F-test values

In the literature many racemates (40) have been tested for in vitro activity. The artemisinin-like enantiomer of each pair was included in the n = 199 database under the assumption that this was the active isomer. In an attempt to better understand the contribution of each enantiomer, these compounds were removed from the dataset, and a third model (n — 202 - 2 high residual compounds - 40 racemates = 160) was developed. The notable predictivity of the model was indicated by a high q2 (0.89) value. Judging by the standard error, r2, and F test values (0.55,0.89, and 251.11, respectively), the model reproduced the input data somewhat better than the n = 199 precursor. 

The fitting procedure is normally carried out consecutively, that is, shells are added to the fitting process as seems appropriate. Every shell should be checked for statistical significance, that is, if the addition of, normally, three parameters (o, R,N) is statistically viable. The check can be made by comparison of the change in the Fit index with F-test values, as outlined in ref. [12]. The oscillations are often very broad and with few features for oxides. In this case good statistical parameters are especially important. 

For the post-ANOVA, pair-wise evaluations, there are procedures to deal with the multiple comparison problem. One such procedure is based on the F-distribution with one and N — k degrees of freedom. This test also relies on the value of sj from ANOVA. The test statistic is F = /sj) [(xi — n + I/M2)], where x.i, x.2 are the means of the n and 2 values for the two lots in the pair-wise comparison. Comparing lot A and lot C F = (1/6.575) [(99.5 - 90.5)2/(1/20 + 1/20)] = 123.2. This far exceeds the critical Fi 75 value at even a 1% level, which is <7.08, based on Fi go, and we therefore reject the hypothesis that lot A and lot C means are equal. Because, the means for lot B and lot D differ from that of lot C by an even greater amoimt, they also are foimd to be statistically different from the lot C mean. By contrast, the comparison of lots A and D, with means of 99.5%i and 100.3%), respectively, have an F-test value of 0.97, far less than the critical 5%o value, which is <4.00. 

Table 3.4 F-Test values for comparison of variance of samples with equal degrees of freedom,...

Each independent variable in the regression model is then re-examined to see if it is still making a significant contribution. Any variable whose partial F-test value is not significant at the 10 level is dropped from the model. This process continues until no more variables enter the model and none are rejected. 

Once the F-test value has been calculated it can be compared with standard tabulated values, using some pre-specified level of significance to check whether it lies in the critical region. If it does not, then there is no evidence to suggest that the samples arise from different sources and the hypothesis that all the values are similar can be accepted. From statistical tables, /o.oi,5,24 = 3.90, and since the experimental value of 1.69 does not exceed this then the result is not significant at the 1 % level and we can accept the hypothesis that there is no difference between the six sets of sub-samples. 

Pumps should be cahbrated with a rotameter [27] prior to and after sampling. Analytical instruments must also be calibrated before measurements. For example, GC/MS must be cahbrated for mass and retention times using reference standard materials [70] and comparison made with the fragmentation patterns of known standards, usually a deuturated compound like toluene-dg. Similarly, the method detection limit must be determined by finding the standard deviation of seven replicate analyses and multiplying it by the f-test value for 99% confidence of seven values [30,62]. It is also usual for internal standards to be added to the samples and to evaluate the correlation coefficients of each standard used when multilevel calibration is employed. For automatic thermal desorption tubes, external and internal standardisations are achieved by injecting solutions of standards into the tubes [27,28] for canisters, solutions of standards are injected into the canisters followed by zero air. 

The correlation coefficient r2 of 0.90 and a standard deviation of residuals of 0.479, together vith a F-test value of F (3,54,0.05) -154.56 seem to indicate that the equation is highly significant. 

What this equation shows is that the F-test and AIC are not independent and that given one of them the other can be determined. These equations also show that sometimes the two criteria can lead to different conclusions. Suppose a modeler fit an Emax model with two estimable parameters and a sigmoid Emax model with three estimable parameters to a data set with 14 observations, such as might be the case when fitting a pharmacodynamic model to individual data. In this case, an F-test greater than 3.84 is required to declare the sigmoid Emax model the superior model at the 0.05 level, which is equivalent to a AAIC of —2.19. An F-test value less than 3.84 is considered to be not statistically significant at the 0.05 level and the reduced model is chosen as the superior model. However, any AAIC less than 0, even values between 0 and —2.19, is still considered to be indicative that the full model is the superior model. Hence, the possibility exists for the two criteria to reach different conclusions. 

The F-test value is extremely high and now 98% of the variation of the biological activities for the 15 compounds in the series can be explained on the basis of the receptor... 

Table 4.8 F-test values— values in bold are significant at the 95% level...

The mean values obtained by the two analysts are compared using a two-tailed f-test. The null and alternative hypotheses are... 

The value of fexp is then compared with a critical value, f(a, v), which is determined by the chosen significance level, a, the degrees of freedom for the sample, V, and whether the significance test is one-tailed or two-tailed. For paired data, the degrees of freedom is - 1. If fexp is greater than f(a, v), then the null hypothesis is rejected and the alternative hypothesis is accepted. If fexp is less than or equal to f(a, v), then the null hypothesis is retained, and a significant difference has not been demonstrated at the stated significance level. This is known as the paired f-test. 

A statistical analysis allows us to determine whether our results are significantly different from known values, or from values obtained by other analysts, by other methods of analysis, or for other samples. A f-test is used to compare mean values, and an F-test to compare precisions. Comparisons between two sets of data require an initial evaluation of whether the data... 

