SimpleStats · Formulas and sources · נוסחאות ומקורות · Formules et sources

Numerical verification

54 checks passed · 2026-10-25T21:04:33.321924+00:00

• t True two-sided — passed
• CI True two-sided — passed
• t True less — passed
• CI True less — passed
• t True greater — passed
• CI True greater — passed
• t False two-sided — passed
• CI False two-sided — passed
• t False less — passed
• CI False less — passed
• t False greater — passed
• CI False greater — passed
• Mann Whitney rank sum — passed
• paired t and dz — passed
• descriptive — passed
• Pearson r and p — passed
• Spearman exhaustive tied permutation — passed
• ANOVA F p eta — passed
• Brown Forsythe F — passed
• chi square and Cramer V — passed
• OLS coefficients SE t p — passed
• HC3 sandwich SE — passed
• HC3 Wald F — passed
• Koenker BP LM — passed
• White LM — passed
• Jarque Bera — passed
• Durbin Watson — passed
• BG lag1 LM — passed
• Cook leverage internal residual — passed
• VIF — passed
• RESET HC3 joint Wald — passed
• Cronbach covariance identity — passed
• Shapiro official fixture W,p — passed
• ttest achieved power 2 — passed
• ttest minimal balanced allocation 2 — passed
• anova achieved power 3 — passed
• anova minimal balanced allocation 3 — passed
• anova achieved power 4 — passed
• anova minimal balanced allocation 4 — passed
• paired duplicate rejection — passed
• regression duplicate rejection — passed
• alpha duplicate rejection — passed
• constant predictor rejection — passed
• Mann Whitney exact p — passed
• Mann Whitney tied asymptotic p — passed
• Spearman seeded repeat — passed
• Spearman large sample p — passed
• complete-pair missing accounting — passed
• one constant group estimable — passed
• infinite values rejected — passed
• en formula and APA attachment — passed
• he formula and APA attachment — passed
• fr formula and APA attachment — passed
• complete methods registry and citations — passed

Verification limits

• Finite fixture suite, not proof for every possible dataset.
• Shapiro uses a published numerical fixture rather than independent reimplementation of AS R94.
• Noncentral power implementations differ by about 1e-6; power comparison tolerance is 2e-6. Shared distribution CDFs are not independently reimplemented.
• Screening cutoffs and design independence are not hypothesis tests.

Intellectus menu coverage

54 entries observed in the authenticated project analysis menu on 2026-10-25. Every listed entry maps to an implemented SimpleStats method with an explicit supported design.

Supported scope

• Matching: 1:1 logit-caliper without replacement, ≤10,000 complete cases; matched-treated target and conditional paired inference.
• Weighting: ATE, ATT and overlap; optional fixed sampling weights. Independent-row variance is not a complex survey design variance.
• Repeated models: one within-subject occasion factor and at most one between-subject grouping factor, in wide complete-subject format; orthonormal time contrasts and GG correction for univariate effects.
• HLM: Gaussian random-intercept model; no arbitrary crossed/random-slope specification.
• Mediation: one continuous mediator in linear models; percentile bootstrap.
• SEM/CFA/path: restricted declarative covariance models. PLS: reflective Mode A, recursive disjoint blocks, factorial inner weighting.
• Time series: nonseasonal user-specified ARIMA.
• This is method coverage, not identical parameter-option or licensed product feature parity.

Extended verification

54 method/variant configurations passed, including all 50 new methods.

29 additional numerical checks passed.

• one sample t, p, CI
• one sample known-SD z
• paired known difference-SD z
• independent known-SD z
• one proportion score z
• two proportion pooled z
• Kendall concordance enumeration
• partial residual correlation
• Friedman rank sums
• Kruskal rank sums
• Fisher hypergeometric sum
• McNemar discordant binomial
• ANCOVA direct group contrast
• repeated ANOVA sums of squares
• binary logit independent likelihood coefficients
• binary logit log likelihood
• linear mediation decomposition
• moderation OLS coefficients
• Kaplan Meier risk product
• matched pair effect
• caliper and no-replacement matching
• ATE normalized weighted contrast
• ATE joint nuisance-adjusted sandwich SE
• ATT normalized weighted contrast
• ATT joint nuisance-adjusted sandwich SE
• overlap normalized weighted contrast
• overlap joint nuisance-adjusted sandwich SE
• SEM MLW covariance discrepancy
• AR1 stationary Gaussian likelihood

All 50 new methods passed preview, save/reload, source CSV fingerprint and Word export checks in English, Hebrew and French.

Descriptive statistics
סטטיסטיקה תיאורית · Statistiques descriptives

mean = Σx/n; s² = Σ(x−mean)²/(n−1); SD = √s²; median = middle ordered value (average of two if n even).

Available cases separately per variable. Sample SD is undefined with one observation. Minimum and maximum are observed extremes.

References · APA

Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Routledge. https://doi.org/10.4324/9780203771587

Student independent t-test
מבחן t בלתי תלוי עם שונות מאוחדת · Test t indépendant à variance commune

sp² = [(n1−1)s1²+(n2−1)s2²]/(n1+n2−2)
t = (mean1−mean2)/√[sp²(1/n1+1/n2)]; df = n1+n2−2
d = (mean1−mean2)/sp

Independent observations, normally distributed errors for exact small-sample inference, common population variance. Group order is first appearance; the sign follows group 1 minus group 2. Automatic selection using Levene is a data-dependent choice, not a guarantee of error control.

