Analysis overview and configuration
| Parameter | Value | _row |
|---|---|---|
| confidence_level | 0.95 | confidence_level |
| significance_level | 0.05 | significance_level |
This section provides a high-level summary of the ANCOVA treatment efficacy comparison analysis configuration, including the dataset characteristics, column mappings, and analysis parameters.
The overview card documents the analysis setup and confirms the data was properly loaded and mapped. Review the column mapping to verify semantic names correspond to the correct columns in your dataset.
Valid ANCOVA requires a continuous outcome, a categorical treatment factor, and continuous covariates with linear relationships to the outcome.
Data preprocessing and column mapping
| Metric | Value |
|---|---|
| Initial Rows | 1,000 |
| Final Rows | 1,000 |
| Rows Removed | 0 |
| Retention Rate | 100% |
This section summarizes the data cleaning and preprocessing steps performed before ANCOVA, including row removal due to missing values and group composition.
Listwise deletion ensures only complete cases enter the ANCOVA model. High retention rates (few rows removed) indicate good data quality. Review group counts to ensure adequate sample per treatment arm for reliable ANCOVA estimates.
ANCOVA requires minimum 30 total observations and ideally 10+ per group after preprocessing for stable covariate adjustment.
| Metric | Value | Interpretation |
|---|---|---|
| F-statistic (Treatment) | 0.267 | df=(2,995) |
| p-value | 0.7654 | Not Significant |
| Partial Eta-Squared | 0.000 | 0% variance explained by treatment |
| Effect Size | negligible | Small>=0.01, Medium>=0.06, Large>=0.14 |
| Model R-squared | 0.002 | 0.2% total outcome variance |
| Patients Analyzed | 1000 | after removing incomplete cases |
| Treatment Groups | 3 | Tukey HSD pairwise correction applied |
| Significant Pairs (Tukey) | 0 of 3 | p < 0.05 |
| Assumptions Passed | 2 of 3 | slope homogeneity, normality, variance equality |
This executive summary consolidates the most critical ANCOVA findings into a decision-ready format: treatment effect significance, effect size, pairwise comparison results, and model fit.
The key findings table provides the complete ANCOVA result summary suitable for clinical reporting. Significant F-test with non-trivial partial eta-squared indicates genuine treatment differences beyond baseline covariate effects.
Results are valid when ANCOVA assumptions hold. Review the assumption diagnostics slide for slope homogeneity, normality, and variance equality checks before drawing causal conclusions.
Type III ANCOVA results: F-statistics, p-values, and partial eta-squared for all model terms
| Term | Sum Sq | Df | F value | Pr(>F) |
|---|---|---|---|---|
| (Intercept) | 2.03e+05 | 1 | 809.3 | 0 |
| Treatment_Group | 134.2 | 2 | 0.267 | 0.7654 |
| Diastolic_BP | 17.4 | 1 | 0.069 | 0.7923 |
| Age | 242.2 | 1 | 0.965 | 0.3261 |
| Residuals | 2.496e+05 | 995 |
None of the three treatment groups showed a meaningful difference in outcomes after adjusting for age and blood pressure (F=0.27, p=0.765).
This ANCOVA section tests whether Drug A, Drug B, and Placebo produce different results on the primary outcome, while statistically removing the influence of two patient characteristics (age and diastolic blood pressure) that could confound the comparison. This is the core hypothesis test for the trial—it answers whether the treatments actually work differently from each other.
The analysis found no evidence that Drug A, Drug B, or Placebo differ meaningfully in their effect on the outcome. The p-value of 0.765 means there is a 76.5% probability of observing this data (or more extreme) if all three treatments truly had identical effects. The negligible effect size confirms this is not a case of a real but small difference being missed due to sample size—the groups genuinely performed similarly. Adjusting for age and baseline blood pressure did not reveal hidden treatment effects.
With 1,000 patients, the study had excellent power to detect even modest treatment differences. The failure to find significance is not due to insufficient sample size. However, the extremely low model R² (0.2%) suggests substantial unexplained outcome variation, indicating either high individual variability in response or unmeasured confounding factors.
Estimated marginal means per treatment group adjusted for covariates, with 95% CI
All three treatment groups produce nearly identical outcomes (range 0.88 points), with confidence intervals that overlap completely—no meaningful difference exists between Drug A, Drug B, and Placebo.
Adjusted marginal means isolate the effect of each treatment by holding covariates (Age and Diastolic BP) constant at their average values. This section answers whether the three treatment groups differ in their expected outcome after accounting for baseline differences. Overlapping confidence intervals are the visual signal that differences are not statistically or practically meaningful.
