Measures of relative importance for health-related quality of life

Abstract

Purpose

In health-related quality of life (HRQOL) studies, data are often collected on multiple domains for two or more groups of study participants. Quantitative measures of relative importance, which are used to rank order the domains based on their ability to discriminate between groups, are an alternative to multiple tests of significance on the group differences. This study describes relative importance measures based on logistic regression (LR) and multivariate analysis of variance (MANOVA) models.

Methods

Relative importance measures are illustrated using data from the Manitoba Inflammatory Bowel Disease (IBD) Cohort Study. Study participants with self-reported active (n = 244) and inactive (n = 105) disease were compared on 12 HRQOL domains from the Inflammatory Bowel Disease Questionnaire (IBDQ) and Medical Outcomes Study 36-item Short-Form (SF-36) Questionnaire.

Results

All but two relative importance measures ranked the IBDQ bowel symptoms and emotional health domains as most important.

Conclusions

MANOVA-based importance measures are recommended for multivariate normal data and when group covariances are equal, while LR measures are recommended for non-normal data and when the correlations among the domains are small. Relative importance measures can be used in exploratory studies to identify a small set of domains for further research.

Appendices

Appendix 1: Additional formulae used in calculating measures of relative importance

Measures based on the LR model

The estimated LR coefficient for the kth domain (k = 1,…, m) can be written as

$$ \hat{\beta }_{k} = {\frac{{r_{{\log {\text{it}}(\hat{p})k}} - R_{( - k)}^{2} R_{k|( - k)}^{2} }}{{1 - R_{k|( - k)}^{2} }}}, $$

(12)

where $r_{{\log {\text{it}}(\hat{p})k}}$ is the correlation between the kth domain and the logit of the predicted probabilities, $R_{( - k)}^{2}$ is the R ² value for a LR model in which the kth domain is excluded, and $R_{k|( - k)}^{2}$ is the R ² value for a model in which the kth domain is regressed on the other (m − 1) domains.

Pratt’s index can also be expressed as,

$$ d_{k} = {\frac{{{\hat{\mathbf{\varvec{\upbeta}}}}^{\text{T}}{\mathbf{X}}^{\text{T}} {\mathbf{QX}}_{[k]} \hat{\beta }_{k}}}{{{\hat{\mathbf{\varvec{\upbeta}}}}^{\text{T}}{\mathbf{X}}^{\text{T}} {{\mathbf{QX}\hat{\varvec{\upbeta}}}}}}},$$

(13)

where X is the N × m data matrix, X _[k] is the N × 1 vector of measurements on the kth domain, ${\hat{\mathbf{\varvec{\upbeta}}}}$ is a m × 1 vector of estimated unstandardized LR coefficients, ${\mathbf{Q}} = {\mathbf{I}}_{N} - \left( {{\mathbf{1}}_{N}{\mathbf{1}}_{N}^{\text{T}} /N} \right)$, I _N is a N × N identity matrix, 1  _N is a N × 1 matrix of ones, and ^T is the transpose operator.

Measures based on the MANOVA model

The vectors of discriminant function coefficients corresponds to the eigenvectors associated with E ⁻¹ H, where

$$ \mathbf{E} =\sum\limits_{j = 1}^{2} {\sum\limits_{i = 1}^{{n_{j} }} {\left( {{\mathbf{X}}_{ij} - {\bar{\mathbf{X}}}_{j}} \right)} \left( {{\mathbf{X}}_{ij} - {\bar{\mathbf{X}}}_{j} }\right)^{\text{T}} } , $$

(14)

is the error sum of squares and cross product matrix, and

$$ {\mathbf{H}} = \sum\limits_{j = 1}^{2} {n_{j} \left( {{\mathbf{X}}_{j} - {\bar{\mathbf{X}}}} \right)\left( {{\mathbf{X}}_{j} - {\bar{\mathbf{X}}}} \right)^{\text{T}} } , $$

(15)

is the hypothesis sum of squares and cross product matrix. The number of statistically significant discriminant functions is c = min (m, g − 1). The discriminant function score, z _ij , for the ith study participant in the jth (i = 1,…, n _j ; j = 1, 2) group is,

$$ z_{ij} = {\hat{\mathbf{a}}\mathbf{X}}_{ij} . $$

(16)

The discriminant function coefficient for the kth variable can also be expressed as

$$ \hat{a}_{k} = - \log \left( {{\frac{{n_{2} }}{{n_{1} }}}}\right) - \frac{1}{2}\left( {{\bar{\mathbf{X}}}_{1} +{\bar{\mathbf{X}}}_{2} } \right)^{T} \text{\bf{S}}^{ - 1} \left({{\bar{\mathbf{X}}}_{1} + {\bar{\mathbf{X}}}_{2} } \right) + u_{k}$$

(17)

where $u_{k}$ is the kth element of $\text{\bf{S}}^{ - 1} \left( {{\bar{\mathbf{X}}}_{1} -{\bar{\mathbf{X}}}_{2} } \right)$, S is the pooled sample covariance matrix, and $\overline{X}_{j}$ is the vector of means for the jth group.An equivalent formula for computing the F-to-remove statistic for the kth domain is

$$ F_{(k)} = {\frac{{k_{2} (\hat{a}_{k} /s_{(kk)} )^{2} }}{{(\bar{z}_{1} - \bar{z}_{2} ) + k_{3} - (\hat{a}_{k} /s_{(kk)} )^{2} }}}. $$

(18)

where $k_{2} = (n_{1} + n_{2} - 2 - m),\,k_{3} = (n_{1} + n_{2} )(n_{1} + n_{2} )/n_{1} n_{2} ,\,\hat{a}_{k}$ is the discriminant function coefficient for the kth domain, $\bar{z}_{1} \,{\text{and}}\,\bar{z}_{2}$ are the group means for the discriminant function score corresponding to ${\hat{\mathbf{a}}}$, and $s_{(kk)}$ is the positive square root of the kth diagonal element of the inverse of E, the error sums of square and cross product matrix.

Appendix 2

See Table 5.

Table 5 Correlations among HRQOL domains for active and inactive disease groups