SMHS ANCOVA - Revision history

Dinov: /* References */

2025-12-09T14:09:15Z

‎References

Dinov: /* References */

2025-12-09T14:08:58Z

‎References

Dinov: /* Problems and Exercises */

2025-12-09T14:08:34Z

‎Problems and Exercises

Dinov: /* Common Issues and Solutions */

2025-12-09T14:05:18Z

‎Common Issues and Solutions

Dinov: /* 4) Assumptions and Diagnostics */

2025-12-09T14:02:45Z

‎4) Assumptions and Diagnostics

Dinov at 13:49, 9 December 2025

2025-12-09T13:49:30Z

Show changes

Dinov: /* MANOVA */

2014-10-29T15:33:08Z

‎MANOVA

Dinov at 15:25, 29 October 2014

2014-10-29T15:25:33Z

Show changes

Clgalla at 15:05, 29 October 2014

2014-10-29T15:05:50Z

Clgalla: /* Scientific Methods for Health Sciences - Analysis of Covariance (ANCOVA) */

2014-10-29T15:00:40Z

‎Scientific Methods for Health Sciences - Analysis of Covariance (ANCOVA)

@@ Line 775: / Line 775: @@
 . Maxwell, S. E., Delaney, H. D., & Kelley, K. (2017). *Designing Experiments and Analyzing Data: A Model Comparison Perspective* (3rd ed.). Routledge.
 . Rutherford, A. (2011). *ANOVA and ANCOVA: A GLM Approach* (2nd ed.). Wiley.
 . Tabachnick, B. G., & Fidell, L. S. (2018). *Using Multivariate Statistics* (7th ed.). Pearson.
 . Miller, G. A., & Chapman, J. P. (2001). Misunderstanding analysis of covariance. *Journal of Abnormal Psychology, 110*(1), 40-48.
 . Stevens, J. P. (2012). *Applied Multivariate Statistics for the Social Sciences* (5th ed.). Routledge.

@@ Line 780: / Line 780: @@
 . Stevens, J. P. (2012). *Applied Multivariate Statistics for the Social Sciences* (5th ed.). Routledge.
-####Online Resources####
+===Online Resources===
 * [https://www.jstatsoft.org/article/view/v033i01 ANCOVA in R Tutorial]
 * [https://cran.r-project.org/web/packages/emmeans/vignettes/emmeans.html Estimated Marginal Means]

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 . '''Applied Problems''':
-    a) Analyze the SOCR [Consumer Price Index dataset](http://wiki.socr.umich.edu/index.php/SOCR_Data_Dinov_021808_ConsumerPriceIndex3Way) using ANCOVA
+    a) Analyze the SOCR [http://wiki.socr.umich.edu/index.php/SOCR_Data_Dinov_021808_ConsumerPriceIndex3Way Consumer Price Index dataset] using ANCOVA
-    b) Conduct a MANCOVA on the [Iris dataset](https://archive.ics.uci.edu/ml/datasets/iris) with species as the factor and sepal length as covariate
+    b) Conduct a MANCOVA on the [https://archive.ics.uci.edu/ml/datasets/iris Iris dataset] with species as the factor and sepal length as covariate
     c) Perform power analysis for a planned study with 4 groups, expecting a medium effect (f² = 0.20), and correlation ρ = 0.6 between covariate and DV

@@ Line 655: / Line 655: @@
 ===Common Issues and Solutions===
-####1) Violation of Homogeneity of Slopes
+====Violation of Homogeneity of Slopes====
 <pre>
 # When slopes differ by group
@@ Line 694: / Line 694: @@
 </pre>
-####2) Dealing with Missing Data
+====Dealing with Missing Data====
 <pre>
 # Multiple imputation for ANCOVA with missing covariate values

@@ Line 202: / Line 202: @@
+}
 </pre>
 <pre>
@@ Line 237: / Line 239: @@
    abline(h = 0, col = "red", lty = 2)
-   # 4. Homogeneity of regression slopes
+   # 4. Homogeneity of regression slopes - FIXED: Properly nested comparison
-   # Create interaction model
+   # Get all terms from the original model
-   interaction_formula <- as.formula(paste(response_var, "~", group_var, "*", covariate))
+   model_terms <- attr(terms(model), "term.labels")
    interaction_model <- lm(interaction_formula, data = data)
    # Test for significant interaction
-   interaction_test <- anova(model, interaction_model)
+   interaction_test <- anova(base_model, interaction_model)
    cat("\n2. Homogeneity of slopes test:\n")
    print(interaction_test)
-   # 5. Plot different slopes if interaction exists
+   # Check if the interaction is significant - FIXED: Handle NA p-values
-   if (interaction_test$`Pr(>F)`[2] < 0.05) {
+   p_value <- interaction_test$`Pr(>F)`[2]
-    plot(data[[covariate]], data[[response_var]],
-         col = as.numeric(data[[group_var]]),
+  if (!is.na(p_value)) {
-         xlab = covariate, ylab = response_var,
+    if (p_value < 0.05) {
-         main = "Different Slopes by Group")
+      cat("\nWARNING: Significant interaction detected (p =",
-    legend("topright", legend = levels(data[[group_var]]),
+          format.pval(p_value, digits = 4),
-           col = 1:length(levels(data[[group_var]])), pch = 1)
+          "). Slopes are not homogeneous.\n")
+   }
-   # 6. Residual histogram
+   # 5. Residual histogram
    hist(residuals, main = "Histogram of Residuals",
         xlab = "Residuals", col = "lightblue")
@@ Line 270: / Line 320: @@
    # Check for outliers (residuals > 3 SD)
-   outlier_count <- sum(abs(scale(residuals)) > 3)
+   if (sd(residuals) > 0) {
-  cat("- Potential outliers (>3 SD):", outlier_count, "\n")
+    outlier_count <- sum(abs(scale(residuals)) > 3, na.rm = TRUE)
+}
 </pre>
 ===Applications===

