Analysis overview and configuration
| Parameter | Value | _row |
|---|---|---|
| confidence_level | 0.95 | confidence_level |
| reference_group | reference_group | |
| covariate_columns | Symptom Severity (1-10),Age,Sleep Quality (1-10) | covariate_columns |
| outcome_column | Treatment Progress (1-10) | outcome_column |
| group_column | Therapy Type | group_column |
| effect_size_type | eta_squared | effect_size_type |
This ANCOVA analysis evaluates whether four therapy types (Cognitive Behavioral, Dialectical Behavioral, Interpersonal, and Mindfulness-Based) produce significantly different treatment outcomes after statistically controlling for baseline symptom severity, age, and sleep quality. The analysis uses 500 patient observations to isolate the effect of therapy type from confounding variables.
After controlling for baseline characteristics, the four therapy types produce statistically indistinguishable treatment outcomes. The neglig
Data preprocessing and column mapping
| Metric | Value |
|---|---|
| Initial Rows | 500 |
| Final Rows | 500 |
| Rows Removed | 0 |
| Retention Rate | 100% |
This section documents the data preprocessing pipeline for the ANCOVA analysis comparing four therapy types. Perfect data retention (100%) indicates no rows were removed during cleaning, meaning all 500 observations proceeded to statistical analysis without exclusions or transformations. This is critical for understanding whether the final results represent the complete dataset or a filtered subset.
The perfect retention rate suggests either exceptionally clean source data or minimal preprocessing criteria applied. This is consistent with the ANCOVA results, where no outliers were detected and all 500 cases contributed to the F-statistic (F=0.324, p=0.808). The complete dataset strengthens statistical power for detecting group differences, though the negligible effect size (partial η²=0.002) reflects genuine similarity between therapy types rather than data limitations.
No train/test split was performed, indicating this is a confirmatory analysis rather than predictive modeling.
| Finding | Value |
|---|---|
| Sample Size | 500 |
| Groups Compared | 4 |
| F-statistic | 0.324 |
| p-value | 0.8083 |
| Effect Size | negligible (η² = 0.002) |
| Assumptions Met | No |
This analysis evaluated whether four therapy types (Cognitive Behavioral, Dialectical Behavioral, Interpersonal, and Mindfulness-Based) produce meaningfully different outcomes in a sample of 500 patients. Using ANCOVA to control for baseline symptom severity, age, and sleep quality, the analysis tests whether therapy selection drives treatment effectiveness after accounting for these confounding factors.
The data provide no evidence that therapy type meaningfully influences patient outcomes in this cohort. After adjusting for baseline differences in symptom severity, age, and sleep quality, all four therapeutic approaches yield statistically indistinguishable results. The negligible effect size indicates therapy selection accounts
Type III ANCOVA results testing group differences after covariate adjustment
| Statistic | Value |
|---|---|
| F-statistic | 0.324 |
| p-value | 0.8083 |
| Partial η² | 0.002 |
| Effect Size | negligible |
| Groups | 4 |
| Sample Size | 500 |
This section presents the Type III ANCOVA omnibus test, which evaluates whether the four therapy types produce significantly different treatment outcomes after statistically controlling for three covariates (Symptom Severity, Age, Sleep Quality). This is the primary hypothesis test determining whether therapy type meaningfully influences the outcome variable.
The analysis found no statistically significant differences among the four therapy types in treatment progress. After adjusting for baseline symptom severity, age, and sleep quality, all therapy groups converged to nearly identical adjusted means (7.33–7.55 range). This suggests therapy type selection does not substantially influence
Estimated marginal means (adjusted group means) with 95% confidence intervals
Adjusted marginal means isolate the effect of therapy type by statistically controlling for baseline differences in symptom severity, age, and sleep quality. This section provides "apples-to-apples" comparisons across the four therapy groups, showing what average progress would be if all participants started from equivalent positions on measured covariates.
