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| In [[statistics]], a '''sum of squares due to lack of fit''', or more tersely a '''lack-of-fit sum of squares''', is one of the components of a partition of the [[Sum of squares (statistics)|sum of squares]] in an [[analysis of variance]], used in the [[numerator]] in an [[F-test]] of the [[null hypothesis]] that says that a proposed model fits well.
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| == Sketch of the idea ==
| | Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. If you are a positive thinker businessman then today you have to put your business online. Some plugins ask users to match pictures or add numbers, and although effective, they appear unprofessional and unnecessary. t need to use the back button or the URL to get to your home page. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience. <br><br>ve labored so hard to publish and put up on their website. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. Whether or not it's an viewers on your web page, your social media pages, or your web page, those who have a present and effective viewers of "fans" are best best for provide provides, reductions, and deals to help re-invigorate their viewers and add to their main point here. Storing write-ups in advance would have to be neccessary with the auto blogs. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. <br><br>Numerous bloggers are utilizing Word - Press and with good reason. php file in the Word - Press root folder and look for this line (line 73 in our example):. Next you'll go by way of to your simple Word - Press site. So, we have to add our social media sharing buttons in website. The popularity of Word - Press has increased the demand for Word - Press themes and these themes sells like hot cake on the internet. <br><br>Here is more info on [http://hgbilder.de/profile/fcvve.html wordpress backup plugin] check out our page. A sitemap is useful for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. s ability to use different themes and skins known as Word - Press Templates or Themes. However, there are a few other Wordpress plugins also for its development which requires adding files in your Wordpress setup. Change the entire appearance of you blog using themes with one click. As for performing online business, websites and blogs are the only medium that are available to interact with customers and Word - Press perform this work with the help of cross-blog communication tools, comments and full user registration plug-ins. |
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| In order for the lack-of-fit sum of squares to differ from the [[Residual sum of squares|sum of squares of residuals]], there must be [[replication (statistics)|more than one]] value of the [[response variable]] for at least one of the values of the set of predictor variables. For example, consider fitting a line
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| : <math> y = \alpha x + \beta \, </math>
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| by the method of [[least squares]]. One takes as estimates of ''α'' and ''β'' the values that minimize the sum of squares of residuals, i.e., the sum of squares of the differences between the observed ''y''-value and the fitted ''y''-value. To have a lack-of-fit sum of squares that differs from the residual sum of squares, one must observe more than one ''y''-value for each of one or more of the ''x''-values. One then partitions the "sum of squares due to error", i.e., the sum of squares of residuals, into two components:
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| : sum of squares due to error = (sum of squares due to "pure" error) + (sum of squares due to lack of fit).
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| The sum of squares due to "pure" error is the sum of squares of the differences between each observed ''y''-value and the average of all ''y''-values corresponding to the same ''x''-value.
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| The sum of squares due to lack of fit is the ''weighted'' sum of squares of differences between each average of ''y''-values corresponding to the same ''x''-value and the corresponding fitted ''y''-value, the weight in each case being simply the number of observed ''y''-values for that ''x''-value.<ref>{{cite book |first=Richard J. |last=Brook |first2=Gregory C. |last2=Arnold |title=Applied Regression Analysis and Experimental Design |publisher=[[CRC Press]] |location= |year=1985 |pages=48–49 |isbn=0824772520 }}</ref><ref>{{cite book |first=John |last=Neter |first2=Michael H. |last2=Kutner |first3=Christopher J. |last3=Nachstheim |first4=William |last4=Wasserman |title=Applied Linear Statistical Models |edition=Fourth |publisher=Irwin |location=Chicago |year=1996 |pages=121–122 |isbn=0256117365 }}</ref> Because it is a property of least squares regression that the vector whose components are "pure errors" and the vector of lack-of-fit components are orthogonal to each other, the following equality holds:
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| : <math>
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| \begin{align}
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| &\sum (\text{observed value} - \text{fitted value})^2 & & \text{(error)} \\
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| &\qquad = \sum (\text{observed value} - \text{local average})^2 & & \text{(pure error)} \\
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| & {} \qquad\qquad {} + \sum \text{weight}\times (\text{local average} - \text{fitted value})^2. & & \text{(lack of fit)}
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| \end{align}
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| </math>
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| Hence the residual sum of squares has been completely decomposed into two components.
