[{"data":1,"prerenderedAt":975},["ShallowReactive",2],{"content-query-AS1B5JITzF":3},{"_path":4,"_dir":5,"_draft":6,"_partial":6,"_locale":7,"title":8,"description":9,"heading":10,"prompt":11,"tags":15,"files":17,"nav":6,"presets":18,"gallery":37,"body":39,"_type":968,"_id":969,"_source":970,"_file":971,"_stem":972,"_extension":973,"sitemap":974},"/tools/moderation-analysis","tools",false,"","Moderation Analysis Calculator","Run moderation analysis online from Excel or CSV data. Test interaction effects, simple slopes, and Johnson-Neyman regions with AI.","Moderation Analysis",{"prefix":12,"label":13,"placeholder":14},"Run moderation analysis","Describe the predictor, moderator, and outcome variables","e.g. X = workload, W = social support (moderator), Y = burnout; test X×W interaction; interaction plot at low/mean/high W; Johnson-Neyman regions of significance",[16],"statistics",true,[19,25,31],{"label":20,"prompt":21,"dataset_url":22,"dataset_title":23,"dataset_citation":24},"Stress × Support → Mental Health","Moderation analysis: X = perceived stress, W = social support, Y = mental health score; test X×W interaction; center X and W before creating product term; interaction plot at −1SD, mean, +1SD of W; Johnson-Neyman plot; report b coefficients with 95% CI","https://data.cdc.gov/api/views/iuq5-y9ct/rows.csv?accessType=DOWNLOAD","NHANES Mental Health Assessment","CDC",{"label":26,"prompt":27,"dataset_url":28,"dataset_title":29,"dataset_citation":30},"Income × Education → Life Satisfaction","Moderation: X = income (log), W = years of education, Y = life satisfaction; mean-center X and W; test interaction; conditional effects at low/mean/high education; plot interaction; F-test for interaction term","https://ourworldindata.org/grapher/happiness-cantril-ladder.csv","Self-Reported Life Satisfaction","Our World in Data",{"label":32,"prompt":33,"dataset_url":34,"dataset_title":35,"dataset_citation":36},"GDP × Governance → Health Outcomes","Moderation analysis across countries: X = GDP per capita, W = governance index, Y = life expectancy; test whether governance moderates the GDP-health relationship; interaction plot; Johnson-Neyman significance regions","https://api.worldbank.org/v2/en/indicator/SP.DYN.LE00.IN?downloadformat=excel","Life Expectancy at Birth","World Bank",[38],"/img/tools/moderation-analysis.png",{"type":40,"children":41,"toc":957},"root",[42,51,84,101,113,119,177,183,378,383,389,555,561,719,725,795,801,845,851,879,910,927],{"type":43,"tag":44,"props":45,"children":47},"element","h2",{"id":46},"what-is-moderation-analysis",[48],{"type":49,"value":50},"text","What Is Moderation Analysis?",{"type":43,"tag":52,"props":53,"children":54},"p",{},[55,61,63,68,70,75,77,82],{"type":43,"tag":56,"props":57,"children":58},"strong",{},[59],{"type":49,"value":60},"Moderation analysis",{"type":49,"value":62}," tests whether the relationship between a predictor X and an outcome Y ",{"type":43,"tag":56,"props":64,"children":65},{},[66],{"type":49,"value":67},"depends on the level of a third variable W",{"type":49,"value":69}," (the moderator). A moderator changes the strength or direction of the X→Y effect — it answers \"for whom?\" or \"under what conditions?\" does X predict Y. Statistically, moderation is tested as a ",{"type":43,"tag":56,"props":71,"children":72},{},[73],{"type":49,"value":74},"multiplicative interaction",{"type":49,"value":76},": Y = b₀ + b₁X + b₂W + b₃(X×W) + ε. The ",{"type":43,"tag":56,"props":78,"children":79},{},[80],{"type":49,"value":81},"interaction term b₃",{"type":49,"value":83}," is the key coefficient — if it is significant, the slope of X on Y is different at different levels of W. A positive b₃ means the X→Y relationship strengthens as W increases; a negative b₃ means it