Optimization of Convex Risk Functions

We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models, and optimality and duality theory for problems involving risk functions

Distribution of SUA and psychiatric outcomes across genotypes of <i>SLC2A9 rs6855911</i> in men and women.

<p>GAD = generalized anxiety disorder; and SUA = serum uric acid.</p><p>The results are expressed as numbers and percentages except for SUA which is expressed as mean and standard deviation.</p><p>Sex-by-genotype interaction were significant for its effect on lifetime and current any anxiety disorders and social phobia (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0076336#pone-0076336-t004" target="_blank">Table 4</a>).</p

Crude and adjusted logistic regression analysis of SUA (per 100 µmol/L) and psychiatric disorders in overall sample.

<p>MDD = major depressive disorder; GAD = generalized anxiety disorder; and SUA = serum uric acid.</p><p>If P value for the quadratic term (SUA²) is significant, ORs derived from models with quadratic term are presented.</p><p>Adjusted for age, sex, socio-economic status, alcohol consumption, smoking, diabetes, hypertension, GFR (calculated according to Modification in Diet in Renal Disease equation) and drugs that influence uric acid and additionally for anxiety (in the association between SUA and depression) and depression (in the association between SUA and anxiety).</p><p>P_interaction = P value for multiplicative interaction parameter between SUA and sex; Interaction for models with a quadratic term was assessed using a likelihood ratio test (LRT) comparing a model which included SUA*sex and SUA²* sex and a model without; and §indicates P value of the LRT.</p>*<p>Logistic regression not possible due to zero prevalence; NA = not available.</p

Distribution of psychiatric disorders across sex-specific quintiles of SUA.

<p>MDD =  major depressive disorder; GAD = generalized anxiety disorder; and SUA =  serum uric acid.</p>*<p>P-value <0.05 for quadratic trend tested by a crude logistic model which included a quadratic term (SUA squared) for continuous value of SUA.</p

Crude and adjusted logistic regression analysis of <i>SLC2A9 rs6855911</i> and psychiatric disorders according to sex.

<p>GAD = generalized anxiety disorder.</p><p>ORs represent the effect of <i>SLC2A9 rs6855911</i> used as a continuous score of 0, 1 and 2 corresponding to GG, AG and AA genotype respectively with AA being the genotype associated with elevated levels of serum uric acid.</p><p>Adjusted for age, sex, socio-economic status, alcohol consumption, smoking, diabetes, hypertension, GFR (calculated according to Modification in Diet in Renal Disease equation) and drugs that influence uric acid and additionally for anxiety (in the association between SUA and depression) and depression (in the association between SUA and anxiety).</p><p>P_interaction = P value for sex-by-genotype interaction for its effect on psychiatric disorders.</p

Impact of adding different adipocytokine, hepatic or inflammatory markers as quartiles in the ability of a clinical + biological risk score to predict type 2 diabetes, using a 23% probability threshold to define high risk subjects.

<p>Results are expressed as percentage and (95% confidence interval). PPV, positive predictive value; NPV, negative predictive value; NRI, net reclassification improvement; IDI, integrated discrimination improvement; IL-1β, interleukin 1 beta; IL-6, interleukin 6; TNF-α, tumour necrosis factor alpha; hs-CRP, high sensitive C reactive protein; γGT, gamma glutamyl transpeptidase. Data from 208 participants who developed type 2 diabetes mellitus and 3634 controls. § p-value 0.052; *, p-value<0.01.</p

Associations between lifetime agoraphobia status at baseline and inflammatory measures (composite score) at follow-up, serially adjusted for covariates.