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| == Mathematical details ==
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| Consider fitting a line with one predictor variable. Define ''i'' as an index of each of the ''n'' distinct ''x'' values, ''j'' as an index of the response variable observations for a given ''x'' value, and ''n''<sub>''i''</sub> as the number of ''y'' values associated with the ''i'' <sup>th</sup> ''x'' value. The value of each response variable observation can be represented by
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| : <math> Y_{ij} = \alpha x_i + \beta + \varepsilon_{ij},\qquad i = 1,\dots, n,\quad j = 1,\dots,n_i.</math>
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| Let
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| : <math> \widehat\alpha, \widehat\beta \,</math>
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| be the [[least squares]] estimates of the unobservable parameters ''α'' and ''β'' based on the observed values of ''x''<sub> ''i''</sub> and ''Y''<sub> ''i j''</sub>.
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| Let
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| : <math> \widehat Y_i = \widehat\alpha x_i + \widehat\beta \,</math>
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| be the fitted values of the response variable. Then | |
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| : <math> \widehat\varepsilon_{ij} = Y_{ij} - \widehat Y_i \,</math>
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| are the [[errors and residuals in statistics|residuals]], which are observable estimates of the unobservable values of the error term ''ε''<sub> ''ij''</sub>. Because of the nature of the method of least squares, the whole vector of residuals, with
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| :<math> N = \sum_{i=1}^n n_i </math>
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| scalar components, necessarily satisfies the two constraints
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| : <math> \sum_{i=1}^n \sum_{j=1}^{n_i} \widehat\varepsilon_{ij} = 0 \,</math>
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| : <math> \sum_{i=1}^n \left(x_i \sum_{j=1}^{n_i} \widehat\varepsilon_{ij} \right) = 0. \,</math>
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| It is thus constrained to lie in an (''N'' − 2)-dimensional subspace of '''R'''<sup> ''N''</sup>, i.e. there are ''N'' − 2 "[[degrees of freedom (statistics)|degrees of freedom]] for error".
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| Now let
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| : <math> \overline{Y}_{i\bullet} = \frac{1}{n_i} \sum_{j=1}^{n_i} Y_{ij} </math>
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| be the average of all ''Y''-values associated with the ''i'' <sup>th</sup> ''x''-value.
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| We partition the sum of squares due to error into two components:
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| :<math>
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| \begin{align}
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| & \sum_{i=1}^n \sum_{j=1}^{n_i} \widehat\varepsilon_{ij}^{\,2}
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| = \sum_{i=1}^n \sum_{j=1}^{n_i} \left( Y_{ij} - \widehat Y_i \right)^2 \\
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| & = \underbrace{ \sum_{i=1}^n \sum_{j=1}^{n_i} \left(Y_{ij} - \overline Y_{i\bullet}\right)^2 }_\text{(sum of squares due to pure error)}
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| + \underbrace{ \sum_{i=1}^n n_i \left( \overline Y_{i\bullet} - \widehat Y_i \right)^2. }_\text{(sum of squares due to lack of fit)}
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| \end{align}
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| </math>
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| == Probability distributions ==
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| === Sums of squares ===
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| Suppose the [[errors and residuals in statistics|error terms]] ''ε''<sub> ''i j''</sub> are [[statistical independence|independent]] and [[normal distribution|normally distributed]] with [[expected value]] 0 and [[variance]] ''σ''<sup>2</sup>. We treat ''x''<sub> ''i''</sub> as constant rather than random. Then the response variables ''Y''<sub> ''i j''</sub> are random only because the errors ''ε''<sub> ''i j''</sub> are random.
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| It can be shown to follow that if the straight-line model is correct, then the '''sum of squares due to error''' divided by the error variance,
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| : <math> \frac{1}{\sigma^2}\sum_{i=1}^n \sum_{j=1}^{n_i} \widehat\varepsilon_{ij}^{\,2} </math>
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| has a [[chi-squared distribution]] with ''N'' − 2 degrees of freedom.
