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A Methodology for the Health Sciences - part 6

Hệ số tương quan nhiều như vậy, đã liên kết với nó cùng một mức độ tự do như sự phân bố F: l và n - k - 1. Thử nghiệm ý nghĩa thống kê cho R 2 được dựa trên các kiểm tra ý nghĩa thống kê của F-thống kê hồi quy. Ở mức ý nghĩa α, bác bỏ giả thuyết của sự kết hợp tuyến tính không giữa Y và X | 438 ASSOCIATION AND PREDICTION MULTIPLE REGRESSION ANALYSIS The multiple correlation coefficient thus has associated with it the same degrees of freedom as the F distribution k and n k 1. Statistical significance testing for R2 is based on the statistical significance test of the F-statistic of regression. At significance level a reject the null hypothesis of the no linear association between Y and X1 . Xk if r2 kFk n k 1 1 a kFk n k 1 1 a n k 1 where Fk n k 1 1 a is the 1 a percentile for the F-distribution with k and n k 1 degrees of freedom. For any of the examples considered above it is easy to compute R2. Consider the last part of Example 11.3 the active female exercise test data where duration VO2 MAX and the maximal heart rate were used to explain the subject s age. The value for R2 is given by 2256.97 4399.16 0.51 that is 51 of the variability in Y age is explained by the three explanatory or predictor variables. The multiple regression coefficient or positive square root is 0.72. The multiple regression coefficient has the same limitations as the simple correlation coefficient. In particular if the explanatory variables take values picked by an experimenter and the variability about the regression line is constant the value of R2 may be increased by taking a large spread among the explanatory variables X1 . Xk. The value for R2 or R may be presented when the data do not come from a multivariate sample in this case it is an indicator of the amount of the variability in the dependent variable explained by the covariates. It is then necessary to remember that the values do not reflect something inherent in the relationship between the dependent and independent variables but rather reflect a quantity that is subject to change according to the value selection for the independent or explanatory variables. Example 11.4. Gardner 1973 considered using environmental factors to explain and predict mortality. He studied the relationship between a number of .