{"example":"Range Bar: Dosing Windows","labtools_version":"0.14.3","generated":"2026-09-23","calls":[{"label":"stats.describe","method":"stats.describe","params":{},"result":{"columns":[{"excess_kurtosis":1.4999999999999982,"iqr":2.0,"max":32.0,"mean":30.0,"median":30.5,"min":27.0,"n":4,"n_missing":0,"name":"Regimen A","q1":29.25,"q3":31.25,"range":5.0,"sd":2.160246899469287,"sem":1.0801234497346435,"skewness":-1.1903401282789945,"sum":120.0,"variance":4.666666666666667},{"excess_kurtosis":-3.869005302721286,"iqr":4.75,"max":43.0,"mean":39.25,"median":39.0,"min":36.0,"n":4,"n_missing":0,"name":"Regimen B","q1":36.75,"q3":41.5,"range":7.0,"sd":3.304037933599835,"sem":1.6520189667999174,"skewness":0.22872775858729724,"sum":157.0,"variance":10.916666666666666},{"excess_kurtosis":3.4834710743801622,"iqr":1.5,"max":57.0,"mean":53.0,"median":52.0,"min":51.0,"n":4,"n_missing":0,"name":"Regimen C","q1":51.75,"q3":53.25,"range":6.0,"sd":2.70801280154532,"sem":1.35400640077266,"skewness":1.8128019580592636,"sum":212.0,"variance":7.333333333333333}],"n_columns":3,"notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality."],"source":{"kind":"arrays"}},"error":null},{"label":"stats.normality","method":"stats.normality","params":{"method":"shapiro_wilk"},"result":{"all_normal":true,"alpha":0.05,"method":"shapiro_wilk","notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality."],"results":[{"excess_kurtosis":1.4999999999999982,"group":"Regimen A","interpretation":"Data is consistent with normality (W = 0.9271, p = 0.5774)","n":4,"normality_rejected":false,"p_value":0.5773549021744702,"skewness":-1.1903401282789945,"statistic":0.9270816792875027,"test":"Shapiro-Wilk"},{"excess_kurtosis":-3.869005302721286,"group":"Regimen B","interpretation":"Data is consistent with normality (W = 0.9157, p = 0.5130)","n":4,"normality_rejected":false,"p_value":0.5130400846566809,"skewness":0.22872775858729724,"statistic":0.9156860669810516,"test":"Shapiro-Wilk"},{"excess_kurtosis":3.4834710743801622,"group":"Regimen C","interpretation":"Data is consistent with normality (W = 0.7729, p = 0.0618)","n":4,"normality_rejected":false,"p_value":0.06184652087897266,"skewness":1.8128019580592636,"statistic":0.7729072344676293,"test":"Shapiro-Wilk"}],"source":{"kind":"arrays"}},"error":null},{"label":"stats.normality","method":"stats.normality","params":{"method":"kolmogorov_smirnov"},"result":{"all_normal":false,"alpha":0.05,"method":"kolmogorov_smirnov","notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' rejects normality (p = 0.0295); prefer a rank-based test."],"results":[{"excess_kurtosis":1.4999999999999982,"group":"Regimen A","interpretation":"Data is consistent with normality (D = 0.2500, p = 0.5069)","n":4,"normality_rejected":false,"p_value":0.5068944658627013,"skewness":-1.1903401282789945,"statistic":0.25,"test":"Kolmogorov-Smirnov (Lilliefors)"},{"excess_kurtosis":-3.869005302721286,"group":"Regimen B","interpretation":"Data is consistent with normality (D = 0.2521, p = 0.4934)","n":4,"normality_rejected":false,"p_value":0.4934494397389022,"skewness":0.22872775858729724,"statistic":0.2520594830309536,"test":"Kolmogorov-Smirnov (Lilliefors)"},{"excess_kurtosis":3.4834710743801622,"group":"Regimen C","interpretation":"Data significantly deviates from normality (D = 0.3940, p = 0.0295)","n":4,"normality_rejected":true,"p_value":0.02950057789272331,"skewness":1.8128019580592636,"statistic":0.39403842465390493,"test":"Kolmogorov-Smirnov (Lilliefors)"}],"source":{"kind":"arrays"}},"error":null},{"label":"stats.levene","method":"stats.levene","params":{"method":"levene"},"result":{"alpha":0.05,"df1":2.0,"df2":9.0,"equal_variance_rejected":false,"group_variances":[4.666666666666667,10.916666666666666,7.333333333333333],"groups":[{"excess_kurtosis":1.4999999999999982,"iqr":2.0,"max":32.0,"mean":30.0,"median":30.5,"min":27.0,"n":4,"n_missing":0,"name":"Regimen