Correlation Coefficient is Significant

When the correlation coefficient calculated, the next step in determining whether a relationship exists between the two variables is to determine if the value for the correlation coefficient is significantly different from zero (the no relationship value). To test whether the correlation coefficient is significantly different from zero, a t-test is performed. The following formula is used to calculate t  

To perform a t-test on a correlation coefficient, one must determine the probability of being correct when concluding that the correlation coefficient is significant. This degree of chance or probability is referred to as the alpha level. A commonly used 

The next step is to determine if the obtained t-test value of 8.00 is significant. This is done by using a t-table which can be found in any basic statistics book. To use the t-table, the degrees of freedom are calculated. In this case, the degrees of freedom are 10(12-2= 10). With the t-table, locate the column for a =. 05 and degrees of freedom (d.f.) = 10. The table for a two-tailed test indicates a value of 2.23. Since our obtained t-test value of 8.00 is greater than the table value, we can conclude that there is a significant correlation between the number of traffic accidents and the number of trips to Big City, USA. 

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Whereas the mean correlation coefficient is significantly lower in the arbitrary data set, the mean skewness and mean kurtosis are similar. Though the latter values do not indicate clearly a difference between the data sets — they just indicate a similar symmetry and flatness of distribution — the deviations from the average behavior describes properly the diversity of the data set The average deviations in skewness and kurtosis are about twice as high in the arbitrary data set as those of the benzene derivatives. The ASD and the combination of deviations in correlation coefficients, skewness, and kurtosis provide the most reliable measure for similarity and diversity of data sets. 

The binding constants of a number of compounds were measured using dialysis, solubility and sorption techniques. The solubility technique was used for compounds which were not radiolabeled. All data was collected at pH = 8.3. The binding constants were then compared to the octanol/water partition coefficients for the compounds and the molar solubilities of the compounds. The data is presented in Table II. The Kow values were taken from the literature.18 22-2 The solubility values were determined in this research with the exception of DDT and Lindane, which were taken from the literature. A plot of log Kc vs. log Kow is presented in Figure 5. The slope of this line is 0.71, the intercept is 0.75 and the value of the correlation coefficient is 0.9258. The regression is highly significant... 

The following description and corresponding MathCad Worksheet allows the user to test if two correlation coefficients are significantly different based on the number of sample pairs (N) used to compute each correlation. For the Worksheet, the user enters the confidence level for the test (e.g., 0.95), two comparative correlation coefficients, r, and r2, and the respective number of paired (X, Y) samples as N and N2. The desired confidence level is entered and the corresponding z statistic and hypothesis test is performed. A Test result of 0 indicates a significant difference between the correlation coefficients a Test result of 1 indicates no significant difference in the correlation coefficients at the selected confidence level. 

Since the correlation coefficient is an already-existing and known statistical function, why is there a need to create a new calculation for the purpose of assessing nonlinearity First, the correlation coefficient s roots in Statistics direct the mind to the random aspects of the data that it is normally used for. In contrast, therefore, using the ratio of the sum of squares helps keep us reminded that we are dealing with a systematic effect whose magnitude we are trying to measure, rather than a random effect for which we want to ascertain statistical significance. 

The value of r is zero when there is no correlation and reaches a value of 1 when there is perfect correlation. The sign of r is the same as that of the slope b and has no significance in assessing the value of the fit. Further interpretation of the correlation coefficient is not straightforward without more detailed knowledge of the statistical properties of the random error in the data sample considered. In an approximate way, the value of gives the percent of the observed variation in y with x that has been explained by the correlation obtained. 

The regression coefficient of NBBrol in the correlation equation is significantly larger than the regression coefficient of NSGrot. 

Carbon-carbon double bonds (C=C) in a trans configuration along the chain backbone cause a significant reduction of a. This reduction is accounted for by the parameter trans. The correlation coefficient between o and ans for the entire dataset is only -0.4212. This correlation coefficient is small because only three of the polymers in the dataset have C=C bonds in a tram configuration. 2 trans KS nonetheless, an important parameter. Its use in the linear regression systematically corrects for most of the large difference between the o values of the same polymer with tram versus cis isomerization for C=C bonds in the chain backbone. 

Another quick and dirty test for heteroscedasticity suggested by Carroll and Ruppert (1988) is to compute the Spearman rank correlation coefficient between absolute studentized residuals and predicted values. If Spearman s correlation coefficient is statistically significant, this is indicative of increasing variance, but in no manner should the degree of correlation be taken as a measure of the degree of heteroscedasticity. They also suggest that a further refinement to any residual plot would be to overlay a nonparametric smoothed curve, such as a LOESS or kernel fit. 

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