# Leverages Computes leverage measures from a fitted model object. ```r leverages(model, ...) ## S3 method for class 'lm' leverages(model, infl = lm.influence(model, do.coef = FALSE), ...) ## S3 method for class 'nls' leverages(model, ...) ## S3 method for class 'ols' leverages(model, ...) ## S3 method for class 'ridge' leverages(model, ...) ## S3 method for class 'nls' hatvalues(model, ...) ## S3 method for class 'ols' hatvalues(model, ...) ## S3 method for class 'ridge' hatvalues(model, ...) ``` ## Arguments - `model`: an object, returned by `lm`, `nls`, `ols` or `ridge`. - `infl`: influence structure as returned by `lm.influence`. - ``...``: further arguments passed to or from other methods. ## Returns A vector containing the diagonal of the prediction (or hat ) matrix. For linear regression (i.e., for `"lm"` or `"ols"` objects) the prediction matrix assumes the form $$ \bold{H} = \bold{X}(\bold{X}^T\bold{X})^{-1}\bold{X}^T, $$ in which case, $h_{ii} = \bold{x}_i^T(\bold{X}^T\bold{X})^{-1}\bold{x}_i$ for $i=1,\dots,n$. Whereas for ridge regression, the prediction matrix is given by $$ \bold{H}(\lambda) = \bold{X}(\bold{X}^T\bold{X} + \lambda\bold{I})^{-1}\bold{X}^T, $$ where $\lambda$ represents the ridge parameter. Thus, the diagonal elements of $\bold{H}(\lambda)$, are $h_{ii}(\lambda) = \bold{x}_i^T(\bold{X}^T\bold{X} + \lambda\bm{I})^{-1}\bold{x}_i$, $i=1,\dots,n$. In nonlinear regression, the tangent plane leverage matrix is given by $$ \hat{\bold{H}} = \hat{\bold{F}}(\hat{\bold{F}}^T\hat{\bold{F}})^{-1}\hat{\bold{F}}^T, $$ where $\bold{F} = \bold{F}(\bold{\beta})$ is the $n\times p$ local model matrix with **i**th row $\partial f_i(\bold{\beta})/\partial\bold{\beta}$ and $\hat{\bold{F}} = \bold{F}(\hat{\bold{\beta}})$. ## Note This function never creates the prediction matrix and only obtains its diagonal elements from the singular value decomposition of $\bold{X}$ or $\hat{\bold{F}}$. Function `hatvalues` only is a wrapper for function `leverages`. ## References Chatterjee, S., Hadi, A.S. (1988). **Sensivity Analysis in Linear Regression**. Wiley, New York. Cook, R.D., Weisberg, S. (1982). **Residuals and Influence in Regression**. Chapman and Hall, London. Ross, W.H. (1987). The geometry of case deletion and the assessment of influence in nonlinear regression. **The Canadian Journal of Statistics** 15 , 91-103. tools:::Rd_expr_doi("10.2307/3315198") St. Laurent, R.T., Cook, R.D. (1992). Leverage and superleverage in nonlinear regression. **Journal of the Amercian Statistical Association** 87 , 985-990. tools:::Rd_expr_doi("10.1080/01621459.1992.10476253") Walker, E., Birch, J.B. (1988). Influence measures in ridge regression. **Technometrics** 30 , 221-227. tools:::Rd_expr_doi("10.1080/00401706.1988.10488370") ## Examples ```r for linear regression fm <- ols(stack.loss ~ ., data = stackloss) lev <- leverages(fm) cutoff <- 2 * mean(lev) plot(lev, ylab = "Leverages", ylim = c(0,0.45)) abline(h = cutoff, lty = 2, lwd = 2, col = "red") text(17, lev[17], label = as.character(17), pos = 3) for ridge regression data(portland) fm <- ridge(y ~ ., data = portland) lev <- leverages(fm) cutoff <- 2 * mean(lev) plot(lev, ylab = "Leverages", ylim = c(0,0.7)) abline(h = cutoff, lty = 2, lwd = 2, col = "red") text(10, lev[10], label = as.character(10), pos = 3) for nonlinear regression data(skeena) model <- recruits ~ b1 * spawners * exp(-b2 * spawners) fm <- nls(model, data = skeena, start = list(b1 = 3, b2 = 0)) lev <- leverages(fm) plot(lev, ylab = "Leverages", ylim = c(0,0.25)) obs <- c(1,9) text(obs, lev[obs], label = as.character(obs), pos = 3) ```

leverages function