A","q1":29.25,"q3":31.25,"range":5.0,"sd":2.160246899469287,"sem":1.0801234497346435,"skewness":-1.1903401282789945,"sum":120.0,"variance":4.666666666666667},{"excess_kurtosis":-3.869005302721286,"iqr":4.75,"max":43.0,"mean":39.25,"median":39.0,"min":36.0,"n":4,"n_missing":0,"name":"Regimen B","q1":36.75,"q3":41.5,"range":7.0,"sd":3.304037933599835,"sem":1.6520189667999174,"skewness":0.22872775858729724,"sum":157.0,"variance":10.916666666666666},{"excess_kurtosis":3.4834710743801622,"iqr":1.5,"max":57.0,"mean":53.0,"median":52.0,"min":51.0,"n":4,"n_missing":0,"name":"Regimen C","q1":51.75,"q3":53.25,"range":6.0,"sd":2.70801280154532,"sem":1.35400640077266,"skewness":1.8128019580592636,"sum":212.0,"variance":7.333333333333333}],"interpretation":"Variances are homogeneous across groups (F = 0.7282, p = 0.5092)","method":"levene","notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Non-significant. This test has low power on small samples, so treat this as 'no evidence against equal variances', not as evidence for them."],"p_value":0.5091996996441339,"significant":false,"source":{"kind":"arrays"},"statistic":0.7281553398058251,"test":"Levene's Test","variance_ratio":2.339285714285714},"error":null},{"label":"stats.levene","method":"stats.levene","params":{"method":"bartlett"},"result":{"alpha":0.05,"df1":2.0,"df2":null,"equal_variance_rejected":false,"group_variances":[4.666666666666667,10.916666666666666,7.333333333333333],"groups":[{"excess_kurtosis":1.4999999999999982,"iqr":2.0,"max":32.0,"mean":30.0,"median":30.5,"min":27.0,"n":4,"n_missing":0,"name":"Regimen A","q1":29.25,"q3":31.25,"range":5.0,"sd":2.160246899469287,"sem":1.0801234497346435,"skewness":-1.1903401282789945,"sum":120.0,"variance":4.666666666666667},{"excess_kurtosis":-3.869005302721286,"iqr":4.75,"max":43.0,"mean":39.25,"median":39.0,"min":36.0,"n":4,"n_missing":0,"name":"Regimen B","q1":36.75,"q3":41.5,"range":7.0,"sd":3.304037933599835,"sem":1.6520189667999174,"skewness":0.22872775858729724,"sum":157.0,"variance":10.916666666666666},{"excess_kurtosis":3.4834710743801622,"iqr":1.5,"max":57.0,"mean":53.0,"median":52.0,"min":51.0,"n":4,"n_missing":0,"name":"Regimen C","q1":51.75,"q3":53.25,"range":6.0,"sd":2.70801280154532,"sem":1.35400640077266,"skewness":1.8128019580592636,"sum":212.0,"variance":7.333333333333333}],"interpretation":"Variances are homogeneous across groups (χ² = 0.4614, p = 0.7940)","method":"bartlett","notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Non-significant. This test has low power on small samples, so treat this as 'no evidence against equal variances', not as evidence for them.","Bartlett's test assumes normality and is badly inflated when that fails — confirm with stats.normality, or use method='levene'."],"p_value":0.7939720083374214,"significant":false,"source":{"kind":"arrays"},"statistic":0.4614141446790681,"test":"Bartlett's Test","variance_ratio":2.339285714285714},"error":null},{"label":"stats.outliers","method":"stats.outliers","params":{"method":"grubbs"},"result":{"alpha":0.05,"method":"grubbs","notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality."],"results":[{"critical_value":1.4812499999999993,"group":"Regimen A","interpretation":"No outliers detected (G = 1.389, critical = 1.481)","n":4,"outlier_count":0,"outliers":[],"test":"Grubbs"},{"critical_value":1.4812499999999993,"group":"Regimen B","interpretation":"No outliers detected (G = 1.135, critical = 1.481)","n":4,"outlier_count":0,"outliers":[],"test":"Grubbs"},{"critical_value":1.4812499999999993,"group":"Regimen C","interpretation":"No outliers detected (G = 1.477, critical = 