Original Article
Ordinal regression model and the linear regression model were superior to the logistic regression models

https://doi.org/10.1016/j.jclinepi.2005.09.007Get rights and content

Abstract

Objective

Ordinal scales often generate scores with skewed data distributions. The optimal method of analyzing such data is not entirely clear. The objective was to compare four statistical multivariable strategies for analyzing skewed health-related quality of life (HRQOL) outcome data. HRQOL data were collected at 1 year following catheterization using the Seattle Angina Questionnaire (SAQ), a disease-specific quality of life and symptom rating scale.

Study Design and Setting

In this methodological study, four regression models were constructed. The first model used linear regression. The second and third models used logistic regression with two different cutpoints and the fourth model used ordinal regression. To compare the results of these four models, odds ratios, 95% confidence intervals, and 95% confidence interval widths (i.e., ratios of upper to lower confidence interval endpoints) were assessed.

Results

Relative to the two logistic regression analysis, the linear regression model and the ordinal regression model produced more stable parameter estimates with smaller confidence interval widths.

Conclusion

A combination of analysis results from both of these models (adjusted SAQ scores and odds ratios) provides the most comprehensive interpretation of the data.

Introduction

In addition to relieving clinical symptoms and prolonging survival, a primary objective of any health care intervention is the enhancement of quality of life and well-being [1]. Consequently, instruments known as health-related quality of life (HRQOL) questionnaires that characterize and measure what subjects experience as a result of receiving medical care are increasingly being used as outcome measures in clinical research. It could be argued, in fact, that for persons with a chronic disease such as coronary artery disease (CAD), where cure is not attainable and therapy is ongoing, HRQOL is likely to be an essential outcome measure.

The theory and practice of psychological measurement and scaling methods, and the research accumulated over the past 50 to 60 years provides the foundation for the measurement of HRQOL [2]. HRQOL scales, represent numerical quantities of an attribute, and ordinal scales (e.g., Likert scales) are frequently used to structure items comprising HRQOL measures. As a result, most HRQOL items not only use discrete scales but are also highly skewed [2]. For this reason, Fayers and Machin have concluded that there are two factors that contribute to the common finding of nonnormality of HRQOL data. First, depending on the disease, some of the items are likely to take extreme values. For example, patients who are revascularized for CAD will frequently experience significant improvements in their functional status but those who do not are likely to continue to be functionally limited. Second, HRQOL questionnaires using Likert-type response scales with possible responses that include ‘not limited,’ ‘somewhat limited,’ ‘very limited,’ and ‘extremely limited’ will yield unequal interval scales and skewness can result from there being an unequal distance between values designated for different categories. It is therefore not surprising to find that HRQOL items follow highly skewed, nonnormal distributions [3].

The Seattle Angina Questionnaire (SAQ) is being used with increasing frequency in clinical research to address the HRQOL outcomes of patients with CAD. The SAQ is a 19-item self-administered questionnaire. Five dimensions of HRQOL are measured, generating five independent scales for exertional capacity, anginal stability, anginal frequency, disease perception, and treatment satisfaction. Based on results of psychometric tests, the SAQ was determined to be a valid, responsive and reliable instrument [4]. The Medical Outcomes Trust adopted the SAQ as a HRQOL measure for patients with CAD. The SAQ has been translated into at least 16 languages for use in Europe, Scandinavia, the Middle East, and North America [5] and is in widespread use worldwide (J.A. Spertus, personal communication).

Results of a recent systematic review identifying all published studies analyzing the SAQ demonstrated that the assumptions of the statistical tests used for the analysis of SAQ scores are regularly violated [6]. Furthermore, results of a preliminary analysis of a large population-based cohort of patients undergoing cardiac catheterization for CAD [7] indicated that up to 35% of patients selected 100 (the best possible score) for each of the dimensions [8]. The resulting ceiling effect produced a strongly skewed dataset, with graphically nonnormal distributions in all five SAQ dimensions (Fig. 1). Transformation of the data using log, square, and square root transformations also failed to yield normally distributed data. The distribution of these data led us to embark on a methodological exploration of the most appropriate multivariable statistical analysis to use when analyzing SAQ HRQOL data.

The central-limit theorem indicates that when one has a large dataset (large number of cases), despite the nonnormality of the raw responses and the residuals, statistical inferences can be made based on the approximate normality of the regression estimates. Recognizing this theorem, and using skewed SAQ HRQOL data, the objective of this study was to compare (a) linear regression analysis applied to the SAQ scoring method set out by Spertus et al. [4], (b) logistic regression analysis using patients who scored 100 (the best score) vs. patients who scored less than 100 as the outcome variable, (c) logistic regression analysis using patients who scored at or above the median vs. patients who scored below the median as the outcome variable, and (d) ordinal regression analysis whereby dimensional scores were categorized into ordered categories, with those in the lowest category having the lowest HRQOL scores and those in the highest category having the highest HRQOL scores.

Section snippets

Data source and variables

The cohort for this study comprised 3,243 adults from the Alberta Provincial Project for Outcome Assessment in Coronary Heart Disease (APPROACH) registry who underwent cardiac catheterization and responded to the 1-year follow-up questionnaire. APPROACH is a province-wide inception cohort of all adult Alberta residents undergoing cardiac catheterization for ischemic heart disease. APPROACH has been described previously [7]. The APPROACH database contains detailed clinical information on adult

Models

Four regression models were constructed. For this comparison, all potential predictor variables collected at catheterization were assessed separately in bivariate analyses with each of the four dependent variables. Recent simulation studies have demonstrated that stepwise approaches may yield unstable models that are highly sample dependent [10]. Therefore, variables that were significantly associated with the outcome variables at the P ≤ .10 level, across all four models were selected to be

Results

A total of 2,135 patients (respondents without missing data for EC scale) with a mean age of 64.6 years (standard deviation = 10.1 years) were used for these analyses. Table 1 describes the study population. The β coefficients from the linear regression have been converted to odds ratios to create a common metric for comparison across models. Table 2 presents the odds ratios and 95% confidence intervals for all of the variables analyzed in each of the three logistic models and the linear

Discussion

Globally, all four modeling methods produced acceptable parameters used for measuring model performance. Analysis of the residuals of the linear regression model showed that, as specified by the central limit theorem, specifically owing to the large sample size, the residuals follow an approximately normal distribution and therefore meet the assumptions for the use of parametric linear regression. Findings from both logistic regression models indicated that both models had similar

References (20)

There are more references available in the full text version of this article.

Cited by (0)

View full text