Logistic regression models for ordinal response: A study of self-efficacy in colorectal cancer screening

The ordinal logistic regression models are used to analyze the dependant variable with multiple outcomes that can be ranked, but have been underutilized. In this study, we describe four logistic regression models for analyzing the ordinal response variable. ^ In this methodological study, the four regression models are proposed. The first model uses the multinomial logistic model. The second is adjacent-category logit model. The third is the proportional odds model and the fourth model is the continuation-ratio model. We illustrate and compare the fit of these models using data from the survey designed by the University of Texas, School of Public Health research project PCCaSO (Promoting Colon Cancer Screening in people 50 and Over), to study the patient’s confidence in the completion colorectal cancer screening (CRCS). ^ The purpose of this study is two fold: first, to provide a synthesized review of models for analyzing data with ordinal response, and second, to evaluate their usefulness in epidemiological research, with particular emphasis on model formulation, interpretation of model coefficients, and their implications. Four ordinal logistic models that are used in this study include (1) Multinomial logistic model, (2) Adjacent-category logistic model [9], (3) Continuation-ratio logistic model [10], (4) Proportional logistic model [11]. We recommend that the analyst performs (1) goodness-of-fit tests, (2) sensitivity analysis by fitting and comparing different models.^

An ordinal logistic regression model with misclassification of the outcome variable and categorical covariate

Ordinal outcomes are frequently employed in diagnosis and clinical trials. Clinical trials of Alzheimer's disease (AD) treatments are a case in point using the status of mild, moderate or severe disease as outcome measures. As in many other outcome oriented studies, the disease status may be misclassified. This study estimates the extent of misclassification in an ordinal outcome such as disease status. Also, this study estimates the extent of misclassification of a predictor variable such as genotype status. An ordinal logistic regression model is commonly used to model the relationship between disease status, the effect of treatment, and other predictive factors. A simulation study was done. First, data based on a set of hypothetical parameters and hypothetical rates of misclassification was created. Next, the maximum likelihood method was employed to generate likelihood equations accounting for misclassification. The Nelder-Mead Simplex method was used to solve for the misclassification and model parameters. Finally, this method was applied to an AD dataset to detect the amount of misclassification present. The estimates of the ordinal regression model parameters were close to the hypothetical parameters. β1 was hypothesized at 0.50 and the mean estimate was 0.488, β2 was hypothesized at 0.04 and the mean of the estimates was 0.04. Although the estimates for the rates of misclassification of X1 were not as close as β1 and β2, they validate this method. X 1 0-1 misclassification was hypothesized as 2.98% and the mean of the simulated estimates was 1.54% and, in the best case, the misclassification of k from high to medium was hypothesized at 4.87% and had a sample mean of 3.62%. In the AD dataset, the estimate for the odds ratio of X 1 of having both copies of the APOE 4 allele changed from an estimate of 1.377 to an estimate 1.418, demonstrating that the estimates of the odds ratio changed when the analysis includes adjustment for misclassification. ^

A multivariate frailty model for disease recurrences and survival

A multivariate frailty hazard model is developed for joint-modeling of three correlated time-to-event outcomes: (1) local recurrence, (2) distant recurrence, and (3) overall survival. The term frailty is introduced to model population heterogeneity. The dependence is modeled by conditioning on a shared frailty that is included in the three hazard functions. Independent variables can be included in the model as covariates. The Markov chain Monte Carlo methods are used to estimate the posterior distributions of model parameters. The algorithm used in present application is the hybrid Metropolis-Hastings algorithm, which simultaneously updates all parameters with evaluations of gradient of log posterior density. The performance of this approach is examined based on simulation studies using Exponential and Weibull distributions. We apply the proposed methods to a study of patients with soft tissue sarcoma, which motivated this research. Our results indicate that patients with chemotherapy had better overall survival with hazard ratio of 0.242 (95% CI: 0.094 - 0.564) and lower risk of distant recurrence with hazard ratio of 0.636 (95% CI: 0.487 - 0.860), but not significantly better in local recurrence with hazard ratio of 0.799 (95% CI: 0.575 - 1.054). The advantages and limitations of the proposed models, and future research directions are discussed. ^

Association of maternal sphinganine:sphingosine ratio and folate levels with neural tube defects along the South Texas border

Neural tube defects (NTDs) remain elevated in Hispanic women along the South Texas Border, despite folate supplementation and folate fortification of cereal products. Missmer et al. examined the relationships between fumonisins, a class of corn mycotoxin, and NTDs in Hispanic women who ate corn tortillas and found increased odds ratios with increasing exposure, as measured by serum sphinganine:sphingosine (sa:so) ratios. This study examined the interactions between categorized maternal serum folate levels and stratified sa:so ratios and the resultant odds ratios of NTDs, stratified by type (anencephaly and spina bifida). The hypothesis was that the above normal folate category would have lower odds ratios of NTDs at given sa:so ratio categories and that there would be a difference in odds ratio patterns for anencephaly and spina bifida. Methods. Data for 406 Hispanic women were obtained from the Missmer case-control study. Sa:so ratios were calculated and subjects were stratified into “below normal,” “normal,” and above normal range for folate. A logistic regression model was applied, controlling for BMI, serum B12, lab batch, and conception date. Results. While OR’s of NTDs increased for increasing sa:so ratios, OR’s for “above normal” folate were not decreased at any sa:so ratio and there was no statistically significant difference between OR’s of anencephaly and spina bifida. Conclusion. Folate does not appear to be protective against the potential teratogenic effect of fumonisins and did not differ in effect on OR’s of NTD by type. More research is necessary to determine the extent of fumonisin exposure in Hispanic women along the South Texas Border.^

