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In [[statistics|statistical data analysis]] the '''total sum of squares''' (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. It is defined as being the sum, over all observations, of the squared differences of each observation from the overall [[mean]].<ref>Everitt, B.S. (2002) ''The Cambridge Dictionary of Statistics'', CUP, ISBN 0-521-81099-X</ref> | |||
In [[statistics|statistical]] [[linear model]]s, (particularly in standard [[regression model]]s), the '''TSS''' is the [[sum]] of the [[square (algebra)|square]]s of the difference of the dependent variable and its [[mean]]: | |||
:<math>\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^2</math> | |||
where <math>\bar{y}</math> is the mean. | |||
For wide classes of linear models, the total sum of squares equals the [[explained sum of squares]] plus the [[residual sum of squares]]. For a proof of this in the multivariate OLS case, see [[Explained sum of squares#Partitioning in the general OLS model|partitioning in the general OLS model]]. | |||
In [[analysis of variance]] (ANOVA) the total sum of squares is the sum of the so-called "within-samples" sum of squares and "between-samples" sum of squares, i.e., partitioning of the sum of squares. | |||
In [[multivariate analysis of variance]] (MANOVA) the following equation applies<ref name="MardiaK1979Multivariate">{{Cite book | |||
| author = [[K. V. Mardia]], J. T. Kent and J. M. Bibby | |||
| title = Multivariate Analysis | |||
| publisher = [[Academic Press]] | |||
| year = 1979 | |||
| isbn = 0-12-471252-5 | |||
}} Especially chapters 11 and 12.</ref> | |||
:<math>\mathbf{T} = \mathbf{W} + \mathbf{B},</math> | |||
where '''T''' is the total sum of squares and products (SSP) [[Matrix (mathematics)|matrix]], '''W''' is the within-samples SSP matrix and '''B''' is the between-samples SSP matrix. | |||
Similar terminology may also be used in [[linear discriminant analysis]], where '''W''' and '''B''' are respectively referred to as the within-groups and between-groups SSP matrics.<ref name="MardiaK1979Multivariate"/> | |||
==See also== | |||
*[[Sum of squares (statistics)]] | |||
*[[Lack-of-fit sum of squares]] | |||
==References== | |||
{{Reflist}} | |||
[[Category:Regression analysis]] | |||
[[Category:Least squares]] | |||
Revision as of 18:54, 7 January 2014
In statistical data analysis the total sum of squares (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. It is defined as being the sum, over all observations, of the squared differences of each observation from the overall mean.[1]
In statistical linear models, (particularly in standard regression models), the TSS is the sum of the squares of the difference of the dependent variable and its mean:
For wide classes of linear models, the total sum of squares equals the explained sum of squares plus the residual sum of squares. For a proof of this in the multivariate OLS case, see partitioning in the general OLS model.
In analysis of variance (ANOVA) the total sum of squares is the sum of the so-called "within-samples" sum of squares and "between-samples" sum of squares, i.e., partitioning of the sum of squares. In multivariate analysis of variance (MANOVA) the following equation applies[2]
where T is the total sum of squares and products (SSP) matrix, W is the within-samples SSP matrix and B is the between-samples SSP matrix. Similar terminology may also be used in linear discriminant analysis, where W and B are respectively referred to as the within-groups and between-groups SSP matrics.[2]
See also
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑ Everitt, B.S. (2002) The Cambridge Dictionary of Statistics, CUP, ISBN 0-521-81099-X
- ↑ 2.0 2.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Especially chapters 11 and 12.