Description Usage Arguments Details Value Author(s) References Examples
Calculates χ^2, reduced χ_{ν}^2 and the χ^2 fit probability for objects of class pcrfit
, lm
, glm
, nls
or any other object with a call
component that includes formula
and data
.
The function checks for replicated data (i.e. multiple same predictor values). If replicates are not given, the function needs error values, otherwise NA
's are returned.
1 |
object |
a single model of class 'pcrfit', a 'replist' or any fitted model of the above. |
error |
in case of a model without replicates, a single error for all response values or a vector of errors for each response value. |
The variance of a fit s^2 is also characterized by the statistic χ^2 defined as followed:
χ^2 \equiv ∑_{i=1}^n \frac{(y_i - f(x_i))^2}{σ_i^2}
The relationship between s^2 and χ^2 can be seen most easily by comparison with the reduced χ^2:
χ_ν^2 = \frac{χ^2}{ν} = \frac{s^2}{\langle σ_i^2 \rangle}
whereas ν = degrees of freedom (N - p), and \langle σ_i^2 \rangle is the weighted average of the individual variances. If the fitting function is a good approximation to the parent function, the value of the reduced chi-square should be approximately unity, χ_ν^2 = 1. If the fitting function is not appropriate for describing the data, the deviations will be larger and the estimated variance will be too large, yielding a value greater than 1. A value less than 1 can be a consequence of the fact that there exists an uncertainty in the determination of s^2, and the observed values of χ_ν^2 will fluctuate from experiment to experiment. To assign significance to the χ^2 value, we can use the integral probability
P_χ(χ^2;ν) = \int_{χ^2}^∞ P_χ(x^2, ν)dx^2
which describes the probability that a random set of n data points sampled from the parent distribution would yield a value of χ^2 equal to or greater than the calculated one. This is calculated by 1 - pchisq(χ^2, ν).
A list with the following items:
chi2 |
the χ^2 value. |
chi2.red |
the reduced χ_ν^2. |
p.value |
the fit probability as described above. |
Andrej-Nikolai Spiess
Data Reduction and Error Analysis for the Physical Sciences.
Bevington PR & Robinson DK.
McGraw-Hill, New York (2003).
Applied Regression Analysis.
Draper NR & Smith H.
Wiley, New York, 1998.
1 2 3 4 5 6 7 8 |
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.