per-category/threshold coefficients and significance.

Quantile

When. The question is about a conditional quantile rather than the mean — the median as an outlier-resistant centre, or an upper/lower quantile because the effect of a predictor differs across the response distribution (what raises the 90th percentile of blood pressure may not be what raises its median).

Dialog. Dependent, Predictors, and the quantile τ between 0 and 1 (0.5, the median, by default). The fit minimizes the asymmetric check loss.

Output. One Quantile Regression table: the coefficients with asymptotic standard errors (iid sparsity estimate), t and significance, and a pseudo-R¹ goodness of fit in the note. Run it at several τ and compare the slopes to see whether an effect is uniform across the distribution.

Example. SystolicBP on Age and BMI at τ = 0.9 — what predicts the high end.

Generalized Estimating Equations

When. Correlated or clustered observations — repeated measures on a subject, patients within clinics — and the target is the population-averaged (marginal) effect of the predictors. (For subject-specific effects with a random intercept, see the Generalized Linear Mixed Model.)

Dialog. Dependent, Predictors, the Subject (cluster) variable that groups correlated observations, the Family — Gaussian (identity link, the default), Binomial (logit) or Poisson (log) — and the working correlation: Exchangeable (the default), Independence or AR(1).

Output. GEE Parameter Estimates — B with robust (sandwich) standard errors, z, significance and confidence intervals — and, in the note, the estimated correlation parameter α, the cluster and observation counts. The sandwich errors are valid even if the working correlation is misspecified; the choice of structure mainly affects efficiency.

Example. Hypertensive (binomial) on Age and BMI, subjects clustered by StudyZone, exchangeable correlation.

Loglinear

When. Two or more categorical variables and the question is about the association structure of their cross-classification — a multi-way contingency table modelled as Poisson counts. The main-effects model is the test of mutual independence; the saturated model reproduces the table and shows every interaction.

Dialog. The Factors (two or more categorical variables) and the Model: Independence (main effects) — the default — or Saturated (all interactions).

Output. Loglinear Parameter Estimates (B, standard error, Wald, df, significance, confidence interval; Exp(B) is the multiplicative effect on the cell count) and Goodness-of-Fit — the deviance and Pearson χ² with df and significance: a small significance rejects the model (for the main-effects model, it rejects independence of the factors).

Example. SmokingStatus × Sex × Residence: does the independence model fit, or do the factors interact?

Mediation

Asks how an independent variable X affects an outcome Y — how much of its effect runs through one or more mediators M (the path X → M → Y) versus directly. Pick the Outcome (Y), the Independent (X), one or more parallel Mediators, and any Covariates entered into every model. It fits, by OLS, the mediator model(s) M ~ X for the a path, the outcome model Y ~ X + M for the b paths and the direct effect c′, and the total model Y ~ X for the total effect c (with c = c′ + Σ a·b).

Two tables come back: the path coefficients with standard errors, t and confidence intervals; and the indirect effect(s) — each a·b, plus a total indirect when there is more than one mediator. Because the indirect effect is a product of coefficients (its sampling distribution is skewed), its interval comes from a nonparametric bootstrap — Percentile or BCa — over B resamples, reproducible for a fixed Seed. An indirect interval that excludes zero is the evidence of mediation. The bootstrap is unweighted (a Weight Cases variable is noted, not applied); Select Cases and Split File are honored.

Moderated mediation

Mediation answers "does X affect Y through M?"; the question that follows in practice is "for whom?" — does the indirect effect differ by group, by severity, by dose. Pick an optional Moderator (W) and the analysis becomes moderated mediation, following the PROCESS (Hayes) model shapes. The Moderated path choice decides where W acts:

• a: X → M (PROCESS model 7) — W changes how strongly X moves the mediator. The mediator model gains W and X×W.
• b: M → Y (model 14) — W changes how strongly the mediator moves the outcome. The outcome model gains W and M×W.
• Both paths (model 58) — both of the above.

The output changes accordingly:

• Path coefficients now include the moderated model's W and interaction terms (a1/a2/a3, b1/b2/b3), and the total-effect row disappears — with a moderator, c = c′ + a·b no longer holds unconditionally, so a "total" row would misreport.

• Conditional Effects — the moderated path's simple slope at each probe value of W, with normal-theory inference. A moderator with at most five distinct values is treated as categorical and probed at each value; otherwise at its mean − 1 SD, mean, and mean + 1 SD.

