Compare Means

Analyze → Compare Means. These procedures ask whether the mean of a scale variable differs: between groups, against a fixed value, or across paired measurements.

Means

When to use it. To get summary statistics for a scale variable broken down by the categories of one or more factors (a descriptive breakdown, not a test).

Dialog. A Dependent scale variable and one or more Independent (grouping) variables.

Output. A table of the dependent's mean, N and standard deviation within each group (and overall).

Example. Mean SystolicBP by StudyZone and by SmokingStatus.

T Tests

All three live under Compare Means and each can be run one- or two-tailed (set in the Options section).

One-Sample T Test

When. Compare a variable's mean against a known test value.

Dialog. The Test Variable(s) and the Test Value.

Output. The mean, the t statistic, degrees of freedom, the significance, and the confidence interval of the difference.

Example. Is mean BMI different from 25 (the overweight threshold)?

Independent-Samples T Test

When. Compare the means of two independent groups.

Dialog. A Test Variable and a Grouping Variable (define the two groups by their values).

Output. Group statistics, Levene's test for equal variances, and the t-test reported both ways (equal and unequal variances assumed) with the CI of the difference.

Example. Does mean FastingGlucose_mmolL differ between Diabetic = Yes and No?

Permutation test

Tick Permutation test to also test the mean difference by randomization: the group labels are reshuffled over and over, and the permutation p is simply the share of relabelings whose mean difference is at least as extreme as the one observed. Nothing is assumed about distributions (no normality, no equal variances), which makes it the methods-course answer for small samples and odd-shaped data, while still testing the mean difference (unlike Mann–Whitney, which tests a rank-based shift).

How it runs, automatically:

• Small samples are exact: when every possible group assignment can be enumerated (up to a million arrangements; balanced designs up to about n = 22), the table says Exact with the arrangement count, and the p is the exact share, not an estimate. An exact p can never be 0: the observed assignment is always one of the arrangements.
• Larger samples are Monte Carlo: the table says Monte Carlo with the resample count (default 10,000, adjustable in the dialog), the p is estimated as (b+1)/(m+1), and Lower/Upper bracket the simulation uncertainty of that p at 99%: how much it could move with a different seed, not an interval for the effect. A fixed Seed makes the run exactly reproducible; rerunning with the same seed gives the same p to the last digit.

The test direction follows the dialog's tails setting (one-tailed counts extremes in the observed direction, like the t test's one-tailed p). Permutation is about rearranging the observed cases, so it is not available while Weight Cases is on.

Paired-Samples T Test

When. Compare two measurements on the same cases.

Dialog. One or more variable pairs.

Output. The paired differences' mean, the t statistic, df, significance and CI.

Example. SystolicBP vs DiastolicBP (illustrative paired comparison).

Comparing two measurement methods on the same subjects? The paired t answers only whether the average difference is zero. The agreement question (how far apart the methods get case by case) is the Bland-Altman analysis's job.

Equivalence T Test (TOST)

Analyze → Compare Means → Equivalence T Test (TOST)…

When to use it. To show two means are practically equivalent. A t test cannot answer this question: "not significantly different" is not "equivalent". You declare how big a difference would still be negligible (the equivalence bounds), and the TOST procedure (two one-sided tests) tests whether the true difference lies inside those bounds. Clinical and, increasingly, psychology journals ask for exactly this when a study's point is that a difference is absent or ignorable.

Dialog. The same three designs as the classical t tests, behind a One-Sample / Paired / Independent selector, plus:

• Bounds: the lower and upper equivalence bounds (lower < upper; they need not be symmetric). For a one-sample test they are centered on the Test Value: bounds of −1.5 to 1.5 with test value 100 declare "anything between 98.5 and 101.5 is equivalent to 100".
• Bound units: Raw units (the default: the variable's own scale) or Cohen's d, converted to raw units per design with the sample standardizer. Raw bounds are preferred when the scale is meaningful: d-scale bounds move with the sample's own spread, a caution the output note repeats.
• Alpha: the level of each one-sided test (default 0.05). The interval shown is the 1 − 2α confidence interval (90% at the default), the standard TOST presentation.

Output. Two tables:

• Equivalence Test (TOST) shows, per variable (or pair; the independent design keeps the familiar two rows, equal variances assumed / not assumed): the mean difference, its standard error and df, both one-sided tests (t and one-sided significance against each bound), the TOST Sig. (the larger of the two one-sided p's) and a printed Decision: Equivalent when TOST Sig. < α, else Not equivalent.
• Equivalence Bounds and Confidence Interval shows the bounds in raw units (as converted, when declared in Cohen's d) beside the 1 − 2α CI of the difference. The two views agree by construction: the difference is equivalent at α exactly when the interval lies inside the bounds.

Example. A generic drug's mean SystolicBP effect vs the brand compound, with equivalence declared as ±3 mmHg: TOST Sig. = 0.02 → Equivalent: the 90% CI of the difference sits entirely inside −3…3.

Reading it with the t test. The four corners are all possible: a difference can be significant yet equivalent (precisely estimated and trivially small), or non-significant yet not equivalent (too little data to claim anything). Run both when the audience expects both.

One-Way ANOVA

When to use it. Compare the means of a scale variable across three or more groups.

Dialog. A Dependent scale variable and a Factor (the grouping variable). The Options section adds:

• Levene's test of homogeneity of variances;
• the Welch robust test (for unequal variances);
• post-hoc comparisons: Bonferroni, Scheffé, Tukey HSD;
• a-priori contrasts (custom weighted comparisons).