References · APA

Student. (1908). The probable error of a mean. Biometrika, 6(1), 1–25. https://doi.org/10.1093/biomet/6.1.1

Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Routledge. https://doi.org/10.4324/9780203771587

Welch independent t-test
מבחן t של ולש · Test t de Welch

a=s1²/n1; b=s2²/n2
t=(mean1−mean2)/√(a+b); df=(a+b)²/[a²/(n1−1)+b²/(n2−1)]

Allows unequal population variances. Observations must be independent. Signed Cohen d still uses pooled sample SD and is distinct from the Welch standard error.

References · APA

Welch, B. L. (1947). The generalization of ‘student's’ problem when several different population variances are involved. Biometrika, 34(1–2), 28–35. https://doi.org/10.1093/biomet/34.1-2.28

Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Routledge. https://doi.org/10.4324/9780203771587

Paired t-test
מבחן t מזווג · Test t apparié

di=xi−yi; t=mean(d)/(sd(d)/√n); df=n−1; dz=mean(d)/sd(d)

Complete pairs only. Pairs must be independent of other pairs; exact t inference assumes normal differences, not normality of each original measurement.

References · APA

Student. (1908). The probable error of a mean. Biometrika, 6(1), 1–25. https://doi.org/10.1093/biomet/6.1.1

Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Routledge. https://doi.org/10.4324/9780203771587

Mean / difference confidence interval
רווח סמך לממוצע או להפרש · Intervalle de confiance de moyenne ou différence

CI = estimate ± t(1−α/2,df) × SE
p(two-sided)=2×P(Tdf≥|t|); p(less)=P(Tdf≤t); p(greater)=P(Tdf≥t)

Test intervals are two-sided at 1−alpha, even for a one-sided test. Mean charts use fixed 95% t intervals, not simultaneous intervals. Confidence is a repeated-sampling property.

References · APA

Student. (1908). The probable error of a mean. Biometrika, 6(1), 1–25. https://doi.org/10.1093/biomet/6.1.1

Median-centered Levene / Brown–Forsythe
לווין ממורכז בחציון / בראון–פורסיית · Levene centré sur la médiane / Brown–Forsythe

zij=|xij−median(group j)|
W = [Σnj(mean(zj)−mean(z))²/(k−1)] / [ΣΣ(zij−mean(zj))²/(N−k)]
W ~ F(k−1,N−k) approximately

Tests equality of group variances. A nonsignificant result does not establish equal variances. Undefined when deviations have no residual variation.

References · APA

Brown, M. B., & Forsythe, A. B. (1974). Robust tests for the equality of variances. Journal of the American Statistical Association, 69(346), 364–367. https://doi.org/10.1080/01621459.1974.10482955

Shapiro–Wilk
שפירו–וילק · Shapiro–Wilk

W = (Σ ai x(i))² / Σ(xi−mean(x))²
a = mᵀV⁻¹ / √(mᵀV⁻¹V⁻¹m)

Ordered observations x(i); m and V are expected normal order statistics and their covariance. SciPy uses numerical coefficient/p-value approximations. Applied up to 5,000 observations; residual tests are diagnostic and affected by fitted-model constraints.

References · APA

Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3–4), 591–611. https://doi.org/10.1093/biomet/52.3-4.591

Mann–Whitney U
מאן–ויטני U · Mann–Whitney U

U1=R1−n1(n1+1)/2; U2=n1n2−U1
E(U)=n1n2/2
Var(U)=n1n2[(N+1)−Σ(tj³−tj)/(N(N−1))]/12

R1 is the group-1 rank sum; tj are tie-block sizes. SciPy auto uses exact inference when the smaller sample is at most 8 and no ties, otherwise tie-corrected normal approximation with 0.5 continuity correction. Tests distributions; a median-shift interpretation needs comparable shape.

References · APA

Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. The Annals of Mathematical Statistics, 18(1), 50–60. https://doi.org/10.1214/aoms/1177730491

Pearson correlation
מתאם פירסון · Corrélation de Pearson

r=Σ(x−mean(x))(y−mean(y))/√[Σ(x−mean(x))²Σ(y−mean(y))²]
t=r√[(n−2)/(1−r²)]; df=n−2

Complete pairs. Exact null p-value assumes independent bivariate-normal observations and zero population correlation; coefficient measures linear association, not causation.

References · APA

Student. (1908). The probable error of a mean. Biometrika, 6(1), 1–25. https://doi.org/10.1093/biomet/6.1.1

Spearman correlation
מתאם ספירמן · Corrélation de Spearman

ρs = PearsonCorrelation(averageRank(x), averageRank(y))
p_exact=min(1,2 min[Prperm(T≤Tobs),Prperm(T≥Tobs)])
p_MC=min(1,2 min[(b_lower+1)/(B+1),(b_upper+1)/(B+1)])

Average ranks handle ties. Up to n=8 enumerate all pairings; n=9–500 use B=9,999 permutations and seed 20261025; above 500 use t approximation with n−2 df. Permutations require exchangeable observations under independence; this is stronger than merely zero rank correlation.

References · APA

Spearman, C. (1904). The proof and measurement of association between two things. The American Journal of Psychology, 15(1), 72–101. https://doi.org/10.2307/1412159

Phipson, B., & Smyth, G. K. (2010). Permutation p-values should never be zero: calculating exact p-values when permutations are randomly drawn. Statistical Applications in Genetics and Molecular Biology, 9(1). https://doi.org/10.2202/1544-6115.1585

Classical one-way ANOVA
ניתוח שונות חד־כיווני · ANOVA classique à un facteur

SSB=Σnj(meanj−grandMean)²; SSW=ΣΣ(xij−meanj)²
F=[SSB/(k−1)]/[SSW/(N−k)]; η²=SSB/(SSB+SSW)

Independent observations, equal population variances and normal errors for exact inference. Omnibus comparison; no automatic pairwise post-hoc tests. Eta squared is observed variance explained, not causal effect.