weakens.",{"type":43,"tag":52,"props":85,"children":86},{},[87,92,94,99],{"type":43,"tag":56,"props":88,"children":89},{},[90],{"type":49,"value":91},"Mean-centering",{"type":49,"value":93}," X and W before computing the product term X×W is strongly recommended: it eliminates non-essential multicollinearity between the main effects and the interaction term, and makes the coefficients b₁ and b₂ more interpretable (they become the conditional effect of X at the mean of W, and vice versa). Without centering, b₁ is the effect of X when W = 0, which may be outside the observed range and have no meaningful interpretation. The ",{"type":43,"tag":56,"props":95,"children":96},{},[97],{"type":49,"value":98},"interaction plot",{"type":49,"value":100}," (also called a simple slopes plot) visualizes the X→Y regression line at three representative levels of W — conventionally at W = −1 SD, W = mean, and W = +1 SD — providing an intuitive picture of how the relationship changes across W.",{"type":43,"tag":52,"props":102,"children":103},{},[104,106,111],{"type":49,"value":105},"A concrete example: a workplace study measures workload (X), burnout (Y), and social support (W) in 300 employees. The interaction term b₃ = −0.43 (p \u003C .001) indicates that high social support buffers the workload-burnout relationship — the interaction plot shows a steep positive slope between workload and burnout for employees with low support, a flat slope for those with average support, and a slightly negative slope for those with high support. The ",{"type":43,"tag":56,"props":107,"children":108},{},[109],{"type":49,"value":110},"Johnson-Neyman technique",{"type":49,"value":112}," identifies the exact value of W (social support score = 0.56) above which the effect of workload on burnout is no longer statistically significant — providing a precise threshold for the buffering effect.",{"type":43,"tag":44,"props":114,"children":116},{"id":115},"how-it-works",[117],{"type":49,"value":118},"How It Works",{"type":43,"tag":120,"props":121,"children":122},"ol",{},[123,134,150],{"type":43,"tag":124,"props":125,"children":126},"li",{},[127,132],{"type":43,"tag":56,"props":128,"children":129},{},[130],{"type":49,"value":131},"Upload your data",{"type":49,"value":133}," — provide a CSV or Excel file with one column each for the predictor (X), moderator (W), and outcome (Y), plus any covariates. All variables should be numeric and continuous (for a binary moderator, describe it as a dummy-coded 0/1 variable).",{"type":43,"tag":124,"props":135,"children":136},{},[137,142,144],{"type":43,"tag":56,"props":138,"children":139},{},[140],{"type":49,"value":141},"Describe the analysis",{"type":49,"value":143}," — e.g. ",{"type":43,"tag":145,"props":146,"children":147},"em",{},[148],{"type":49,"value":149},"\"X = workload, W = social support, Y = burnout; mean-center X and W; test X×W interaction; interaction plot at low/mean/high W; Johnson-Neyman regions of significance; report all coefficients with 95% CI\"",{"type":43,"tag":124,"props":151,"children":152},{},[153,158,160,167,169,175],{"type":43,"tag":56,"props":154,"children":155},{},[156],{"type":49,"value":157},"Get full results",{"type":49,"value":159}," — the AI writes Python code using ",{"type":43,"tag":161,"props":162,"children":164},"a",{"href":163},"https://www.statsmodels.org/",[165],{"type":49,"value":166},"statsmodels",{"type":49,"value":168}," and ",{"type":43,"tag":161,"props":170,"children":172},{"href":171},"https://plotly.com/python/",[173],{"type":49,"value":174},"Plotly",{"type":49,"value":176}," to mean-center predictors, fit the interaction model, test the interaction term, plot conditional slopes, and produce the Johnson-Neyman significance region