Coefficients of determination and correlation

If the factors have very little effect on the response, we would expect that the sum of squares removed from by would be small, and therefore SS, would be 

ANOVA table for linear models containing a Pq term. 

Source Degrees of freedom Sum of squares Mean square 

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Although the coefficients of determination and the correlation coefficients are conceptually simple and attractive, and are frequently used as a measure of how well a model fits a set of data, they are not, by themselves, a good measure of the effectiveness of the factors as they appear in the model, primarily because they do not take into account the degrees of freedom. Thus, the value of R can usually be increased by adding another parameter to the model (until p =J), but this increased R value does not necessarily mean that the expanded model offers a significantly better fit. It should also be noted that the coefficient of determination gives no indication of whether the lack of perfect prediction is caused by an inadequate model or by purely experimental uncertainty. 

Calculate the coefficient of determination, r2, and the coefficient of correlation, r, for the model and data of Exercise 9.6. What is the difference between the coefficient of determination and the coefficient of multiple determination ... 

According to all things said for the coefficient of determination, the correlation coefficient itself is a measure of the strength of relationship and it takes values between -1 and +1. When the correlation coefficient nears one the linear relationship between variables is strong, and when it is close to zero it means that there is no linear relationship between variables. This, however, does not mean that there is no relationship between variables, which might even be strong, of a certain curved shape. We point out that the correlation coefficient is an indefinite number, i.e. it does not depend on the units the variables have been expressed in. 

How well the estimated straight line fits the experimental data can be assessed by determining the coefficient of determination and the correlation coefficient. 

Program PROG 13 calculates (i) the coefficients for each degree of the polynomial, (ii) the variance, (iii) error sum of squares, (iv) total sum of squares, (v) coefficient of determination, and (vi) the correlation coefficient. The program shows that the fourth degree gives the lowest value of the variance, and therefore shows the best fit. The results are ... 

When there is more than one observation (whether it be replicate values or more than one reportable value over a number of assays) at each level of the analyte, then a lack-of-fit analysis can be conducted. This analysis tests whether the average response at each level of the analyte is a significantly better model for average assay response than the linear model. A significant lack-of-fit can exist even with a high correlation coefficient (or high coefficient of determination) and the maximum deviation of response from the predicted value of the line should be assessed for practical significance. 

According to the rules of statistical analysis, the fraction of the variation of a property accounted for by a correlation is equal to the square of the correlation coefficient. The square of the correlation coefficient is hence sometimes referred to as the "coefficient of determination". A correlation coefficient of 0.9957 indicates that Equation 3.9 accounts for approximately 99.15% of the variation of the Vw values. The addition of extra terms proportional to °%v and... 

Three goodness of fit metrics bear particular attention the coefficient of determination, the correlation coefficient, and the concordance coefficient. The coefficient of determination (R2) is simply... 

If the regression parameters are estimated by the least squares method, the square root of the coefficient of determination and the multiple correlation coefficient will be identical. 

The multiple correlation is represented by R. It is in fact the correlation of Y with Y, where Y a linear combination of the X s. Of course, the X s may individually have correlations with Y of either sign. Hence R is arbitrarily defined as being positive. Direct practical interpretation of R is difficult. Two transformations help to improve interpretation. One is R. As with the simple model, R is the coefficient of determination and represents the fraction of SSY accounted for by the model (R = SSReg/SSY). For orthogonal predictors, R = jy For P = 2, i R > r y, + andXjrepresent a resolving pair, where R < rfy + r y, X and X2 are confounded. These relationships were shown in Table 6 and evaluated in Table 7. A second transformation is "%Sy removed. The percentage reduction in is related to R as follows ... 

A correlation procedure yielded a correlation coefficient of. 89. What is the coefficient of determination and what does it represent ... 

Khangarot and Ray (1989) developed correlation coefficients to describe the relationship between the toxicity of 11 unspecified cations and 2-day Daphnia magna immobilization EC50 values. Their correlation coefficients were converted to coefficients of determination and the statistics for their linear regression analysis were r2=0.876 and p < 0.001. 

Figure 2.2. Examples of correlations with high and low coefficients of determination. Data were simulated for combinations of various levels of noise (a = 1,5, 25, top to bottom) and sample size (n - 10, 20, 40, left to right). The residual standard deviation follows the noise level (for example, 0.9, 5.7, 24.7, from top to bottom). Note that the coefficient 0.9990 in the top left panel is on the low side for many analytical calibrations where the points so exactly fit the theoretical line that > 0.999 even for low n and small calibration ranges.

Canonical Correlation Analysis (CCA) is perhaps the oldest truly multivariate method for studying the relation between two measurement tables X and Y [5]. It generalizes the concept of squared multiple correlation or coefficient of determination, R. In Chapter 10 on multiple linear regression we found that is a measure for the linear association between a univeiriate y and a multivariate X. This R tells how much of the variance of y is explained by X = y y/yV = IlylP/llylP. Now, we extend this notion to a set of response variables collected in the multivariate data set Y. 

Here, the notation (, I C, X2) stands for the squared multiple correlation coefficient (or coefficient of determination) of the multiple regression of y, on Xj and X2. The improvement is quite modest, suggesting once more that there is only a weak (linear) relation between the two sets of data. 

Franke, U. Munk, A. Wiese, M., Ionization constants and distribution coefficients of phenothiazines and calcium channel antagonists determined by a pH-metric method and correlation with calculated partition coefficients, J. Pharm. Sci. 88, 89-95... 

Alternatively, NIR spectroscopy has been applied to relate NIR data to mechanical properties [4], A multivariate data analysis was performed on a series of commercial ethene copolymers with 1-butene and 1-octene. For the density correlation, a coefficient of determination better than 99% was obtained, whereas this was 97.7% for the flexural modulus, and only 85% for the tensile strength. 

Y as a function of a change in X. These include, but are not limited to correlation (r), the coefficient of determination (R2), the slope (, ), intercept (K0), the z-statistic, and of course the respective confidence limits for these statistical parameters. The use of graphical representation is also a powerful tool for discerning the relationships between X and Y paired data sets. 

The attached worksheet from MathCad ( 1986-2001 MathSoft Engineering Education, Inc., 101 Main Street Cambridge, MA 02142-1521) is used for computing the statistical parameters and graphics discussed in Chapters 58 through 61, in references [b-l-b-4]. It is recommended that the statistics incorporated into this series of Worksheets be used for evaluations of goodness of fit statistics such as the correlation coefficient, the coefficient of determination, the standard error of estimate and the useful range of calibration standards used in method development. If you would like this Worksheet sent to you, please request this by e-mail from the authors. 

The RGB composite of the coefficients of determination of the individual linear correlation coefficients (Figure 2.26) shows that for the northern hemisphere high correlations of volatilisation rate and wind speed in the Atlantic Ocean can be found in the Gulf Stream and low values in the Labrador Sea and the adjacent Davis Strait. High correlations with the sea surface temperature are located near 45 °N close to the eastern coast of the American continent, in the Baltic Sea, North Sea and in... 

In the northern hemisphere the coefficient of determination of the partial correlation between the pollutant concentration in the dissolved phase and the volatilisation rate excluding wind speed and SST, R2V ut, is very low in comparison to both coefficients that omit the pollutant concentration (Figure 2.27). Hence the apparently high correlation between pollutant concentration and volatilisation rate, shown as yellow to green colour in Figure 2.26 in some areas in the Pacific Ocean is not caused by a causal relation between them, but spurious. Both partial correlation coefficients are much lower there. The differences between values of R, uc andR2v ut (Figure 2.27c)... 

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