References · APA

Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Routledge. https://doi.org/10.4324/9780203771587

Pearson chi-square and Cramér V
חי־בריבוע של פירסון ו־V של קרמר · Khi-deux de Pearson et V de Cramér

Eij=rowTotali×columnTotalj/N
χ²=ΣΣ(Oij−Eij)²/Eij; df=(r−1)(c−1)
V=√[χ²/(N min(r−1,c−1))]

Independent count observations, mutually exclusive categories. No Yates correction and no bias correction for V. Sparse expected cells trigger a warning; the app does not substitute an exact test.

References · APA

Pearson, K. (1900). on the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 50(302), 157–175. https://doi.org/10.1080/14786440009463897

Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Routledge. https://doi.org/10.4324/9780203771587

Ordinary least squares
ריבועים פחותים רגילים · Moindres carrés ordinaires

β̂=(XᵀX)⁻¹Xᵀy; e=y−Xβ̂; s²=eᵀe/(n−p)
Cov(β̂)=s²(XᵀX)⁻¹; t=β̂j/SEj; df=n−p
R²=1−SSE/SST; adjusted R²=1−(1−R²)(n−1)/(n−p)
F=[(SST−SSE)/(p−1)]/[SSE/(n−p)]

X includes the intercept; p counts all coefficients. Full-rank predictors required. Conventional t/F inference assumes correct linear conditional mean, independent homoskedastic normal errors for exact finite-sample inference. No causal interpretation without a justified design.

References · APA

Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Routledge. https://doi.org/10.4324/9780203771587

HC3 robust inference
הסקה עמידה HC3 · Inférence robuste HC3

A=(XᵀX)⁻¹; h=diag(XAXᵀ)
V_HC3=A Xᵀ diag[e²/(1−h)²] X A
tj=β̂j/√Vjj; df=n−p
F_Wald=(Rβ̂)ᵀ(RVRᵀ)⁻¹(Rβ̂)/q

R selects the q non-intercept coefficients. t and F reference distributions are finite-sample approximations for heteroskedasticity-robust inference. HC3 does not correct serial/cluster dependence or misspecified mean functions.

References · APA

MacKinnon, J. G., & White, H. (1985). Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics, 29(3), 305–325. https://doi.org/10.1016/0304-4076(85)90158-7

Ramsey RESET with HC3
RESET של רמזי עם HC3 · RESET de Ramsey avec HC3

Augment X with fitted(y)² and fitted(y)³
F=(Rβ̂)ᵀ(R V_HC3 Rᵀ)⁻¹(Rβ̂)/q; q=rank of tested restrictions

Joint test of added fitted-value powers; rejects some functional-form errors but cannot identify the correct alternative model. Robust Wald F approximation. Unavailable when augmented design is not estimable.

References · APA

Ramsey, J. B. (1969). Tests for specification errors in classical linear least-squares regression analysis. Journal of the Royal Statistical Society Series B: Statistical Methodology, 31(2), 350–371. https://doi.org/10.1111/j.2517-6161.1969.tb00796.x

MacKinnon, J. G., & White, H. (1985). Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics, 29(3), 305–325. https://doi.org/10.1016/0304-4076(85)90158-7

Koenker–Breusch–Pagan
קנקר–ברוש–פייגן · Koenker–Breusch–Pagan

Regress e² on X (including intercept)
LM=n×R²_aux; LM ~ χ²(p−1)

Studentized Koenker version, not original normality-dependent BP statistic. Tests variance related to regressors; asymptotic reference.

References · APA

Koenker, R. (1981). A note on studentizing a test for heteroscedasticity. Journal of Econometrics, 17(1), 107–112. https://doi.org/10.1016/0304-4076(81)90062-2

Breusch, T. S., & Pagan, A. R. (1979). A simple test for heteroscedasticity and random coefficient variation. Econometrica, 47(5), 1287–1294. https://doi.org/10.2307/1911963

White heteroskedasticity test
מבחן וייט לשונות לא קבועה · Test d’hétéroscédasticité de White

Z={all unique XjXk, j≤k}; regress e² on Z
LM=n×R²_aux; df=rank(Z)−1; LM ~ χ²(df)

Includes squares and cross-products; skipped if nominal quadratic term count is at least min(n−2,500). Rank determines degrees of freedom.

References · APA

White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48(4), 817–838. https://doi.org/10.2307/1912934

Jarque–Bera
ז׳ארק–ברה · Jarque–Bera

mk=mean[(e−mean(e))^k]; S=m3/m2^(3/2); K=m4/m2²
JB=n[S²/6+(K−3)²/24]; JB ~ χ²(2)

Asymptotic normality diagnostic using unadjusted central moments; finite-sample residual constraints affect calibration.

References · APA

Jarque, C. M., & Bera, A. K. (1980). Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Economics Letters, 6(3), 255–259. https://doi.org/10.1016/0165-1765(80)90024-5

Variance inflation factor
גורם ניפוח שונות · Facteur d’inflation de variance

VIFj=1/(1−Rj²)
Rj² from regressing predictor j on remaining predictors plus intercept

VIF above 5 is a screening heuristic, not a hypothesis test or mandatory rejection. Predictor correlation inverse gives equivalent values for larger designs.

References · APA

Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Routledge. https://doi.org/10.4324/9780203771587

Cook distance, leverage, studentized residual
מרחק קוק, מינוף ושארית מתוקננת · Distance de Cook, levier, résidu studentisé

H=X(XᵀX)⁻¹Xᵀ; hi=Hii
ri=ei/[s√(1−hi)]
Di=ri² hi/[p(1−hi)]

Internal studentization uses full-sample residual variance. Screening: Cook D>4/n, leverage>2p/n, |r|>3; not calibrated p-values. Source row numbers are retained; nothing is automatically removed.

