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Standard Error of Estimate
Standard error is the estimated standard deviation of an estimate. It measures the uncertainty associated with the estimate. Compared with the standard deviations of the underlying distribution, which are usually unknown, standard errors can be calculated from observed data.
First, this entry will explain how estimates are obtained. Then, the entry discusses how the standard errors of estimates are derived, with an emphasis on the differences between standard errors and standard deviations. Two examples are used to illustrate the calculation of standard errors of a parameter estimate and standard errors of a future outcome estimate, respectively. Finally, the relationship between standard errors and confidence intervals is discussed.
Estimate
One main purpose of research designs is to make inferences about a population. Many population characteristics are described by numerical measures or can be translated into numerical values. Numeric descriptive measures of a population are called parameters of the population. Common examples of parameters include the percentage of voters supporting candidate Mr. Jones, the average weight of 1-month-old babies, the variance or standard deviation of the annual income of fresh college graduates in year 2008, the correlation coefficient between annual income and years of education, and the increase of average annual income associated with 1 more year of education. In many cases, it is infeasible to sample the whole population, and the sampled information might not be 100% accurate; thus, the true values of the parameters are unknown and need to be estimated. An estimate is an approximation of the true parameter value using specific rules when sampled information is incomplete, noisy, or uncertain. The rule to calculate an estimate is called an estimator, which could be a formula or a procedure. Different estimators give different estimate values for the same parameter, and many statistical methods have been developed to evaluate the good and bad qualities of estimators.
Another main purpose of research design is to make predictions for future events. Often, future events can be characterized by numeric values, such as the highest temperature of an area next week, the life expectancy of someone who has already lived to 80, and the cure rate if a new treatment is applied. The true values of the measurements of future events are unknown until the event happens. Statistical regression models study the relationship between the specific measurements (outcome) and other related variables (predictors) based on historic data, and then estimate the future values. The estimate of a future value is called a prediction. Although the true values of the regression coefficients are fixed, different models give different predictions for the same future value. Even with the same model the predictions will vary because the regression coefficients, which measure the relationship between the predictors and the outcome, are estimated from a particular sample and vary from sample to sample.
Standard Error of Estimate
The differences between the true value and its estimates are called errors. The exact value of an estimate depends on the estimating procedure as well as the sample used to calculate the estimate. Because the sampled data are a random sample of the underlying population, the estimates vary from sample to sample even when the same estimating procedure is used, especially when the sample size is small. When the sample size goes to infinity, the estimate converges to its expectation, which might or might not be the same as the true value being estimated. The difference between the expectation of an estimate and the true value is called bias. Some estimating procedures produce unbiased estimates, whereas others do not. Both randomness in the sample and bias lead to difference between estimates and the corresponding true value, that is, errors.
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