Computing Confidence Bounds for Power and Sample Size of the General Linear Univariate Model

Heterogenous transmission and seroprevalence of SARS-CoV-2 in two demographically diverse populations with low vaccination uptake in Kenya, March and June 2021

Background

SARS-CoV-2 has extensively spread in cities and rural communities, and studies are needed to quantify exposure in the population. We report seroprevalence of SARS-CoV-2 in two well-characterized populations in Kenya at two time points. These data inform the design and delivery of public health mitigation measures.

Methods

Leveraging on existing population based infectious disease surveillance (PBIDS) in two demographically diverse settings, a rural site in western Kenya in Asembo, Siaya County, and an urban informal settlement in Kibera, Nairobi County, we set up a longitudinal cohort of randomly selected households with serial sampling of all consenting household members in March and June/July 2021. Both sites included 1,794 and 1,638 participants in the March and June/July 2021, respectively. Individual seroprevalence of SARS-CoV-2 antibodies was expressed as a percentage of the seropositive among the individuals tested, accounting for household clustering and weighted by the PBIDS age and sex distribution.

Results

Overall weighted individual seroprevalence increased from 56.2% (95%CI: 52.1, 60.2%) in March 2021 to 63.9% (95%CI: 59.5, 68.0%) in June 2021 in Kibera. For Asembo, the seroprevalence almost doubled from 26.0% (95%CI: 22.4, 30.0%) in March 2021 to 48.7% (95%CI: 44.3, 53.2%) in July 2021. Seroprevalence was highly heterogeneous by age and geography in these populations-higher seroprevalence was observed in the urban informal settlement (compared to the rural setting), and children aged <10 years had the lowest seroprevalence in both sites. Only 1.2% and 1.6% of the study participants reported receipt of at least one dose of the COVID-19 vaccine by the second round of serosurvey-none by the first round.

Conclusions

In these two populations, SARS-CoV-2 seroprevalence increased in the first 16 months of the COVID-19 pandemic in Kenya. It is important to prioritize additional mitigation measures, such as vaccine distribution, in crowded and low socioeconomic settings.

Seroprevalence and risk factors of SARS-CoV-2 infection in an urban informal settlement in Nairobi, Kenya, December 2020

Introduction: Urban informal settlements may be disproportionately affected by the COVID-19 pandemic due to overcrowding and other socioeconomic challenges that make adoption and implementation of public health mitigation measures difficult. We conducted a seroprevalence survey in the Kibera informal settlement, Nairobi, Kenya, to determine the extent of SARS-CoV-2 infection.

Methods: Members of randomly selected households from an existing population-based infectious disease surveillance (PBIDS) provided blood specimens between 27 ^th November and 5 ^th December 2020. The specimens were tested for antibodies to the SARS-CoV-2 spike protein. Seroprevalence estimates were weighted by age and sex distribution of the PBIDS population and accounted for household clustering. Multivariable logistic regression was used to identify risk factors for individual seropositivity.  

Results: Consent was obtained from 523 individuals in 175 households, yielding 511 serum specimens that were tested. The overall weighted seroprevalence was 43.3% (95% CI, 37.4 – 49.5%) and did not vary by sex. Of the sampled households, 122(69.7%) had at least one seropositive individual. The individual seroprevalence increased by age from 7.6% (95% CI, 2.4 – 21.3%) among children (<5 years), 32.7% (95% CI, 22.9 – 44.4%) among children 5 – 9 years, 41.8% (95% CI, 33.0 – 51.1%) for those 10-19 years, and 54.9%(46.2 – 63.3%) for adults (≥20 years). Relative to those from medium-sized households (3 and 4 individuals), participants from large (≥5 persons) households had significantly increased odds of being seropositive, aOR, 1.98(95% CI, 1.17 – 1.58), while those from small-sized households (≤2 individuals) had increased odds but not statistically significant, aOR, 2.31 (95% CI, 0.93 – 5.74). 

Conclusion: In densely populated urban settings, close to half of the individuals had an infection to SARS-CoV-2 after eight months of the COVID-19 pandemic in Kenya. This highlights the importance to prioritize mitigation measures, including COVID-19 vaccine distribution, in the crowded, low socioeconomic settings.

