The following methods for the detection of kinetic outliers are implemented
uni1: KOD method according to Bar et al. (2003). Outliers are defined by removing the sample efficiency from the replicate group and testing it against the remaining samples' efficiencies using a Z-test:
$$P = 2 \cdot \left[1 - \Phi\left(\frac{e_i - \mu_{train}}{\sigma_{train}}\right)\right] < 0.05$$
uni2: This method from the package author is more or less a test on sigmoidal structure for the individual curves. It is different in that there is no comparison against other curves from a replicate set. The test is simple: The difference between first and second derivative maxima should be less than 10 cycles:
$$\left(\frac{\partial^3 F(x;a,b,...)}{\partial x^3} = 0\right) - \left(\frac{\partial^2 F(x;a,b...)}{\partial x^2} = 0\right) < 10$$
Sounds astonishingly simple, but works: Runs are defines as 'outliers' that really failed to amplify, i.e. have no sigmoidal structure or are very shallow. It is the default setting in modlist.
multi1: KOD method according to Tichopad et al. (2010). Assuming two vectors with first and second derivative maxima \(t_1\) and \(t_2\) from a 4-parameter sigmoidal fit within a window of points around the first derivative maximum, a linear model \(t_2 = t_1 \cdot b + a + \tau\) is made. Both \(t_1\) and the residuals from the fit \(\tau = t_2 - \hat{t_2}\) are Z-transformed:
$$t_1(norm) = \frac{t_1 - \bar{t}_1}{{\sigma_t}_1}, \; {\tau_1}_{norm} = \frac{\tau_1 - \bar{\tau}_1}{{\sigma_\tau}_1}$$
Both \(t_1\) and \(\tau\) are used for making a robust covariance matrix. The outcome is plugged into a mahalanobis distance analysis using the 'adaptive reweighted estimator' from package 'mvoutlier' and p-values for significance of being an 'outlier' are deduced from a \(\chi^2\) distribution. If more than two parameters are supplied, princomp is used instead.
multi2: Second KOD method according to Tichopad et al. (2010), mentioned in the paper. Uses the same pipeline as multi1, but with the slope at the first derivative maximum and maximum fluorescence as parameters:
$$\frac{\partial F(x;a,b,...)}{\partial x}, F_{max}$$
multi3: KOD method according to Sisti et al. (2010). Similar to multi2, but uses maximum fluorescence, slope at first derivative maximum and y-value at first derivative maximum as fixpoints:
$$\frac{\partial F(x;a,b,...)}{\partial x}, F\left(\frac{\partial^2 F(x;a,b,...)}{\partial x^2} = 0\right), F_{max}$$
All essential parameters for the methods can be tweaked by parKOD. See there and in 'Examples'.