# Cook's distances Cook's distance is a measure to assess the influence of the **i**th observation on the model parameter estimates. This function computes the Cook's distance based on leave-one-out cases deletion for ordinary least squares, nonlinear least squares, lad and ridge regression. ```r ## S3 method for class 'lad' cooks.distance(model, ...) ## S3 method for class 'nls' cooks.distance(model, ...) ## S3 method for class 'ols' cooks.distance(model, ...) ## S3 method for class 'ridge' cooks.distance(model, type = "cov", ...) ``` ## Arguments - `model`: an object, returned by `ols`, `nls`, `lad` or `ridge`. - `type`: only required for `'ridge'` objects, options available are `"1st"`, `"cov"` and `"both"` to obtain the Cook's distance based on Equation (2.5), (2.6) or both by Walker and Birch (1988), respectively. - ``...``: further arguments passed to or from other methods. ## Returns A vector whose **i**th element contains the Cook's distance, $$ D_i(\bold{M},c) = \frac{(\hat{\bold{\beta}}_{(i)} - \hat{\bold{\beta}})^T\bold{M}(\hat{\bold{\beta}}_{(i)} - \hat{\bold{\beta}})}{c}, $$ for $i = 1,\dots,n$, with $\bold{M}$ a positive definite matrix and $c > 0$. Specific choices of $\bold{M}$ and $c$ are done for objects of class `ols`, `nls`, `lad` and `ridge`. The Cook's distance for nonlinear regression is based on linear approximation, which may be inappropriate for expectation surfaces markedly nonplanar. ## References Cook, R.D., Weisberg, S. (1980). Characterizations of an empirical influence function for detecting influential cases in regression. **Technometrics** 22 , 495-508. tools:::Rd_expr_doi("10.1080/00401706.1980.10486199") 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") Sun, R.B., Wei, B.C. (2004). On influence assessment for LAD regression. **Statistics & Probability Letters** 67 , 97-110. tools:::Rd_expr_doi("10.1016/j.spl.2003.08.018") 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) CD <- cooks.distance(fm) plot(CD, ylab = "Cook's distances", ylim = c(0,0.8)) text(21, CD[21], label = as.character(21), pos = 3) for LAD regression fm <- lad(stack.loss ~ ., data = stackloss) CD <- cooks.distance(fm) plot(CD, ylab = "Cook's distances", ylim = c(0,0.4)) text(17, CD[17], label = as.character(17), pos = 3) for ridge regression data(portland) fm <- ridge(y ~ ., data = portland) CD <- cooks.distance(fm) plot(CD, ylab = "Cook's distances", ylim = c(0,0.5)) text(8, CD[8], label = as.character(8), 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)) CD <- cooks.distance(fm) plot(CD, ylab = "Cook's distances", ylim = c(0,0.35)) obs <- c(5, 6, 9, 19, 25) text(obs, CD[obs], label = as.character(obs), pos = 3) ```

cooks.distance function