plot",{"type":43,"tag":44,"props":178,"children":180},{"id":179},"required-data-format",[181],{"type":49,"value":182},"Required Data Format",{"type":43,"tag":184,"props":185,"children":186},"table",{},[187,211],{"type":43,"tag":188,"props":189,"children":190},"thead",{},[191],{"type":43,"tag":192,"props":193,"children":194},"tr",{},[195,201,206],{"type":43,"tag":196,"props":197,"children":198},"th",{},[199],{"type":49,"value":200},"Column",{"type":43,"tag":196,"props":202,"children":203},{},[204],{"type":49,"value":205},"Description",{"type":43,"tag":196,"props":207,"children":208},{},[209],{"type":49,"value":210},"Example",{"type":43,"tag":212,"props":213,"children":214},"tbody",{},[215,258,298,338],{"type":43,"tag":192,"props":216,"children":217},{},[218,229,234],{"type":43,"tag":219,"props":220,"children":221},"td",{},[222],{"type":43,"tag":223,"props":224,"children":226},"code",{"className":225},[],[227],{"type":49,"value":228},"X",{"type":43,"tag":219,"props":230,"children":231},{},[232],{"type":49,"value":233},"Predictor variable",{"type":43,"tag":219,"props":235,"children":236},{},[237,243,245,251,252],{"type":43,"tag":223,"props":238,"children":240},{"className":239},[],[241],{"type":49,"value":242},"workload",{"type":49,"value":244},", ",{"type":43,"tag":223,"props":246,"children":248},{"className":247},[],[249],{"type":49,"value":250},"dose",{"type":49,"value":244},{"type":43,"tag":223,"props":253,"children":255},{"className":254},[],[256],{"type":49,"value":257},"income",{"type":43,"tag":192,"props":259,"children":260},{},[261,270,275],{"type":43,"tag":219,"props":262,"children":263},{},[264],{"type":43,"tag":223,"props":265,"children":267},{"className":266},[],[268],{"type":49,"value":269},"W",{"type":43,"tag":219,"props":271,"children":272},{},[273],{"type":49,"value":274},"Moderator variable",{"type":43,"tag":219,"props":276,"children":277},{},[278,284,285,291,292],{"type":43,"tag":223,"props":279,"children":281},{"className":280},[],[282],{"type":49,"value":283},"support",{"type":49,"value":244},{"type":43,"tag":223,"props":286,"children":288},{"className":287},[],[289],{"type":49,"value":290},"age",{"type":49,"value":244},{"type":43,"tag":223,"props":293,"children":295},{"className":294},[],[296],{"type":49,"value":297},"condition",{"type":43,"tag":192,"props":299,"children":300},{},[301,310,315],{"type":43,"tag":219,"props":302,"children":303},{},[304],{"type":43,"tag":223,"props":305,"children":307},{"className":306},[],[308],{"type":49,"value":309},"Y",{"type":43,"tag":219,"props":311,"children":312},{},[313],{"type":49,"value":314},"Outcome variable",{"type":43,"tag":219,"props":316,"children":317},{},[318,324,325,331,332],{"type":43,"tag":223,"props":319,"children":321},{"className":320},[],[322],{"type":49,"value":323},"burnout",{"type":49,"value":244},{"type":43,"tag":223,"props":326,"children":328},{"className":327},[],[329],{"type":49,"value":330},"response",{"type":49,"value":244},{"type":43,"tag":223,"props":333,"children":335},{"className":334},[],[336],{"type":49,"value":337},"satisfaction",{"type":43,"tag":192,"props":339,"children":340},{},[341,350,355],{"type":43,"tag":219,"props":342,"children":343},{},[344],{"type":43,"tag":223,"props":345,"children":347},{"className":346},[],[348],{"type":49,"value":349},"covariate",{"type":43,"tag":219,"props":351,"children":352},{},[353],{"type":49,"value":354},"Optional: control variables",{"type":43,"tag":219,"props":356,"children":357},{},[358,364,365,371,372],{"type":43,"tag":223,"props":359,"children":361},{"className":360},[],[362],{"type":49,"value":363},"gender",{"type":49,"value":244},{"type":43,"tag":223,"props":366,"children":368},{"className":367},[],[369],{"type":49,"value":370},"tenure",{"type":49,"value":244},{"type":43,"tag":223,"props":373,"children":375},{"className":374},[],[376],{"type":49,"value":377},"baseline",{"type":43,"tag":52,"props":379,"children":380},{},[381],{"type":49,"value":382},"Any