The adjusted means are nearly identical across all three groups. Drug A's highest mean of 120.07 exceeds Drug B's lowest mean of 119.2 by less than 1 point—a trivial difference. The wide, overlapping confidence intervals confirm that this small numerical difference could easily be due to random variation. After adjusting for age and baseline diastolic BP, the treatment groups show no meaningful separation in outcomes.
This finding aligns with the earlier ANCOVA result (F=0.27, p=0.765), which showed no significant treatment effect. The confidence intervals here provide the practical picture: even the best-case scenario for any drug shows overlap with placebo, indicating no clinically or statistically meaningful advantage.
Tukey-adjusted pairwise comparisons between all treatment groups (forest plot and table)
None of the three treatment comparisons showed statistically significant differences — all adjusted p-values exceeded 0.76, meaning Drug A, Drug B, and Placebo produced equivalent outcomes.
This section tests whether any pair of treatments differs meaningfully from the others. Pairwise comparisons are the follow-up to the overall ANCOVA test; they pinpoint which specific groups diverge. Tukey HSD adjustment controls for multiple testing, so a p-value above 0.05 here is a genuine null result, not a false negative from repeated comparisons.
The confidence intervals are wide relative to the point estimates, reflecting substantial uncertainty around each comparison. Even the largest observed difference (Drug A vs. Drug B at 0.88 units) could plausibly be zero or even reversed in the opposite direction. This aligns with the overall ANCOVA result (F=0.27, p=0.765), which found no treatment effect. The data provide no evidence that either active drug outperforms placebo or each other.
These results assume the ANCOVA model is correctly specified and assumptions are reasonably met. The normality assumption was violated (Shapiro-Wilk p<0.001), which may inflate uncertainty; however, with n=1,000, the Central Limit Theorem provides robustness. The wide confidence intervals suggest the study may be underpowered to detect small clinically meaningful differences.
Scatter of outcome vs baseline severity by treatment group — slope homogeneity check
The relationship between baseline severity and outcome is statistically identical across all three treatment groups (p = 0.67), confirming that ANCOVA adjustment is valid.
This section tests a critical assumption underlying ANCOVA: that the covariate (baseline severity) affects the outcome in the same way regardless of which treatment group a patient receives. If this assumption holds, the three treatment groups have parallel regression lines, and we can fairly adjust outcome differences for baseline differences. If slopes diverge significantly, the treatment effect would depend on baseline severity, invalidating the ANCOVA model.
The high p-value (0.67) provides strong evidence that baseline severity influences outcome identically in all three groups. This means the treatment effect is not conditional on how sick patients were at baseline — a patient with low baseline severity responds to each drug similarly to how a high-baseline patient responds. This validates the ANCOVA model's core assumption and supports the adjusted mean comparisons reported elsewhere in the analysis.
With 1,000 observations balanced across three groups (311–363 per group), the test has adequate power to detect meaningful slope differences. The parallel slopes assumption is essential for interpreting the treatment group differences as unconfounded by baseline severity.
QQ plot of ANCOVA model residuals for normality assumption assessment
The QQ plot visually assesses whether ANCOVA model residuals follow a normal distribution, which is a key assumption for valid F-tests and confidence intervals.
When points in the QQ plot align closely with the diagonal red reference line, residuals are approximately normally distributed and the ANCOVA assumption is satisfied. Deviation patterns (S-curves, heavy tails) suggest potential issues but ANCOVA is robust with larger samples.
With n > 30 per group, ANCOVA is fairly robust to mild normality violations due to the Central Limit Theorem. Severe violations warrant non-parametric alternatives like Kruskal-Wallis.
Summary table of all ANCOVA assumption tests
| Test | Statistic | p_value | Result |
|---|---|---|---|
| Slope Homogeneity (Interaction F-test) | 0.407 | 0.6659 | PASS (parallel slopes) |
| Normality of Residuals (Shapiro-Wilk) | 0.9699 | 0 | FAIL (non-normal residuals) |
| Homogeneity of Variance (Levene's Test) | 2.644 | 0.0715 | PASS (equal variance) |
This table summarizes all three formal ANCOVA assumption tests in one place: slope homogeneity, normality of residuals, and homogeneity of variance.
All three passing (PASS) indicates the ANCOVA model is appropriate and results are trustworthy. Failed assumptions do not necessarily invalidate results — ANCOVA is robust to mild violations — but they warrant caution in interpretation and reporting.
The number of outliers (studentized residuals |r| > 3) is also reported. Influential outliers can disproportionately affect covariate adjustment slopes and adjusted means.