<p>Values for inflammatory measures are given as unadjusted log10-transformed means ± standard deviation, values for interleukin-1β are given as prevalence.</p><p>HDL, high-density lipoprotein; CRP, C-reactive protein; TNF, tumor necrosis factor; OR, odds ratio; 95CI, 95% confidence interval.</p><p>^a Duration between somatic evaluation at baseline and somatic evaluation at follow-up.</p><p>^b A value of “3” represents an SES of III (middle class) on the Hollingshead Scale.</p><p>^c Systolic blood pressure ≥ 140 mmHg and/or diastolic blood pressure ≥ 90 mmHg.</p><p>^d Physically active at least or more than 20 minutes twice a week.</p><p>^e Generalized anxiety disorder, panic disorder, posttraumatic stress disorder, and/or social phobia.</p><p>^f Marijuana, cocaine, solvent, hallucinogen, stimulant, sedative or/and narcotic dependence.</p><p>^g Multiple regression with log10 transformed cytokine or CRP or adiponectin.</p><p>^h Logistic regression with interleukin-1β concentration dichotomized at the median.</p><p>ⁱ Composite score of inflammatory markers includes CRP, interleukin-6 and TNF-α. Only participants who had data available for all three of these markers were included in these analyses.</p><p>Associations between lifetime agoraphobia status at baseline and inflammatory measures (composite score) at follow-up, serially adjusted for covariates.</p

Baseline levels of adipocytokine, hepatic and inflammatory markers for participants who developed type 2 diabetes (n = 208) and those who remained free from it (n = 3634) after 5 years of follow-up.

<p>LOD, lower limit of detection; CRP, C-reactive protein; γGT, gamma-glutamyl transpeptidase. Results are expressed as % of column total. Statistical analysis by Student's t-test chi-square or Fisher's exact test.</p

Impact of adding different adipocytokine, hepatic or inflammatory markers as quartiles in the predictive capacity of a clinical + biological (C+B) risk score for type 2 diabetes.

<p>Statistical analysis by logistic regression. Each line shows the results of the original model (first line with HL, Hosmer-Lemeshow goodness-of-fit test (only p-values are reported); AIC, Akaike's information criterion; BIC, Bayesian information criterion; AROC, area under the ROC curve; IL-1β, interleukin 1 beta; IL-6, interleukin 6; TNF-α, tumour necrosis factor alpha; hs-CRP, high sensitive C reactive protein; γGT, gamma glutamyl transpeptidase. <b>§</b> using the type 2 diabetes risk predicted by the model as a continuous variable; <b>§§</b> splitting the type 2 diabetes risk into two categories (not at risk and at risk). Data from 208 participants who developed type 2 diabetes mellitus and 3634 controls. ** significantly different (p<0.01) from the baseline model (Kahn's C+B score).</p

Associations between lifetime agoraphobia status at baseline and inflammatory measures (adiponectin) at follow-up, serially adjusted for covariates.

<p>Values for inflammatory measures are given as unadjusted log10-transformed means ± standard deviation, values for interleukin-1β are given as prevalence.</p><p>HDL, high-density lipoprotein; CRP, C-reactive protein; TNF, tumor necrosis factor; OR, odds ratio; 95CI, 95% confidence interval.</p><p>^a Duration between somatic evaluation at baseline and somatic evaluation at follow-up.</p><p>^b A value of “3” represents an SES of III (middle class) on the Hollingshead Scale.</p><p>^c Systolic blood pressure ≥ 140 mmHg and/or diastolic blood pressure ≥ 90 mmHg.</p><p>^d Physically active at least or more than 20 minutes twice a week.</p><p>^e Generalized anxiety disorder, panic disorder, posttraumatic stress disorder, and/or social phobia.</p><p>^f Marijuana, cocaine, solvent, hallucinogen, stimulant, sedative or/and narcotic dependence.</p><p>^g Multiple regression with log10 transformed cytokine or CRP or adiponectin.</p><p>^h Logistic regression with interleukin-1β concentration dichotomized at the median.</p><p>ⁱ Composite score of inflammatory markers includes CRP, interleukin-6 and TNF-α. Only participants who had data available for all three of these markers were included in these analyses.</p><p>Associations between lifetime agoraphobia status at baseline and inflammatory measures (adiponectin) at follow-up, serially adjusted for covariates.</p