An overview of variance inflation factors for sample-size calculation

For power and sample-size calculations, most practicing researchers rely on power and sample-size software programs to design their studies. There are many factors that affect the statistical power that, in many situations, go beyond the coverage of commercial software programs. Factors commonly known as design effects influence statistical power by inflating the variance of the test statistics. The authors quantify how these factors affect the variances so that researchers can adjust the statistical power or sample size accordingly. The authors review design effects for factorial design, crossover design, cluster randomization, unequal sample-size design, multiarm design, logistic regression, Cox regression, and the linear mixed model, as well as missing data in various designs. To design a study, researchers can apply these design effects, also known as variance inflation factors to adjust the power or sample size calculated from a two-group parallel design using standard formulas and software.     

Sample-size requirements for comparisons of two groups on repeated observations of a binary outcome

When preparing a research protocol, an investigator must be as careful in projecting sample-size requirements as in specifying hypotheses. In this article, tables are presented that provide estimates of sample-size requirements for statistical power of 0.80 with two-tailed alpha-levels of 0.05 in studies with a balanced design that plan to compare two groups on time-averaged, repeated observations of a binary outcome. The estimates, which are based on the algorithm of Diggle, Heagerty, Liang, and Zeger, are a function of several features of the study, including the response rates for each group, the number of repeated observations per participant, and the strength of the association among observations within participants as quantified with an intraclass correlation coefficient.     

L1 penalized continuation ratio models for ordinal response prediction using high-dimensional datasets

Health status and outcomes are frequently measured on an ordinal scale. For high-throughput genomic datasets, the common approach to analyzing ordinal response data has been to break the problem into one or more dichotomous response analyses. This dichotomous response approach does not make use of all available data and therefore leads to loss of power and increases the number of type I errors. Herein we describe an innovative frequentist approach that combines two statistical techniques, L(1) penalization and continuation ratio models, for modeling an ordinal response using gene expression microarray data. We conducted a simulation study to assess the performance of two computational approaches and two model selection criteria for fitting frequentist L(1) penalized continuation ratio models. Moreover, we empirically compared the approaches using three application datasets, each of which seeks to classify an ordinal class using microarray gene expression data as the predictor variables. We conclude that the L(1) penalized constrained continuation ratio model is a useful approach for modeling an ordinal response for datasets where the number of covariates (p) exceeds the sample size (n) and the decision of whether to use Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC) for selecting the final model should depend upon the similarities between the pathologies underlying the disease states to be classified.     

Ordinal response prediction using bootstrap aggregation,with application to a high‐throughput methylation data set

Many investigators conducting translational research are performing high‐throughput genomic experiments and then developing multigenic classifiers using the resulting high‐dimensional data set. In a large number of applications, the class to be predicted may be inherently ordinal. Examples of ordinal outcomes include tumor‐node‐metastasis (TNM) stage (I, II, III, IV); drug toxicity evaluated as none, mild, moderate, or severe; and response to treatment classified as complete response, partial response, stable disease, or progressive disease. While one can apply nominal response classification methods to ordinal response data, in doing so some information is lost that may improve the predictive performance of the classifier. This study examined the effectiveness of alternative ordinal splitting functions combined with bootstrap aggregation for classifying an ordinal response. We demonstrate that the ordinal impurity and ordered twoing methods have desirable properties for classifying ordinal response data and both perform well in comparison to other previously described methods. Developing a multigenic classifier is a common goal for microarray studies, and therefore application of the ordinal ensemble methods is demonstrated on a high‐throughput methylation data set. Copyright © 2009 John Wiley & Sons, Ltd.     