column names work — describe them in your prompt. All variables should be numeric. For binary moderators (e.g., gender), code as 0/1 and interpret the interaction as the difference in X→Y slopes between groups. For categorical moderators with 3+ levels, dummy-code them and test a series of two-way interactions.",{"type":43,"tag":44,"props":384,"children":386},{"id":385},"interpreting-the-results",[387],{"type":49,"value":388},"Interpreting the Results",{"type":43,"tag":184,"props":390,"children":391},{},[392,408],{"type":43,"tag":188,"props":393,"children":394},{},[395],{"type":43,"tag":192,"props":396,"children":397},{},[398,403],{"type":43,"tag":196,"props":399,"children":400},{},[401],{"type":49,"value":402},"Output",{"type":43,"tag":196,"props":404,"children":405},{},[406],{"type":49,"value":407},"What it means",{"type":43,"tag":212,"props":409,"children":410},{},[411,427,443,459,475,491,507,523,539],{"type":43,"tag":192,"props":412,"children":413},{},[414,422],{"type":43,"tag":219,"props":415,"children":416},{},[417],{"type":43,"tag":56,"props":418,"children":419},{},[420],{"type":49,"value":421},"b₁ (main effect of X)",{"type":43,"tag":219,"props":423,"children":424},{},[425],{"type":49,"value":426},"Effect of X on Y when W = 0 (or W = mean, if centered)",{"type":43,"tag":192,"props":428,"children":429},{},[430,438],{"type":43,"tag":219,"props":431,"children":432},{},[433],{"type":43,"tag":56,"props":434,"children":435},{},[436],{"type":49,"value":437},"b₂ (main effect of W)",{"type":43,"tag":219,"props":439,"children":440},{},[441],{"type":49,"value":442},"Effect of W on Y when X = 0 (or X = mean, if centered)",{"type":43,"tag":192,"props":444,"children":445},{},[446,454],{"type":43,"tag":219,"props":447,"children":448},{},[449],{"type":43,"tag":56,"props":450,"children":451},{},[452],{"type":49,"value":453},"b₃ (interaction X×W)",{"type":43,"tag":219,"props":455,"children":456},{},[457],{"type":49,"value":458},"Change in the X→Y slope for each 1-unit increase in W — the key moderation coefficient",{"type":43,"tag":192,"props":460,"children":461},{},[462,470],{"type":43,"tag":219,"props":463,"children":464},{},[465],{"type":43,"tag":56,"props":466,"children":467},{},[468],{"type":49,"value":469},"F-test for interaction",{"type":43,"tag":219,"props":471,"children":472},{},[473],{"type":49,"value":474},"Significance test for b₃ — p \u003C .05 indicates significant moderation",{"type":43,"tag":192,"props":476,"children":477},{},[478,486],{"type":43,"tag":219,"props":479,"children":480},{},[481],{"type":43,"tag":56,"props":482,"children":483},{},[484],{"type":49,"value":485},"Conditional slope of X at W levels",{"type":43,"tag":219,"props":487,"children":488},{},[489],{"type":49,"value":490},"b₁ + b₃×W — the X→Y slope at specific W values (−1 SD, mean, +1 SD)",{"type":43,"tag":192,"props":492,"children":493},{},[494,502],{"type":43,"tag":219,"props":495,"children":496},{},[497],{"type":43,"tag":56,"props":498,"children":499},{},[500],{"type":49,"value":501},"Simple slope t-tests",{"type":43,"tag":219,"props":503,"children":504},{},[505],{"type":49,"value":506},"Test whether the conditional slope is significantly different from zero at each W level",{"type":43,"tag":192,"props":508,"children":509},{},[510,518],{"type":43,"tag":219,"props":511,"children":512},{},[513],{"type":43,"tag":56,"props":514,"children":515},{},[516],{"type":49,"value":517},"Johnson-Neyman region",{"type":43,"tag":219,"props":519,"children":520},{},[521],{"type":49,"value":522},"Range of W values where the X→Y effect is statistically significant (p \u003C .05)",{"type":43,"tag":192,"props":524,"children":525},{},[526,534],{"type":43,"tag":219,"props":527,"children":528},{},[529],{"type":43,"tag":56,"props":530,"children":531},{},[532],{"type":49,"value":533},"Interaction plot",{"type":43,"tag":219,"props":535,"children":536},{},[537],{"type":49,"value":538},"Visualization of X→Y regression lines at low, mean, and high W",{"type":43,"tag":192,"props":540,"children":541},{},[542,550],{"type":43,"tag":219,"props":543,"children":544},{},[545],{"type":43,"tag":56,"props":546,"children":547},{},[548],{"type":49,"value":549},"ΔR²",{"type":43,"tag":219,"props":551,"children":552},{},[553],{"type":49,"value":554},"Change in R² when adding the interaction term — effect size of the moderation",{"type":43,"tag":44,"props":556,"children":558},{"id":557},"example-prompts",[559],{"type":49,"value":560},"Example Prompts",{"type":43,"tag":184,"props":562,"children":563},{},[564,580],{"type":43,"tag":188,"props":565,"children":566},{},[567],{"type":43,"tag":192,"props":568,"children":569},{},[570,575],{"type":43,"tag":196,"props":571,"children":572},{},[573],{"type":49,"value":574},"Scenario",{"type":43,"tag":196,"props":576,"children":577},{},[578],{"type":49,"value":579},"What to type",{"type":43,"tag":212,"props":581,"children":582},{},[583,600,617,634,651,668,685,702],{"type":43,"tag":192,"props":584,"children":585},{},[586,591],{"type":43,"tag":219,"props":587,"children":588},{},[589],{"type":49,"value":590},"Basic moderation",{"type":43,"tag":219,"props":592,"children":593},{},[594],{"type":43,"tag":223,"props":595,"children":597},{"className":596},[],[598],{"type":49,"value":599},"X = stress, W = resilience, Y = wellbeing; mean-center X and W; test interaction; interaction plot; JN regions",{"type":43,"tag":192,"props":601,"children":602},{},[603,608],{"type":43,"tag":219,"props":604,"children":605},{},[606],{"type":49,"value":607},"Binary moderator",{"type":43,"tag":219,"props":609,"children":610},{},[611],{"type":43,"tag":223,"props":612,"children":614},{"className":613},[],[615],{"type":49,"value":616},"W is binary (0=female, 1=male); test if gender moderates X→Y; report conditional slopes for each group",{"type":43,"tag":192,"props":618,"children":619},{},[620,625],{"type":43,"tag":219,"props":621,"children":622},{},[623],{"type":49,"value":624},"Covariates",{"type":43,"tag":219,"props":626,"children":627},{},[628],{"type":43,"tag":223,"props":629,"children":631},{"className":630},[],[632],{"type":49,"value":633},"moderation controlling for age and baseline; include as covariates in the model; report adjusted interaction effect",{"type":43,"tag":192,"props":635,"children":636},{},[637,642],{"type":43,"tag":219,"props":638,"children":639},{},[640],{"type":49,"value":641},"Curvilinear moderation",{"type":43,"tag":219,"props":643,"children":644},{},[645],{"type":43,"tag":223,"props":646,"children":648},{"className":647},[],[649],{"type":49,"value":650},"test both linear and quadratic moderation: add X², W², and X²×W terms; compare model fit",{"type":43,"tag":192,"props":652,"children":653},{},[654,659],{"type":43,"tag":219,"props":655,"children":656},{},[657],{"type":49,"value":658},"ΔR² effect size",{"type":43,"tag":219,"props":660,"children":661},{},[662],{"type":43,"tag":223,"props":663,"children":665},{"className":664},[],[666],{"type":49,"value":667},"report ΔR² when adding interaction term; report f² (Cohen's f-squared) for the interaction",{"type":43,"tag":192,"props":669,"children":670},{},[671,676],{"type":43,"tag":219,"props":672,"children":673},{},[674],{"type":49,"value":675},"Pick-a-point probing",{"type":43,"tag":219,"props":677,"children":678},{},[679],{"type":43,"tag":223,"props":680,"children":682},{"className":681},[],[683],{"type":49,"value":684},"probe interaction at W = 10, 20, 30 (specific meaningful values); simple slope at each; t-test and CI",{"type":43,"tag":192,"props":686,"children":687},{},[688,693],{"type":43,"tag":219,"props":689,"children":690},{},[691],{"type":49,"value":692},"Moderated mediation",{"type":43,"tag":219,"props":694,"children":695},{},[696],{"type":43,"tag":223,"props":697,"children":699},{"className":698},[],[700],{"type":49,"value":701},"moderated mediation: X→M→Y with W moderating the a path (X→M); conditional indirect effects at 3 W levels",{"type":43,"tag":192,"props":703,"children":704},{},[705,710],{"type":43,"tag":219,"props":706,"children":707},{},[708],{"type":49,"value":709},"Three-way interaction",{"type":43,"tag":219,"props":711,"children":712},{},[713],{"type":43,"tag":223,"props":714,"children":716},{"className":715},[],[717],{"type":49,"value":718},"X × W × Z three-way interaction; test if Z moderates the X×W moderation; plot at 4 combinations of W and Z",{"type":43,"tag":44,"props":720,"children":722},{"id":721},"assumptions-to-check",[723],{"type":49,"value":724},"Assumptions to Check",{"type":43,"tag":726,"props":727,"children":728},"ul",{},[729,739,748,765,775,785],{"type":43,"tag":124,"props":730,"children":731},{},[732,737],{"type":43,"tag":56,"props":733,"children":734},{},[735],{"type":49,"value":736},"Linearity of main effects",{"type":49,"value":738}," — moderation analysis assumes the main effects of X and W on Y are linear; inspect residual plots and scatterplots before adding the interaction; if either X or W has a non-linear relationship with Y, the interaction test may be confounded by the non-linearity",{"type":43,"tag":124,"props":740,"children":741},{},[742,746],{"type":43,"tag":56,"props":743,"children":744},{},[745],{"type":49,"value":91},{"type":49,"value":747}," — always center X and W (subtract their means) before computing the product term X×W; failing to center does not affect the interaction coefficient b₃ or its significance, but causes multicollinearity between main effects and the interaction term, inflating standard errors and making main effect coefficients uninterpretable",{"type":43,"tag":124,"props":749,"children":750},{},[751,756,758,763],{"type":43,"tag":56,"props":752,"children":753},{},[754],{"type":49,"value":755},"Homoscedasticity",{"type":49,"value":757}," — interaction models are particularly sensitive to heteroscedasticity; the variance of Y may increase with X×W; inspect residual plots and consider using ",{"type":43,"tag":56,"props":759,"children":760},{},[761],{"type":49,"value":762},"heteroscedasticity-consistent (HC) standard errors",{"type":49,"value":764}," (White correction) if variance is non-constant",{"type":43,"tag":124,"props":766,"children":767},{},[768,773],{"type":43,"tag":56,"props":769,"children":770},{},[771],{"type":49,"value":772},"No extreme outliers",{"type":49,"value":774}," — leverage points at the extremes of X×W can strongly influence the interaction estimate; identify observations with high leverage (hat values) and Cook's distance; consider robust regression if outliers are influential",{"type":43,"tag":124,"props":776,"children":777},{},[778,783],{"type":43,"tag":56,"props":779,"children":780},{},[781],{"type":49,"value":782},"Sample size for interactions",{"type":49,"value":784}," — detecting interaction effects requires substantially more power than detecting main effects; the interaction effect size (ΔR²) is typically much smaller than main effect R²; rule of thumb: n ≥ 200 for typical interaction effects (b₃ corresponding to ΔR² ≈ 0.01–0.02); simulation studies suggest many published moderation studies are severely underpowered",{"type":43,"tag":124,"props":786,"children":787},{},[788,793],{"type":43,"tag":56,"props":789,"children":790},{},[791],{"type":49,"value":792},"Measurement reliability",{"type":49,"value":794}," — unreliability