Confidence Interval for a Contrast of Covariate-Adjusted Cell Means in a Factorial ANCOVA

Given a fitted lm or aov object for a factorial analysis of covariance (one or more crossed factors plus one or more covariates) and a numeric contrast vector over the cells of the factorial design, contrast_adjusted() forms the contrast of the covariate-adjusted cell means, \(\hat{\psi} = \sum_j c_j \, \hat{\bar{Y}}_j\), where each \(\hat{\bar{Y}}_j\) is the model's predicted mean for cell \(j\) evaluated at the mean of every covariate (the adjusted, or least-squares, cell mean). It returns the point estimate, a t confidence interval on the model's residual degrees of freedom, and the accompanying t statistic and two-sided p-value for \(H_0\!: \psi = 0\).

Usage

contrast_adjusted(model, contrast, conf_level = 0.95)

Arguments

model: A fitted lm or aov object for a factorial ANCOVA: one or more crossed factors and one or more numeric covariates on the right-hand side of the formula.
contrast: A numeric vector of contrast weights, one weight per cell of the factorial design (the crossing of the model's factors). Its length must equal the number of cells. The weights typically sum to zero.
conf_level: The confidence level for the interval (default 0.95).

Value

A five-row dmar_tbl (a data.frame with columns term and value). The term values are "contrast" (the point estimate \(\hat{\psi}\) of the contrast of adjusted cell means), "lower_limit" and "upper_limit" (the confidence limits), "t" (the t statistic), and "p" (the two-sided p-value). The stored value column is numeric at full precision.

Details

The adjusted cell means are the means the ANCOVA actually tests: the predicted outcome for each combination of factor levels, holding every covariate at its sample mean. Writing \(L\) for the linear map that sends the model coefficients to that contrast of adjusted means (built by evaluating the model's design matrix at each cell with the covariates set to their means and combining the rows with the contrast weights), the point estimate is \(\hat{\psi} = L' \hat{\beta}\) and its standard error is the square root of the quadratic form \(L' \, \mathrm{vcov}(\hat{\beta}) \, L\). The interval is \(\hat{\psi} \pm t_{1 - \alpha/2,\, \nu}\, \mathrm{SE}\), with \(\nu\) the residual degrees of freedom of the fitted model.

Because \(L\) is read off the fitted model's own design matrix, the function is agnostic to how the factors are parameterized: a cell-means parameterization (y ~ 0 + cell + x) and the crossed-factor parameterization (y ~ A * B + x) give the same contrast estimate and standard error, provided the contrast vector is ordered to match the cells of the reference grid (see Note).

The t interval returned here has exact per-comparison coverage for a single contrast chosen in advance. For a family of contrasts examined together, adjust the critical value for multiplicity (for example the Scheffe critical value for the full cell space, cv_scheffe, or a Bryant–Paulson simultaneous interval, ci_c_ancova_bp).

Note

The contrast weights are matched to the cells of the reference grid, which is the crossing of the model's factors in the order the factors appear in the model formula, with the first factor varying fastest (the order expand.grid produces over the factor levels). For a single factor this is simply the order of its levels. When in doubt, fit the cell-means form y ~ 0 + cell + covariates with cell = interaction(A, B, ...) and order the weights to match levels(cell).

References

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 9 on designs with covariates.)

Author

Ken Kelley kkelley@nd.edu

Examples

# A 2 x 2 factorial ANCOVA with one covariate.
set.seed(113)
d <- data.frame(
  A = factor(rep(c("a1", "a2"), each = 40)),
  B = factor(rep(rep(c("b1", "b2"), each = 20), 2)),
  x = rnorm(80)
)
d$y <- 5 + 2 * (d$A == "a2") + 1.5 * (d$B == "b2") +
  0.8 * d$x + rnorm(80)
fit <- lm(y ~ A * B + x, data = d)

# Cells in reference-grid order: (a1,b1), (a2,b1), (a1,b2), (a2,b2).
# Main effect of A, averaged over B: mean(a2 cells) - mean(a1 cells).
contrast_adjusted(fit, contrast = c(-0.5, 0.5, -0.5, 0.5))
#>  term        value   
#>  contrast    1.9     
#>  lower_limit 1.49    
#>  upper_limit 2.31    
#>  t           9.18    
#>  p           < 0.0001
#> 
#> Confidence level: 95%