POSC for a composite of several predictors (bivariate normal)

Computes the probability of outcome superiority curve for the optimally weighted (regression) composite of several predictors, from a correlation matrix.

Usage

posc_multi(
  r_mat,
  n,
  outcome = ncol(r_mat),
  adjust = TRUE,
  level = 0.95,
  max_delta = 3,
  predictor_name = "Predicted Y",
  outcome_name = NULL,
  outcome_idx = NULL
)

Arguments

r_mat: Correlation matrix of the predictors and the outcome.
n: Sample size used to estimate r_mat.
outcome: Row/column of r_mat holding the outcome, as an index or a name. Defaults to the last one.
adjust: Use the adjusted multiple correlation? See Details.
level: Confidence level of the band.
max_delta: Largest difference in the composite (in SD units) to evaluate.
predictor_name, outcome_name: Labels used in printed output and plots. outcome_name defaults to the outcome's name in r_mat, if it has one.
outcome_idx: Deprecated; use outcome.

Value

An object of class "posc": a list with elements

curve: data frame with columns delta, p, lower, upper, over a symmetric grid of differences.
method, r, n, level, sd_x, max_delta, labels, details: information about how the curve was computed.

Use plot(), summary() and predict() to work with it.

Details

The composite that best predicts the outcome correlates with it at the multiple correlation $$R = \sqrt{\mathbf{r}_{xy}^\top \mathbf{R}_{xx}^{-1} \mathbf{r}_{xy}},$$ where $\mathbf{R}_{xx}$ is the predictor intercorrelation matrix and $\mathbf{r}_{xy}$ the predictor-outcome correlations. The POSC is then computed as in posc_uni(), with differences in the composite expressed in its standard-deviation units.

A sample multiple correlation is biased upwards as an estimate of the population multiple correlation, because the weights are fitted to the same sample. With adjust = TRUE (the default) $R$ is replaced by the square root of the adjusted $R^2$, $1 - (1 - R^2)(n - 1)/(n - k - 1)$ for $k$ predictors, which corrects most of this bias. (How well weights estimated in this sample would predict in a new sample is lower still.)

The confidence band uses a Fisher-$z$ interval for $R$ with standard error $1/\sqrt{n - k - 2}$, truncated at zero, and should be treated as approximate.

Examples

r_mat <- matrix(c(1.0, 0.4, 0.3, 0.2,
                  0.4, 1.0, -0.3, 0.2,
                  0.3, -0.3, 1.0, 0.3,
                  0.2, 0.2, 0.3, 1.0), 4, 4,
                dimnames = list(c("X1", "X2", "X3", "Y"),
                                c("X1", "X2", "X3", "Y")))
fit <- posc_multi(r_mat, n = 100, outcome = "Y")
fit
#> Probability of Outcome Superiority Curve
#>   Method:    bivariate normal, optimally weighted composite
#>   Predictor: Predicted Y    Outcome: Y
#>   r = 0.400, 95% CI [0.219, 0.555], n = 100
#>   (r adjusted for the number of predictors; unadjusted r = 0.431)
#> 
#>   P(higher Y | Predicted Y higher by delta):
#>  delta     p lower upper
#>  0.500 0.561 0.532 0.593
#>  1.000 0.621 0.563 0.681
#>  2.000 0.732 0.625 0.827
plot(fit)