References · APA

Cook, R. D. (1977). Detection of influential observation in linear regression. Technometrics, 19(1), 15–18. https://doi.org/10.1080/00401706.1977.10489493

Durbin–Watson
דרבין–ווטסון · Durbin–Watson

DW=Σ(i=2..n)(ei−e(i−1))² / Σei²

Uses current complete-case row order. Descriptive statistic only; no universal pass threshold or p-value. Does not establish independence between people or clusters.

References · APA

Durbin, J., & Watson, G. S. (1950). Testing for serial correlation in least squares regression. i. Biometrika, 37(3–4), 409–428. https://doi.org/10.1093/biomet/37.3-4.409

Breusch–Godfrey lag 1
ברוש–גודפרי בפיגור 1 · Breusch–Godfrey au retard 1

Auxiliary regression: ei ~ X + e(i−1)
LM=n×R²_aux; LM ~ χ²(1)

First lagged residual is zero-padded by statsmodels. Meaningful row ordering required; asymptotic test. No proof of study-design independence.

References · APA

Godfrey, L. G. (1978). Testing for higher order serial correlation in regression equations when the regressors include lagged dependent variables. Econometrica, 46(6), 1303–1310. https://doi.org/10.2307/1913830

Breusch, T. S. (1978). Testing for autocorrelation in dynamic linear models*. Australian Economic Papers, 17(31), 334–355. https://doi.org/10.1111/j.1467-8454.1978.tb00635.x

Raw Cronbach alpha
אלפא גולמית של קרונבך · Alpha brut de Cronbach

α=k/(k−1) × [1−Σsj²/Var(Σitems)]

Complete-case rows; k≥2 items. Raw, unstandardized alpha can be negative. Reverse coding is not automatic. Alpha is not evidence of unidimensionality or validity; reliability interpretation requires additional measurement assumptions.

References · APA

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297–334. https://doi.org/10.1007/BF02310555

Independent t-test power
עוצמה למבחן t בלתי תלוי · Puissance du test t indépendant

df=2m−2; δ=d√(m/2); c=t(1−α/2,df)
Power=P(Tdf,δ<−c)+P(Tdf,δ>c)

Solve balanced per-group m, round up, report achieved power. Two-sided equal-variance t design; positive Cohen d. No attrition or cluster design effect.

References · APA

Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Routledge. https://doi.org/10.4324/9780203771587

Balanced ANOVA power
עוצמה לניתוח שונות מאוזן · Puissance de l’ANOVA équilibrée

N=km; df1=k−1; df2=N−k; λ=Nf²
c=F(1−α;df1,df2); Power=P(Fdf1,df2,λ>c)
f=√[Σπj(μj−μ)²/σ²]

Cohen f, equal group sizes πj=1/k. Solve total N, then round per-group m upward and return N=k×m plus achieved power. No repeated-measures or unequal-allocation design.

References · APA

Cohen, J. (2013). Statistical power analysis for the behavioral sciences. Routledge. https://doi.org/10.4324/9780203771587

Shapiro–Wilk test
מבחן שפירו–וילק · Test de Shapiro–Wilk

W=(Σ ai x(i))² / Σ(xi−mean(x))²; coefficients ai are based on normal order-statistic moments.

Normality is not established by a nonsignificant result. Estimated normal parameters require Lilliefors rather than the ordinary KS null. Outlier cutoffs are screening rules, not p-values.

References · APA

SciPy developers. (n.d.). Shapiro–Wilk test [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/generated/scipy.stats.shapiro.html

Normality: Kolmogorov–Smirnov / Lilliefors
נורמליות: קולמוגורוב–סמירנוב / ליליפורס · Normalité : Kolmogorov–Smirnov / Lilliefors

D=maxx |Fn(x)−Φ((x−mean(x))/s)|; Lilliefors null calibration accounts for estimated mean and sample SD.

Normality is not established by a nonsignificant result. Estimated normal parameters require Lilliefors rather than the ordinary KS null. Outlier cutoffs are screening rules, not p-values.

References · APA

Statsmodels developers. (n.d.). Normality: Kolmogorov–Smirnov / Lilliefors [Software documentation]. https://www.statsmodels.org/v0.14.6/generated/statsmodels.stats.diagnostic.lilliefors.html

Univariate outliers
תצפיות חריגות חד־משתניות · Valeurs atypiques univariées

IQR=Q3−Q1; lower=Q1−1.5 IQR; upper=Q3+1.5 IQR; zi=(xi−mean(x))/s. Flags are not p-values.

Normality is not established by a nonsignificant result. Estimated normal parameters require Lilliefors rather than the ordinary KS null. Outlier cutoffs are screening rules, not p-values.

References · APA

SciPy developers. (n.d.). Univariate outliers [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/stats.html

Levene / Brown–Forsythe test
מבחן לווין / בראון–פורסיית · Test de Levene / Brown–Forsythe

zij=|xij−median(group j)|; W=[Σnj(mean(zj)−mean(z))²/(k−1)]/[ΣΣ(zij−mean(zj))²/(N−k)].

Normality is not established by a nonsignificant result. Estimated normal parameters require Lilliefors rather than the ordinary KS null. Outlier cutoffs are screening rules, not p-values.

References · APA

SciPy developers. (n.d.). Levene / Brown–Forsythe test [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/generated/scipy.stats.levene.html

One-sample t-test
מבחן t למדגם יחיד · Test t à un échantillon

t=(mean−μ0)/(s/√n); df=n−1

Two-sided one-sample mean test with a two-sided confidence interval.

References · APA

SciPy developers. (n.d.). One-sample t-test [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/generated/scipy.stats.ttest_1samp.html

One-sample z-test
מבחן z למדגם יחיד · Test z à un échantillon

z=(estimate−null)/SE; SE=σ/√n or √(σ1²/n1+σ2²/n2)

Population standard deviations must be known and supplied. Paired z uses the known SD of differences, not the SD of each separate measurement.