Family-based association tests for ordinal traits adjusting for covariates

We present a class of family-based association tests (FBATs) for ordinal traits that adjust for the effects of covariates. For complex diseases, especially mental health conditions including nicotine dependence and substance use, the outcome variables are often recorded in an ordinal rather than quantitative scale. The naturally recorded ordinal traits are commonly analyzed either as quantitative traits or are dichotomized. It has been demonstrated repeatedly in recent studies that these commonly used approaches to dealing with ordinal traits are inadequate and result in loss of power. In this report, we make use of conditional likelihood to derive score test statistics that belong to a general class of FBATs. We conducted simulation studies to compare the type I error and power of our proposed test with existing tests. The empirical result suggests that our test produces reasonable type I errors and has power far exceeding (often doubling) those of existing tests. We applied our proposed test to a data set on alcohol dependence and found that six single nucleotide polymorphisms (SNPs) are significantly associated (P-values < or =0.001) with alcohol dependence after adjusting for gender and age. Three of the SNPs (rs619, rs1972373, and rs1571423) or their tightly linked regions have been suggested in the literature from the analysis of the same data, demonstrating the consistent findings between various methods. The other three SNPs (rs485874, rs718251, and rs1869907) are identified for the first time using this data set, underscoring the potential power of our proposed test.     

Ordinal regression models for epidemiologic data

Health status is often measured in epidemiologic studies on an ordinal scale, but data of this type are generally reduced for analysis to a single dichotomy. Several statistical models have been developed to make full use of information in ordinal response data, but have not been much used in analyzing epidemiologic studies. The authors discuss two of these statistical models--the cumulative odds model and the continuation ratio model. They may be interpreted in terms of odds ratios, can account for confounding variables, have clear and testable assumptions, and have parameters that may be estimated and hypotheses that may be tested using available statistical packages. However, calculations of asymptotic relative efficiency and results of simulations showed that simple logistic regression applied to dichotomized responses can in some realistic situations have more than 75% of the efficiency of ordinal regression models, but only if the ordinal scale is collapsed into a dichotomy close to the optimal point. The application of the proposed models to data from a study of chest x-rays of workers exposed to mineral fibers confirmed that they are easy to use and interpret, but gave results quite similar to those obtained using simple logistic regression after dichotomizing outcome in the conventional way.     

Estimates,power and sample size calculations for two-sample ordinal outcomes under before-after study designs

Sample size calculations are given for comparing two groups of subjects, typically referring to active and non-active intervention groups, on an ordinal outcome in experiments where the subjects are measured before and after intervention. These calculations apply to log-odds models with random intercepts, treatment, time and treatment-by-time interaction terms, the latter being the term of interest. The assumed forms of the odds ratios are flexible, allowing for proportional odds, adjacent categories, or other conditional models for ordinal responses. Simulations studies show that, for given sample sizes, the nominal and actual powers of the proposed test are similar.     

Modelling ordinal responses from co-twin control studies

The co-twin control design has been widely used in studying the effects of environmental factors on the development of diseases. For binary outcomes that arise from co-twin control studies, the conditional likelihood method is commonly used. This approach, however, does not readily extend to ordinal response data because the standard conditional likelihood does not exist for cumulative logit or proportional odds models. In this paper, we investigate the applicability of the random-effects and GEE approaches in analysing ordinal response data from co-twin control studies. Using both approaches, we re-analyse data from a co-twin control study of the impact of military services during the Vietnam era on post-traumatic stress disorders (PTSD). The ordinal models have considerably increased power in detecting the effects of exposure when compared to the analyses using a dichotomized response. We discuss the interpretation of the estimates from GEE and random-effects models in the context of the twin data. © 1998 John Wiley & Sons, Ltd.     

