Statistical test Fisher

The practical consequence from this is that in the study type under consideration, always the dam/litter rather than the individual fetus is the basic statistical unit (see Chapters 23, 33, 34 and 35). Six malformed fetuses from six different litters in a treated group of dams is much more likely to constitute a teratogenic effect of the test substance than ten malformed fetuses all from the same litter. It is, therefore, important to report all fetal observations in this context and to select appropriate statistical tests (e.g., Fisher s exact test with Bonferroni correction) based on litter frequency. For continuous data, a procedure to calculate the mean value over the litter means (e.g., ANOVA followed by Dunnet s test) is preferred. An increase in variance (e.g., standard deviation), even without a change in the mean, may indicate that some animals were more susceptible than others, and may indicate the onset of a critical effect. 

The primary statistical tests used in the studies described in this text are based on the chi-square tests which are in turn derived from the chi-square distribution which is based on the chi distribution. These tests include the chi-square test for goodness of fit, the chi-square test of independence, and Fisher s Exact Test. There are also corrections to some of the tests that account for small number deviations, Yates Correction for Continuity, and for multiple studies attempting to verify the same procedures or processes, Bonferroni s correction. 

There is rather significant scattering of data and some of the eight experiments may be regarded as outliers. Therefore we have tested the null hypothesis H0 of equality of the lowest mean of corrosion rate in experiment No 1 and the highest mean in experiment No 8. The calculated value F=s12/s82=5.92/5.02=1.4 for standard deviations was compared with Fisher distribution statistical test values... 

Statistical theory must also be held not only with respect but also with healthy skepticism. It should be remembered that the development of statistics, as they have come to be applied to clinical trials, has arisen from a variety of nonmammalian biological sources. Experimental agriculture stimulated the early giants (Drs. Fisher and Yates) to explore probability density functions. While epidemiological studies have confirmed much that is similar in human populations, it is unknown whether these probability density functions apply uniformly to all disease states. Any statistical test that we employ makes assumptions that are usually not stated. 

This analysis allows us to split the variability observed for B into contributions due to different factors. The probability (p-value) provides a measure of the statistical significance (at a confidence level of 95%) of each factor. Overall at least one of the factors has had a significant effect (p = 0.0001) on the measured level of B. This is in a good agreement with previous observations. Multiple range tests (Fisher s least significant difference (LSD)) was performed to determine which of the treatment means were significantly different from each other, and the results are summarized in Table 4.5.5. 

The po-wer and Type II error rate (beta) Power and its related Type II error rate are probably the most neglected aspects of a statistical test. Power refers to the probability that a relationship in the population will be detected when one in fact exists. Therefore, power might reflect what R. A. Fisher called the "sensitivity" of an experiment (Fisher, 1942). Importantly, the power of an obtained statistical test reflects the probability that such a result can be replicated (Goodman, 1992). The effects of low statistical power on the reproducibility of research findings has been well documented (Goodman, 1992 Harris, 1997). The Type II error (beta) occurs when one fails to reject a felse null hypothesis. 

Thus, in order to apply the t-test, we must first test for differences between the two variances Fisher s F-test is used for this purpose. Again we shall omit all reference to the derivation of this test statistic and the rationale for statistical tests based on it. Instead we focus here on its application to testing two data sets to determine the confidence level at which we can assert that the variances i and Vx,2 (Equation [8.2b]) of the two are indistinguishable (the null hypothesis F(, for the test). The value of the experimental test statistic F is calculated simply as ... 

The quality of the fitted polynomial model was expressed by the determination coefficient (R ) and its statistical significance was performed with the Fisher s statistical test for analysis of variance (ANOVA). 

In such a contingency table we would typically be interested in whether there were differences in the probabilities or rates of failures which result in the observed numbers of failures rii j,i,j =1,2,3 between the different locations or systems. Thereare two main statistical tests which could be used to do this the Chi-Squared Test and Fisher s Exact Test (Pearson 1900, Fisher 1922). 

Fisher statistic, Fisher value ratio of variances for two models to be compared. It can be overall or partial F value. The overall Fisher statistic tests the entire equation, whether all coefficients are significant in the model. The partial F value is used to test whether the incriminated variable is significant in the model. 

The statistical significance of the experimental results was determined by the Student s t-test (Fisher and Yates, 1943). For all analysis, p<0.05 was accepted as a significant probability level. 

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. 

