“analysis of variance”的概念、定义、翻译、参考文献-科学参考

analysis of variance

Mathematics

A general procedure for partitioning the overall variability in a set of data into components due to specified causes and random variation. It involves calculating such quantities as the ‘between-groups sum of squares’ and the ‘residual sum of squares’ and dividing by the degrees of freedom to give so-called ‘mean squares’. The results are usually presented in an ANOVA table, the name being derived from the opening letters of the words ‘analysis of variance’. Such a table provides a concise summary from which the influence of the explanatory variables can be estimated and hypotheses can be tested, usually by means of F-tests.
http://www.bmj.com/cgi/content/full/312/7044/1472 A description of the analysis and its interpretation in a medical context (subscription).

Statistics

See ANOVA.

Chemical Engineering

One of a number of statistical techniques used to resolve and observe the variance between sets of statistical data into components. These techniques are used to determine whether the difference between samples is explicable as random sampling variation from within the same statistical populations. ANOVA techniques are used in quality control.

Computer

A technique, originally developed by R. A. Fisher, whereby the total variation in a vector of numbers y₁ … y_n, expressed as the sum of squares about the mean
$\underset{l}{Σ} {(y_{i} - y .)}^{2},$
is split up into component sums of squares ascribable to the effects of various classifying factors that may index the data. Thus if the data consist of a two-way m×n array, classified by factors A and B and indexed by
$i = 1, \dots, m j = 1, \dots, n$
then the analysis of variance gives the identity
$\underset{\begin{matrix} Total \end{matrix}}{\underset{i j}{Σ} {(y_{i j} - y .)}^{2}} \equiv \underset{\begin{matrix} A \\ main effect \end{matrix}}{\underset{i j}{Σ} {(y_{i .} - y_{.})}^{2}} + \underset{\begin{matrix} B \\ main effect \end{matrix}}{\underset{i j}{Σ} {(y_{. j} - y_{.})}^{2}} + \underset{\begin{matrix} A .B \\ interaction \end{matrix}}{\underset{i j}{Σ} {(y_{i j} - y_{i .} - y_{. j} + y_{.})}^{2}}$
Geometrically the analysis of variance becomes the successive projections of the vector y, considered as a point in n-dimensional space, onto orthogonal hyperplanes within that space. The dimensions of the hyperplanes give the degrees of freedom for each term; in the above example these are
$m n - 1 \equiv (m - 1) + (n - 1) + (m - 1) (n - 1)$
A statistical model applied to the data allows mean squares, equal to (sum of squares)/(degrees of freedom), to be compared with an error mean square that measures the background ‘noise’. Large mean squares indicate large effects of the factors concerned. The above processes can be much elaborated (see experimental design, regression analysis).

Economics

A statistical technique based on decomposing the total variance of some characteristic of a population into parts correlated with other characteristics, and residual variation. In particular, analysis of variance is used to test whether sections of a population appear to differ significantly in some property. For example, if y_i is the personal income of individual i, analysis of variance can be used to test whether there are significant regional differences in mean income. The total variance of the population is decomposed into the part due to differences within regions, and the part due to differences between regional means. The larger the proportion of total variance due to differences between group means, the more likely it is that the groups are systematically different, whereas the higher the proportion of total variance due to within-group variance, the more likely it is that apparent differences between group means arise from sampling error.

单词	analysis of variance
释义	analysis of variance Mathematics A general procedure for partitioning the overall variability in a set of data into components due to specified causes and random variation. It involves calculating such quantities as the ‘between-groups sum of squares’ and the ‘residual sum of squares’ and dividing by the degrees of freedom to give so-called ‘mean squares’. The results are usually presented in an ANOVA table, the name being derived from the opening letters of the words ‘analysis of variance’. Such a table provides a concise summary from which the influence of the explanatory variables can be estimated and hypotheses can be tested, usually by means of F-tests. http://www.bmj.com/cgi/content/full/312/7044/1472 A description of the analysis and its interpretation in a medical context (subscription). Statistics See ANOVA. Chemical Engineering One of a number of statistical techniques used to resolve and observe the variance between sets of statistical data into components. These techniques are used to determine whether the difference between samples is explicable as random sampling variation from within the same statistical populations. ANOVA techniques are used in quality control. Computer A technique, originally developed by R. A. Fisher, whereby the total variation in a vector of numbers y₁ … y_n, expressed as the sum of squares about the mean $\underset{l}{Σ} {(y_{i} - y .)}^{2},$ is split up into component sums of squares ascribable to the effects of various classifying factors that may index the data. Thus if the data consist of a two-way m×n array, classified by factors A and B and indexed by $i = 1, \dots, m j = 1, \dots, n$ then the analysis of variance gives the identity $\underset{\begin{matrix} Total \end{matrix}}{\underset{i j}{Σ} {(y_{i j} - y .)}^{2}} \equiv \underset{\begin{matrix} A \\ main effect \end{matrix}}{\underset{i j}{Σ} {(y_{i .} - y_{.})}^{2}} + \underset{\begin{matrix} B \\ main effect \end{matrix}}{\underset{i j}{Σ} {(y_{. j} - y_{.})}^{2}} + \underset{\begin{matrix} A .B \\ interaction \end{matrix}}{\underset{i j}{Σ} {(y_{i j} - y_{i .} - y_{. j} + y_{.})}^{2}}$ Geometrically the analysis of variance becomes the successive projections of the vector y, considered as a point in n-dimensional space, onto orthogonal hyperplanes within that space. The dimensions of the hyperplanes give the degrees of freedom for each term; in the above example these are $m n - 1 \equiv (m - 1) + (n - 1) + (m - 1) (n - 1)$ A statistical model applied to the data allows mean squares, equal to (sum of squares)/(degrees of freedom), to be compared with an error mean square that measures the background ‘noise’. Large mean squares indicate large effects of the factors concerned. The above processes can be much elaborated (see experimental design, regression analysis). Economics A statistical technique based on decomposing the total variance of some characteristic of a population into parts correlated with other characteristics, and residual variation. In particular, analysis of variance is used to test whether sections of a population appear to differ significantly in some property. For example, if y_i is the personal income of individual i, analysis of variance can be used to test whether there are significant regional differences in mean income. The total variance of the population is decomposed into the part due to differences within regions, and the part due to differences between regional means. The larger the proportion of total variance due to differences between group means, the more likely it is that the groups are systematically different, whereas the higher the proportion of total variance due to within-group variance, the more likely it is that apparent differences between group means arise from sampling error.
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