1.481)","n":4,"outlier_count":0,"outliers":[],"test":"Grubbs"}],"source":{"kind":"arrays"},"total_outliers":0},"error":null},{"label":"stats.outliers","method":"stats.outliers","params":{"method":"generalized_esd","max_outliers":1},"result":{"alpha":0.05,"method":"generalized_esd","notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality."],"results":[{"critical_value":1.4812499999999993,"group":"Regimen A","interpretation":"No outliers detected (tested up to 1 outliers)","n":4,"outlier_count":0,"outliers":[],"test":"Generalized ESD"},{"critical_value":1.4812499999999993,"group":"Regimen B","interpretation":"No outliers detected (tested up to 1 outliers)","n":4,"outlier_count":0,"outliers":[],"test":"Generalized ESD"},{"critical_value":1.4812499999999993,"group":"Regimen C","interpretation":"No outliers detected (tested up to 1 outliers)","n":4,"outlier_count":0,"outliers":[],"test":"Generalized ESD"}],"source":{"kind":"arrays"},"total_outliers":0},"error":null},{"label":"stats.outliers","method":"stats.outliers","params":{"method":"dixon_q"},"result":{"alpha":0.05,"method":"dixon_q","notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality."],"results":[{"critical_value":0.941,"group":"Regimen A","interpretation":"No outliers detected (Q_low = 0.600, Q_high = 0.200, critical = 0.941)","n":4,"outlier_count":0,"outliers":[],"test":"Dixon Q"},{"critical_value":0.941,"group":"Regimen B","interpretation":"No outliers detected (Q_low = 0.143, Q_high = 0.286, critical = 0.941)","n":4,"outlier_count":0,"outliers":[],"test":"Dixon Q"},{"critical_value":0.941,"group":"Regimen C","interpretation":"No outliers detected (Q_low = 0.167, Q_high = 0.833, critical = 0.941)","n":4,"outlier_count":0,"outliers":[],"test":"Dixon Q"}],"source":{"kind":"arrays"},"total_outliers":0},"error":null},{"label":"stats.outliers","method":"stats.outliers","params":{"method":"modified_z"},"result":{"alpha":0.05,"method":"modified_z","notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Flagged points are candidates, not verdicts, and nothing has been removed. Look for a physical cause before excluding anything; if there is none, keep the point and use a robust test instead. Any exclusion must be reported."],"results":[{"critical_value":3.5,"group":"Regimen A","interpretation":"No outliers detected (threshold = 3.50)","n":4,"outlier_count":0,"outliers":[],"test":"Modified Z-Score"},{"critical_value":3.5,"group":"Regimen B","interpretation":"No outliers detected (threshold = 3.50)","n":4,"outlier_count":0,"outliers":[],"test":"Modified Z-Score"},{"critical_value":3.5,"group":"Regimen C","interpretation":"Detected 1 outlier(s) using modified Z-score (threshold = 3.50)","n":4,"outlier_count":1,"outliers":[{"index":0,"test_statistic":6.745,"value":57.0}],"test":"Modified Z-Score"}],"source":{"kind":"arrays"},"total_outliers":1},"error":null},{"label":"stats.outliers","method":"stats.outliers","params":{"method":"iqr"},"result":{"alpha":0.05,"method":"iqr","notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Flagged points are candidates, not verdicts, and nothing has been removed. Look for a physical cause before excluding anything; if there is none, keep the point and use a robust test instead. Any exclusion must be reported.","The IQR rule is descriptive — it has no p-value and flags roughly 0.7% of a perfectly normal sample by construction."],"results":[{"critical_value":1.5,"group":"Regimen A","interpretation":"No outliers detected (bounds: [26.250, 34.250], k = 1.5)","n":4,"outlier_count":0,"outliers":[],"test":"IQR Method"},{"critical_value":1.5,"group":"Regimen B","interpretation":"No outliers detected (bounds: [29.625, 48.625], k = 1.5)","n":4,"outlier_count":0,"outliers":[],"test":"IQR Method"},{"critical_value":1.5,"group":"Regimen C","interpretation":"Detected 1 outlier(s) outside [49.500, 55.500] (k = 1.5)","n":4,"outlier_count":1,"outliers":[{"index":0,"test_statistic":2.5,"value":57.0}],"test":"IQR Method"}],"source":{"kind":"arrays"},"total_outliers":1},"error":null},{"label":"stats.ci","method":"stats.ci","params":{"conf_level":0.95},"result":{"conf_level":0.95,"estimate":30.0,"interpretation":"Over repeated sampling, intervals built this way contain the true mean at the stated rate. It is not the probability that this particular interval contains it.","kind":"mean","lower":26.562565118141748,"margin_of_error":3.437434881858252,"method":"t","n":4,"name":"data","notes":["n = 4. The t interval assumes the underlying distribution is roughly normal, and at this size that assumption cannot be checked from the data."],"sd":2.160246899469287,"source":{"kind":"arrays_or_dataset"},"upper":33.43743488185825},"error":null},{"label":"stats.ci","method":"stats.ci","params":{"conf_level":0.95},"result":{"conf_level":0.95,"estimate":39.25,"interpretation":"Over repeated sampling, intervals built this way contain the true mean at the stated rate. It is not the probability that this particular interval contains it.","kind":"mean","lower":33.99253834284908,"margin_of_error":5.25746165715092,"method":"t","n":4,"name":"data","notes":["n = 4. The t interval assumes the underlying distribution is roughly normal, and at this size that assumption cannot be checked from the data."],"sd":3.304037933599835,"source":{"kind":"arrays_or_dataset"},"upper":44.50746165715092},"error":null},{"label":"stats.ci","method":"stats.ci","params":{"conf_level":0.95},"result":{"conf_level":0.95,"estimate":53.0,"interpretation":"Over repeated sampling, intervals built this way contain the true mean at the stated rate. It is not the probability that this particular interval contains it.","kind":"mean","lower":48.69094733253063,"margin_of_error":4.309052667469373,"method":"t","n":4,"name":"data","notes":["n = 4. The t interval assumes the underlying distribution is roughly normal, and at this size that assumption cannot be checked from the data."],"sd":2.70801280154532,"source":{"kind":"arrays_or_dataset"},"upper":57.30905266746937},"error":null},{"label":"stats.anova","method":"stats.anova","params":{},"result":{"alpha":0.05,"assumptions":{"alpha":0.05,"homogeneity_of_variance":{"equal_variance_rejected":false,"p_value":0.5091996996441339,"statistic":0.7281553398058251,"test":"Levene's Test"},"independence":"Assumed, not tested — observations must come from independent sampling units. Repeated measures on the same unit require a paired or mixed-model analysis instead.","normality":[{"group":"Regimen A","normality_rejected":false,"p_value":0.5773549021744702,"statistic":0.9270816792875027,"test":"Shapiro-Wilk"},{"group":"Regimen B","normality_rejected":false,"p_value":0.5130400846566809,"statistic":0.9156860669810516,"test":"Shapiro-Wilk"},{"group":"Regimen C","normality_rejected":false,"p_value":0.06184652087897266,"statistic":0.7729072344676293,"test":"Shapiro-Wilk"}]},"design":"one_way","df_between":2.0,"df_within":9.0,"effect_size":{"cohens_f":3.9478417579045075,"estimate":0.9201432581550673,"eta_squared":0.9397062047796536,"interpretation":"eta-squared is the variance explained in THIS sample and is upward-biased; omega-squared estimates the population value and is the one to report.","magnitude":"large","name":"omega_squared","omega_squared":0.9201432581550673},"groups":[{"excess_kurtosis":1.4999999999999982,"iqr":2.0,"max":32.0,"mean":30.0,"median":30.5,"min":27.0,"n":4,"n_missing":0,"name":"Regimen A","q1":29.25,"q3":31.25,"range":5.0,"sd":2.160246899469287,"sem":1.0801234497346435,"skewness":-1.1903401282789945,"sum":120.0,"variance":4.666666666666667},{"excess_kurtosis":-3.869005302721286,"iqr":4.75,"max":43.0,"mean":39.25,"median":39.0,"min":36.0,"n":4,"n_missing":0,"name":"Regimen