Pearson confidence intervals

The relationship between renal impairment and the absorption and disposition of HMR1964 (insulin glulisine) will be assessed by regressing pharmacokinetic parameters onto CLcr. Regression parameter estimates ( standard error) with confidence intervals and coefficients of correlation (Pearson) with p-values for test of difference from zero will be reported. Scatter plots of the concentration time profiles and pharmacokinetic parameters against creatinine clearance will be produced. 

Values of r can vary between -1 and 1, with 1 indicating a perfect linear correlation, -1 being a perfect inverse linear correlation, and 0 indicating an absence of correlation. The definition of what constitutes a good value for r is somewhat subjective and situation dependent, and could easily fill an entire chapter or even a book. As we will see in subsequent sections, the dynamic range of the data being considered can have a dramatic effect on Pearson s r. We will also see that when comparing values of Pearson s r for dilferent models, we must consider the confidence intervals around r. 

While the calculation of confidence intervals for a correlation is straightforward, it is rarely used in the cheminformatics literature. As such, we will provide a brief review of the method for calculating a confidence interval on a Pearson r. Since values of Pearson s r cannot exceed 1, its distribution is not normal. The distribution is closer to normal for lower values of r and becomes more skewed as r approaches 1. In order to calculate a confidence interval, values of r must be converted to Fisher s i distribution using Equation 1.10.1. 

For this example, let us assume we are dealing with a dataset where we are predicting the solubility of 23 molecules. According to standard tables of z values, we should use a value of 2.08 for a t test at 95% confidence for a two-tailed distribution with a sample size of 23. The value of cr, would then be 0.23. For a Pearson r of 0.7, the confidence intervals on z would be... 

We will now consider the case of two predictive models. Model A and Model B, for aqueous solubility. We will use R to compare the performance of these models when tested on 25, 50, and 100 compounds. Listing 9 provides an example of how this comparison can be performed in R. In this listing, we first calculate the Pearson r and the upper and lower 95% confidence intervals for the Pearson r. Table 1.6 and Figure 1.5 show the correlations and associated bar plots. The bar plots show the value of Pearson r for each subset and the associated whiskers show the upper and lower limits of the 95% confidence interval. 

TABLE 1.6 Pearson r, Upper and Lower Bound for 95% Confidence Intervals for Regression Models... 

Data processing ras performed using the IBM SPSS Statistics software version 22.0 for Windows operating system. The verification of research hypotheses it was used the statistical t-student test for independent samples, U of Mann-Whitney and Pearson correlation coefficient The interpretation of statistical tests ras carried out using a significance level of p-value < 0.05 with a confidence interval of 95%. 

Speed [pixels/frame] s.d. Mean of the direction [°] confidence interval Pearson s coefficient r of directionality... 

If we wish to say something about the difference which obtains, then it is better to quote a so-called point estimate of the true treatment effect, together with associated confidence limits. The point estimate (which in the simplest case would be the difference between the two sample means) gives a value of the treatment effect supported by the observed data in the absence of any other information. It does not, of course, have to obtain. The upper and lower 1 — a confidence limits define an interval of values which, were we to adopt them as the null hypothesis for the treatment effect, would not be rejected by a hypothesis test of size a. If we accept the general Neyman-Pearson framework and if we wish to claim any single value as the proven treatment effect, then it is the lower confidence limit, rather than any value used in the power calculation, which fulfills this role. (See Chapter 4.)... 

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