mean squared error

(noun)

A measure of the average of the squares of the "errors"; the amount by which the value implied by the estimator differs from the quantity to be estimated.

Related Terms

Examples of mean squared error in the following topics:

  • Computing R.M.S. Error

    • Root-mean-square (RMS) error, also known as RMS deviation, is a frequently used measure of the differences between values predicted by a model or an estimator and the values actually observed.
    • Root-mean-square error serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power.
    • RMS error is the square root of mean squared error (MSE), which is a risk function corresponding to the expected value of the squared error loss or quadratic loss.
    • MSE measures the average of the squares of the "errors. " The MSE is the second moment (about the origin) of the error, and thus incorporates both the variance of the estimator and its bias.
    • RMS error is simply the square root of the resulting MSE quantity.

  • Estimation

    • Bias leads to a sample mean that is either lower or higher than the true mean .
    • The mean squared error (MSE) of $\hat { \theta }$ is defined as the expected value of the squared errors.
    • In this case, high MSE means the average distance of the arrows from the bull's-eye is high, and low MSE means the average distance from the bull's-eye is low.
    • This generalized error in the mean is the square root of the sample variance (treated as a population) times $\frac{1+(N-1)\rho}{(N-1)(1-\rho)}$.
    • The $\rho = 0$ line is the more familiar standard error in the mean for samples that are uncorrelated.

  • Model Assumptions

    • This means, for example, that the predictor variables are assumed to be error-free; that is, they are not contaminated with measurement errors.
    • This means that the mean of the response variable is a linear combination of the parameters (regression coefficients) and the predictor variables.
    • This means that different response variables have the same variance in their errors, regardless of the values of the predictor variables.
    • In effect, residuals appear clustered and spread apart on their predicted plots for larger and smaller values for points along the linear regression line, and the mean squared error for the model will be wrong.
    • Typically, for example, a response variable whose mean is large will have a greater variance than one whose mean is small.

  • Standard Error

    • In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the underlying errors.
    • As mentioned, the standard error of the mean (SEM) is the standard deviation of the sample-mean's estimate of a population mean.
    • It can also be viewed as the standard deviation of the error in the sample mean relative to the true mean, since the sample mean is an unbiased estimator.
    • This is due to the fact that the standard error of the mean is a biased estimator of the population standard error.
    • Paraphrase standard error, standard error of the mean, standard error correction and relative standard error.

  • Mean Squares and the F-Ratio

    • The variance is also called the variation due to error or unexplained variation.
    • $MS$ means "mean square. " $MS_{\text{between}}$ is the variance between groups and $MS_{\text{within}}$ is the variance within groups.
    • Equation for errors within samples ($df$'s for the denominator): $df_{\text{within}} = n-k$
    • Mostly just sampling errors would contribute to variations away from one.
    • Demonstrate how sums of squares and mean squares produce the $F$-ratio and the implications that changes in mean squares have on it.

  • Partitioning the Sums of Squares

    • Partition sum of squares Y into sum of squares predicted and sum of squares error
    • The last column contains the squares of these errors of prediction.
    • Recall that SSY is the sum of the squared deviations from the mean.
    • SSY can be partitioned into two parts: the sum of squares predicted (SSY') and the sum of squares error (SSE).
    • The sum of squares error is the sum of the squared errors of prediction.

  • Estimating the Accuracy of an Average

    • The standard error of the mean is the standard deviation of the sample mean's estimate of a population mean.
    • The standard error of the mean (i.e., standard error of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all possible samples (of a given size) drawn from the population.
    • As mentioned, the standard error of the mean (SEM) is the standard deviation of the sample-mean's estimate of a population mean.
    • It can also be viewed as the standard deviation of the error in the sample mean relative to the true mean, since the sample mean is an unbiased estimator.
    • Generally, the SEM is the sample estimate of the population standard deviation (sample standard deviation) divided by the square root of the sample size:

  • Examining the standard error formula

    • The formula for the standard error of the difference in two means is similar to the formula for other standard errors.
    • Recall that the standard error of a single mean, $\bar{x}_1$, can be approximated by
    • The standard error of the difference of two sample means can be constructed from the standard errors of the separate sample means:
    • 5.14: The standard error squared represents the variance of the estimate.

  • Least-Squares Regression

    • The criteria for determining the least squares regression line is that the sum of the squared errors is made as small as possible.
    • The criteria for the best fit line is that the sum of squared errors (SSE) is made as small as possible.
    • Therefore, this best fit line is called the least squares regression line.
    • Under these conditions, the method of OLS provides minimum-variance, mean-unbiased estimation when the errors have finite variances.
    • Under the additional assumption that the errors are normally distributed, OLS is the maximum likelihood estimator.

  • Unequal Sample Sizes

    • SPSS calls them estimated marginal means, whereas SAS and SAS JMP call them least squares means.
    • That is, if you add up the sums of squares for Diet, Exercise, D x E, and Error, you get 902.625.
    • Maxwell and Delaney (2003) caution that such an approach could result in a Type II error in the test of the interaction.
    • This, in turn, would increase the Type I error rate for the test of the main effect.
    • Type III sums of squares weight the means equally and, for these data, the marginal means for b1 and b2 are equal: