Statistical analysis error

Simulation runs are typically short (t 10 - 10 MD or MC steps, correspondmg to perhaps a few nanoseconds of real time) compared with the time allowed in laboratory experiments. This means that we need to test whether or not a simulation has reached equilibrium before we can trust the averages calculated in it. Moreover, there is a clear need to subject the simulation averages to a statistical analysis, to make a realistic estimate of the errors. 

When designing and evaluating an analytical method, we usually make three separate considerations of experimental error. First, before beginning an analysis, errors associated with each measurement are evaluated to ensure that their cumulative effect will not limit the utility of the analysis. Errors known or believed to affect the result can then be minimized. Second, during the analysis the measurement process is monitored, ensuring that it remains under control. Finally, at the end of the analysis the quality of the measurements and the result are evaluated and compared with the original design criteria. This chapter is an introduction to the sources and evaluation of errors in analytical measurements, the effect of measurement error on the result of an analysis, and the statistical analysis of data. 

In this experiment students measure the length of a pestle using a wooden meter stick, a stainless-steel ruler, and a vernier caliper. The data collected in this experiment provide an opportunity to discuss significant figures and sources of error. Statistical analysis includes the Q-test, f-test, and F-test. 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

Guedens, W. J. Yperman, J. Mullens, J. et al. Statistical Analysis of Errors A Practical Approach for an Undergraduate Ghemistry Lab, Part 1. The Goncept, /. Chem. Educ. 1993, 70, 776-779 Part 2. Some Worked Examples, /. Chem. Educ. 1993, 70, 838-841. 

Statistical errors of dynamic properties could be expressed by breaking a simulation up into multiple blocks, taking the average from each block, and using those values for statistical analysis. In principle, a block analysis of dynamic properties could be carried out in much the same way as that applied to a static average. However, the block lengths would have to be substantial to make a reasonably accurate estimate of the errors. This approach is based on the assumption that each block is an independent sample. 

The goal of any statistical analysis is inference concerning whether on the basis of available data, some hypothesis about the natural world is true. The hypothesis may consist of the value of some parameter or parameters, such as a physical constant or the exact proportion of an allelic variant in a human population, or the hypothesis may be a qualitative statement, such as This protein adopts an a/p barrel fold or I am currently in Philadelphia. The parameters or hypothesis can be unobservable or as yet unobserved. How the data arise from the parameters is called the model for the system under study and may include estimates of experimental error as well as our best understanding of the physical process of the system. 

Statistical tests for the detection of errors other than sc are discussed in C. A. Bennett and N, L, Franklin, Statistical Analysis. 

The first option is unworkable, though, because this other technology is unlikely to have ( 4) the same cut-off characteristics around 564 /xm, or (B) measure the same characteristics (e.g., volume instead of length). The third options falls out of favor for the simple reason that such a model is not available, and if it were, errors of extrapolation would be propagated into any result obtained from the statistical analysis. 

The precision stated in Table 10 is given by the standard deviations obtained from a statistical analysis of the experimental data of one run and of a number of runs. These parameters give an indication of the internal consistency of the data of one run of measurements and of the reproducibility between runs. The systematic error is far more difficult to discern and to evaluate, which causes an uncertainty in the resulting values. Such an estimate of systematic errors or uncertainties can be obtained if the measuring method can also be applied under circumstances where a more exact or a true value of the property to be determined is known from other sources. 

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

Grimm JW, Lynch JA. 1991. Statistical analysis of errors in estimating wet deposition using five surface estimation algorithms. Atmos Environ 25(2) 317-327. 

Scale bar 200 pm. Red arrows indicate the avascular zones. Quantification of digital analysis of the fluorescence angiography images number of branching points (mm2) (B) and mean mesh size (102 pm2) (C) as markers of vessel density for CAM. P < 0.05 was considered to be statistically significant. Error bars represent standard error of the mean. 

In principle, FCS can also measure very slow processes. In this limit the measurements are constrained by the stability of the system and the patience of the investigator. Because FCS requires the statistical analysis of many fluctuations to yield an accurate estimation of rate parameters, the slower the typical fluctuation, the longer the time required for the measurement. The fractional error of an FCS measurement, expressed as the root mean square of fluorescence fluctuations divided by the mean fluorescence, varies as 1V-1/2, where N is the number of fluctuations that are measured. If the characteristic lifetime of a fluctuation is r, the duration of a measurement to achieve a fractional error of E = N l,/- is T = Nr. Suppose, for example, that r = 1 s. If 1% accuracy is desired, N = 104 and so T = 104 s. 