• Conditional Indirect Effects — the indirect effect at each probe value, each with a bootstrap interval. A conditional indirect interval that excludes zero is evidence of mediation at that level of W.

• Index of Moderated Mediation — the slope of the indirect effect as a linear function of W (a3·b or a·b3), with a bootstrap interval: an index interval that excludes zero is evidence the mediation itself is moderated. With both paths moderated the indirect effect is quadratic in W, no single index exists, and the output says so instead.

• Interaction Plot (the Interaction plot checkbox, on by default) — the moderated path's fitted simple-slope lines, one per probe level of W, every other predictor held at its mean. Lines that fan apart are the moderation; parallel lines mean W changes nothing.

• Johnson–Neyman Regions (the Johnson–Neyman regions checkbox) — probing at three values answers "is the effect significant here?"; Johnson–Neyman answers "where along W is it significant?" For each moderated path the table lists the boundary values of W at which the path's conditional effect crosses significance (at the interval's level, on the model's residual df) — the effect there equals ±t·SE, shown beside it — with the percent of cases below and above each boundary, and the note states the region in words ("significant for W < a and for W > b", "for W > a", "over the whole observed range", "nowhere"). Only boundaries inside the observed range of W are listed; a non-significant interaction usually yields none. The Johnson–Neyman Plot draws the conditional effect with its confidence band across the observed range, the zero line, and the boundaries as vertical markers — where the band clears zero is the region. The regions are for the path's conditional effect, not for the conditional indirect effect (whose inference is bootstrap-only), the same choice PROCESS makes.

• Moderator 2 (Z) — a three-way interaction. Sometimes the question is "does the moderation itself depend on something else?" — W's effect on the mediation differs by group, say. Pick a second moderator and Z joins W in the moderated model with every product up to X×W×Z (the coefficients are labelled a1…a7 / b1…b7). The conditional effects and conditional indirect effects are then probed on the W × Z grid (each moderator by its own rule — every level of a categorical one, mean ± 1 SD of a continuous one — listed Z by Z), the interaction plot is drawn once per Z level, and two new tables appear when one path is moderated: the Index of Moderated Moderated Mediation (a7·b or a·b7 — how much W's moderation of the indirect effect changes per unit of Z; an interval excluding zero says Z moderates the moderation) and the Conditional Index of Moderated Mediation (the W-moderation index at each Z level). Johnson–Neyman regions, if requested, are solved for W at each Z level.

Serial mediation

Parallel mediators are assumed not to affect each other. When the theory says the first mediator causes the second — X raises self-efficacy, which raises effort, which raises performance — tick Serial mediation (M1 → M2) with exactly two mediators; the first in variable order is M1. The models become M1 ~ X, M2 ~ X + M1 and Y ~ X + M1 + M2, the path table gains d21 (M1 → M2), and the indirect table reports three specific indirect effects — through M1 alone (a1·b1), through M2 alone (a2·b2) and through both in sequence (a1·d21·b2) — plus their total, each with a bootstrap interval; c = c′ + total indirect still holds exactly. A moderator W may enter both a-paths, both b-paths or both (d21 itself is not moderated); each specific indirect effect, the serial one included, is then conditional on W with its own index of moderated mediation. A second moderator is not offered with serial mediation.

Scope: a continuous outcome; single, parallel or two serial mediators; one or two moderators (one with serial) on the a-path, b-path or both, with Johnson–Neyman regions on request.

General Linear Model

Analyze → General Linear Model. A factorial between-subjects ANOVA / ANCOVA with Type III sums of squares: several factors, their interactions, and optional continuous covariates. Add post-hoc comparisons and contrasts as in One-Way ANOVA.

Example. SystolicBP by Sex and SmokingStatus (and their interaction), adjusting for Age as a covariate.

Linear Mixed Model

Analyze → Linear Mixed Model. For clustered or repeated data: a model with a random intercept and optional random slopes, using variance-components or unstructured covariance, estimated by REML (the default) or ML. Reports fixed effects and the estimated variance components.

Generalized Linear Mixed Model

Analyze → Generalized Linear Mixed Model. The mixed-model idea for a binary or count outcome: a random-intercept logistic (0/1) or Poisson (count) regression for clustered or repeated data — patients within clinics, repeated yes/no measures — estimated by adaptive Gauss–Hermite quadrature. The effects are subject-specific (conditional on the cluster); for population-averaged effects use GEE.

Dialog. The Outcome (0/1 for binomial, a non-negative count for Poisson), the Predictors (fixed effects), the Grouping variable that gets the random intercept, and the Family — Binomial — logistic (0/1) or Poisson — counts (log).

