In the following example, we model the probability of being enrolled in an honors program (not enrolled vs enrolled) predicted by gender, math score and reading score. As odds ratios are simple non-linear transformations of the regression coefficients, we can use the delta method to obtain their standard errors. One such tranformation is expressing logistic regression coefficients as odds ratios. However, other transformations of regrssion coefficients that predict cannot readily handle are often useful to report. Indeed, if you only need standard errors for adjusted predictions on either the linear predictor scale or the response variable scale, you can use predict and skip the manual calculations. Example 2: Odds ratioĮxample 1 was somewhat trivial given that the predict function calculates delta method standard errors for adjusted predictions. Notice that in this example since \(G(B)\) is a linear function, delta method is not an exact method and not an approximations. We would like to calculate the standard error of the adjusted prediction of y at the mean of x, 5.5, from the linear regression of y on x: Adjusted predictions are functions of the regression coefficients, so we can use the delta method to approximate their standard errors. For example, we can get the predicted value of an “average” respondent by calculating the predicted value at the mean of all covariates. Example 1: Adjusted predictionĪdjusted predictions, or adjusted means, are predicted values of the response calculated at a set of covariate values. For a random variable \(X\) with known variance-covariance matrix \(Cov(X)\), the variance of the transformation of \(X\), \(G(X)\) is approximated by: Variance of this approximation to estimate the variance of \(G(X)\) and thus the standard error ofĪ transformed parameter. Where \(\nabla G(\mu_X)\) is the gradient of \(G(X)\) at \(X = \mu_X\), or a vector of partial derivatives of \(G(X)\) at point \(\mu_X\). The first two terms of the Taylor expansion are then an approximation for \(G(X)\), Let \(G\) be the transformation function and \(\mu_X\) be the mean vector of random variables (X=(x1,x2,…)). ![]() We, thus, first get the Taylor series approximation of the function using the first two terms of the Taylor expansion of the transformation function about the mean of of the random variable. Although the delta method is often appropriate to use with large samples, this page is by no means an endorsement of the use of the delta method over other methods to estimate standard errors, such as bootstrapping.Įssentially, the delta method involves calculating the variance of the Taylor series approximation of a function. Regression coefficients are themselves random variables, so we can use the delta method to approximate the standard errors of their transformations. The delta method approximates the standard errors of transformations of random variable using a first-order Taylor approximation. Well approximated using the delta method. Point estimates of our desired values, but the standardĮrrors of these point estimates are not so easily calculated. Often in addition to reporting parameters fit by a model, we need to report ![]() Version info: Code for this page was tested in R version 3.1.1 ()
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