in X or W attenuates the interaction coefficient (b₃) more than it attenuates main effects; if X or W are measured with error, the true interaction may be substantially larger than the observed b₃",{"type":43,"tag":44,"props":796,"children":798},{"id":797},"related-tools",[799],{"type":49,"value":800},"Related Tools",{"type":43,"tag":52,"props":802,"children":803},{},[804,806,812,814,819,821,827,829,835,837,843],{"type":49,"value":805},"Use the ",{"type":43,"tag":161,"props":807,"children":809},{"href":808},"/tools/mediation-analysis",[810],{"type":49,"value":811},"Mediation Analysis Calculator",{"type":49,"value":813}," when X affects Y through M (mechanism question) rather than when W changes the X→Y relationship (context question) — mediation and moderation address different research questions, though they can be combined as ",{"type":43,"tag":56,"props":815,"children":816},{},[817],{"type":49,"value":818},"moderated mediation",{"type":49,"value":820},". Use the ",{"type":43,"tag":161,"props":822,"children":824},{"href":823},"/tools/multiple-regression",[825],{"type":49,"value":826},"Multiple Regression",{"type":49,"value":828}," calculator to fit the regression model without the interaction term and compare R² — the ΔR² attributable to the interaction term is the effect size of the moderation. Use the ",{"type":43,"tag":161,"props":830,"children":832},{"href":831},"/tools/two-way-anova",[833],{"type":49,"value":834},"Online two-way ANOVA calculator",{"type":49,"value":836}," when both X and W are categorical — two-way ANOVA is a special case of moderation with categorical factors. Use the ",{"type":43,"tag":161,"props":838,"children":840},{"href":839},"/tools/partial-correlation",[841],{"type":49,"value":842},"Partial Correlation Calculator",{"type":49,"value":844}," to examine whether the X-Y correlation changes substantially when W is controlled — a large change suggests W may act as a moderator or mediator.",{"type":43,"tag":44,"props":846,"children":848},{"id":847},"frequently-asked-questions",[849],{"type":49,"value":850},"Frequently Asked Questions",{"type":43,"tag":52,"props":852,"children":853},{},[854,859,864,866,871,873,877],{"type":43,"tag":56,"props":855,"children":856},{},[857],{"type":49,"value":858},"What is the difference between moderation and mediation?",{"type":43,"tag":56,"props":860,"children":861},{},[862],{"type":49,"value":863},"Mediation",{"type":49,"value":865}," asks \"how?\" or \"through what mechanism?\" — X affects M, which in turn affects Y; the mediator M is on the causal pathway from X to Y. ",{"type":43,"tag":56,"props":867,"children":868},{},[869],{"type":49,"value":870},"Moderation",{"type":49,"value":872}," asks \"when?\" or \"for whom?\" — W changes the strength or direction of the X→Y relationship; the moderator W is not on the X→Y causal path but interacts with X to affect Y. Example: stress (X) → burnout (Y) could be mediated by poor sleep (M: stress causes poor sleep, which causes burnout) or moderated by social support (W: the stress-burnout relationship is weaker when support is high). The two can be combined: ",{"type":43,"tag":56,"props":874,"children":875},{},[876],{"type":49,"value":818},{"type":49,"value":878}," tests whether the mediated path (X→M→Y) is stronger at some levels of W than others.",{"type":43,"tag":52,"props":880,"children":881},{},[882,887,889,894,896,901,903,908],{"type":43,"tag":56,"props":883,"children":884},{},[885],{"type":49,"value":886},"Why is the interaction term often non-significant even when I expect moderation?",{"type":49,"value":888},"\nModeration effects are typically small — interaction terms account for ΔR² ≈ 0.01–0.03 in most social science and health research, compared to main effects that may account for