Output. Fixed Effects (B, S.E., Wald, df, significance, Exp(B) — the conditional odds or rate ratio — with its Wald interval), Random Effects (the intercept variance between groups, its SD and the SD's standard error) and Model Fit (−2 log-likelihood, AIC, BIC; the note gives the group and observation counts).

Example. Hypertensive on Age and BMI with a random intercept for StudyZone — how much does the odds of hypertension vary between zones after the individual predictors?

The within-subjects / multivariate family

Also under the modelling menus:

• Repeated Measures ANOVA — for a factor measured several times on the same cases, with Mauchly's test of sphericity and the Greenhouse-Geisser / Huynh-Feldt corrections.
• One-Way MANOVA — several dependent variables at once, reporting Pillai's trace, Wilks' lambda, Hotelling-Lawley and Roy's largest root.

Weighting. Repeated Measures ANOVA (its within-subjects partition) and MANOVA honour the Weight Cases setting: the case weight is a frequency weight, so a weight-2 case counts exactly as two — in the within-subjects sums of squares and the error/hypothesis scatter matrices alike.

How this is computed

This section names which method each procedure prints — the estimator, the statistic, its degrees of freedom, and the tie, missing-value and weight rules — and links the recorded convention that explains the choice. The full index is the convention inventory; every number here is validated against independent packages per the validation suite.

• Correlate — Pearson r with t on n − 2 df; Spearman on mid-ranks with the t approximation (not R's exact form); Kendall's τ-b with the normal approximation; pairwise deletion by default, listwise on request; no CI on r (correlation). Partial correlation on n − 2 − k df (partial).
• Linear — least squares by QR, a scale-invariant singularity verdict, t intervals on the residual df, standardized β = B·sd(x)/sd(y), Durbin–Watson on case order, tolerance / VIF (QR · scale invariance); through the origin reports the uncentred R² and F (origin); stepwise enters at p ≤ .05 and removes at p ≥ .10 (stepwise).
• Regularized — standardized predictors, a 100-point λ path, 10-fold CV with the minimum and 1-SE λ (regularized). Robust — Huber c = 1.345 / Tukey c = 4.685 on the MAD/0.6745 scale, sandwich SEs (robust). Nonlinear — Levenberg–Marquardt, Gauss–Newton covariance (nonlinear). 2SLS — σ̂² on the structural residuals, the first-stage F (validation).
• Logistic / Probit / Poisson / Negative binomial / Gamma — IRLS through the QR (IRLS); Wald z intervals, Exp(B) = exp(B ± z·SE) (R's confint.glm is profile-likelihood — use confint.default) (Wald); Hosmer–Lemeshow on ten deciles of risk, g − 2 df (validation); NB2 with α by ML (tolerances); the Gamma dispersion is the Pearson estimate and the likelihood is evaluated at it (R's logLik uses D/n) (Gamma); the diagnostic residuals are randomized quantile residuals, seeded (quantile residuals).
• Multinomial / Ordinal — the lowest category is the baseline; the ordinal model is logit P(Y ≤ c) = θ_c − β·x (SPSS PLUM, polr) (reference and sign). Quantile — IRLS on the check loss, Koenker iid SEs; GEE — sandwich covariance under the chosen working correlation; Loglinear — a Poisson GLM on the counts (validation · loglinear).
• Mediation — OLS paths on the PROCESS model shapes, Aiken–West probes at mean ∓ 1 SD, percentile / BCa bootstrap intervals from a seeded PRNG, Johnson–Neyman regions (mediation · tolerances).
• General Linear Model — Type III sums of squares under sum-to-zero contrasts (Type III); MANOVA's Pillai / Wilks / Hotelling–Lawley / Roy and the SPSS partial η²; repeated measures with Mauchly's test and the Greenhouse–Geisser / Huynh–Feldt ε (validation).
• Linear Mixed Model — REML by default (untick for ML when comparing fixed effects by likelihood ratio), Swamy-style variance components, the ICC (mixed models). GLMM — a random intercept integrated by Gauss–Hermite quadrature (same section).

Choosing among them

• One outcome, continuous → Linear.
• One outcome, categorical → the matching Logistic / Poisson / Multinomial / Ordinal member.
• Several categorical factors (± covariates) on one continuous outcome → GLM.
• Repeated or clustered structure → Mixed or Repeated Measures.
• Several outcomes together → MANOVA.

See Correlation vs regression if you are unsure whether you want association or prediction.