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 **MANOVA is an extension of ANOVA, though, unlike ANOVA, it uses the variance-covariance between variables in testing the statistical significance of the mean difference. It is similar to ANOVA, but allows adding of interval independents as covariates. Several specific use-cases for MANOVA: (1) to compare groups formed by categorical independent variables on group differences in a set of dependent variables;  (2) to use lack of difference for a set of dependent variables as a criterion for reducing a set of independent variables to a smaller, more easily modeled number of variables; (3) to identify the independent variables which differentiate a set of dependent variables the most.
 **Analogous to ANOVA, MANOVA is based on the product of model variance matrix, $\Sigma_{model}$ and inverse of the error variance matrix, $\Sigma_{res}^{-1}, or A=\Sigma_{model} * \Sigma_{res}^{-1}$. The hypothesis that $\Sigma_{model} = \Sigma_{residual}$ implies that the product $A \sim I$. Invariance considerations imply the MANOVA statistic should be a measure of magnitude of the singular value decomposition of this matrix product, there is no unique choice owing to the multi-dimensional nature of the alternative hypothesis.
-**MANOVA calculations closely resemble the ANOVA calculations, except that they are in vector and matrix forms. Assume that instead of a single dependent variance in the one-way ANOVA, there are three dependent variables as in our neuroimaging example above. Under the null hypothesis, it is assumed that scores on the three variables for each of the four groups are sampled from a tri-variate normal distribution mean vector $\mu =(\mu_{1}, \mu_{2}, \mu_{3})^{T}$ and variance-covariance matrix $\Sigma=\bigl(\begin{smallmatrix} \sigma_{1}^{2} & \rho_{1,2}\sigma_{1}\sigma{2}  &\rho_{1,3}\sigma_{1}\sigma{3} \\ \rho_{2,1}\sigma_{2}\sigma_{1} & \sigma_{2}^{2}  & \rho_{2,3}\sigma_{2}\sigma{3} \\\rho_{3,1}\sigma_{3}\sigma_{1}&\rho_{3,2}\sigma_{3}\sigma_{2} &\sigma_{3}^{2}\end{smallmatrix}\bigr)$. Where the covariance between variables 1 and 2 is expressed in terms of their correlation $(\rho_{1,2})$ and individual variances $(\sigma_{1}$ and $\sigma_{2})$. Under the null hypothesis, the scores for all subjects in groups 1, 2 and 3 are sampled from the same distribution.
+**MANOVA calculations closely resemble the ANOVA calculations, except that they are in vector and matrix forms. Assume that instead of a single dependent variance in the one-way ANOVA, there are three dependent variables as in our neuroimaging example above. Under the null hypothesis, it is assumed that scores on the three variables for each of the four groups are sampled from a tri-variate normal distribution mean vector $\mu =(\mu_{1}, \mu_{2}, \mu_{3})^{T}$ and variance-covariance matrix $\Sigma=\bigg ( \begin{smallmatrix} \sigma_{1}^{2} & \rho_{1,2}\sigma_{1}\sigma{2}  &\rho_{1,3}\sigma_{1}\sigma_{3} \\ \rho_{2,1}\sigma_{2}\sigma_{1} & \sigma_{2}^{2}  & \rho_{2,3}\sigma_{2}\sigma_{3} \\\rho_{3,1}\sigma_{3}\sigma_{1}&\rho_{3,2}\sigma_{3}\sigma_{2} &\sigma_{3}^{2}\end{smallmatrix}\bigg )$. Where the covariance between variables 1 and 2 is expressed in terms of their correlation $(\rho_{1,2})$ and individual variances $(\sigma_{1}$ and $\sigma_{2})$. Under the null hypothesis, the scores for all subjects in groups 1, 2 and 3 are sampled from the same distribution.
 Example: a $2*2$ factorial design with medication as one factor and type of therapy as the second factor. The matrix of the data looks includes the patient ID, the drug-treatment (vitamin-E or Placebo), Therapy (Cognitive/physical), MMSE, CDR, Imaging. It's better when the study the design is balanced with equal numbers of patients in all four conditions, as this avoid potential problems of sample-size-driven effects (e.g., variance estimates). Recall that a univariate ANOVA (on any single outcome measure) would contain three types of effects -- a main effect for therapy, a mean effect for medication, and an interaction between therapy and medication. Similarly, MANOVA will contain the same three effects: main effects: (1) Therapy: The univariate ANOVA main effect for therapy tells whether the physical vs. cognitive therapy groups have different means, irrespective of their medications. The MANOVA main effect for therapy tells whether the physical vs. cognitive therapy group have different mean vectors irrespective of their