# Lack-of-fit sum of squares

**error sum of squaressum of squared errors**

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 of residuals 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.wikipedia

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### Partition of sums of squares

**sum of squaressums of squaresSum of squares (statistics)**

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 of residuals 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.

Note that the residual sum of squares can be further partitioned as the lack-of-fit sum of squares plus the sum of squares due to pure error.

### F-test

**F''-testF testF-statistic**

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 of residuals 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.

### Analysis of variance

**ANOVAanalysis of variance (ANOVA)corrected the means**

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 of residuals 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.

See also Lack-of-fit sum of squares.

### Residual sum of squares

**sum of squared residualssum of squares of residualsresidual sum-of-squares**

In order for the lack-of-fit sum of squares to differ from the sum of squares of residuals, there must be more than one value of the response variable for at least one of the values of the set of predictor variables.

### Errors and residuals

**residualserror termresidual**

are the residuals, which are observable estimates of the unobservable values of the error term ε ij. Suppose the error terms ε i j are independent and normally distributed with expected value 0 and variance σ 2.

### Goodness of fit

**goodness-of-fitfitgoodness-of-fit test**

The assumptions of normal distribution of errors and independence can be shown to entail that this lack-of-fit test is the likelihood-ratio test of this null hypothesis.

In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares.

### Linear regression

**regression coefficientmultiple linear regressionregression**

### Statistics

**statisticalstatistical analysisstatistician**

### Fraction (mathematics)

**denominatorfractionsfraction**

### Null hypothesis

**nullnull hypotheseshypothesis**

### Replication (statistics)

**replicationreplicatereplicates**

In order for the lack-of-fit sum of squares to differ from the sum of squares of residuals, there must be more than one value of the response variable for at least one of the values of the set of predictor variables.

### Dependent and independent variables

**dependent variableindependent variableexplanatory variable**

In order for the lack-of-fit sum of squares to differ from the sum of squares of residuals, there must be more than one value of the response variable for at least one of the values of the set of predictor variables. The pure-error sum of squares is the sum of squared deviations of each value of the dependent variable from the average value over all observations sharing its independent variable value(s).

### Least squares

**least-squaresmethod of least squaresleast squares method**

by the method of least squares.

### Degrees of freedom (statistics)

**degrees of freedomdegree of freedomEffective degrees of freedom**

It is thus constrained to lie in an (N − 2)-dimensional subspace of R N, i.e. there are N − 2 "degrees of freedom for error".

### Normal distribution

**normally distributedGaussian distributionnormal**

The assumptions of normal distribution of errors and independence can be shown to entail that this lack-of-fit test is the likelihood-ratio test of this null hypothesis. Suppose the error terms ε i j are independent and normally distributed with expected value 0 and variance σ 2.

### Expected value

**expectationexpectedmean**

Suppose the error terms ε i j are independent and normally distributed with expected value 0 and variance σ 2.

### Variance

**sample variancepopulation variancevariability**

Suppose the error terms ε i j are independent and normally distributed with expected value 0 and variance σ 2.

### Chi-squared distribution

**chi-squaredchi-square distributionchi square distribution**

has a chi-squared distribution with N − 2 degrees of freedom.

### Noncentral chi-squared distribution

**noncentral chi-squarednon-central chi-squared distributionnon-central Chi-squared**

But the numerator then has a noncentral chi-squared distribution, and consequently the quotient as a whole has a non-central F-distribution.

### Noncentral F-distribution

**noncentral ''F''-distributionnon-central F-distributionnoncentral ''F''-distributed**

But the numerator then has a noncentral chi-squared distribution, and consequently the quotient as a whole has a non-central F-distribution.

### Stochastic ordering

**stochastically largerstochastic orderstochastically dominates**

Since the non-central F-distribution is stochastically larger than the (central) F-distribution, one rejects the null hypothesis if the F-statistic is larger than the critical F value.

### Cumulative distribution function

**distribution functionCDFcumulative probability distribution function**

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 1 = (n − p) and d 2 = (N − n).

### F-distribution

**F distributionF''-distributionF'' distribution**

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 1 = (n − p) and d 2 = (N − n). has an F-distribution with the corresponding number of degrees of freedom in the numerator and the denominator, provided that the model is correct.

### Confidence interval

**confidence intervalsconfidence levelconfidence**

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 1 = (n − p) and d 2 = (N − n).