Once a significant difference has been demonstrated by an analysis of variance, a modified version of the f-test, known as Fisher s least significant difference, can be used to determine which analyst or analysts are responsible for the difference. The test statistic for comparing the mean values Xj and X2 is the f-test described in Chapter 4, except that Spool is replaced by the square root of the within-sample variance obtained from an analysis of variance. 

The same conclusion can be drawn from another statistical test for model comparison namely, through the use of Aikake s information criteria (AIC) calculations. This is often preferred, especially for automated data fitting, since it is more simple than F tests and can be used with a wider variety of models. In this test, the data is fit to the various models and the SSq determined. The AIC value is then calculated with the following formula... 

Student s f-test. This is a test1 used for small samples its purpose is to compare the mean from a sample with some standard value and to express some level of confidence in the significance of the comparison. It is also used to test the difference between the means of two sets of data x, and x2. 

F-test 140, (T) 841 Faraday constant 60, 504 Faraday s laws 503, 504 Fast sulphon black F 319 Fats D. of saponification value, (ti) 308 Ferric alum indicator see Ammonium iron(III) sulphate... 

Let s compare these plots of the REV s to the plot in Figure 52. Notice that these REV s do not exhibit ideal behavior. Ideally, as rank increases, the REV s would drop to some minimum value and then remain at that level. These REV s begin to tail back up. This sort of non-ideal behavior is not uncommon when working with actual data. Unfortunately, it can complicate matters when we use the 2-way F-test to see which REV s represent basis vectors and which ones represent noise vectors. 

Let s see what the 2-way F-test tells us. Table 9 contains the REV s and F ratios for the data in the two training sets A1 and A2. The details of calculating the F ratios, and determining the values for the numerator and denominator are discussed in Appendix D. 

The process of calculation becomes more complicated on adding further terms. Coats and Redfem [555] effectively put (U-2)/U equal to a constant value and the relationship is equivalent to that already given for In g i/T2 from the single term expansion. They assumed that f(q) = (1 — q)" and determined n by testing values which have significance in solid state decomposition reactions (i.e. n = 0, 0.5, 0.67 and 1.00). Sharp [75,556] has shown that the approach may be applied to other functions of g(q). If it is assumed that the zero-order equation applied at low a, as q -> 0, then g(q) == a. 

Sei f-Test 6.4A A small piece of calcium carbonate was placed in the same calorimeter, and 0.100 L of dilute hydrochloric acid was poured over it. The temperature of the calorimeter rose by 3.57°C. What is the value of AU ... 

Alternatively, the experimental error can be given a particular value for each reaction of the series, or for each temperature, based on statistical evaluation of the respective kinetic experiment. The rate constants are then taken with different weights in further calculations (205,206). Although this procedure seems to be more exact and more profoundly based, it cannot be quite generally recommended. It should first be statistically proven by the F test (204) that the standard errors in fact differ because of the small number of measurements, it can seldom be done on a significant level. In addition, all reactions of the series are a priori of the same importance, and it is always a... 

Does the found standard deviation, Sx, correspond to expectations The expected value E sx) is a (Greek sigma), again either a theoretical value or an experimental average. This question is answered by the F-test explained in Section 1.7.1. Proving Sx to be different from a is not easily accomplished, especially if n is small. 

For standard deviations, an analogous confidence interval CI(.9jr) can be derived via the F-test. In contrast to Cl(Xmean), ClCij ) is not symmetrical around the most probable value because by definition can only be positive. The concept is as follows an upper limit, on is sought that has the quality of a very precise measurement, that is, its uncertainty must be very small and therefore its number of degrees of freedom / must be very large. The same logic applies to the lower limit. s/ ... 

ANOVA) if the standard deviations are indistinguishable, an ANOVA test can be carried out (simple ANOVA, one parameter additivity model) to detect the presence of significant differences in data set means. The interpretation of the F-test is given (the critical F-value for p = 0.05, one-sided test, is calculated using the algorithm from Section 5.1.3). 

The F-test indicates the dependence of die dependent variables widi the independent variables, P level indicates the statistical significance of the correlafion(Table 4). The F-test results for the relation of the amount of ortho methylol phenols with F/P molar ratio and the reaction temperature were low, however, for the OH/P wt %, the F-test result was very significant, indicating a clear dependence of ortho methylol phenols on the OH/P wt %. It can also be seen that P level values for the relation between the amount of ortho methylol phenols and both F/P molar ratio and reaction temperature are above the set P value of 0.05, while for the OH/P wt%, the P value is under the set value. This data indicated that the relations of dependent variables ortho methylol phenols with independent variable OH/P wt% is statistically significant at the 0.05 significmitx level, while tiie relation of dependent variables ortho methylol phenols with F/P molar ratio and reaction temperature are not statistically significant. 

F-test results and P level values for all variables of 10 PF resins. 

Elkins, M.P. and Rabat, H.F. "Drug Induced Modifications of Laboratory Test Values". Amer. J. Hosp. Pharm. (1968),... 

In Table 31.9 we represent the results of Malinowski s F-test as computed from the eigenvalues of the transformed retention times in Table 31.2. The second eigenvector produces an F-value which still exceeds the critical F-statistic (6.6 with 1 and 5 degrees of freedom) at the 0.05 level of probability. Hence, from this evidence we conclude again that there are two, possibly three, structural eigenvectors in this data set. 

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