Ordered multinomial regression for genetic association analysis of ordinal phenotypes at Biobank scale

Logistic regression is the primary analysis tool for binary traits in genome-wide association studies (GWAS). Multinomial regression extends logistic regression to multiple categories. However, many phenotypes more naturally take ordered, discrete values. Examples include (a) subtypes defined from multiple sources of clinical information and (b) derived phenotypes generated by specific phenotyping algorithms for electronic health records (EHR). GWAS of ordinal traits have been problematic. Dichotomizing can lead to a range of arbitrary cutoff values, generating inconsistent, hard to interpret results. Using multinomial regression ignores trait value hierarchy and potentially loses power. Treating ordinal data as quantitative can lead to misleading inference. To address these issues, we analyze ordinal traits with an ordered, multinomial model. This approach increases power and leads to more interpretable results. We derive efficient algorithms for computing test statistics, making ordinal trait GWAS computationally practical for Biobank scale data. Our method is available as a Julia package OrdinalGWAS.jl. Application to a COPDGene study confirms previously found signals based on binary case–control status, but with more significance. Additionally, we demonstrate the capability of our package to run on UK Biobank data by analyzing hypertension as an ordinal trait.     

The impact of ignoring measurement error when estimating sample size for epidemiologic studies

The author presents two examples illustrating the bias in sample-size estimates that can result from ignoring measurement error among study variables. The first example examines the impact of ignoring misclassification of the study's outcome variable on the accuracy of sample-size estimates. In addition, the author outlines a simple yet effective means of adjusting sample-size estimates to account for outcome misclassification. In the second example, the author illustrates the potential for severe underestimation of required sample size in studies using linear regression to evaluate associations between the outcome of interest and an independent variable subject to classical measurement error. The author concludes with a discussion of pertinent literature that might be helpful to study planners interested in adjusting sample-size estimates to account for measurement errors in both outcome and predictor variables.     

Power and sample size evaluation for the Cochran-Mantel-Haenszel mean score (Wilcoxon rank sum) test and the Cochran-Armitage test for trend

The power of a chi‐square test, and thus the required sample size, are a function of the noncentrality parameter that can be obtained as the limiting expectation of the test statistic under an alternative hypothesis specification. Herein, we apply this principle to derive simple expressions for two tests that are commonly applied to discrete ordinal data. The Wilcoxon rank sum test for the equality of distributions in two groups is algebraically equivalent to the Mann–Whitney test. The Kruskal–Wallis test applies to multiple groups. These tests are equivalent to a Cochran–Mantel–Haenszel mean score test using rank scores for a set of C‐discrete categories. Although various authors have assessed the power function of the Wilcoxon and Mann–Whitney tests, herein it is shown that the power of these tests with discrete observations, that is, with tied ranks, is readily provided by the power function of the corresponding Cochran–Mantel–Haenszel mean scores test for two and R > 2 groups. These expressions yield results virtually identical to those derived previously for rank scores and also apply to other score functions. The Cochran–Armitage test for trend assesses whether there is an monotonically increasing or decreasing trend in the proportions with a positive outcome or response over the C‐ordered categories of an ordinal independent variable, for example, dose. Herein, it is shown that the power of the test is a function of the slope of the response probabilities over the ordinal scores assigned to the groups that yields simple expressions for the power of the test. Copyright © 2011 John Wiley & Sons, Ltd.     

Surveying physicians to determine the minimal important difference: implications for sample-size calculation.

The minimal important difference (MID) is the smallest benefit of treatment that would result in clinicians recommending it to their patients. The MID is necessary to calculate sample size for randomized clinical trials, but its chosen value is often arbitrary. This study set out to determine the practicability of surveying physicians to elicit the MID for clinical trial sample-size calculation. Using a mail survey, we elicited the MID of different physician specialties (family medicine, internal medicine, vascular surgery) for using propranolol to slow abdominal aortic aneurysm (AAA) growth assuming that propranolol was efficacious in this condition. We used different outcome measures (growth rate or proportion of patients requiring surgery) and different methods of data presentation for the proportion of patients requiring surgery (absolute risk reduction or number needed to treat). The MID varied significantly by physician specialty, experience with AAA and propranolol, and the method used to elicit the MID. Consequently, sample-size calculations using these various MIDs varied from 116 to 3015. Future attempts to elicit the MID need to consider carefully who is surveyed, how data are presented, and how opinions are elicited.     