B","q1":36.75,"q3":41.5,"range":7.0,"sd":3.304037933599835,"sem":1.6520189667999174,"skewness":0.22872775858729724,"sum":157.0,"variance":10.916666666666666},{"excess_kurtosis":3.4834710743801622,"iqr":1.5,"max":57.0,"mean":53.0,"median":52.0,"min":51.0,"n":4,"n_missing":0,"name":"Regimen C","q1":51.75,"q3":53.25,"range":6.0,"sd":2.70801280154532,"sem":1.35400640077266,"skewness":1.8128019580592636,"sum":212.0,"variance":7.333333333333333}],"k_groups":3,"ms_between":535.75,"ms_within":7.638888888888889,"n_total":12,"notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality."],"p_value":3.24509038133982e-06,"post_hoc":"A significant F is an omnibus result. Run pairwise stats.ttest comparisons and pass their p-values through stats.adjust before reporting any specific pair.","significant":true,"source":{"kind":"arrays"},"statistic":70.13454545454545},"error":null},{"label":"stats.kruskal","method":"stats.kruskal","params":{},"result":{"alpha":0.05,"assumptions":{"note":"Distribution-free. Groups must be independent; similar group shapes are needed to read the result as a difference in medians rather than in distribution."},"df":2,"effect_size":{"estimate":0.8982456140350877,"interpretation":"Proportion of rank variance explained by group; 0..1.","name":"epsilon_squared"},"groups":[{"excess_kurtosis":1.4999999999999982,"iqr":2.0,"max":32.0,"mean":30.0,"median":30.5,"min":27.0,"n":4,"n_missing":0,"name":"Regimen A","q1":29.25,"q3":31.25,"range":5.0,"sd":2.160246899469287,"sem":1.0801234497346435,"skewness":-1.1903401282789945,"sum":120.0,"variance":4.666666666666667},{"excess_kurtosis":-3.869005302721286,"iqr":4.75,"max":43.0,"mean":39.25,"median":39.0,"min":36.0,"n":4,"n_missing":0,"name":"Regimen B","q1":36.75,"q3":41.5,"range":7.0,"sd":3.304037933599835,"sem":1.6520189667999174,"skewness":0.22872775858729724,"sum":157.0,"variance":10.916666666666666},{"excess_kurtosis":3.4834710743801622,"iqr":1.5,"max":57.0,"mean":53.0,"median":52.0,"min":51.0,"n":4,"n_missing":0,"name":"Regimen C","q1":51.75,"q3":53.25,"range":6.0,"sd":2.70801280154532,"sem":1.35400640077266,"skewness":1.8128019580592636,"sum":212.0,"variance":7.333333333333333}],"interpretation":"At least one group significantly differs (H = 9.881, df = 2, p = 0.0072)","n_total":12,"notes":["Group 'Regimen A' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen B' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'Regimen C' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality."],"p_value":0.007152088427287769,"post_hoc":"A significant H is an omnibus result. Run pairwise stats.mannwhitney comparisons and pass their p-values through stats.adjust (benjamini_hochberg or holm) before reporting any specific pair.","sample_sizes":[4,4,4],"significant":true,"source":{"kind":"arrays"},"statistic":9.880701754385965,"test":"Kruskal-Wallis"},"error":null},{"label":"stats.ttest","method":"stats.ttest","params":{"test":"welch","check_assumptions":false},"result":{"alpha":0.05,"alternative":"two_sided","assumptions":{"checked":false},"ci_of_difference":{"conf_level":0.95,"lower":-14.274401969170711,"method":"t (Welch)","upper":-4.225598030829289},"df":5.168596344287335,"effect_size":{"ci":{"conf_level":0.95,"lower":-4.838034296578368,"upper":-0.8492820325745262},"cohens_d":-3.3138019588472534,"estimate":-2.881566920736742,"interpretation":"Standardised mean difference (group 1 minus group 2), small-sample-bias corrected.","magnitude":"very