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Teaching and Learning in Nursing, 9, 53— Carnegie Mellon n. Case studies [Fact sheet]. In contrast, the spline was able to model the irregularity and hence to represent the curve's behavior more precisely. The raw colour intensities black circles , measured over time x-axis, hours were fitted by both a parametric model thick blue line and a model-free spline thick red line. The thin dashed lines indicate the maximum slope of each approach thin dashed black line corresponds to the model fitting approach, the thin dashed red line to the spline, respectively.
In the left panel the irregularity is better customized by the spline fit, whereas the model straightens it with the consequence of underestimating the maximum height A.
Note the particularly broad CIs for these parameter estimates in Fig. The example corresponding to almost no respiration Fig. Ideally, non-respiration would result in a horizontal line, and, hence, non-convergence for modeling approaches. However, the linearly increasing noise allowed a model to be fitted to the data which apparently resulted in biologically unreasonable parameter estimates via extrapolation; i.
In contrast, although it exhibited overfitting behavior, the spline approach was able to follow the data more precisely, apparently without the need to extrapolate. These estimation problems were also indicated by the particularly broad CIs for these parameters if inferred from the spline.
These curves only differed regarding the beginning of the respiration reaction. Left, a plot of the raw respiration data illustrates their courses individually for each of the ten repetitions. A non-overlap of the CIs of different curves indicates a difference of a statistically detectable amount, and the distance between two intervals provides information about the expected minimum difference.
Although all differences were statistically detectable, the user had the additional information of the effect sizes and thus was, in principle, able to use background information to decide whether the detected differences were biologically relevant.
The results from the time series approach in the third dataset are shown in Fig. Curve 20, the fourth repetition from time point Again, the user was now free to decide whether these differences should be regarded as biologically relevant. The upper panels show the plot of the respiration curves of E. The blue lines highlight the position of the upper and lower limits of the CIs from repetition no.
Regarding the comparison of group means, Fig.
The mean CIs can be used analogously to the strategy described above: overlapping CIs indicate no detectable difference between the groups, while non-overlapping ones indicate such differences. The upper panels show the results from the preliminary calculation, a simple calculation of group means of confidence limits and point estimators.
The groups, here the distinct pretreatments cultivation times t1 to t9 , are given on the y-axis. For orientation, the blue lines highlight the position of upper and lower limit of CIs from repetition no. Since these are CIs for the differences between the means, a non-overlap with zero indicates a statistically detectable difference between the considered group means of the examined curve parameters.
To examine whether it is valid to subtract the negative controls A01 from the measurements from all other wells before estimating curve parameters, we compared the parameter values for maximum height A from the A01 with that from selected wells with a negative reaction. Our findings suggest that the negative control might display a reproducible, strain-specific growth-like behavior, and even though these curves are shallower than unambiguously positive reactions, their maximum height can well be larger than that of typical negative reactions on the same plate.
This makes it impossible to regard it as an approximation of an error term to be subtracted from the measurements from each other well. These findings are described in detail in File S9. Analysis of archetypes Fig. For five as predefined number, the resulting curve archetypes insert in Fig.
The outer figure is a scree plot in which the residual sums of squares RSS, y-axis are plotted against the corresponding predefined numbers of archetypes x-axis. The insert upper right is a parallel coordinates plot showing the original measurements gray lines as well as the optimal archetypes green, black, blue, violet and red lines obtained if five archetypes are requested. On the x-axis, the names of the curve parameters are indicated.
The minima and maxima of the four y-axes are also indicated. For an interpretation of the archetypes, see the main text. When facing huge and complicatedly structured datasets such as the PM ones discussed here or that commonly occurring in other -omics analyses, the only way to get a comprehensive insight into the experimental results is a suitable graphical raw data representation. Such exploratory graphics have to be comprehensible in short time but also be highly informative .
The convenience of an exploratory graphical representation depends mainly on its flexibility. Hence, the graphics should be easily adjustable to individual users' requirements to enable them to discover potentially all interesting and important features of the data.
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In this study we explored an open-source solution for these specifications and showed that curve kinetics offer a more powerful data visualization than level plots . By using the function xyplot from the lattice package  highly structured graphics can easily be produced while retaining flexibility by systematically decoupling the various elements of a display. Itemization by substrate, tested strain or even repetition number was quite simple and constraints regarding the number of displayed curves or the position of the subpanel were not imposed at all.
We thus recommend this or equivalent visualization approaches for PM data. A potential improvement compared to Fig. The information content of the longitudinal PM raw data is a multiple of what an endpoint measurement could ever provide. A suitable analysis strategy thus has to be able to summarize this information and eliminate noise. These requirements can be met by model-fitting and spline-fitting approaches aiming on both dimension reduction and noise reduction  — .
With grofit , the result is a set of four parameters sufficient for comprehensively describing the curves' shape. The main goals of a subsequent data evaluation would be the determination of the influence of different substrates, organisms investigated, or pretreatments, via the comparative characterization of respiration over time. The aim was to find a reliable estimation method that was able to deal adequately with curves' potential deviations from the common sigmoid shape .
Our results indicate, however, that the parameter estimation procedures perform best if applied to curves that follow the typical sigmoidal shape. When comparing the two main approaches for curve description, it turned out that the spline smoother is flexible enough to follow even extreme curve shapes and is therefore superior for general parameter estimation, while the model-fitting approach appeared to be more constrained by the underlying model equations and straightened the curves to much.
In this study, the default parameters for the smoothing spline and the number of knots were used, since the evaluation of best-performing parameters was beyond the scope of this study. However, the selection of these two kinds of parameters is the critical step in this method . Also, other spline families and generalized additive model frameworks would exhibit interesting features for curve fitting by imposing monotonicity constraints on smooth effects and on ordinal, categorical variables .
We cannot exclude that as yet unimplemented models would outperform the ones considered here or even the spline fit, but in the current situation we regard the use of splines as the best recommendation that can be provided to users interested in fitting PM curves with R. As the native PM software represents the curves as series of rectangles, this deviation is most likely an overestimation and is expected to increase if more steep curves are encountered.
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Based on these results we favour the spline-based approach to parameter estimation over the native PM software not only because it provides CIs but also because its point estimates are less prone to bias due to the described irregularities in curve shapes. However, the spline-based approach exhibits overfitting behavior in the case of certain curves that strongly deviate from a sigmoidal curve shape. This appears to occur especially when almost no reaction takes place, as shown in Fig. One way out could be the selection of more suitable smoothing parameters.
Alternatively, the methods for the extraction of the parameters from the spline could be revised. It is well known that phenomena such as autocorrelation which is usual for growth curves and non-homoscedasticity of the residuals violate the underlying assumptions of model- and spline-fitting  , . When dealing with high-throughput datasets such as the PM ones, however, the detailed assessment of a potential violation of the assumptions made when fitting each curve is not practicable. Moreover, while for instance the spline might overfit the data in such situations, it is here only used for smoothing each curve before extracting the four abstract parameters of interest.
It is thus unlikely that potential violations of the underlying assumptions of the fit adversely affect the unbiasedness of the parameter estimates. This might explain why the spline appears more robust than the other methods if applied to PM data.