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'.