Significance of the regression

The F-test for the significance of the factor effects is usually called the test for the significance of the regression. 

It has been suggested that it is unfair to judge the effectiveness of the factors (i.e., the significance of the regression) on the basis of s /s] one of the components of is and the factor effects should not be asked (or expected) to account for imprecision. An alternative F-test might be s f. Comment. 

Sometimes the GOF test is called test of significance of the regression . 

The significance of the regression can also be tested by determining whether the slope is significantly different from zero by means of... 

The problem of the significance of the regression coefficients can be examined only if the statistical data take into consideration the following conditions [5.19] ... 

In statistics, the reproducibility variance is a random variable having a number of degrees of freedom equal to u = N(m — 1). Without the reproducibility variances or any other equivalent variance, we cannot estimate the significance of the regression coefficients. It is important to remember that, for the calculation of this variance, we need to have new statistical data or, more precisely, statistical data not used in the procedures of the identification of the coefficients. This requirement explains the division of the statistical data of Fig. 5.3 into two parts one sigmficant part for the identification of the coefficients and one small part for the reproducibility variance calculation. 

Table 5.18 contains the calculation concerning the significance of the regression coefficients from relation (5.110). However, respect to table 5.6, the rejection condition of the hypothesis has been changed so that we can compare the computed t value (tj) with the t value corresponding to the accepted significance level (t /2)-... 

The correlation coefficient r is a measure of quality of fit of the model. It constitutes the variance in the data. In an ideal situation one would want the correlation coefficient to be equal to or approach 1, but in reality because of the complexity of biological data, any value above 0.90 is adequate. The standard deviation is an absolute measure of the quality of fit. Ideally s should approach zero, but in experimental situations, this is not so. It should be small but it cannot have a value lower than the standard deviation of the experimental data. The magnitude of s may be attributed to some experimental error in the data as well as imperfections in the biological model. A larger data set and a smaller number of variables generally lead to lower values of s. The F value is often used as a measure of the level of statistical significance of the regression model. It is defined as denoted in Equation 1.27. 

The significance of the Regression Line is estimated by comparing its mean square with the residual in the usual way. Here it lies between the 20% and 5% levels. 

An alternative method is described by backward elimination. This technique starts with a full equation containing every measured variate and successively deletes one variable at each step. The variables are dropped from the equation on the basis of testing the significance of the regression coefficients, i.e. for each variable is the coefficient zero The F-statistic is referred to as the computed F-to-remove. The procedure is terminated when all variables remaining in the model are considered significant. 

Under the null hypothesis, F is distributed as an F-distribution with p, n-p degrees of freedom. If F > FP n P a the null hypothesis is rejected. This is called the analysis of variance approach to regression. The power of this approach comes in when multiple covariates are available (see Multiple Linear Regression later in the chapter). The F-test then becomes an overall test of the significance of the regression model. 

In similar fashion, a QSAR can be developed for the boiling point, although the mathematical relation is more complicated, perhaps a logarithmic form. In both cases, however, one may proceed to use the mathematical equations to interpolate and extrapolate or to attempt to invert the process and determine the molecular structure that corresponds to a given property value. It is clearly seen here that the physical significance of the regression correlate, number of carbon atoms, is molecular structure. That is, what is encoded in the number of carbon atoms adequately expresses what we know about the normal alkanes in terms of molecular structure for this particular data set. 

F test for statistical significance of the regression—MS i/MS, F test for lack of fit=MSLf/MSp5... 

It is clear that it is the former case that applies here, as F = 17.45. Note that in the absence of repeated experiments the significance of the regression is calculated by means of the test F = MSg Q /MS/ siD but the validity of the model may not be tested. 

As elsewhere, it is generally better to find a compromise between the two solutions. A possible approach, allowing estimation of the experimental error, the significance of the regression and testing of the model s goodness of fit would be ... 

The p descriptor variables were chosen without recourse to using the information in the response variable i.e., they were chosen in an unsupervised manner. This point is crucial as it allows us to use the usual E-test and corresponding p-values to assess the significance of the regression. There will be much more on this later in the chapter. 

---