ΔR² > 0.10. This means interaction tests require large samples (n ≥ 200–500) to achieve adequate power. Additional reasons for low power: (1) ",{"type":43,"tag":56,"props":890,"children":891},{},[892],{"type":49,"value":893},"measurement error",{"type":49,"value":895}," in X or W attenuates the interaction coefficient substantially; (2) ",{"type":43,"tag":56,"props":897,"children":898},{},[899],{"type":49,"value":900},"range restriction",{"type":49,"value":902}," — if X or W have limited variance in your sample, the interaction cannot emerge; (3) ",{"type":43,"tag":56,"props":904,"children":905},{},[906],{"type":49,"value":907},"non-linear moderation",{"type":49,"value":909}," — if the true moderation is not linear (e.g., only extreme values of W moderate X→Y), the linear interaction term will miss it. Before concluding no moderation exists, consider whether your design and sample size had adequate power to detect the expected effect size.",{"type":43,"tag":52,"props":911,"children":912},{},[913,918,920,925],{"type":43,"tag":56,"props":914,"children":915},{},[916],{"type":49,"value":917},"What is the Johnson-Neyman technique and when should I use it?",{"type":49,"value":919},"\nThe ",{"type":43,"tag":56,"props":921,"children":922},{},[923],{"type":49,"value":924},"Johnson-Neyman (J-N) technique",{"type":49,"value":926}," finds the exact value(s) of the moderator W at which the conditional effect of X on Y transitions between statistically significant and non-significant. Instead of probing the interaction at arbitrary W levels (−1 SD, mean, +1 SD), J-N provides the precise boundary. For example, if the J-N boundary is at W = 12.4, the X→Y effect is significant for all W \u003C 12.4 (assuming a negative interaction) but non-significant for W ≥ 12.4. Always check what proportion of your sample falls in each region — if 90% of participants are in the non-significant region, the significant effect at low W may be driven by a small fraction of observations. Use J-N alongside the interaction plot for complete interpretation.",{"type":43,"tag":52,"props":928,"children":929},{},[930,935,937,942,944,948,950,955],{"type":43,"tag":56,"props":931,"children":932},{},[933],{"type":49,"value":934},"Should I report b₃, simple slopes, or ΔR² as my primary effect size?",{"type":49,"value":936},"\nReport all three: (1) ",{"type":43,"tag":56,"props":938,"children":939},{},[940],{"type":49,"value":941},"b₃",{"type":49,"value":943}," with 95% CI and p-value (unstandardized) gives the raw interaction coefficient; standardized b₃ (when X, W, Y are all z-scored) allows comparison across studies; (2) ",{"type":43,"tag":56,"props":945,"children":946},{},[947],{"type":49,"value":549},{"type":49,"value":949}," is the variance-explained effect size for the interaction — small ≈ 0.005, medium ≈ 0.01, large ≈ 0.025 (Frazier et al., 2004); (3) ",{"type":43,"tag":56,"props":951,"children":952},{},[953],{"type":49,"value":954},"simple slopes",{"type":49,"value":956}," at meaningful W levels give practical interpretation of the conditional effects. For meta-analyses, the standardized interaction coefficient is most useful. For applied reporting to non-statisticians, the interaction plot with labeled simple slopes is often most accessible.",{"title":7,"searchDepth":958,"depth":958,"links":959},2,[960,961,962,963,964,965,966,967],{"id":46,"depth":958,"text":50},{"id":115,"depth":958,"text":118},{"id":179,"depth":958,"text":182},{"id":385,"depth":958,"text":388},{"id":557,"depth":958,"text":560},{"id":721,"depth":958,"text":724},{"id":797,"depth":958,"text":800},{"id":847,"depth":958,"text":850},"markdown","content:tools:077.moderation-analysis.md","content","tools/077.moderation-analysis.md","tools/077.moderation-analysis","md",{"loc":4},1775502472568]