References · APA

SciPy developers. (n.d.). One-sample z-test [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/stats.html

Paired z-test
מבחן z מזווג · Test z apparié

z=(estimate−null)/SE; SE=σ/√n or √(σ1²/n1+σ2²/n2)

Population standard deviations must be known and supplied. Paired z uses the known SD of differences, not the SD of each separate measurement.

References · APA

SciPy developers. (n.d.). Paired z-test [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/stats.html

Independent z-test
מבחן z בלתי תלוי · Test z indépendant

z=(estimate−null)/SE; SE=σ/√n or √(σ1²/n1+σ2²/n2)

Population standard deviations must be known and supplied. Paired z uses the known SD of differences, not the SD of each separate measurement.

References · APA

SciPy developers. (n.d.). Independent z-test [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/stats.html

One-proportion z-test
מבחן z לשיעור יחיד · Test z à une proportion

z=(p̂−p0)/√[p0(1−p0)/n]

Normal score test; Wilson confidence interval for the observed proportion. Sparse expected counts are flagged.

References · APA

Statsmodels developers. (n.d.). One-proportion z-test [Software documentation]. https://www.statsmodels.org/v0.14.6/generated/statsmodels.stats.proportion.proportions_ztest.html

Two-proportions z-test
מבחן z לשני שיעורים · Test z à deux proportions

z=(p̂1−p̂0)/√[p̂pool(1−p̂pool)(1/n1+1/n0)]

Pooled null standard error for testing; Newcombe interval for the difference.

References · APA

Statsmodels developers. (n.d.). Two-proportions z-test [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Kendall tau-b
טאו b של קנדל · Tau-b de Kendall

τb=(concordant−discordant)/√[(C+D+Tx)(C+D+Ty)]

Complete cases; no causal conclusion. Partial correlations adjust only for the selected measured covariates.

References · APA

SciPy developers. (n.d.). Kendall tau-b [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/generated/scipy.stats.kendalltau.html

Partial correlation
מתאם חלקי · Corrélation partielle

r=cor(resid(x~Z),resid(y~Z)); t=r√[(n−q−2)/(1−r²)]

Complete cases; no causal conclusion. Partial correlations adjust only for the selected measured covariates.

References · APA

SciPy developers. (n.d.). Partial correlation [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/stats.html

Point-biserial correlation
מתאם נקודתי־ביסריאלי · Corrélation point-bisériale

rpb=Pearson(y,indicator(group=first category))

Complete cases; no causal conclusion. Partial correlations adjust only for the selected measured covariates.

References · APA

SciPy developers. (n.d.). Point-biserial correlation [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/generated/scipy.stats.pointbiserialr.html

Factorial ANOVA
ניתוח שונות פקטוריאלי · ANOVA factorielle

F=[trace(H)/dfH]/[trace(E)/dfE]; adjusted df = ε×df for within-subject effects

Classical between-subject OLS or orthonormal within-subject contrasts. Repeated designs use Greenhouse–Geisser correction; report corrected and uncorrected df. Covariates are centered. Complete subjects, balanced measurement occasions.

References · APA

Statsmodels developers. (n.d.). Factorial ANOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

ANCOVA
ניתוח שונות משותפת · ANCOVA

F=[trace(H)/dfH]/[trace(E)/dfE]; adjusted df = ε×df for within-subject effects

Classical between-subject OLS or orthonormal within-subject contrasts. Repeated designs use Greenhouse–Geisser correction; report corrected and uncorrected df. Covariates are centered. Complete subjects, balanced measurement occasions.

References · APA

Statsmodels developers. (n.d.). ANCOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Repeated-measures ANOVA
ניתוח שונות למדידות חוזרות · ANOVA à mesures répétées

F=[trace(H)/dfH]/[trace(E)/dfE]; adjusted df = ε×df for within-subject effects
GG ε=trace(S)²/[(t−1)trace(S²)], where S is covariance of orthonormal time-contrast residuals.

Classical between-subject OLS or orthonormal within-subject contrasts. Repeated designs use Greenhouse–Geisser correction; report corrected and uncorrected df. Covariates are centered. Complete subjects, balanced measurement occasions.

References · APA

Statsmodels developers. (n.d.). Repeated-measures ANOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Repeated-measures ANCOVA
ניתוח שונות משותפת למדידות חוזרות · ANCOVA à mesures répétées

F=[trace(H)/dfH]/[trace(E)/dfE]; adjusted df = ε×df for within-subject effects
GG ε=trace(S)²/[(t−1)trace(S²)], where S is covariance of orthonormal time-contrast residuals.

Classical between-subject OLS or orthonormal within-subject contrasts. Repeated designs use Greenhouse–Geisser correction; report corrected and uncorrected df. Covariates are centered. Complete subjects, balanced measurement occasions.

References · APA

Statsmodels developers. (n.d.). Repeated-measures ANCOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Mixed-design ANOVA
ניתוח שונות במערך מעורב · ANOVA mixte

F=[trace(H)/dfH]/[trace(E)/dfE]; adjusted df = ε×df for within-subject effects
GG ε=trace(S)²/[(t−1)trace(S²)], where S is covariance of orthonormal time-contrast residuals.

Classical between-subject OLS or orthonormal within-subject contrasts. Repeated designs use Greenhouse–Geisser correction; report corrected and uncorrected df. Covariates are centered. Complete subjects, balanced measurement occasions.

