Coefficient Hypothesis Testing

Because the researcher undoubtedly will be faced with describing regression functions via the correlation coefficient, r, which is such a popular statistic, we develop its use further. (Note The correlation coefficient can be used to determine if r = 0, and if r = 0, then bi also equals 0.) 

This hypothesis test of r = 0 can be performed applying the six-step procedure. 

Step 3 Write out the test statistic, which is a t-test (Equation 2.45)  

If kcl haiXn-i) reject //o at a. Step 5 Perform the computation (step 3). 

Example 2.6 Using Example 2.1, the problem can be done as follows. 

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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. 

The null hypothesis test for this problem is stated as follows are two correlation coefficients rx and r2 statistically the same (i.e., rx = r2)l The alternative hypothesis is then rj r2. If the absolute value of the test statistic Z(n) is greater than the absolute value of the z-statistic, then the null hypothesis is rejected and the alternative hypothesis accepted - there is a significant difference between rx and r2. If the absolute value of Z(n) is less than the z-statistic, then the null hypothesis is accepted and the alternative hypothesis is rejected, thus there is not a significant difference between rx and r2. Let us look at a standard example again (equation 60-22). 

The canonical correlation coefficients can also be used for hypothesis testing. The most important test is a test for uncorrelatedness of the x- and y-variables. This corresponds to testing the null hypothesis that the theoretical covariance matrix between the x- and y-variables is a zero matrix (of dimension mx x mY). Under the assumption of multivariate normal distribution, the test statistic... 

Computer packages such as SAS can fit these models, provide estimates of the values of the b coefficients together with standard errors, and give p-values associated with the hypothesis tests of interest. These hypotheses will be exactly as Hqj, Hq2 and Hq3 in Section 6.3. Methods of stepwise regression are also available for the identification of a subset of the baseline variables/factors that are predictive of outcome. 

The null hypothesis tested with the F-ratio is a general hypothesis stating that the true coefficients are all zero (note that b, is not included). The / "-ratio has an F-distribution with df= m and [Pg.126]

Confidence Interval and Hypothesis Tests for Regression Coefficients... 

The significance of any calculated correlation coefficient is evaluated by adopting the usual two-sided hypothesis test. 

And Z(n) = 0.89833, therefore Z(n), the test statistic, is less than 1.96, the z-statistic, and the null hypothesis is accepted - there is not a significant difference between the correlation coefficients. 

ApA < 1. In Fig. 2 the region of curvature is much broader and extends beyond — 4 < ApA < + 4. One explanation for the poor agreement between the predictions in Fig. 3 and the behaviour observed for ionisation of acetic acid is that in the region around ApA = 0, the proton-transfer step in mechanism (8) is kinetically significant. In order to test this hypothesis and attempt to fit (9) and (10) to experimental data, it is necessary to assume values for the rate coefficients for the formation and breakdown of the hydrogen-bonded complexes in mechanism (8) and to propose a suitable relationship between the rate coefficients of the proton-transfer step and the equilibrium constant for the reaction. There are various ways in which the latter can be achieved. Experimental data for proton-transfer reactions are usually fitted quite well by the Bronsted relation (17). In (17), GB is a... 

An advantage of LR in comparison to LDA is the fact that statistical inference in the form of tests and confidence intervals for the regression parameters can be derived (compare Section 4.3). It is thus possible to test whether the /th regression coefficient bj = 0. If the hypothesis can be rejected, the jth regressor variable xj... 

The experimental results for dispersion coefficients in gases show that they can be satisfactorily represented as Peclet number expressed as a function of particle Reynolds number, and that similar correlations are obtained, irrespective of the gases used. However, it might be expected that the Schmidt number would be an important variable, but it is not possible to test this hypothesis with gases as the values of Schmidt number are all approximately the same and equal to about unity. 

The Hausman test was used to test the null hypothesis that the coefficients estimated by the efficient random-effect model are the same as the ones estimated by the consistent fixed-effect model. If this null hypothesis cannot be rejected (insignificant P-value in general, it is larger than 0.05), then the random-effect model is more appropriate. 

The coefficient c measures the impact that treatment has on pr(y= 1). If c = 0 then pr(y = I) is unaffected by which treatment group the patients are in there is no treatment effect. Having fitted this model to the data and in particular obtained an estimate of c and its standard error then we can test the hypothesis Hq c = 0 in the usual way through the signal-to-noise ratio. 

Several criteria can be used to select the best models, such as the F-test on regression, the adjusted correlation coefficient (R ad) and the PRESS [20] (Predictive error sum of squares). In general, even only adequate models show significant F values for regression, which means that the hypothesis that the independent variables have no influence on the dependent variables may not be accepted. The F value is less practical for further selection of the best model terms since it hardly makes any distinction between different predictive models. 

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