This homogeneity value will become positive when the null hypothesis is not rejected by Fisher s E-test (F < Ei a)Vl)V2) and will be closer to 1 the more homogeneous the material is. If inhomogeneity is statistically proved by the test statistic F > Fi a>VuV2, the homogeneity value becomes negative. In the limiting case F = hom(A) becomes zero. 

Statistical analysis For statistical analysis of the behavioral tests an analysis of variance (two-way ANOVA) was used. For the symptomatology a Fisher exact probability test or an unpaired t-test with Welch s correction was used. In all tests p values <0.05 were considered significant. 

A statistically significant increase in hepatocellular carcinomas was seen in male and female mice that were dosed with 590 and 1,179 mg/kg/day hexachloroethane in com oil by gavage for 78 continuous weeks (Weisburger 1977). The incidence of tumors in the exposed mice was greater than that in controls on the basis of both the Fisher Exact test and the Cochran-Armitage test. There were no hepatic tumors in male or female rats with chronic exposure to doses of 10-423 mg/kg/day (NTP 1977, 1989 Weisburger 1977). 

The test material, test cells, used, method of treatment, harvesting of cells, cytotoxicity assay, and so on, should be clearly stated as well as the statistical methods used. Richardson et al. (1989) recommend that comparison be made between the frequencies in control cells and at each dose level using Fisher s Exact Test. 

Mendal et al. (1993) compared eight tests of normality to detect a mixture consisting of two normally distributed components with different means but equal variances. Fisher s skewness statistic was preferable when one component comprised less than 15% of the total distribution. When the two components comprised more nearly equal proportions (35-65%) of the total distribution, the Engelman and Hartigan test (1969) was preferable. For other mixing proportions, the maximum likelihood ratio test was best. Thus, the maximum likelihood ratio test appears to perform very well, with only small loss from optimality, even when it is not the best procedure. 

In this equation, n denotes the number of observations (i.e., data points), r is the correlation coefficient, i is the standard deviation of the residuals, F is the value of the Fisher test for the significance of the equation, and the P-value denotes the statistical significance level. The values in parenthesis of this QSPR equation indicate the 95% confidence inferval of fhe estimated coefficients. From the statistical parameters it can be derived that this QSPR model is significant at the 99% confidence level. 

Statistical Analysis. Analysis of variance (ANOVA) of toxicity data was conducted using SAS/STAT software (version 8.2 SAS Institute, Cary, NC). All toxicity data were transformed (square root, log, or rank) before ANOVA. Comparisons among multiple treatment means were made by Fisher s LSD procedure, and differences between individual treatments and controls were determined by one-tailed Dunnett s or Wilcoxon tests. Statements of statistical significance refer to a probability of type 1 error of 5% or less (p s 0.05). Median lethal concentrations (LCjq) were determined by the Trimmed Spearman-Karber method using TOXSTAT software (version 3.5 Lincoln Software Associates, Bisbee, AZ). 

For statistical analysis, fetal abnormality values belong to two types those where at least 50% of litters have one or more fetuses affected, and those where most litters have no affected fetuses. For the first type, the incidences (percentage of affected fetuses within that litter) are analyzed by the Kruskal-Wallis test (13) for the second type, the number of litters with affected fetuses is compared with the number with no affected fetuses by Fisher s Exact test (14). 

Instead of the imivariate Fisher ratio, SLDA considers the ratio between the generalized within-category dispersion (the determinant of the pooled within-category covariance matrix) and total dispersion (the determinant of the generalized covariance matrix). This ratio is called Wilks lambda, and the smaller it is, the better the separation between categories. The selected variable is that that produces the maximum decrease of Wilks lambda, tested by a suitable F statistic for the input of a new variable or for the deletion of a previously selected one. 

Cancer. Increased incidences of relatively rare renal tubular cell adenomas and carcinomas were observed in male rats, but the increases were not statistically significant by the Fisher Exact test or the Cochran-Armitage test (NTP 1986). When adjusted for mortality, however, the increased incidences were significantly different from control in the high-dose males when analyzed by the Lifetable test and significant for dose-related trend by the Lifetable and the Incidental Tumor tests. 

Given two sets of repeated measurements, the question of whether the data come from populations having equal variances might arise. This is tested by calculating the Fisher F statistic, which is defined as... 

The hypergeometric distribution can be generalized to a multivariable form, the multivariate hypergeometric distribution, which can be used to extend Fisher s Exact Test to contingency tables larger than 2 by 2 and to multidimensional contingency tables. There is statistical software available to perform these calculations however, due to the complexity of the calculations and the large number of trial tables whose probability of occurrence must be calculated, this extension has received limited use. 

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