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| Moreover, given the total number of observations ''N'', the number of levels of the independent variable ''n,'' and the number of parameters in the model ''p'':
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| * The sum of squares due to pure error, divided by the error variance ''σ''<sup>2</sup>, has a chi-squared distribution with ''N'' − ''n'' degrees of freedom;
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| * The sum of squares due to lack of fit, divided by the error variance ''σ''<sup>2</sup>, has a chi-squared distribution with ''n'' − ''p'' degrees of freedom (here ''p'' = 2 as there are two parameters in the straight-line model);
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| * The two sums of squares are probabilistically independent.
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| === The test statistic ===
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| It then follows that the statistic
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| : <math>
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| \begin{align}
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| F & = \frac{ \text{lack-of-fit sum of squares} /\text{degrees of freedom} }{\text{pure-error sum of squares} / \text{degrees of freedom} } \\[8pt]
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| & = \frac{\left.\sum_{i=1}^n n_i \left( \overline Y_{i\bullet} - \widehat Y_i \right)^2\right/ (n-p)}{\left.\sum_{i=1}^n \sum_{j=1}^{n_i} \left(Y_{ij} - \overline Y_{i\bullet}\right)^2 \right/ (N - n)}
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| \end{align}
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| </math>
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| has an [[F-distribution]] with the corresponding number of degrees of freedom in the numerator and the denominator, provided that the model is correct. If the model is wrong, then the probability distribution of the denominator is still as stated above, and the numerator and denominator are still independent. But the numerator then has a [[noncentral chi-squared distribution]], and consequently the quotient as a whole has a [[non-central F-distribution]].
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| One uses this F-statistic to test the [[null hypothesis]] that there is no lack of linear fit. Since the non-central F-distribution is [[stochastic order|stochastically larger]] than the (central) F-distribution, one rejects the null hypothesis if the F-statistic is larger than the critical F value. The critical value corresponds to the [[cumulative distribution function]] of the [[F distribution]] with ''x'' equal to the desired [[confidence level]], and degrees of freedom ''d''<sub>1</sub> = (''n'' − ''p'') and ''d''<sub>2</sub> = (''N'' − ''n''). This critical value can be calculated using online tools<ref name=soperds>{{cite web|last=Soper|first=D.S.|title=Critical F-value Calculator (Online Software)|url=http://www.danielsoper.com/statcalc3|work=Statistics Calculators|accessdate=19 April 2012}}</ref> or found in tables of statistical values.<ref name=lowryr>{{cite web|last=Lowry|first=Richard|title=VassarStats|url=http://vassarstats.net/textbook/apx_d.html|work=Concepts and Applications of Inferential Statistics|accessdate=19 April 2012}}</ref>
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| The assumptions of [[normal distribution]] of errors and [[independence (probability theory)|independence]] can be shown to entail that this [[lack-of-fit test]] is the [[likelihood-ratio test]] of this null hypothesis.
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| == See also ==
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| * [[Linear regression]]
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| == Notes ==
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| {{reflist}}
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| [[Category:Analysis of variance]]
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| [[Category:Regression analysis]]
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| [[Category:Design of experiments]]
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| [[Category:Least squares]]
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The main advantage of using the blog is that anyone can use the Word - Press blog and customize the elements in the theme regardless to limited knowledge about internet and website development. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. This is a service where people write articles using a specific keyword you have given them. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. This particular wordpress plugin is essential for not only having the capability where you improve your position, but to enhance your organic searches for your website.
Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. If you are a positive thinker businessman then today you have to put your business online. Some plugins ask users to match pictures or add numbers, and although effective, they appear unprofessional and unnecessary. t need to use the back button or the URL to get to your home page. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience.
ve labored so hard to publish and put up on their website. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. Whether or not it's an viewers on your web page, your social media pages, or your web page, those who have a present and effective viewers of "fans" are best best for provide provides, reductions, and deals to help re-invigorate their viewers and add to their main point here. Storing write-ups in advance would have to be neccessary with the auto blogs. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration.
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