medication. The vectors in this case are the $(3*1)$ column vectors of means (MMSE, CDR and Imaging); (2) Medication: The univariate ANOVA for medication tells whether the placebo group has a different mean from the Vitamin-E group irrespective of the therapy type. The MANOVA main effect for medication tells whether the placebo group has a different mean vector from the VItamin-E group irrespective of therapy; interaction effects: (3) The univariate ANOVA interaction tells whether the four means for a single variable differ from the value predicted from knowledge of the main effects of therapy and medication. The MANOVA interaction term tells whether the four mean vectors differ from the vector predicted from knowledge of the main effects of therapy and medication.
 **Variance partitioning: MANOVA has the same properties as an ANOVA. The only difference is that an ANOVA deals with a $(1*1)$ mean vector for any group (as the response is univatiate). While a MANOVA deals with a $(k*1)$ vector for any group, $k$ being the number of dependent variables, 3 in our example. The variance-covariance matrix for 1 variable is a $(1*1)$ matrix that has only one element, the variance of the variable. What is the variance-covariance matrix for $k$ variables is a $(k*k)$ matrix with the variances on the diagonal and the covariances representing the off diagonal elements. The ANOVA partitions the $(1*1)$ covariance matrix into a part due to error and a part due to the researcher-specified hypotheses (the two main effects and the interaction term). That is: $V_{total} = V_{therapy} + V_{medication} + V_{therapy*medication} + V_{error}.$ Likewise, MANOVA partitions its $(k*k)$ covariance matrix into a part due to research-hypotheses and a part due to error. Thus, in out example, MANOVA will have a $(3*3)$ covariance matrix for total variability, a $(3*3)$ covariance matrix due to therapy, a $(3*3)$ covariance matrix due to medication, a $(3*3)$ covariance matrix due to the interaction of therapy with medication, and finally a $(3*3)$ covariance matrix for the error. That is: $V_{total} = V_{therapy} + V_{medication} + V_{therapy*medication} + V_{error}$. Now,$V$ stands for the appropriate $(3*3)$ matrix, as opposed to $(1*1)$ value, as in ANOVA. The second equation is the matrix-form of the first one. Here is how we interpret these matrices. The error matrix looks like:
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 **Pillai’s Trace: Pillai’s criterion is the pooled effect variances. $Pillai’s\: trace=trace[H(H+E)^{-1}]=\prod_{i=1}^{k}\frac{1}{1+\lambda_{i}}$.
 **Hotelling’s Trace: the pooled ratio of effect variance to error variance:$trace(A)=trace[HE^{-1}]=\sum_{i=1}^{k}\lambda_{i}$.
 **Roy’s largest root: gives an upper bound for the F statistic, $Roy’s\: largest \,root=max(\hat{\lambda_{i}})$.
 ====MANCOVA====

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 |}
 </center>
 *ANOVA conditions: valid if (1) the population from which the samples were obtained must be normally or approximately normally distributed; (2) the samples must be independent; (3) the variances of the populations must be equal; (4) the groups must have the same sample size.

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 **Roy’s largest root: gives an upper bound for the F statistic, $Roy’s\: largest \,root=max(\hat{\lambda_{i}})$.
 ::MANCOVA
 A multivariate analysis of covariance MANCOVA is a statistical an extension of ANCOVA and is designed for cases where there is more than one dependent variable and where the control of concomitant continuous independent variables is required. The process of characterizing a covariate in a data source allows the reduction of the magnitude of the error term, represented in the MANCOVA design as $MS_{error}$. The MANCOVA allows the characterization of the difference in group means in regards to a linear combination of multiple dependent variables, while simultaneously controlling for covariates.
 *Assumptions: (1) normality, for each group, each dependent variable follows a normal distribution and any linear combination of dependent variables are normally distributed; (2) independence of observations from all other observations; (3) homogeneity of variance: each dependent variable demonstrate similar levels of variance across each independent variable; (4) homogeneity of covariance: the intercorrelation matrix between dependent variables equals to each other across all levels of independent variable.