24.6 Single Factor Covariance Model
The single-factor covariance model (Analysis of Covariance, ANCOVA) accounts for both treatment effects and a continuous covariate:
Yij=μ.+τi+γ(Xij−ˉX..)+ϵij
for i=1,…,r (treatments) and j=1,…,ni (observations per treatment).
- μ.: Overall mean response.
- τi: Fixed treatment effects (∑τi=0).
- γ: Fixed regression coefficient (relationship between covariate X and response Y).
- Xij: Observed covariate (fixed, not random).
- ϵij∼iidN(0,σ2): Independent random errors.
If we use γXij directly (without centering), then μ. is no longer the overall mean. Thus, centering the covariate is necessary to maintain interpretability.
Expectation and Variance
E(Yij)=μ.+τi+γ(Xij−ˉX..)var(Yij)=σ2
Since Yij∼N(μij,σ2), we express:
μij=μ.+τi+γ(Xij−ˉX..)
where ∑τi=0. The mean response μij is a regression line with intercept μ.+τi and slope γ for each treatment i.
Key Assumptions
- All treatments share the same slope (γ).
- No interaction between treatment and covariate (parallel regression lines).
- If slopes differ, ANCOVA is not appropriate → use separate regressions per treatment.
A more general model allows multiple covariates:
Yij=μ.+τi+γ1(Xij1−ˉX..1)+γ2(Xij2−ˉX..2)+ϵij
Using indicator variables for treatments:
For treatment i=1: l1={1if case belongs to treatment 1−1if case belongs to treatment r0otherwise
For treatment i=r−1: lr−1={1if case belongs to treatment r−1−1if case belongs to treatment r0otherwise
Defining xij=Xij−ˉX.., the regression model is:
Yij=μ.+τ1lij,1+⋯+τr−1lij,r−1+γxij+ϵij
where Iij,1 is the indicator variable l1 for the j-th case in treatment i.
The treatment effects (τi) are simply regression coefficients for the indicator variables.
24.6.1 Statistical Inference for Treatment Effects
To test treatment effects:
H0:τ1=τ2=⋯=0Ha:Not all τi=0
Full Model (with treatment effects): Yij=μ.+τi+γXij+ϵij
Reduced Model (without treatment effects): Yij=μ.+γXij+ϵij
F-Test for Treatment Effects
The test statistic is:
F=SSE(R)−SSE(F)(N−2)−(N−(r+1))/SSE(F)N−(r+1)
where:
SSE(R): Sum of squared errors for the reduced model.
SSE(F): Sum of squared errors for the full model.
N: Total number of observations.
r: Number of treatment groups.
Under H0, the statistic follows an F-distribution:
F∼F(r−1,N−(r+1))
Comparisons of Treatment Effects
For r=3, we estimate:
Comparison | Estimate | Variance of Estimator |
---|---|---|
τ1−τ2 | ˆτ1−ˆτ2 | var(ˆτ1)+var(ˆτ2)−2cov(ˆτ1,ˆτ2) |
τ1−τ3 | 2ˆτ1+ˆτ2 | 4var(ˆτ1)+var(ˆτ2)−4cov(ˆτ1,ˆτ2) |
τ2−τ3 | ˆτ1+2ˆτ2 | var(ˆτ1)+4var(ˆτ2)−4cov(ˆτ1,ˆτ2) |
24.6.2 Testing for Parallel Slopes
To check if slopes differ across treatments, we use the model:
Yij=μ.+τ1Iij,1+τ2Iij,2+γXij+β1Iij,1Xij+β2Iij,2Xij+ϵij
where:
- β1,β2: Interaction coefficients (slope differences across treatments).
Hypothesis Test
H0:β1=β2=0(Slopes are equal)Ha:At least one β≠0(Slopes differ)
If the F-test fails to reject H0, then we assume parallel slopes.
24.6.3 Adjusted Means
The adjusted treatment means account for covariate effects:
Yi.(adj)=ˉYi.−ˆγ(ˉXi.−ˉX..)
where:
ˉYi.: Observed mean response for treatment i.
ˆγ: Estimated regression coefficient.
ˉXi.: Mean covariate value for treatment i.
ˉX..: Overall mean covariate value.
This provides estimated treatment means after controlling for covariate effects.
This chapter has provided an exploration of ANOVA, a foundational technique for comparing group means across multiple experimental conditions. Beginning with the Completely Randomized Design, we established the basic framework for understanding how ANOVA partitions variability. We then extended the discussion to Nonparametric ANOVA, accommodating situations where the assumptions of traditional ANOVA are violated.
Subsequent sections introduced advanced designs such as Randomized Block and Nested Designs, which offer increased precision and control over variability by accounting for structured sources of heterogeneity. We also addressed Sample Size Planning, emphasizing the importance of statistical power and design efficiency. Finally, we examined the Single Factor Covariance Model, integrating covariates into the ANOVA framework to adjust for confounding variables and improve estimation accuracy.
Together, these topics equip the reader with the methodological rigor needed to design, implement, and interpret experiments involving multiple group comparisons in complex real-world settings.