References · APA

Statsmodels developers. (n.d.). Mixed-design ANOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Mixed-design ANCOVA
ניתוח שונות משותפת במערך מעורב · ANCOVA mixte

F=[trace(H)/dfH]/[trace(E)/dfE]; adjusted df = ε×df for within-subject effects
GG ε=trace(S)²/[(t−1)trace(S²)], where S is covariance of orthonormal time-contrast residuals.

Classical between-subject OLS or orthonormal within-subject contrasts. Repeated designs use Greenhouse–Geisser correction; report corrected and uncorrected df. Covariates are centered. Complete subjects, balanced measurement occasions.

References · APA

Statsmodels developers. (n.d.). Mixed-design ANCOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

MANOVA
ניתוח שונות רב־משתני · MANOVA

Wilks Λ=|E|/|E+H|; Pillai V=trace[H(H+E)⁻¹]

Multivariate linear hypothesis tests; within-subject responses are orthonormal time contrasts. Requires nonsingular residual covariance and sufficient complete subjects. No sphericity assumption for multivariate tests.

References · APA

Statsmodels developers. (n.d.). MANOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

MANCOVA
ניתוח שונות משותפת רב־משתני · MANCOVA

Wilks Λ=|E|/|E+H|; Pillai V=trace[H(H+E)⁻¹]

Multivariate linear hypothesis tests; within-subject responses are orthonormal time contrasts. Requires nonsingular residual covariance and sufficient complete subjects. No sphericity assumption for multivariate tests.

References · APA

Statsmodels developers. (n.d.). MANCOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Repeated-measures MANOVA
ניתוח שונות רב־משתני למדידות חוזרות · MANOVA à mesures répétées

Wilks Λ=|E|/|E+H|; Pillai V=trace[H(H+E)⁻¹]

Multivariate linear hypothesis tests; within-subject responses are orthonormal time contrasts. Requires nonsingular residual covariance and sufficient complete subjects. No sphericity assumption for multivariate tests.

References · APA

Statsmodels developers. (n.d.). Repeated-measures MANOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Repeated-measures MANCOVA
ניתוח שונות משותפת רב־משתני למדידות חוזרות · MANCOVA à mesures répétées

Wilks Λ=|E|/|E+H|; Pillai V=trace[H(H+E)⁻¹]

Multivariate linear hypothesis tests; within-subject responses are orthonormal time contrasts. Requires nonsingular residual covariance and sufficient complete subjects. No sphericity assumption for multivariate tests.

References · APA

Statsmodels developers. (n.d.). Repeated-measures MANCOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Mixed-design MANOVA
ניתוח שונות רב־משתני במערך מעורב · MANOVA mixte

Wilks Λ=|E|/|E+H|; Pillai V=trace[H(H+E)⁻¹]

Multivariate linear hypothesis tests; within-subject responses are orthonormal time contrasts. Requires nonsingular residual covariance and sufficient complete subjects. No sphericity assumption for multivariate tests.

References · APA

Statsmodels developers. (n.d.). Mixed-design MANOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Mixed-design MANCOVA
ניתוח שונות משותפת רב־משתני במערך מעורב · MANCOVA mixte

Wilks Λ=|E|/|E+H|; Pillai V=trace[H(H+E)⁻¹]

Multivariate linear hypothesis tests; within-subject responses are orthonormal time contrasts. Requires nonsingular residual covariance and sufficient complete subjects. No sphericity assumption for multivariate tests.

References · APA

Statsmodels developers. (n.d.). Mixed-design MANCOVA [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Binary logistic regression
רגרסיה לוגיסטית בינארית · Régression logistique binaire

logit Pr(Y=event|X)=Xβ; OR=exp(β)

Categorical predictors are dummy-coded with an explicit reported reference. Ordinal categories must be ordered by the user. Convergence and model rank are checked; separation is not treated as a valid fitted model.

References · APA

Statsmodels developers. (n.d.). Binary logistic regression [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Multinomial logistic regression
רגרסיה לוגיסטית מולטינומיאלית · Régression logistique multinomiale

log[Pr(Y=j)/Pr(Y=reference)]=Xβj

Categorical predictors are dummy-coded with an explicit reported reference. Ordinal categories must be ordered by the user. Convergence and model rank are checked; separation is not treated as a valid fitted model.

References · APA

Statsmodels developers. (n.d.). Multinomial logistic regression [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Ordinal logistic regression
רגרסיה לוגיסטית סודרת · Régression logistique ordinale

logit Pr(Y≤j|X)=θj−Xβ

Categorical predictors are dummy-coded with an explicit reported reference. Ordinal categories must be ordered by the user. Convergence and model rank are checked; separation is not treated as a valid fitted model.

References · APA

Statsmodels developers. (n.d.). Ordinal logistic regression [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Hierarchical linear regression
רגרסיה ליניארית היררכית · Régression linéaire hiérarchique

ΔF=[(SSEreduced−SSEfull)/q]/[SSEfull/dfresid]

Categorical predictors are dummy-coded with an explicit reported reference. Ordinal categories must be ordered by the user. Convergence and model rank are checked; separation is not treated as a valid fitted model.

References · APA

Statsmodels developers. (n.d.). Hierarchical linear regression [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Hierarchical linear model
מודל ליניארי רב־רמתי · Modèle linéaire hiérarchique

Y=Xβ+Zu+ε; u~N(0,G), ε~N(0,σ²I)

Categorical predictors are dummy-coded with an explicit reported reference. Ordinal categories must be ordered by the user. Convergence and model rank are checked; separation is not treated as a valid fitted model.

References · APA

Statsmodels developers. (n.d.). Hierarchical linear model [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Mann–Whitney U
מאן–ויטני U · Mann–Whitney U

U1=R1−n1(n1+1)/2

Exact or tie-aware asymptotic inference is recorded. Distributional differences need not be differences of medians. McNemar requires paired binary measurements with the same category set.

