The General Linear Model (GLM)

 

The Common Foundation: The General Linear Model (GLM)



First, it's crucial to understand that ANOVA, ANCOVA, and MANOVA are all part of the same family: the General Linear Model (GLM). They all test for differences between group means, but they differ in the number and type of variables they can handle.

  • Independent Variable(s): The grouping variable(s) (e.g., drug type, breed, diet). These are categorical (factors).

  • Dependent Variable(s): The outcome you are measuring (e.g., weight gain, blood cell count, enzyme level). These are continuous.

1. ANOVA (Analysis of Variance)

What it is: ANOVA is used to determine if there are any statistically significant differences between the means of three or more independent groups based on one continuous dependent variable.

When to use it:

  • You have one categorical independent variable (factor) with three or more levels.

  • You have one continuous dependent variable.

  • You want to know: "Do the means of my groups differ significantly?"

Key Concept: ANOVA works by partitioning the total variance in the data into two parts:

  1. Variance between groups: Due to the treatment or group effect.

  2. Variance within groups: Due to random error or individual differences.

If the between-group variance is significantly larger than the within-group variance, then the group means are likely different.

Example in Animal Medicine:

  • Research Question: Is there a significant difference in the mean recovery time (days) for dogs with a specific infection after being treated with one of three different antibiotics (Drug A, Drug B, or Drug C)?

  • Experimental Setup:

    • Independent Variable (Factor): Antibiotic Type with 3 levels: A, B, C.

    • Dependent Variable: Recovery Time (measured in days, a continuous variable).

    • 30 dogs are randomly assigned to one of the three drug groups (10 per group).

  • ANOVA Analysis:

    • The ANOVA test will produce an F-statistic and a p-value.

    • Null Hypothesis (H₀): µ_A = µ_B = µ_C (The mean recovery times are all equal).

    • Interpretation: If the p-value is less than the significance level (e.g., p < 0.05), we reject the null hypothesis. We conclude that at least one antibiotic leads to a different mean recovery time.

  • Important Next Step: A significant ANOVA doesn't tell you which groups are different. You must perform a post-hoc test (e.g., Tukey's HSD, Bonferroni) to compare each pair of groups (A vs. B, A vs. C, B vs. C).

2. ANCOVA (Analysis of Covariance)

What it is: ANCOVA is an extension of ANOVA that allows you to control for the effect of a continuous variable that you did not manipulate but that might influence the dependent variable. This control variable is called a covariate.

When to use it:

  • You have one or more categorical independent variables.

  • You have one continuous dependent variable.

  • You have at least one continuous variable (the covariate) that is a potential source of unwanted variation.

Key Concept: ANCOVA "adjusts" the dependent variable for the covariate. It essentially asks: "After accounting for the effect of this other variable (the covariate), are there still significant differences between my groups?"

Example in Animal Medicine:

  • Research Question: Do two different diets (Diet X and Diet Y) lead to a significant difference in the final weight of pigs, after accounting for their initial weight?

  • Why ANCOVA? The initial weight of the pigs at the start of the study will strongly influence their final weight. If the groups aren't perfectly balanced on initial weight (which they often aren't, even with random assignment), this could bias the results.

  • Experimental Setup:

    • Independent Variable (Factor): Diet Type with 2 levels: X, Y.

    • Dependent Variable: Final Weight (kg).

    • Covariate: Initial Weight (kg).

    • 20 piglets are randomly assigned to one of the two diets.

  • ANCOVA Analysis:

    • ANCOVA will first determine if Initial Weight is a significant predictor of Final Weight (it almost certainly will be).

    • It will then statistically "adjust" the Final Weight of each pig to what it would be if they all started with the same Initial Weight (e.g., the mean initial weight of the whole sample).

    • Finally, it tests for a difference in these adjusted final weights between the two diet groups.

    • Interpretation: "After controlling for initial weight, there was a significant effect of diet on final weight (F(1, 17) = 9.85, p = 0.006)." This gives you much greater confidence that any difference is due to the diet itself, not pre-existing size differences.

3. MANOVA (Multivariate Analysis of Variance)

What it is: MANOVA is used when you have multiple related continuous dependent variables. It tests for group differences across this combination of outcomes simultaneously.

When to use it:

  • You have one or more categorical independent variables.

  • You have two or more continuous dependent variables that are correlated with each other.

  • You want to know: "Does the treatment affect the overall profile of outcomes?"

Key Concept: Instead of calculating the variance for a single variable (like ANOVA), MANOVA creates a linear combination (a weighted score) of all the dependent variables and then analyzes the variance of this new composite variable. This protects against Type I errors that can occur when running multiple ANOVAs.

Example in Animal Medicine:

  • Research Question: Does a new anti-inflammatory drug affect a holistic set of health markers in arthritic horses compared to a placebo?

  • Why MANOVA? Inflammation affects multiple body systems. Looking at just one outcome (e.g., lameness score) might miss the full picture. Running separate ANOVAs for lameness, pain score, and plasma fibrinogen increases the chance of a false positive.

  • Experimental Setup:

    • Independent Variable (Factor): Treatment with 2 levels: New Drug, Placebo.

    • Dependent Variables:

      1. Lameness Score (continuous scale)

      2. Pain Score (continuous scale)

      3. Plasma Fibrinogen Concentration (a blood biomarker for inflammation)

    • 24 horses are randomly assigned to a treatment group.

  • MANOVA Analysis:

    • MANOVA will test the null hypothesis that the vector of means for the three dependent variables is the same for both treatment groups.

    • It produces a test statistic (e.g., Wilks' Lambda) and a p-value.

    • Interpretation: A significant MANOVA (p < 0.05) tells you that the drug has a significant overall effect on the combination of the three health markers.

  • Important Next Step: Following a significant MANOVA, you can run post-hoc ANOVAs on each individual dependent variable to see which ones are driving the overall effect. You can also run Discriminant Analysis to understand how the variables combine to differentiate the groups.

Summary Table

FeatureANOVAANCOVAMANOVA
Full NameAnalysis of VarianceAnalysis of CovarianceMultivariate Analysis of Variance
PurposeCompare group means on one outcome.Compare group means on one outcome after controlling for a covariate.Compare groups on multiple related outcomes simultaneously.
Independent VarsOne or more categorical factors.One or more categorical factors.One or more categorical factors.
Dependent VarsOne continuous variable.One continuous variable.Two or more continuous, correlated variables.
Key Extra Element-Covariate(s): Continuous variable(s) to control for.-
Animal Medicine ExampleCompare 3 antibiotics on recovery time.Compare 2 diets on final weight, controlling for initial weight.Compare a drug vs. placebo on lameness, pain, and a blood marker.

Choosing the Right Test

  1. How many dependent variables do you have?

    • One -> Use ANOVA or ANCOVA.

    • Multiple, correlated -> Use MANOVA.

  2. Do you need to control for a confounding variable?

    • Yes, and it's continuous -> Use ANCOVA.

    • No -> Use ANOVA.

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