Statistical analysis experiments

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. 

The following experiments may he used to introduce the statistical analysis of data in the analytical chemistry laboratory. Each experiment is annotated with a brief description of the data collected and the type of statistical analysis used in evaluating the data. 

In this experiment students standardize a solution of HGl by titration using several different indicators to signal the titration s end point. A statistical analysis of the data using f-tests and F-tests allows students to compare results obtained using the same indicator, with results obtained using different indicators. The results of this experiment can be used later when discussing the selection of appropriate indicators. 

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. 

In this experiment students synthesize basic copper(ll) carbonate and determine the %w/w Gu by reducing the copper to Gu. A statistical analysis of the results shows that the synthesis does not produce GUGO3, the compound that many predict to be the product (although it does not exist). Results are shown to be consistent with a hemihydrate of malachite, Gu2(0H)2(G03) I/2H2O, or azurite, GU3(0H)2(G03)2. 

Consistent Data-Recording Procedures. Clear procedures for recording all pertinent data from the experiment must be developed and documented, and unambiguous data recording forms estabUshed. These should include provisions not only for recording the values of the measured responses and the desired experimental conditions, but also the conditions that resulted, if these differ from those plaimed. It is generally preferable to use the values of the actual conditions in the statistical analysis of the experimental results. For example, if a test was supposed to have been conducted at 150°C but was mn at 148.3°C, the actual temperature would be used in the analysis. In experimentation with industrial processes, process equiUbrium should be reached before the responses are measured. This is particularly important when complex chemical reactions are involved. 

If the experiment is conducted in stages, precautions must be taken to ensure that possible differences between the stages do not invaUdate the results. Appropriate procedures to compare the stages must be included, both in the test plan and in the statistical analysis. For example, some standard test conditions, known as controls, may be included in each stage of the experiment. 

C. Lipson and N. J. Sheth, Statistica/Design andAna/ysis of Engineering Experiments, McGraw-HiU, New York, 1972. "This book is written in a relatively simple style so that a reader with a moderate knowledge of mathematics may foUow the subject matter. No prior knowledge of statistics is necessary." Appreciably more discussion is devoted to statistical analysis than to the planning of experiments. Some relatively nonstandard subjects (for an introductory text), such as accelerated experiments, fatigue experiments, and renewal analysis are also included. 

All of these tests, by their nature, need to be repeated several times with different specimens of any polymer sample, in order to ensure that there is enough information for statistical analysis. Although the physicist Lord Rutherford said that if your results need statistics, you ought to have done a better experiment, his dictum cannot be extended to tests on mechanical... 

The conclusion, based on a mixture of physical insight and statistical analysis, is that = 0.515n is close to the truth, but further experiments can be run. 

For the reasons described, no specific test will be advanced here as being superior, but Huber s model and the classical one for z = 2 and z = 3 are incorporated into program HUBER the authors are of the opinion that the best recourse is to openly declare all values and do the analysis twice, once with the presumed outliers included, and once excluded from the statistical analysis in the latter case the excluded points should nonetheless be included in tables (in parentheses) and in graphs (different symbol). Outliers should not be labeled as such solely on the basis of a fixed (statistical) rule the decision should primarily reflect scientific experience. The justification must be explicitly stated in any report cf. Sections 4.18 and 4.19. If the circumstances demand that a mle be set down, it is best to use a robust model such as Huber s its sensitivity for the problem at hand, and the typical rate for false positives, should be investigated by, for example, a Monte... 

In certain instances it will be necessary to subject data obtained firm viable counts to statistical analysis or, more sensibly, experiments should be designed so as to render them amenable to statistical treatment. 

Process validation can be done in different ways, viz. prospectively, by carrying out a planned program of experiments, before routine production is started concurrently, during routine production retrospectively, by statistical analysis of historical data and during scale-up studies (developmental validation). 

Statistical analysis. Values are given as the mean SEM. Data are represented as averages of independent experiments, performed in duplicate or triplicate. Statistical analyses were done using the Student s t-test. P < 0.05 was considered statistically significant. 

Table I is a list of physical properties of materials which were of special concern, along with target values felt to indicate useful levels in a particular application. From the beginning it was predicted that one of the biggest problems would be to balance Properties A and E, usually considered mutually exclusive. It was also assumed that Properties B and E were highly correlated. Statistically designed experiments and data analysis were chosen to determine most efficiently the formulations which would give the best combination of all the target properties.

MINITAB readily produces many useful manipulations of data such as were obtained in this experiment. Figure 2 shows histograms of the responses, indicating that, for the limited number of data points, the experimental values for each response approach a normal distribution. Thus, the statistical analysis was considered valid. Table III shows a copy of the computer printout of a correlation table with all the responses. Clearly, Property A and Property B are negatively correlated, as predicted, but Property B and Property E are not well correlated. 

For each scenario, the statistical analysis of this type of experimental design would be a two-way analysis of variance. This is predicated on the construction of the experiment, which includes some implicit assumptions. These assumptions are... 

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 Results of experiments are represented as M m, where M - means, m - standard deviation of means m= c/Vn, Where, a is the standard deviation and n - is the population size. 

Experiment II. Determination of total phenolics spectrophotometrically Material required, Procedure, Statistical analysis and Precaution are the same as described in Section 3.2. 

Statistical analysis All experiments were repeated three times. The data are analyzed using t-test for dependent variables or when have large sample or more than two combinations one-way ANOVA. 

It should be emphasized, however, that most of our relationships involve, in different combinations, only three or four of the quantities on the right of Eq. (8). To develop an expression to represent a particular property we need a database of experimental values for it. For each compound in the database we compute V(r) and all of the variables in Eq. (8). A statistical analysis package is then used to identify a subset of these to which the experi-... 

Willse, A., Belcher, A.M., Preti, G., Wahl, J.H., Thresher, M., Yang, P., Yamazaki, K. and Beauchamp, G.K. (2005) Identification of major histocompatibility complex-regulated body odorants by statistical analysis of a comparative gas chromatography/mass spectrometry experiment. Anal. Chem. 77, 2348-2361. 

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