Residuals and Diagnostics for Ordinal Regression Models: A Surrogate Approach

<p>Ordinal outcomes are common in scientific research and everyday practice, and we often rely on regression models to make inference. A long-standing problem with such regression analyses is the lack of effective diagnostic tools for validating model assumptions. The difficulty arises from the fact that an ordinal variable has discrete values that are labeled with, but not, numerical values. The values merely represent ordered categories. In this article, we propose a surrogate approach to defining residuals for an ordinal outcome <i>Y</i>. The idea is to define a continuous variable <i>S</i> as a “surrogate” of <i>Y</i> and then obtain residuals based on <i>S</i>. For the general class of cumulative link regression models, we study the residual’s theoretical and graphical properties. We show that the residual has null properties similar to those of the common residuals for continuous outcomes. Our numerical studies demonstrate that the residual has power to detect misspecification with respect to (1) mean structures; (2) link functions; (3) heteroscedasticity; (4) proportionality; and (5) mixed populations. The proposed residual also enables us to develop numeric measures for goodness of fit using classical distance notions. Our results suggest that compared to a previously defined residual, our residual can reveal deeper insights into model diagnostics. We stress that this work focuses on residual analysis, rather than hypothesis testing. The latter has limited utility as it only provides a single <i>p</i>-value, whereas our residual can reveal what components of the model are misspecified and advise how to make improvements. Supplementary materials for this article are available online.</p

Seasonal Variation of 25-Hydroxyvitamin D₃ (bottom panel) and Intact PTH (iPTH) (top panel) Weekly Mean Values.

<p>The maximum seasonal variation of 25(OH)D₃ (peak to trough) was 6.8 ng/mL, reaching its trough in the 8^th week (early March) of each year and its peak in the 34^th week (early September). Peak iPTH values occurred at week 12 (early April) and trough values at week 37 (late September), a pattern that is roughly reciprocal to that of 25(OH)D₃, but lags by 3.5 weeks. Individual points represent the mean of the normalized distribution for each week. The solid lines represent the fit. Dark vertical dashed lines represent 25-hydroxyvitamin D₃ peaks and troughs, and light vertical dashed lines represent the iPTH peaks and troughs. To convert 25-hydroxyvitamin D₃ from ng/mL to nmol/L, multiply by 2.496 (rounded as 2.5).</p

Seasonal Variation in Weekly Mean Values of 25-Hydroxyvitamin D₃ (panels A and B) and Intact PTH (iPTH; panels C and D) by Gender and Region.

<p>A. Women show higher average values for 25(OH)D₃ than men. B. North and Central regions show similar average values for 25(OH)D₃ but the South region shows higher average values in winter, even though all three regions show similar values in summer. C. Men show higher average values than women for iPTH. D. The Central region shows the highest average values for iPTH, followed by the Northern region and then the South region.</p

Categorical percentage of patients for total 25(OH)D for all patients who had detectable 25(OH)D₂.

<p>The percentiles are categorized by month. The upper portion (blue) of each month shows the percentage of patients with sufficient 25(OH)D₃; the central portion (yellow), insufficient but not deficient; and the lower portion (red), deficient.</p

Percentage of patients with elevated iPTH concentrations (>65 pg/mL), by month.

<p>A. All groups. B. The percentage of subjects with iPTH greater than the upper limit of the reference interval (>65 pg/mL) reaches its trough in late summer and its peak in late winter. C. By region and gender. D. Patients with undetectable 25(OH)D₂ E. All patients with detectable 25(OH)D₂, even though the sinusoidal pattern is somewhat diminished, it still fit a sinusoidal function (null hypothesis p<0.001). The seasonal relationship holds across region, gender and presence or absence of detectable 25(OH)D₂. The regions demonstrate similar patterns, but the percentage of subjects with iPTH greater than the reference interval was greater for the central and northern regions than the southern region and greater for men than women.</p

Percentages of patients with 25(OH)D₃ deficiency (<20 ng/mL), insufficiency (20–29 ng/mL), and sufficiency (≥30 ng/mL), by month.

<p>The percentiles are categorized by month. The upper portion (blue) of each month shows the percentage of patients with sufficient 25(OH)D₃; the central portion (yellow), insufficient but not deficient; and the lower portion (red), deficient. The percentage of patients considered deficient or insufficient depends on the season, lower in summer (approximately 50%) and higher in winter (approximately 70%). The sun symbol indicates the summer solstice and the crescent moon symbol, the winter solstice. The central panel shows the seasonal similarity of total 25(OH)D with that of 25(OH)D2. The lower panel shows the rhythmic pattern across regions and gender. The Central and Northern regions were combined because they were similar. Their trough to peak difference is greater than that for the Southern region. Women and men percentages show similar patterns, except that the trough to peak difference is greater for men than women. The sun symbol indicates the summer solstice and the crescent moon, the winter solstice.</p