#Second to get the standard error on response metric #First to get the standard error on the linear metric #The key here is to apply the delta method twice #FOR STANDARD ERROR ON RESPONSE (PROBABILITY) METRIC # so vector of partial derivatives is 1,0,0,50,50 # regression equation is Y = b0*1 + b_prog2*0 + b_prog3*0 +b_math*50 + b_read*50 # here we want standard error of G(b) for males in prog1 with math = 50 and read = 50 # G(b) = b0 + b_prog2*prog2 + b_prog3*prog3 + b_math*math + b_read*read M3 <- glm(female ~ prog + math + read, data=d, family=binomial) #sqrt of variance for delta method standard error # so vector of partial derivatives is 1,1,1,0,50 # regression equation is Y = b0*1 + b_female*1 + b_prog2*1 + b_prog3*0 +b_math*50 # here we want standard error of G(b) for females in prog2 with math = 50 # G(b) = b0 + b_female1*female1 + b_prog2*prog2 + b_prog3*prog3 + b_math*math M2 <- lm(write ~ female + prog + math, data=d) # Delta method standard error for predicted value of Y at set values of We then take the sqrt of the variance to get the standard error of the conditional mean. Now we apply the delta method: var(G(B)) = dG/db’ %*% var(b) %*% dG/db. Example 1: Delta method standard error for conditional mean of Y at mean of Xįirst let’s make up some data and run a very simple linear regression. The predict() function calculates delta-method standard errors for conditional means, but it will not quite work for marginal means. Var(G(B)) = dG/db’ %*% var(b) %*% dG/db,where dG/db is a vector of partial derivatives of G(b) with respect toĮach b coefficient, also known as the Jacobian matrix, and var(b) is the variance-covariance matrix of theī coefficients.
For a function G(b) of a random variable vector, b, the variance of G(b) by the delta method is: As model coefficients are themselves random variables, we can use the delta method to get the variance of conditional and marginal means, because they are functions of the model ceofficients. We can use the delta method to get the variance of a function of random variable. Both conditional and marginal means are functions of the model coefficients. (NOT SURE ABOUT THIS) Marginal means are often calculated to isolate the effect of a single predictor by averaging the predicted response over various levels of the predictor of interest while holding the values of other predictors constant. Additionally, this page will not be updated in the near future.Īdjusted predictions are often calculated to predict the response at a given set of predictor values, usually to get an idea of the response value at representative predictor values.
In the normally distributed sampling distribution, the sample mean, quantiles of the normal distribution and standard error can be used in the calculation of the population mean’s confidence intervals.Please note that this page is incomplete and there may be inconsistencies in the code or explanations. Additionally, the sample standard deviation will also become approximately equal to the population standard deviation with the increase in sample size. In the case of finite population standard deviation, an increase in sample size will eventually reduce the standard error of the sample mean to zero as the population’s estimation will improve. So, standard error helps estimate how far the sample mean from the true population means. Statisticians usually use the sample from a large pool of data as it is difficult to process such a huge data set, and as such, sampling makes the task a lot easier. It is very important to understand the concept of standard error as it predominantly used by statisticians as it allows them to measure the precision of their sampling method. Standard Error = s / √n Relevance and Use of Standard Error Formula Step 5: Finally, the formula for standard error can be derived by dividing the sample standard deviation (step 4) by the square root of the sample size (step 2), as shown below.
Step 4: Next, compute the sample standard deviation (s), which involves a complex calculation that uses each sample variable (step 1), sample mean (step 3) and sample size (step 2) as shown below. It is denoted by, and mathematically it is represented as, Step 3: Next, compute the sample mean, which can be derived by dividing the summation of all the variables in the sample (step 1) by the sample size (step 2). Step 2: Next, determine the sample size, which is the total number of variables in the sample. The sample variables are denoted by x such that x i refers to the i th variable of the sample. Step 1: Firstly, collect the sample variables from the population-based on a certain sampling method. The formula for standard error can be derived by using the following steps: Therefore, the standard error of the sample mean is 0.77. Standard Error is calculated using the formula given below