large","name":"hedges_g"},"groups":[{"excess_kurtosis":1.4999999999999982,"iqr":2.0,"max":32.0,"mean":30.0,"median":30.5,"min":27.0,"n":4,"n_missing":0,"name":"a","q1":29.25,"q3":31.25,"range":5.0,"sd":2.160246899469287,"sem":1.0801234497346435,"skewness":-1.1903401282789945,"sum":120.0,"variance":4.666666666666667},{"excess_kurtosis":-3.869005302721286,"iqr":4.75,"max":43.0,"mean":39.25,"median":39.0,"min":36.0,"n":4,"n_missing":0,"name":"b","q1":36.75,"q3":41.5,"range":7.0,"sd":3.304037933599835,"sem":1.6520189667999174,"skewness":0.22872775858729724,"sum":157.0,"variance":10.916666666666666}],"mean_difference":-9.25,"notes":["Group 'a' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'b' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality."],"p_two_sided":0.00496425735444046,"p_value":0.00496425735444046,"significant":true,"source":{"kind":"arrays"},"statistic":-4.686423673220315,"test":"welch"},"error":null},{"label":"stats.mannwhitney","method":"stats.mannwhitney","params":{},"result":{"alpha":0.05,"alternative":"two_sided","assumptions":{"note":"Distribution-free: no normality assumption. Observations must be independent both within and between groups, and the two distributions are assumed to have similar shape if the result is to be read as a median shift."},"effect_size":{"estimate":-1.0,"interpretation":"P(x > y) - P(x < y) for a random pair, one value from each group. 0 means complete overlap.","magnitude":"large","name":"cliffs_delta","rank_biserial_r":-1.0},"groups":[{"excess_kurtosis":1.4999999999999982,"iqr":2.0,"max":32.0,"mean":30.0,"median":30.5,"min":27.0,"n":4,"n_missing":0,"name":"a","q1":29.25,"q3":31.25,"range":5.0,"sd":2.160246899469287,"sem":1.0801234497346435,"skewness":-1.1903401282789945,"sum":120.0,"variance":4.666666666666667},{"excess_kurtosis":-3.869005302721286,"iqr":4.75,"max":43.0,"mean":39.25,"median":39.0,"min":36.0,"n":4,"n_missing":0,"name":"b","q1":36.75,"q3":41.5,"range":7.0,"sd":3.304037933599835,"sem":1.6520189667999174,"skewness":0.22872775858729724,"sum":157.0,"variance":10.916666666666666}],"interpretation":"Distributions significantly differ (U = 0.0, Z = -2.165, p = 0.0304)","notes":["Group 'a' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'b' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","The normal approximation used for the p-value is unreliable below roughly n = 8 per group; treat the p-value as indicative only."],"p_two_sided":0.030382821976576612,"p_value":0.030382821976576612,"sample_sizes":[4,4],"significant":true,"source":{"kind":"arrays"},"statistic":0.0,"test":"Mann-Whitney U"},"error":null},{"label":"stats.ttest","method":"stats.ttest","params":{"test":"welch","check_assumptions":false},"result":{"alpha":0.05,"alternative":"two_sided","assumptions":{"checked":false},"ci_of_difference":{"conf_level":0.95,"lower":-27.28942947328942,"method":"t (Welch)","upper":-18.71057052671058},"df":5.7176470588235295,"effect_size":{"ci":{"conf_level":0.95,"lower":-12.810086979793772,"upper":-3.5206702328231945},"cohens_d":-9.38971068066885,"estimate":-8.16496580927726,"interpretation":"Standardised mean difference (group 1 minus group 2), small-sample-bias corrected.","magnitude":"very large","name":"hedges_g"},"groups":[{"excess_kurtosis":1.4999999999999982,"iqr":2.0,"max":32.0,"mean":30.0,"median":30.5,"min":27.0,"n":4,"n_missing":0,"name":"a","q1":29.25,"q3":31.25,"range":5.0,"sd":2.160246899469287,"sem":1.0801234497346435,"skewness":-1.1903401282789945,"sum":120.0,"variance":4.666666666666667},{"excess_kurtosis":3.4834710743801622,"iqr":1.5,"max":57.0,"mean":53.0,"median":52.0,"min":51.0,"n":4,"n_missing":0,"name":"b","q1":51.75,"q3":53.25,"range":6.0,"sd":2.70801280154532,"sem":1.35400640077266,"skewness":1.8128019580592636,"sum":212.0,"variance":7.333333333333333}],"mean_difference":-23.0,"notes":["Group 'a' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'b' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality."],"p_two_sided":1.6323465757655916e-05,"p_value":1.6323465757655916e-05,"significant":true,"source":{"kind":"arrays"},"statistic":-13.279056191361393,"test":"welch"},"error":null},{"label":"stats.mannwhitney","method":"stats.mannwhitney","params":{},"result":{"alpha":0.05,"alternative":"two_sided","assumptions":{"note":"Distribution-free: no normality assumption. Observations must be independent both within and between groups, and the two distributions are assumed to have similar shape if the result is to be read as a median shift."