References · APA

SciPy developers. (n.d.). Mann–Whitney U [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/generated/scipy.stats.mannwhitneyu.html

Wilcoxon signed-rank
וילקוקסון לדרגות חתומות · Wilcoxon signé

W=min(sum ranks of positive differences, sum ranks of negative differences)

Exact or tie-aware asymptotic inference is recorded. Distributional differences need not be differences of medians. McNemar requires paired binary measurements with the same category set.

References · APA

SciPy developers. (n.d.). Wilcoxon signed-rank [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/generated/scipy.stats.wilcoxon.html

Friedman test
מבחן פרידמן · Test de Friedman

Q=12 ΣRj²/[nk(k+1)]−3n(k+1); tie-corrected

Exact or tie-aware asymptotic inference is recorded. Distributional differences need not be differences of medians. McNemar requires paired binary measurements with the same category set.

References · APA

SciPy developers. (n.d.). Friedman test [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/generated/scipy.stats.friedmanchisquare.html

Kruskal–Wallis test
מבחן קרוסקל–ואליס · Test de Kruskal–Wallis

H=12 ΣRj²/nj/[N(N+1)]−3(N+1); tie-corrected

Exact or tie-aware asymptotic inference is recorded. Distributional differences need not be differences of medians. McNemar requires paired binary measurements with the same category set.

References · APA

SciPy developers. (n.d.). Kruskal–Wallis test [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/generated/scipy.stats.kruskal.html

Fisher exact test
המבחן המדויק של פישר · Test exact de Fisher

Pr(A=a | margins)=choose(c1,a) choose(N−c1,r1−a)/choose(N,r1)

Exact or tie-aware asymptotic inference is recorded. Distributional differences need not be differences of medians. McNemar requires paired binary measurements with the same category set.

References · APA

SciPy developers. (n.d.). Fisher exact test [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/generated/scipy.stats.fisher_exact.html

McNemar test
מבחן מקנמר · Test de McNemar

Exact binomial test on discordant pairs b,c with null probability 1/2

Exact or tie-aware asymptotic inference is recorded. Distributional differences need not be differences of medians. McNemar requires paired binary measurements with the same category set.

References · APA

Statsmodels developers. (n.d.). McNemar test [Software documentation]. https://www.statsmodels.org/v0.14.6/generated/statsmodels.stats.contingency_tables.mcnemar.html

Chi-square goodness of fit
חי־בריבוע לטיב התאמה · Khi-deux d’ajustement

χ²=Σ(Oj−Ej)²/Ej; df=k−1

Exact or tie-aware asymptotic inference is recorded. Distributional differences need not be differences of medians. McNemar requires paired binary measurements with the same category set.

References · APA

SciPy developers. (n.d.). Chi-square goodness of fit [Software documentation]. https://docs.scipy.org/doc/scipy-1.13.1/reference/generated/scipy.stats.chisquare.html

Mediation analysis
ניתוח תיווך · Analyse de médiation

Indirect=a×b; total=c; direct=c′; percentile CI from resampling complete subjects

Model-specific assumptions and estimation details are reported with the fitted result. Exploratory analyses do not establish causal effects; time-series order must be meaningful.

References · APA

Statsmodels developers. (n.d.). Mediation analysis [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Moderation analysis
ניתוח מיתון · Analyse de modération

Y=β0+β1Xc+β2Mc+β3XcMc+Zγ+ε

Model-specific assumptions and estimation details are reported with the fitted result. Exploratory analyses do not establish causal effects; time-series order must be meaningful.

References · APA

Statsmodels developers. (n.d.). Moderation analysis [Software documentation]. https://www.statsmodels.org/v0.14.6/stats.html

Survival analysis
ניתוח הישרדות · Analyse de survie

Ŝ(t)=Π(1−dj/nj); h(t|X)=h0(t) exp(Xβ)

Model-specific assumptions and estimation details are reported with the fitted result. Exploratory analyses do not establish causal effects; time-series order must be meaningful.

References · APA

Lifelines developers. (n.d.). Survival analysis [Software documentation]. https://lifelines.readthedocs.io/en/latest/Survival%20Regression.html

Exploratory factor analysis
ניתוח גורמים חוקר · Analyse factorielle exploratoire

Σ=ΛΦΛᵀ+Ψ; communality=diag(ΛΦΛᵀ); Φ=I for orthogonal factors, estimated factor correlation for oblique rotation.

Model-specific assumptions and estimation details are reported with the fitted result. Exploratory analyses do not establish causal effects; time-series order must be meaningful.

References · APA

Factor Analyzer developers. (n.d.). Exploratory factor analysis [Software documentation]. https://factor-analyzer.readthedocs.io/en/latest/factor_analyzer.html

K-means cluster analysis
ניתוח אשכולות K-means · Classification K-means

min Σclusters Σi ||zi−centroid||²

Model-specific assumptions and estimation details are reported with the fitted result. Exploratory analyses do not establish causal effects; time-series order must be meaningful.

References · APA

Scikit-learn developers. (n.d.). K-means cluster analysis [Software documentation]. https://scikit-learn.org/1.6/modules/generated/sklearn.cluster.KMeans.html

Linear discriminant analysis
ניתוח מבחין ליניארי · Analyse discriminante linéaire

δk(x)=xᵀΣ⁻¹μk−½μkᵀΣ⁻¹μk+log πk

Model-specific assumptions and estimation details are reported with the fitted result. Exploratory analyses do not establish causal effects; time-series order must be meaningful.

