24.6 Single Factor Covariance Model

The single-factor covariance model (Analysis of Covariance, ANCOVA) accounts for both treatment effects and a continuous covariate:

$$Y_{ij} = \mu_{.} + \tau_i + \gamma(X_{ij} - \bar{X}_{..}) + \epsilon_{ij}$$

for $i = 1, \dots, r$ (treatments) and $j = 1, \dots, n_i$ (observations per treatment).

• $\mu_{.}$: Overall mean response.
• $\tau_i$: Fixed treatment effects ($\sum \tau_i = 0$).
• $\gamma$: Fixed regression coefficient (relationship between covariate $X$ and response $Y$).
• $X_{ij}$: Observed covariate (fixed, not random).
• $\epsilon_{ij} \sim iid N(0, \sigma^2)$: Independent random errors.

If we use $\gamma X_{ij}$ directly (without centering), then $\mu_{.}$ is no longer the overall mean. Thus, centering the covariate is necessary to maintain interpretability.

Expectation and Variance

$$\begin{aligned} E(Y_{ij}) &= \mu_. + \tau_i + \gamma(X_{ij}-\bar{X}_{..}) \\ var(Y_{ij}) &= \sigma^2 \end{aligned}$$

Since $Y_{ij} \sim N(\mu_{ij},\sigma^2)$, we express:

$$\mu_{ij} = \mu_. + \tau_i + \gamma(X_{ij} - \bar{X}_{..})$$

where $\sum \tau_i = 0$. The mean response $\mu_{ij}$ is a regression line with intercept $\mu_. + \tau_i$ and slope $\gamma$ for each treatment $i$.

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Key Assumptions

1. All treatments share the same slope ($\gamma$).
2. No interaction between treatment and covariate (parallel regression lines).
3. If slopes differ, ANCOVA is not appropriate → use separate regressions per treatment.

A more general model allows multiple covariates:

$$Y_{ij} = \mu_. + \tau_i + \gamma_1(X_{ij1}-\bar{X}_{..1}) + \gamma_2(X_{ij2}-\bar{X}_{..2}) + \epsilon_{ij}$$

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Using indicator variables for treatments:

For treatment $i = 1$: $$l_1 = \begin{cases} 1 & \text{if case belongs to treatment 1} \\ -1 & \text{if case belongs to treatment $r$} \\ 0 & \text{otherwise} \end{cases}$$

For treatment $i = r-1$: $$l_{r-1} = \begin{cases} 1 & \text{if case belongs to treatment $r-1$} \\ -1 & \text{if case belongs to treatment $r$} \\ 0 & \text{otherwise} \end{cases}$$

Defining $x_{ij} = X_{ij}- \bar{X}_{..}$, the regression model is:

$$Y_{ij} = \mu_. + \tau_1 l_{ij,1} + \dots + \tau_{r-1} l_{ij,r-1} + \gamma x_{ij} + \epsilon_{ij}$$

where $I_{ij,1}$ is the indicator variable $l_1$ for the $j$-th case in treatment $i$.

The treatment effects ($\tau_i$) are simply regression coefficients for the indicator variables.

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24.6.1 Statistical Inference for Treatment Effects

To test treatment effects:

$$\begin{aligned} &H_0: \tau_1 = \tau_2 = \dots = 0 \\ &H_a: \text{Not all } \tau_i = 0 \end{aligned}$$

1. Full Model (with treatment effects): $$Y_{ij} = \mu_. + \tau_i + \gamma X_{ij} + \epsilon_{ij}$$

2. Reduced Model (without treatment effects): $$Y_{ij} = \mu_. + \gamma X_{ij} + \epsilon_{ij}$$

F-Test for Treatment Effects

The test statistic is:

$$F = \frac{SSE(R) - SSE(F)}{(N-2)-(N-(r+1))} \Big/ \frac{SSE(F)}{N-(r+1)}$$

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 $H_0$, the statistic follows an $F$-distribution:

$$F \sim F_{(r-1, N-(r+1))}$$

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Comparisons of Treatment Effects

For $r = 3$, we estimate:

Comparison	Estimate	Variance of Estimator
$\tau_1 - \tau_2$	$\hat{\tau}_1 - \hat{\tau}_2$	$var(\hat{\tau}_1) + var(\hat{\tau}_2) - 2cov(\hat{\tau}_1, \hat{\tau}_2)$
$\tau_1 - \tau_3$	$2 \hat{\tau}_1 + \hat{\tau}_2$	$4var(\hat{\tau}_1) + var(\hat{\tau}_2) - 4cov(\hat{\tau}_1, \hat{\tau}_2)$
$\tau_2 - \tau_3$	$\hat{\tau}_1 + 2 \hat{\tau}_2$	$var(\hat{\tau}_1) + 4var(\hat{\tau}_2) - 4cov(\hat{\tau}_1, \hat{\tau}_2)$

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24.6.2 Testing for Parallel Slopes

To check if slopes differ across treatments, we use the model:

$$Y_{ij} = \mu_{.} + \tau_1 I_{ij,1} + \tau_2 I_{ij,2} + \gamma X_{ij} + \beta_1 I_{ij,1}X_{ij} + \beta_2 I_{ij,2}X_{ij} + \epsilon_{ij}$$

where:

• $\beta_1, \beta_2$: Interaction coefficients (slope differences across treatments).

Hypothesis Test

$$\begin{aligned} &H_0: \beta_1 = \beta_2 = 0 \quad (\text{Slopes are equal}) \\ &H_a: \text{At least one } \beta \neq 0 \quad (\text{Slopes differ}) \end{aligned}$$

If the $F$-test fails to reject $H_0$, then we assume parallel slopes.

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24.6.3 Adjusted Means

The adjusted treatment means account for covariate effects:

$$Y_{i.}(\text{adj}) = \bar{Y}_{i.} - \hat{\gamma}(\bar{X}_{i.} - \bar{X}_{..})$$

where:

• $\bar{Y}_{i.}$: Observed mean response for treatment $i$.

• $\hat{\gamma}$: Estimated regression coefficient.

• $\bar{X}_{i.}$: Mean covariate value for treatment $i$.

• $\bar{X}_{..}$: Overall mean covariate value.

This provides estimated treatment means after controlling for covariate effects.

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