Output. The ANOVA table (between/within sums of squares, F, significance), plus any homogeneity test, post-hoc table and contrast results you requested.

Example. Mean SystolicBP across the levels of EducationLevel, with Tukey HSD to see which education levels differ.

Bayesian t test

A Bayesian counterpart to the three t tests, under Analyze → Bayesian → t Test…. Where a classical t test gives a p-value (which can only ever fail to reject the null, never support it), a Bayesian t test reports a Bayes factor that grades the evidence in either direction.

When to use it. Whenever you would run a one-sample, paired or independent-samples t test but want to quantify evidence for the null as well as against it, or want a posterior on the effect size rather than a confidence interval. (To argue equivalence against declared bounds, which is the frequentist route journals usually ask for, see the Equivalence T Test (TOST) above.)

Dialog. A One-Sample / Paired / Independent selector (the same three designs as the classical t tests), the test variables, and:

• Prior scale r: the width of the Cauchy prior on the standardized effect size δ; the default 0.707 (≈ 1/√2) is the standard "medium" scale.
• Credible %: the level for the posterior interval of δ (default 95).

Output. A Bayesian t Test table with, per variable/pair:

• BF₁₀, the evidence for a difference, and BF₀₁ = 1/BF₁₀, the evidence for the null. A BF₁₀ of 6 means the data are 6× more likely under a real effect; a BF₁₀ of 0.2 (BF₀₁ = 5) means they are 5× more likely under the null.
• Evidence: a verbal category for the Bayes factor (Anecdotal, Moderate, Strong, Very strong, Extreme), tagged with the favoured hypothesis.
• Median δ and its credible interval: the posterior for the standardized effect size.

Example. A paired Bayesian t test of Pre vs Post: a BF₁₀ of 0.3 ("Moderate (H0)") is positive evidence that the intervention made no difference, a conclusion a non-significant p-value could not have supported.

Bayesian correlation

Also under Analyze → Bayesian → Correlation…: the Bayesian counterpart to bivariate correlation. Pick two or more numeric variables; each pair is reported with its Pearson r, the BF₁₀ / BF₀₁ (evidence for / against a correlation) and evidence category, and the posterior median and credible interval of ρ.

• Prior width κ: the stretched-beta prior on ρ; the default 1 is a uniform prior on −1…1. (Smaller κ pulls the prior toward 0.)
• Credible %: the level for the posterior interval of ρ.

A BF₁₀ well above 1 is evidence the two variables are correlated; a BF₀₁ above 1 is evidence they are not. A significance test on r cannot deliver the latter.

Bayesian ANOVA

Under Analyze → Bayesian → ANOVA…: the Bayesian counterpart to the one-way ANOVA. Pick one or more numeric dependent variables and a factor; each is reported with its group count, η² (the variance explained), and the BF₁₀ / BF₀₁ for the group-means model against the no-effect null, i.e. whether the factor matters at all.

• Prior scale r: the Cauchy prior on the standardized group effects; the default 0.5 is the standard "medium" fixed-effect scale.

A BF₁₀ above 1 is evidence the groups differ; a BF₀₁ above 1 is evidence they do not. That is direct support for a null that an ANOVA F test cannot provide.

Bayesian linear regression

Under Analyze → Bayesian → Linear Regression…: the Bayesian counterpart to linear regression. Pick a dependent variable and one or more numeric predictors. It reports:

• Coefficients: each predictor's posterior mean (the OLS estimate under the diffuse prior) and its credible interval.
• Model summary: N, the predictor count, R², and the residual variance posterior (mean and credible interval).
• Bayes factor: BF₁₀ / BF₀₁ for the full model against the intercept-only null (whether the predictors collectively matter).

Controls: a Prior scale r (the g-prior on the effects; default √2/4 ≈ 0.354) and a Credible % for the intervals. A BF₁₀ above 1 favours the predictors; a BF₀₁ above 1 is evidence that an intercept-only model suffices.

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. Each line 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.

• T tests: one-sample, paired and independent t with t confidence intervals; the independent test prints both the pooled row (df n₁ + n₂ − 2) and the Welch row (Welch–Satterthwaite df) beside Levene's test, and never switches for you (both rows). Cohen's d uses the pooled n − 1 SD (paired: d_z on the SD of the differences), Hedges' g the factor 1 − 3/(4df − 1), and the CI on d the noncentral t (effect sizes).
• Levene's test: mean-centred, the original form (R and SciPy default to the median-centred Brown–Forsythe; set the centre to reproduce) (Levene).
• Equivalence (TOST): bounds centred on the test value, the 1 − 2α interval, Welch by default (TOST).
• One-way ANOVA: the classic between / within partition (k − 1, N − k df); η², ε² and ω² in the Olejnik–Algina forms (ANOVA effect sizes); Welch's robust F*; planned contrasts both pooled and Welch–Satterthwaite; post hoc Bonferroni, Scheffé and Tukey HSD on the pooled within MS (post hoc).
• Bayesian t test / correlation / ANOVA / regression: JZS Bayes factors with the Cauchy prior scale r = 0.707, the stretched-beta prior width κ = 1 for ρ, the Zellner–Siow g-prior (validation).

Choosing among them

If a normality or variance assumption looks shaky (check it in Explore), consider the nonparametric equivalents, or keep the mean as the statistic and add the permutation test.