References · APA

Scikit-learn developers. (n.d.). Linear discriminant analysis [Software documentation]. https://scikit-learn.org/1.6/modules/generated/sklearn.discriminant_analysis.LinearDiscriminantAnalysis.html

Time series ARIMA
סדרות עתיות ARIMA · Séries temporelles ARIMA

φ(B)(1−B)^d yt=c+θ(B)εt

Model-specific assumptions and estimation details are reported with the fitted result. Exploratory analyses do not establish causal effects; time-series order must be meaningful.

References · APA

Statsmodels developers. (n.d.). Time series ARIMA [Software documentation]. https://www.statsmodels.org/v0.14.6/generated/statsmodels.tsa.arima.model.ARIMA.html

Path analysis
ניתוח נתיבים · Analyse de chemins

SEM: Σ(θ)=Λ(I−B)⁻¹ Ψ(I−B)⁻ᵀΛᵀ+Θ
PLS: weighted composite scores followed by structural regressions

SEM/CFA/path uses covariance-based maximum likelihood and reports convergence and model fit. PLS is composite-based Mode A path weighting, not covariance SEM. Model specification accepts only variable names and documented relation operators; no executable code.

References · APA

Semopy developers. (n.d.). Path analysis [Software documentation]. https://semopy.com/

Confirmatory factor analysis
ניתוח גורמים מאשש · Analyse factorielle confirmatoire

SEM: Σ(θ)=Λ(I−B)⁻¹ Ψ(I−B)⁻ᵀΛᵀ+Θ
PLS: weighted composite scores followed by structural regressions

SEM/CFA/path uses covariance-based maximum likelihood and reports convergence and model fit. PLS is composite-based Mode A path weighting, not covariance SEM. Model specification accepts only variable names and documented relation operators; no executable code.

References · APA

Semopy developers. (n.d.). Confirmatory factor analysis [Software documentation]. https://semopy.com/

Structural equation model
מודל משוואות מבניות · Modèle d’équations structurelles

SEM: Σ(θ)=Λ(I−B)⁻¹ Ψ(I−B)⁻ᵀΛᵀ+Θ
PLS: weighted composite scores followed by structural regressions

SEM/CFA/path uses covariance-based maximum likelihood and reports convergence and model fit. PLS is composite-based Mode A path weighting, not covariance SEM. Model specification accepts only variable names and documented relation operators; no executable code.

References · APA

Semopy developers. (n.d.). Structural equation model [Software documentation]. https://semopy.com/

PLS path modeling
מידול נתיבים בריבועים פחותים חלקיים · Modélisation PLS

Mode A: wj ∝ Zjᵀ innerScorej; scores standardized each iteration. Factorial inner weights are neighboring-score correlations. Structural β=(ScoreXᵀScoreX)⁻¹ScoreXᵀScoreY; bootstrap percentile CI.

Reflective Mode A composite PLS with factorial inner weighting, not covariance SEM. Recursive models with disjoint indicator blocks; complete-case standardized indicators. Bootstrap refits all outer weights and inner paths.

References · APA

Semopy developers. (n.d.). PLS path modeling [Software documentation]. https://semopy.com/

Propensity score matching
התאמה לפי ציון נטייה PSM · Appariement par score de propension

e(X)=Pr(T=1|X); distance=|logit(ei)−logit(ej)|; ATTmatch=mean(Ytreated−Ycontrol)
SMD=(weightedMean1−weightedMean0)/√[(baselineWeightedVariance1+baselineWeightedVariance0)/2]
ESS=(Σw)²/Σw²; the SMD denominator is fixed at baseline, before matching/weighting.
For m selected pairs, Δi=Ytreated,i−Ycontrol,i; θ=ΣwiΔi/Σwi
SE²=m/(m−1) × Σwi²(Δi−θ)²/(Σwi)²; CI=θ±t(1−α/2,m−1)SE (conditional on fitted scores/matches).

Requires measured baseline confounders, consistency, positivity and conditional exchangeability. Balance does not rule out unmeasured confounding. Matching uses 1:1 logit-caliper without replacement; weighting offers ATE, ATT and overlap targets. Optional sampling weights are preserved and reported as such.

References · APA

Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1), 41–55. https://doi.org/10.1093/biomet/70.1.41

Austin, P. C., & Stuart, E. A. (2015). Moving towards best practice when using inverse probability of treatment weighting (IPTW) using the propensity score to estimate causal treatment effects in observational studies. Statistics in Medicine, 34(28), 3661–3679. https://doi.org/10.1002/sim.6607

Propensity score weighting
שקלול לפי ציון נטייה · Pondération par score de propension

ATE weights=T/e+(1−T)/(1−e); ATT weights=T+(1−T)e/(1−e); overlap weights=T(1−e)+(1−T)e
SMD=(weightedMean1−weightedMean0)/√[(baselineWeightedVariance1+baselineWeightedVariance0)/2]
ESS=(Σw)²/Σw²; the SMD denominator is fixed at baseline, before matching/weighting.
Joint score ψi=[si Xi(Ti−ei), si w1i(Yi−μ1), si w0i(Yi−μ0)]
A=−mean(∂ψ/∂θ), B=mean(ψψᵀ), Cov(θ)=A⁻¹ B A⁻ᵀ/n; contrast=μ1−μ0. si are fixed existing sampling weights (or 1).

Requires measured baseline confounders, consistency, positivity and conditional exchangeability. Balance does not rule out unmeasured confounding. Matching uses 1:1 logit-caliper without replacement; weighting offers ATE, ATT and overlap targets. Optional sampling weights are preserved and reported as such.

References · APA

Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1), 41–55. https://doi.org/10.1093/biomet/70.1.41

Austin, P. C., & Stuart, E. A. (2015). Moving towards best practice when using inverse probability of treatment weighting (IPTW) using the propensity score to estimate causal treatment effects in observational studies. Statistics in Medicine, 34(28), 3661–3679. https://doi.org/10.1002/sim.6607