Calculates \(\chi^2\), reduced \(\chi_{\nu}^2\) and the \(\chi^2\) 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.
fitchisq(object, error = NULL)
a single model of class 'pcrfit', a 'replist' or any fitted model of the above.
in case of a model without replicates, a single error for all response values or a vector of errors for each response value.
A list with the following items:
the \(\chi^2\) value.
the reduced \(\chi_\nu^2\).
the fit probability as described above.
The variance of a fit \(s^2\) is also characterized by the statistic \(\chi^2\) defined as followed: $$\chi^2 \equiv \sum_{i=1}^n \frac{(y_i - f(x_i))^2}{\sigma_i^2}$$ The relationship between \(s^2\) and \(\chi^2\) can be seen most easily by comparison with the reduced \(\chi^2\): $$\chi_\nu^2 = \frac{\chi^2}{\nu} = \frac{s^2}{\langle \sigma_i^2 \rangle}$$ whereas \(\nu\) = degrees of freedom (N - p), and \(\langle \sigma_i^2 \rangle\) 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, \(\chi_\nu^2 = 1\). 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 \(\chi_\nu^2\) will fluctuate from experiment to experiment. To assign significance to the \(\chi^2\) value, we can use the integral probability $$P_\chi(\chi^2;\nu) = \int_{\chi^2}^\infty P_\chi(x^2, \nu)dx^2$$ which describes the probability that a random set of n data points sampled from the parent distribution would yield a value of \(\chi^2\) equal to or greater than the calculated one. This is calculated by \(1 - pchisq(\chi^2, \nu)\).
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.
# NOT RUN {
## Using replicates by making a 'replist'.
ml1 <- modlist(reps, fluo = 2:5)
rl1 <- replist(ml1, group = c(1, 1, 1, 1))
fitchisq(rl1[[1]])
## Using single model with added error.
m1 <- pcrfit(reps, 1, 2, l5)
fitchisq(m1, 0.1)
# }
Run the code above in your browser using DataLab