# Likelihood Displacement Compute the likelihood displacement influence measure based on leave-one-out cases deletion for linear models, lad and ridge regression. ```r logLik.displacement(model, ...) ## S3 method for class 'lm' logLik.displacement(model, pars = "full", ...) ## S3 method for class 'nls' logLik.displacement(model, ...) ## S3 method for class 'ols' logLik.displacement(model, pars = "full", ...) ## S3 method for class 'lad' logLik.displacement(model, method = "quasi", pars = "full", ...) ## S3 method for class 'ridge' logLik.displacement(model, pars = "full", ...) ``` ## Arguments - `model`: an object, returned by `lm`, `nls`, `ols`, `lad` or `ridge`. - `pars`: should be considered the whole vector of parameters (`pars = "full"`), or only the vector of coefficients (`pars = "coef"`). This option is not used for `nls` objects. - `method`: only required for `'lad'` objects, options available are `"quasi"` and `"BR"` to obtain the likelihood displacement based on Sun and Wei (2004) and Elian et al. (2000) approaches, respectively. - ``...``: further arguments passed to or from other methods. ## Returns A vector whose **i**th element contains the distance between the likelihood functions, $$ LD_i(\bold{\beta},\sigma^2) = 2\{l(\hat{\bold{\beta}},\hat{\sigma}^2) -l(\hat{\bold{\beta}}_{(i)},\hat{\sigma}^2_{(i)})\}, $$ for `pars = "full"`, where $\hat{\bold{\beta}}_{(i)}$ and $\hat{\sigma}^2_{(i)}$ denote the estimates of $\bold{\beta}$ and $\sigma^2$ when the **i**th observation is removed from the dataset. If we are interested only in $\bold{\beta}$ (i.e. `pars = "coef"`) the likelihood displacement becomes $$ LD_i(\bold{\beta}|\sigma^2) = 2\{l(\hat{\bold{\beta}},\hat{\sigma}^2) -\max_{\sigma^2} l(\hat{\bold{\beta}}_{(i)},\hat{\sigma}^2)\}. $$ ## References Cook, R.D., Weisberg, S. (1982). **Residuals and Influence in Regression**. Chapman and Hall, London. Cook, R.D., Pena, D., Weisberg, S. (1988). The likelihood displacement: A unifying principle for influence measures. **Communications in Statistics - Theory and Methods** 17 , 623-640. tools:::Rd_expr_doi("10.1080/03610928808829645") Elian, S.N., Andre, C.D.S., Narula, S.C. (2000). Influence measure for the L1 regression. **Communications in Statistics - Theory and Methods** 29 , 837-849. tools:::Rd_expr_doi("10.1080/03610920008832518") Ogueda, A., Osorio, F. (2025). Influence diagnostics for ridge regression using the Kullback-Leibler divergence. **Statistical Papers** 66 , 85. tools:::Rd_expr_doi("10.1007/s00362-025-01701-1") 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") ## Examples ```r # Likelihood displacement for linear regression fm <- ols(stack.loss ~ ., data = stackloss) LD <- logLik.displacement(fm) plot(LD, ylab = "Likelihood displacement", ylim = c(0,9)) text(21, LD[21], label = as.character(21), pos = 3) # Likelihood displacement for LAD regression fm <- lad(stack.loss ~ ., data = stackloss) LD <- logLik.displacement(fm) plot(LD, ylab = "Likelihood displacement", ylim = c(0,1.5)) text(17, LD[17], label = as.character(17), pos = 3) # Likelihood displacement for ridge regression data(portland) fm <- ridge(y ~ ., data = portland) LD <- logLik.displacement(fm) plot(LD, ylab = "Likelihood displacement", ylim = c(0,4)) text(8, LD[8], label = as.character(8), pos = 3) # Likelihood displacement for nonlinear regression data(skeena) model <- recruits ~ b1 * spawners * exp(-b2 * spawners) fm <- nls(model, data = skeena, start = list(b1 = 3, b2 = 0)) LD <- logLik.displacement(fm) plot(LD, ylab = "Likelihood displacement", ylim = c(0,0.7)) obs <- c(5, 6, 9, 19, 25) text(obs, LD[obs], label = as.character(obs), pos = 3) ```

logLik.displacement function