Despite covariate adjustment, the four therapy approaches yield nearly identical adjusted outcomes, with differences of ≤0.21 points. The overlapping confidence intervals confirm the ANCOVA's non-significant F-statistic (p = 0.808), indicating that therapy type explains negligible variance in progress
Tukey-adjusted pairwise comparisons between all therapy groups
This section tests whether any therapy type produces meaningfully different outcomes compared to others. By examining all 6 possible pairwise comparisons with Tukey adjustment, the analysis controls for multiple testing errors while identifying which specific therapy pairs differ significantly. This directly addresses whether therapy type influences the outcome measure.
Despite comparing four distinct therapy approaches, no pairwise differences reached statistical significance after Tukey correction. The largest observed difference (Interpersonal vs. Mindfulness-Based: 0.21 points) remains non-significant (p = 0.76). All confidence intervals encompass zero, indicating the true population differences are statistically indistinguishable from zero. This aligns with the overall ANCOVA finding (F = 0.324, p
Linearity assumption check - scatterplot of outcome vs primary covariate by group
This section validates the linearity assumption required for ANCOVA—that each covariate maintains a linear (straight-line) relationship with the outcome across all therapy groups. Violations of linearity reduce statistical power and can bias treatment effect estimates, making this check essential before interpreting therapy type differences.
The tight clustering of fitted values relative to raw outcome variability suggests the linear model captures the central trend well, with residual scatter reflecting genuine outcome variation rather than systematic non-linearity. The slope homogeneity test (p=0.243, PASS) confirms that covariate-outcome relationships do not differ meaningfully across therapy groups, supporting the validity of parallel regression lines assumed in
Normality of residuals assumption (Shapiro-Wilk test + QQ plot)
This section evaluates whether the residuals from the ANCOVA model follow a normal distribution—a key assumption for valid statistical inference. The Shapiro-Wilk test and QQ plot together provide evidence about whether the model's prediction errors are normally distributed, which affects the reliability of p-values and confidence intervals in comparing therapy types.
The residuals exhibit non-normal behavior, with observed values deviating more from the mean than a normal distribution would predict. However, with n=500, the ANCOVA analysis remains robust to this violation due to the Central Limit Theorem. The negligible effect sizes and non-significant therapy comparisons (all p > 0.76) are unlikely to be artifacts of this assumption violation, as the large sample size provides protection against moderate departures from normality.
This violation does not invalidate the main findings regarding therapy type differences. The homogeneity of variance assumption
Complete summary of assumption tests and diagnostic checks
| test | statistic | p_value | conclusion |
|---|---|---|---|
| Slope Homogeneity | F = 1.397 | 0.243 | PASS (p > 0.05) |
| Normality (Shapiro-Wilk) | W = 0.941 | 3.62e-13 | FAIL (p < 0.05) |
| Homogeneity of Variance (Levene) | F = 0.311 | 0.817 | PASS (p > 0.05) |
| Outliers (|studentized| > 3) | 0 outliers | — | PASS (no outliers) |
This section validates whether the ANCOVA model meets its four critical statistical assumptions: homogeneity of regression slopes, normality of residuals, homogeneity of variance, and independence of observations. These assumptions are essential for ensuring that the comparison of therapy type effects on outcomes is statistically valid and reliable.
The analysis reveals a critical violation: residuals are non-normally distributed despite adequate sample size (n=500) and absence of outliers. This suggests the outcome variable may have inherent distributional properties that violate ANCOVA's normality requirement. However, the non-significant main effect (F=0.324
Individual covariate contributions to the model
| covariate | f_stat | p_value | eta_sq |
|---|---|---|---|
| Symptom Severity (1-10) | 0.048 | 0.826 | 0 |
| Age | 0.11 | 0.741 | 0 |
| Sleep Quality (1-10) | 3.62 | 0.0577 | 0.007 |
This section isolates the independent contribution of each baseline characteristic (symptom severity, age, sleep quality) to the outcome, after accounting for therapy type and other covariates. Significant covariates act as confounders that may differ across therapy groups; controlling for them ensures observed therapy differences reflect true treatment effects rather than pre-existing baseline imbalances.
None of the three covariates reached statistical significance (all p > 0.05), indicating that baseline differences in symptom severity, age, and sleep quality do not substantially confound the therapy comparison. Sleep quality approached significance but still failed to meet the 0.05 threshold. This suggests the therapy groups were reasonably balanced on these characteristics, and