},"effect_size":{"estimate":-1.0,"interpretation":"P(x > y) - P(x < y) for a random pair, one value from each group. 0 means complete overlap.","magnitude":"large","name":"cliffs_delta","rank_biserial_r":-1.0},"groups":[{"excess_kurtosis":1.4999999999999982,"iqr":2.0,"max":32.0,"mean":30.0,"median":30.5,"min":27.0,"n":4,"n_missing":0,"name":"a","q1":29.25,"q3":31.25,"range":5.0,"sd":2.160246899469287,"sem":1.0801234497346435,"skewness":-1.1903401282789945,"sum":120.0,"variance":4.666666666666667},{"excess_kurtosis":3.4834710743801622,"iqr":1.5,"max":57.0,"mean":53.0,"median":52.0,"min":51.0,"n":4,"n_missing":0,"name":"b","q1":51.75,"q3":53.25,"range":6.0,"sd":2.70801280154532,"sem":1.35400640077266,"skewness":1.8128019580592636,"sum":212.0,"variance":7.333333333333333}],"interpretation":"Distributions significantly differ (U = 0.0, Z = -2.178, p = 0.0294)","notes":["Group 'a' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'b' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","The normal approximation used for the p-value is unreliable below roughly n = 8 per group; treat the p-value as indicative only."],"p_two_sided":0.029401048190338726,"p_value":0.029401048190338726,"sample_sizes":[4,4],"significant":true,"source":{"kind":"arrays"},"statistic":0.0,"test":"Mann-Whitney U"},"error":null},{"label":"stats.ttest","method":"stats.ttest","params":{"test":"welch","check_assumptions":false},"result":{"alpha":0.05,"alternative":"two_sided","assumptions":{"checked":false},"ci_of_difference":{"conf_level":0.95,"lower":-19.025840160220334,"method":"t (Welch)","upper":-8.474159839779666},"df":5.77727363983136,"effect_size":{"ci":{"conf_level":0.95,"lower":-6.419486333010133,"upper":-1.4385729096326003},"cohens_d":-4.5518323873130235,"estimate":-3.958115119402629,"interpretation":"Standardised mean difference (group 1 minus group 2), small-sample-bias corrected.","magnitude":"very large","name":"hedges_g"},"groups":[{"excess_kurtosis":-3.869005302721286,"iqr":4.75,"max":43.0,"mean":39.25,"median":39.0,"min":36.0,"n":4,"n_missing":0,"name":"a","q1":36.75,"q3":41.5,"range":7.0,"sd":3.304037933599835,"sem":1.6520189667999174,"skewness":0.22872775858729724,"sum":157.0,"variance":10.916666666666666},{"excess_kurtosis":3.4834710743801622,"iqr":1.5,"max":57.0,"mean":53.0,"median":52.0,"min":51.0,"n":4,"n_missing":0,"name":"b","q1":51.75,"q3":53.25,"range":6.0,"sd":2.70801280154532,"sem":1.35400640077266,"skewness":1.8128019580592636,"sum":212.0,"variance":7.333333333333333}],"mean_difference":-13.75,"notes":["Group 'a' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality.","Group 'b' has n = 4; normality screening has very low power at this size, so a non-significant Shapiro–Wilk is weak evidence of normality."],"p_two_sided":0.0007721418690956446,"p_value":0.0007721418690956446,"significant":true,"source":{"kind":"arrays"},"statistic":-6.4372630957871815,"test":"welch"},"error":null},{"label":"stats.mannwhitney","method":"stats.mannwhitney","params":{},"result":{"alpha":0.05,"alternative":"two_sided","assumptions":{"note":"Distribution-free: no normality assumption. 