Genome-wide association study of LDL-cholesterol over ApoB ratio

Genome-Wide Association Study analytical (GWAS) methods were applied in a large biracial sample of individuals to investigate variation across the genome for its association with a surrogate low-density lipoprotein (LDL) particle size phenotype, the ratio of LDL-cholesterol level over ApoB level. Genotyping was performed on the Affymetrix 6.0 GeneChip with approximately one million single nucleotide polymorphisms (SNPs). The ratio of LDL cholesterol to ApoB was calculated, and association tests used multivariable linear regression analysis with an additive genetic model after adjustment for the covariates sex, age and BMI. Association tests were performed separately in African Americans and Caucasians. There were 9,562 qualified individuals in the Caucasian group and 3,015 qualified individuals in the African American group. Overall, in Caucasians two statistically significant loci were identified as being associated with the ratio of LDL-cholesterol over ApoB: rs10488699 (p<5 x10-8, 11q23.3 near BUD13) and the SNP rs964184 (p<5 x10-8 11q23.3 near ZNF259). We also found rs12286037 ((p<4x10-7) (11q23.3) near APOA5/A4/C3/A1 with suggestive associate in the Caucasian sample. In exploratory analyses, a difference in the pattern of association between individuals taking and not taking LDL-cholesterol lowering medications was observed. Individuals who were not taking medications had smaller p-value than those taking medication. In the African-American group, there were no significant (p<5x10-8) or suggestive associations (p<4x10-7) with the ratio of LDL-cholesterol over ApoB after adjusting for age, BMI, and sex and comparing individuals with and without LDL-cholesterol lowering medication. Conclusions: There were significant and suggestive associations between SNP genotype and the ratio of LDL-cholesterol to ApoB in Caucasians, but these associations may be modified by medication treatment.^

Effects of informative censoring on the hazard function: Application to the proportional hazards regression model

The standard analyses of survival data involve the assumption that survival and censoring are independent. When censoring and survival are related, the phenomenon is known as informative censoring. This paper examines the effects of an informative censoring assumption on the hazard function and the estimated hazard ratio provided by the Cox model.^ The limiting factor in all analyses of informative censoring is the problem of non-identifiability. Non-identifiability implies that it is impossible to distinguish a situation in which censoring and death are independent from one in which there is dependence. However, it is possible that informative censoring occurs. Examination of the literature indicates how others have approached the problem and covers the relevant theoretical background.^ Three models are examined in detail. The first model uses conditionally independent marginal hazards to obtain the unconditional survival function and hazards. The second model is based on the Gumbel Type A method for combining independent marginal distributions into bivariate distributions using a dependency parameter. Finally, a formulation based on a compartmental model is presented and its results described. For the latter two approaches, the resulting hazard is used in the Cox model in a simulation study.^ The unconditional survival distribution formed from the first model involves dependency, but the crude hazard resulting from this unconditional distribution is identical to the marginal hazard, and inferences based on the hazard are valid. The hazard ratios formed from two distributions following the Gumbel Type A model are biased by a factor dependent on the amount of censoring in the two populations and the strength of the dependency of death and censoring in the two populations. The Cox model estimates this biased hazard ratio. In general, the hazard resulting from the compartmental model is not constant, even if the individual marginal hazards are constant, unless censoring is non-informative. The hazard ratio tends to a specific limit.^ Methods of evaluating situations in which informative censoring is present are described, and the relative utility of the three models examined is discussed. ^

Extension of k-ratio methods to rank-based analyses

An extension of k-ratio multiple comparison methods to rank-based analyses is described. The new method is analogous to the Duncan-Godbold approximate k-ratio procedure for unequal sample sizes or correlated means. The close parallel of the new methods to the Duncan-Godbold approach is shown by demonstrating that they are based upon different parameterizations as starting points.^ A semi-parametric basis for the new methods is shown by starting from the Cox proportional hazards model, using Wald statistics. From there the log-rank and Gehan-Breslow-Wilcoxon methods may be seen as score statistic based methods.^ Simulations and analysis of a published data set are used to show the performance of the new methods. ^

A multiplicative model for standardizing vital rates without using external standards

Standardization is a common method for adjusting confounding factors when comparing two or more exposure category to assess excess risk. Arbitrary choice of standard population in standardization introduces selection bias due to healthy worker effect. Small sample in specific groups also poses problems in estimating relative risk and the statistical significance is problematic. As an alternative, statistical models were proposed to overcome such limitations and find adjusted rates. In this dissertation, a multiplicative model is considered to address the issues related to standardized index namely: Standardized Mortality Ratio (SMR) and Comparative Mortality Factor (CMF). The model provides an alternative to conventional standardized technique. Maximum likelihood estimates of parameters of the model are used to construct an index similar to the SMR for estimating relative risk of exposure groups under comparison. Parametric Bootstrap resampling method is used to evaluate the goodness of fit of the model, behavior of estimated parameters and variability in relative risk on generated sample. The model provides an alternative to both direct and indirect standardization method. ^