The primary goal of this series of chapters is to describe the statistical tests required to determine the magnitude of the random (i.e., precision and accuracy) and systematic (i.e., bias) error contributions due to choosing Analytical METHODS A or B, and/or the location/operator where each standard method is performed. The statistical analysis for this series of articles consists of five main parts as ... 

Biochips produce huge data sets. Data collected from microarray experiments are random snapshots with errors, inherently noisy and incomplete. Extracting meaningful information from thousands of data points by means of bioinformatics and statistical analysis is sophisticated and calls for collaboration among researchers from different disciplines. An increasing number of image and data analysis tools, in part freely accessible ( ) to academic researchers and non-profit institutions, is available in the web. Some examples are found in Tables 3 and 4. 

Statistical analysis In each treatment, 10 microspores were used to measure the maximal fluorescence. The means and their standard errors are determined, if the investigator has microspectrofluorimeter or microspectrofluorimeter having special statictic t- test programme. 

Statistical analysis Statistical analysis consists of determination of Standard error of means (SEM) or the Student- t-testing. The measurement error was about 1-2% (10 spectra per one variant, n=10). Counting was performed in four or five replicates (the number of Petri dishes per a treatment). SEM for the fluorescence spectra was 2 %. 

Indeterminate errors, also called random errors, on the other hand, are errors that are not specifically identified and are therefore impossible to avoid. Since the errors cannot be specifically identified, results arising from such errors cannot be immediately rejected or compensated for as in the case of determinate errors. Rather, a statistical analysis must be performed to determine whether the results are far enough off-track to merit rejection. 

It is important to appreciate that the statistical significance of the results is wholly dependent on the quality of the data obtained from the trial. Data that contain obvious gross errors should be removed prior to statistical analysis. It is essential that participants inform the trial co-ordinator of any gross error that they know has occurred during the analysis and also if any deviation from the method as written has taken place. The statistical parameters calculated and the outlier tests performed are those used in the internationally agreed Protocol for the Design, Conduct and Interpretation of Collaborative Studies.14... 

Equation 2.67 indicates that the standard enthalpy and entropy of reaction 2.64 derived from Kc data may be close to the values obtained with molality equilibrium constants. Because Ar// is calculated from the slope of In AT versus l/T, it will be similar to the value derived with Km data provided that the density of the solution remains approximately constant in the experimental temperature range. On the other hand, the error in ArSj calculated with Kc data can be roughly estimated as R In p (from equations 2.57 and 2.67). In the case of water, this is about zero for most solvents, which have p in the range of 0.7-2 kg dm-3, the corrections are smaller (from —3 to 6 J K-1 mol-1) than the usual experimental uncertainties associated with the statistical analysis of the data. 

In any book, there are relevant issues that are not covered. The most obvious in this book is probably a lack of in-depth statistical analysis of the results of model-based and model-free analyses. Data fitting does produce standard deviations for the fitted parameters, but translation into confidence limits is much more difficult for reasonably complex models. Also, the effects of the separation of linear and non-linear parameters are, to our knowledge, not well investigated. Very little is known about errors and confidence limits in the area of model-free analysis. 

The statistical analysis is problematic for this example as the residuals are obviously not normally distributed. Nonetheless, the high errors in the parameters and the large standard deviation of the residuals indicate a bad fit. 

Macdonald (144) analyzed several equations of state which had a variety of mathematical forms including the Tammann equation and the secant bulk modulus equation chosen by Hayward. (In his statistical analysis, Macdonald used the PVT data of Kell and Whalley (26) which has been shown to be in error (29) Thus, the conclusions of Macdonald may be questionable.) He disagreed with Hayward and selected the Murnaghan equation to be superior to either the Tammann equation or the linear secant modulus equation chosen by Hayward. If, however, the Tammann equation and the Murnaghan equation were both expanded to second order in pressure, then Macdonald found that the results obtained from both equations would agree. As shown earlier, the expansion of the Tammann equation to second order is equivalent to the bulk modulus form of the original Tait equation. 

Our statistical analysis reveals a large improvement from cc-pCV(DT)Z to cc-pCV(TQ)Z see Fig. 1.4. In fact, the cc-pCV(TQ)Z calculations are clearly more accurate than their much more expensive cc-pcV6Z counterparts and nearly as accurate as the cc-pcV(56)Z extrapolations.The cc-pCV(TQ)Z extrapolations yield mean and maximum absolute errors of 1.7 and 4.0 kJ/mol, respectively, compared with those of 0.8 and 2.3 kJ/mol at the cc-pcV(56)Z level. Chemical accuracy is thus obtained at the cc-pCV(TQ)Z level, greatly expanding the range of molecules for which ab initio electronic-structure calculations will afford thermochemical data of chemical accuracy. 

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