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Report the adjusted values."],"results":[{"adjusted":0.08820314457101618,"index":0,"label":"Regimen A vs Regimen B","p_value":0.030382821976576612,"significant":false},{"adjusted":0.08820314457101618,"index":1,"label":"Regimen A vs Regimen C","p_value":0.029401048190338726,"significant":false},{"adjusted":0.08820314457101618,"index":2,"label":"Regimen B vs Regimen C","p_value":0.029401048190338726,"significant":false}],"source":{"kind":"p_values"}},"error":null},{"label":"stats.adjust","method":"stats.adjust","params":{"method":"benjamini_hochberg"},"result":{"alpha":0.05,"controls":"false_discovery_rate","method":"benjamini_hochberg","n_significant_adjusted":3,"n_significant_raw":3,"n_tests":3,"notes":[],"results":[{"adjusted":0.030382821976576612,"index":0,"label":"Regimen A vs Regimen B","p_value":0.030382821976576612,"significant":true},{"adjusted":0.030382821976576612,"index":1,"label":"Regimen A vs Regimen C","p_value":0.029401048190338726,"significant":true},{"adjusted":0.030382821976576612,"index":2,"label":"Regimen B vs Regimen C","p_value":0.029401048190338726,"significant":true}],"source":{"kind":"p_values"}},"error":null},{"label":"stats.power","method":"stats.power","params":{"n1":4,"n2":4,"mode":"mde","power":0.8},"result":{"alpha":0.05,"effect_size":2.3807542132711887,"interpretation":"Power is the probability of rejecting the null when an effect of this size is real. Computing it from the effect you already observed ('observed power') is circular — plan with an effect size from a pilot or the literature, and report the MDE after.","mode":"mde","n1":4,"n2":4,"n_total":8,"notes":["With n1 = 4 and n2 = 4 at alpha = 0.05, an effect smaller than d = 2.381 would be missed more than 20% of the time. A null result from this design rules out effects LARGER than that, not effects in general."],"power":0.8,"source":{"kind":"parameters"},"test":"two-sample t-test, two-sided"},"error":null},{"label":"stats.power","method":"stats.power","params":{"n1":4,"n2":4,"effect_size":0.5},"result":{"alpha":0.05,"effect_size":0.5,"interpretation":"Power is the probability of rejecting the null when an effect of this size is real. Computing it from the effect you already observed ('observed power') is circular — plan with an effect size from a pilot or the literature, and report the MDE after.","mode":"power","n1":4,"n2":4,"n_total":8,"notes":["Power is 0.09, below the conventional 0.8: this design would miss a true effect of d = 0.5 more than 91% of the time."],"power":0.09228237923828697,"source":{"kind":"parameters"},"test":"two-sample t-test, two-sided"},"error":null},{"label":"stats.power","method":"stats.power","params":{"n1":4,"n2":4,"effect_size":0.8},"result":{"alpha":0.05,"effect_size":0.8,"interpretation":"Power is the probability of rejecting the null when an effect of this size is real. Computing it from the effect you already observed ('observed power') is circular — plan with an effect size from a pilot or the literature, and report the MDE after.","mode":"power","n1":4,"n2":4,"n_total":8,"notes":["Power is 0.16, below the conventional 0.8: this design would miss a true effect of d = 0.8 more than 84% of the time."],"power":0.1601134409476368,"source":{"kind":"parameters"},"test":"two-sample t-test, two-sided"},"error":null},{"label":"stats.power","method":"stats.power","params":{"n1":4,"n2":4,"effect_size":1.2},"result":{"alpha":0.05,"effect_size":1.2,"interpretation":"Power is the probability of rejecting the null when an effect of this size is real. Computing it from the effect you already observed ('observed power') is circular — plan with an effect size from a pilot or the literature, and report the MDE after.","mode":"power","n1":4,"n2":4,"n_total":8,"notes":["Power is 0.30, below the conventional 0.8: this design would miss a true effect of d = 1.2 more than 70% of the time."],"power":0.29900512864890283,"source":{"kind":"parameters"},"test":"two-sample t-test, two-sided"},"error":null}]}