Modelling ordered categorical data: recent advances and future challenges.

This article summarizes recent advances in the modelling of ordered categorical (ordinal) response variables. We begin by reviewing some models for ordinal data introduced in the literature in the past 25 years. We then survey recent extensions of these models and related methodology for special types of applications, such as for repeated measurement and other forms of clustering. We also survey other aspects of ordinal modelling, such as small-sample analyses, power and sample size considerations, and availability of software. Throughout, we suggest problem areas for future research and we highlight challenges for statisticians who deal with ordinal data.     

Generalized multifactor dimensionality reduction approaches to identification of genetic interactions underlying ordinal traits

The manifestation of complex traits is influenced by gene–gene and gene–environment interactions, and the identification of multifactor interactions is an important but challenging undertaking for genetic studies. Many complex phenotypes such as disease severity are measured on an ordinal scale with more than two categories. A proportional odds model can improve statistical power for these outcomes, when compared to a logit model either collapsing the categories into two mutually exclusive groups or limiting the analysis to pairs of categories. In this study, we propose a proportional odds model-based generalized multifactor dimensionality reduction (GMDR) method for detection of interactions underlying polytomous ordinal phenotypes. Computer simulations demonstrated that this new GMDR method has a higher power and more accurate predictive ability than the GMDR methods based on a logit model and a multinomial logit model. We applied this new method to the genetic analysis of low-density lipoprotein (LDL) cholesterol, a causal risk factor for coronary artery disease, in the Multi-Ethnic Study of Atherosclerosis, and identified a significant joint action of the CELSR2, SERPINA12, HPGD, and APOB genes. This finding provides new information to advance the limited knowledge about genetic regulation and gene interactions in metabolic pathways of LDL cholesterol. In conclusion, the proportional odds model-based GMDR is a useful tool that can boost statistical power and prediction accuracy in studying multifactor interactions underlying ordinal traits.     

Sample size calculations for population- and family-based case-control association studies on marker genotypes

Most previous sample size calculations for case-control studies to detect genetic associations with disease assumed that the disease gene locus is known, whereas, in fact, markers are used. We calculated sample sizes for unmatched case-control and sibling case-control studies to detect an association between a biallelic marker and a disease governed by a putative biallelic disease locus. Required sample sizes increase with increasing discrepancy between the marker and disease allele frequencies, and with less-than-maximal linkage disequilibrium between the marker and disease alleles. Qualitatively similar results were found for studies of parent offspring triads based on the transmission disequilibrium test (Abel and Müller-Myhsok, 1998, Am. J. Hum. Genet. 63:664-667; Tu and Whittemore, 1999, Am. J. Hum. Genet. 64:641-649). We also studied other factors affecting required sample size, including attributable risk for the disease allele, inheritance mechanism, disease prevalence, and for sibling case-control designs, extragenetic familial aggregation of disease and recombination. The large sample-size requirements represent a formidable challenge to studies of this type.     

A novel modeling framework for ordinal data defined by collapsed counts

Adolescent alcohol use is a serious public health concern. Despite advances in the theoretical conceptualization of pathways to alcohol use, researchers are limited by the statistical techniques currently available. Researchers often fit linear models and restrictive categorical models (e.g., proportional odds models) to ordinal data with many response categories defined by collapsed count data (0 drinking days, 1–2days, 3–6days, etc.). Consequently, existing models ignore the underlying count process, resulting in disjoint between the construct of interest and the models being fitted. Our proposed ordinal modeling approach overcomes this limitation by explicitly linking ordinal responses to a suitable underlying count distribution. In doing so, researchers can use maximum likelihood estimation to fit count models to ordinal data as if they had directly observed the underlying discrete counts. The usefulness of our ordinal negative binomial and ordinal zero‐inflated negative binomial models is verified by simulation studies. We also demonstrate our approach using real empirical data from the 2010 National Survey of Drug Use and Health. Results show the benefit of the proposed ordinal modeling framework compared with existing methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Random-effect models for ordinal responses: application to self-reported disability among older persons

BACKGROUND: Longitudinal studies with ordinal repeated outcomes are now widespread in epidemiology and clinical research. The statistical analysis of these studies combines two difficulties: the choice of the best ordinal model and taking into account correlations for within-subject responses. METHODS: Random-effect models are of particular value in this context and we propose here a fitting strategy. The various ordinal models extended to the case of repeated responses are detailed. We explain how the choice of model constrains the random effect structure. Model selection criteria and goodness-of-fit measures are also presented. These issues are dealt with by using an example of self-reported disability in older women assessed annually over a period of seven years. RESULTS: The proportionality of the odds ratios was validated for the covariables "age" and "gait speed". In contrast the impact of the covariable "pain" differs according to the levels of disability. The restricted partial proportional odds model was found to have a goodness of fit equivalent to the full generalized ordered logit model while the stereotype model appeared to give poorer fit. CONCLUSIONS: The random-effects models presented in this paper allow taking into account the ordinal nature of the outcome in longitudinal studies. Furthermore the impact of the risk factors can be modeled according to the response levels. This approach can be useful for a better understanding of complex processes of evolution.     

A comparison of stratified and adjusted trend tests for binomial proportions

For the problem of testing for a monotonic trend between an ordinal exposure and a binary response while controlling for categorical cofactors, we derived stratified and adjusted versions of two single contrast tests. Using simulations, we compared their statistical properties to those of the commonly used stratified Mantel-extension trend (MET) test. All tests had high power in case of monotonic relationships between exposure and response. In case of non-monotonic relationships, the stratified and adjusted contrast tests were markedly less powerful than the stratified MET-test particularly for five exposure levels, a favourable feature when the aim is to test for the existence of a monotonic dose-response relationship. These results are illustrated by an example.     

A practical approach to computing power for generalized linear models with nominal, count, or ordinal responses

Data analysts facing study design questions on a regular basis could derive substantial benefit from a straightforward and unified approach to power calculations for generalized linear models. Many current proposals for dealing with binary, ordinal, or count outcomes are conceptually or computationally demanding, limited in terms of accommodating covariates, and/or have not been extensively assessed for accuracy assuming moderate sample sizes. Here, we present a simple method for estimating conditional power that requires only standard software for fitting the desired generalized linear model for a non-continuous outcome. The model is fit to an appropriate expanded data set using easily calculated weights that represent response probabilities given the assumed values of the parameters. The variance-covariance matrix resulting from this fit is then used in conjunction with an established non-central chi square approximation to the distribution of the Wald statistic. Alternatively, the model can be re-fit under the null hypothesis to approximate power based on the likelihood ratio statistic. We provide guidelines for constructing a representative expanded data set to allow close approximation of unconditional power based on the assumed joint distribution of the covariates. Relative to prior proposals, the approach proves particularly flexible for handling one or more continuous covariates without any need for discretizing. We illustrate the method for a variety of outcome types and covariate patterns, using simulations to demonstrate its accuracy for realistic sample sizes.     

Are ordinal models useful for classification?

There is recent interest in classification procedures intended for use only when the response is ordinal. Ordinal response, however, is evident in the parameters estimated by either multinomial logistic or normal discriminant analyses, both of which classify either ordinal or non-ordinal responses. Further, there may be harm in applying ordinal models inappropriately and ample opportunity to assume mistakenly ordinality in real data sets. Therefore, it becomes important to ascertain whether there is benefit obtained in the appropriate application of ordinal models. This paper presents the results of a simulation study designed to compare classification accuracy of various models. We show that ordinal models classify less accurately than the multinomial logistic and normal discriminant procedures under a variety of circumstances. Until further studies become available, we presently conclude that ordinal models confer no advantage when the main purpose of the analysis is classification.