Quantifying the Relative Importance of Predictors in Multiple Linear Regression Analyses for Public Health Studies (original) (raw)

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Quantifying the relative importance of predictors in multiple linear regression analysis is crucial for understanding the influence of different factors on outcome variables. This paper examines various definitions of predictor relative importance, focusing on dispersion importance, which reflects the proportion of variance in the response variable accounted for by each predictor. Using data from public health studies, the authors present methodologies for calculating relative importance indices, comparing their effectiveness through examples and comprehensive statistical analyses.

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To cite this Article: Chao, Yi-Chun E., Zhao, Yue, Kupper, Lawrence L. and Nylander-French, Leena A. (2008) ‘Quantifying the Relative Importance of Predictors in Multiple Linear Regression Analyses for Public Health Studie: Journal of Occupational and Environmental Hygiene, 5:8, 519 — 529

To cite this Article: Chao, Yi-Chun E., Zhao, Yue, Kupper, Lawrence L. and Nylander-French, Leena A. (2008) ‘Quantifying the Relative Importance of Predictors in Multiple Linear Regression Analyses for Public Health Studie: Journal of Occupational and Environmental Hygiene, 5:8, 519 — 529

FIGURE 1. Graphic representation of Johnson’s Relative Weight for a regression model with three predictors (adapted from Refer- ence 9). The original predictors x}, x7, and x are transformed to their maximally related orthogonal counterparts, z;, z2, and 23. The regression coefficients of y* on z;, z2, and z3 are represented by By, B3, and B, respectively, and the regression coefficients of x7 on z, are represented by jx, where j=1,2,3 and k=1, 2, 3. For example, the regression coefficients of wf or x; on Z1, 22, and z3 are represented by Ait, Aiz2, and A13, respectively. Johnson’s Relative Weight for xy or x; would then be calculated gece. — )2 Bx2 172 px2 1 72- ao 7-192.

FIGURE 1. Graphic representation of Johnson’s Relative Weight for a regression model with three predictors (adapted from Refer- ence 9). The original predictors x}, x7, and x are transformed to their maximally related orthogonal counterparts, z;, z2, and 23. The regression coefficients of y* on z;, z2, and z3 are represented by By, B3, and B, respectively, and the regression coefficients of x7 on z, are represented by jx, where j=1,2,3 and k=1, 2, 3. For example, the regression coefficients of wf or x; on Z1, 22, and z3 are represented by Ait, Aiz2, and A13, respectively. Johnson’s Relative Weight for xy or x; would then be calculated gece. — )2 Bx2 172 px2 1 72- ao 7-192.

Note: Example 1, Styrene exposure level as the outcome variable, n = 214. AThe value of the relative importance index estimated from the original data set. ® The rank is determined by comparing the estimated relative importance values. The rank of | indicates the most important predictor, the rank of 2 indicates the second most important predictor, etc. TABLE I. Measures of Relative Importance for Example 1

Note: Example 1, Styrene exposure level as the outcome variable, n = 214. AThe value of the relative importance index estimated from the original data set. ® The rank is determined by comparing the estimated relative importance values. The rank of | indicates the most important predictor, the rank of 2 indicates the second most important predictor, etc. TABLE I. Measures of Relative Importance for Example 1

Note: Example 1, 1500 bootstrap replications. AThe single value above the 95% Cl is the value of the relative importance index estimated from the original data set 8 The 95% bootstrap confidence interval for the corresponding relative importance index. TABLE Il. Bootstrap Standard Errors (SE) and 95% Confidence Intervals (Cl) for Three Relative Importance Indices Used in Example 1

Note: Example 1, 1500 bootstrap replications. AThe single value above the 95% Cl is the value of the relative importance index estimated from the original data set 8 The 95% bootstrap confidence interval for the corresponding relative importance index. TABLE Il. Bootstrap Standard Errors (SE) and 95% Confidence Intervals (Cl) for Three Relative Importance Indices Used in Example 1

Notes: Example 1, 1500 bootstrap replications. The rank of | indicates the most important predictor, the rank of 2 indicates the second most important predictor, etc.

Notes: Example 1, 1500 bootstrap replications. The rank of | indicates the most important predictor, the rank of 2 indicates the second most important predictor, etc.

Note: Example 2, aortic cholesterol concentration as the outcome variable (n = 153). AThe value of the relative importance index estimated from the original data set. ® The rank is determined by comparing the estimated relative importance values. The rank of | indicates the most important predictor, the rank of 2 indicates the second most important predictor, etc. TABLE IV. Measures of Relative Importance for Example 2

Note: Example 2, aortic cholesterol concentration as the outcome variable (n = 153). AThe value of the relative importance index estimated from the original data set. ® The rank is determined by comparing the estimated relative importance values. The rank of | indicates the most important predictor, the rank of 2 indicates the second most important predictor, etc. TABLE IV. Measures of Relative Importance for Example 2

Note: Example 2, 1500 bootstrap replications. AThe single value above the 95% Cl is the value of the relative importance index estimated from the original data set ® The 95% bootstrap confidence interval for the corresponding relative importance index. TABLE V. Bootstrap Standard Errors (SE) and 95% Confidence Intervals (Cl) for Three Relative Importance Indices Used in Example 2

Note: Example 2, 1500 bootstrap replications. AThe single value above the 95% Cl is the value of the relative importance index estimated from the original data set ® The 95% bootstrap confidence interval for the corresponding relative importance index. TABLE V. Bootstrap Standard Errors (SE) and 95% Confidence Intervals (Cl) for Three Relative Importance Indices Used in Example 2

Notes: Example 2, 1500 bootstrap replications. The rank of | indicates the most important predictor, the rank of 2 indicates the second most important predictor, etc. TABLE VI. Bootstrap Relative Frequency Distribu- tion for Predictor Rankings Based on Three Impor- tance Indices Used in Example 2

Notes: Example 2, 1500 bootstrap replications. The rank of | indicates the most important predictor, the rank of 2 indicates the second most important predictor, etc. TABLE VI. Bootstrap Relative Frequency Distribu- tion for Predictor Rankings Based on Three Impor- tance Indices Used in Example 2

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