Bonferroni corrections

In a next step, we correlated oestrogen levels with the movement parameters. The Bonferroni corrected correlations show that only females confronted with a male stimulus change their behaviour quality. Single females make slower and more complex movements. Both single and paired females show more information per time in their movements with increasing oestrogen levels (Table 3). 

Fig. 7 Transcriptome effects of T3 administration on the developing zebrafish embryo, (a) Heatmap of microarray results from 652 probes showing significant differences between T3-treated and control samples. Fold induction values (in log scale) are represented by different shades of color (scale shown in the far-right bar.), (b) Distribution of overrepresented (red), underrepresented (blue), and unchanged/undetected (ivory) transcripts in T3-treated embryos belonging to the functional categories ossification, visual processes, and oxygen transport. The significance of the observed variations (p values) was calculated by the hypergeometric distribution with the Bonferroni correction...

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. 

Note that performing multiple hypothesis tests (e.g., Student s t-test) may inflate the false positive rate. That is, the number of genes detected as active by chance alone will increase with the number of genes tested. For example, a microarray with 7000 features would require at least 7000 hypothesis tests per treatment comparison. Several methods have been developed to control the false positive rate, such as the conservative Bonferroni correction and the FDR control method (49). 

Bonferroni keeps the rate of type I errors down to 5 percent... In many instances, omnibus tests are not possible. If there are several comparisons to be made, each involving different groups of individuals and different endpoints, there may be no choice but to use a series of discrete statistical tests. In these circumstances, the Bonferroni correction may be applied to maintain the overall risk of false positives at the standard level of 5 per cent. 

What the Bonferroni correction does is to raise the standard of proof for all the individual tests. Each test is then less likely to produce a false positive and the complete series of analyses will jointly generate a 5 per cent risk. 

Bonferroni correction - there is no such thing as a free lunch... 

Data should not be broken down into multiple sub-groups unless either a Bonferroni correction or a distinction between primary and secondary analyses is used to provide additional protection. 

The Bonferroni correction raises the standard of proof required for each individual test. This has the effect of maintaining the overall risk of any false positives at 5 per cent, but reduces statistical power. 

One conservative approach to controlling Type I error rate is to use Bonferroni-corrected a-levels. Under this paradigm, reject the null hypothesis if the observed p-value is less than a/k, where k is the kth hypothesis tested. Clearly, however, for many hypothesis tests, it would be nearly impossible to find a statistically significant model. Therefore, most modelers fix a at some small value, usually 0.01 or 0.001, in the hopes of reducing the Type I error rate. 

The changes of observed parameters were evaluated using repeated-measures ANOVA with Bonferronis corrected J>ost hoc r-test for multiple comparisons of dependent variables. Pearsons correlation coefficient and test were used for the evaluation of dependence of quantitative variables, -values less than 0.05 were considered significant. All statistic analyses were performed using the software package GraphPad Prism 4.0. 

Where many outcomes are being compared, Bonferroni corrections should be performed... 

Figure 10.1 Power of individual tests given the Bonferroni correction, as a function of w, the...

Second, the situation may nevertheless arise where rather many tests will be significant at level a but none will be significant at level a/n. We may then formally be unable to declare the treatment as significant while strongly believing that this is so. (This suggests that in practice we do not necessarily come to conclusions about the efficacy of treatments by looking at results individually.) Third, the Bonferroni correction is rather pessimistic and will be conservative where, as may usually be expected to be the case, clinical outcomes are positively correlated. 

Now, if we apply the Bonferroni correction, the power is approximately given by... 

A.2 The power of at least one test being significant when Bonferroni corrections are employed... 

Table F presents corrected jackknife residual values, which essentially are Bonferroni corrections on the jackknife residuals. For example, let a = 0.05, k = the number of bi values in the model, excluding bo say = 1 and n = 20. In this case. Table F shows that a jackknife residual greater than 3.54 in absolute value, r(, ) > 3.54, would be considered an outlier.

Bonferroni Corrected Jackknife Residual Critical Values... 

TABLE G (continued) Bonferroni Corrected Jackknife Residual Critical Values Level of Significance a = 0.05 ... 

TABLE K Cook s Distance Table Critical Values for the Maximum of n Values of Cook s d(i) x (n - k - 1) (Bonferroni Correction Used) Observations and k Predictors Level of Significance o = 0.1 n... 

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