Seroprevalence and risk factors of SARS-CoV-2 infection in an urban informal settlement in Nairobi, Kenya, December 2020

Introduction: Urban informal settlements may be disproportionately affected by the COVID-19 pandemic due to overcrowding and other socioeconomic challenges that make adoption and implementation of public health mitigation measures difficult. We conducted a seroprevalence survey in the Kibera informal settlement, Nairobi, Kenya, to determine the extent of SARS-CoV-2 infection. Methods: Members of randomly selected households from an existing population-based infectious disease surveillance (PBIDS) provided blood specimens between 27th November and 5th December 2020. The specimens were tested for antibodies to the SARS-CoV-2 spike protein. Seroprevalence estimates were weighted by age and sex distribution of the PBIDS population and accounted for household clustering. Multivariable logistic regression was used to identify risk factors for individual seropositivity.   Results: Consent was obtained from 523 individuals in 175 households, yielding 511 serum specimens that were tested. The overall weighted seroprevalence was 43.3% (95% CI, 37.4 – 49.5%) and did not vary by sex. Of the sampled households, 122(69.7%) had at least one seropositive individual. The individual seroprevalence increased by age from 7.6% (95% CI, 2.4 – 21.3%) among children (<5 years), 32.7% (95% CI, 22.9 – 44.4%) among children 5 – 9 years, 41.8% (95% CI, 33.0 – 51.1%) for those 10-19 years, and 54.9%(46.2 – 63.3%) for adults (≥20 years). Relative to those from medium-sized households (3 and 4 individuals), participants from large (≥5 persons) households had significantly increased odds of being seropositive, aOR, 1.98(95% CI, 1.17 – 1.58), while those from small-sized households (≤2 individuals) had increased odds but not statistically significant, aOR, 2.31 (95% CI, 0.93 – 5.74).  Conclusion: In densely populated urban settings, close to half of the individuals had an infection to SARS-CoV-2 after eight months of the COVID-19 pandemic in Kenya. This highlights the importance to prioritize mitigation measures, including COVID-19 vaccine distribution, in the crowded, low socioeconomic settings.

Power calculation for overall hypothesis testing with high-dimensional commensurate outcomes

The complexity of system biology means that any metabolic, genetic, or proteomic pathway typically includes so many components (e.g., molecules) that statistical methods specialized for overall testing of high-dimensional and commensurate outcomes are required. While many overall tests have been proposed, very few have power and sample size methods. We develop accurate power and sample size methods and software to facilitate study planning for high-dimensional pathway analysis. With an account of any complex correlation structure between high-dimensional outcomes, the new methods allow power calculation even when the sample size is less than the number of variables. We derive the exact (finite-sample) and approximate non-null distributions of the 'univariate' approach to repeated measures test statistic, as well as power-equivalent scenarios useful to generalize our numerical evaluations. Extensive simulations of group comparisons support the accuracy of the approximations even when the ratio of number of variables to sample size is large. We derive a minimum set of constants and parameters sufficient and practical for power calculation. Using the new methods and specifying the minimum set to determine power for a study of metabolic consequences of vitamin B6 deficiency helps illustrate the practical value of the new results. Free software implementing the power and sample size methods applies to a wide range of designs, including one group pre-intervention and post-intervention comparisons, multiple parallel group comparisons with one-way or factorial designs, and the adjustment and evaluation of covariate effects.

: Online Power Computation for Linear Models with and without a Baseline Covariate

GLIMMPSE is a free, web-based software tool that calculates power and sample size for the general linear multivariate model with Gaussian errors (http://glimmpse.SampleSizeShop.org/). GLIMMPSE provides a user-friendly interface for the computation of power and sample size. We consider models with fixed predictors, and models with fixed predictors and a single Gaussian covariate. Validation experiments demonstrate that GLIMMPSE matches the accuracy of previously published results, and performs well against simulations. We provide several online tutorials based on research in head and neck cancer. The tutorials demonstrate the use of GLIMMPSE to calculate power and sample size.

Confidence regions for repeated measures ANOVA power curves based on estimated covariance

Background

Using covariance or mean estimates from previous data introduces randomness into each power value in a power curve. Creating confidence intervals about the power estimates improves study planning by allowing scientists to account for the uncertainty in the power estimates. Driving examples arise in many imaging applications.

Methods

We use both analytical and Monte Carlo simulation methods. Our analytical derivations apply to power for tests with the univariate approach to repeated measures (UNIREP). Approximate confidence intervals and regions for power based on an estimated covariance matrix and fixed means are described. Extensive simulations are used to examine the properties of the approximations.

Results

Closed-form expressions are given for approximate power and confidence intervals and regions. Monte Carlo simulations support the accuracy of the approximations for practical ranges of sample size, rank of the design matrix, error degrees of freedom, and the amount of deviation from sphericity. The new methods provide accurate coverage probabilities for all four UNIREP tests, even for small sample sizes. Accuracy is higher for higher power values than for lower power values, making the methods especially useful in practical research conditions. The new techniques allow the plotting of power confidence regions around an estimated power curve, an approach that has been well received by researchers. Free software makes the new methods readily available.

Conclusions

The new techniques allow a convenient way to account for the uncertainty of using an estimated covariance matrix in choosing a sample size for a repeated measures ANOVA design. Medical imaging and many other types of healthcare research often use repeated measures ANOVA.

Sample size re-estimation in a breast cancer trial

BACKGROUND

During the recruitment phase of a randomized breast cancer trial, investigating the time to recurrence, we found a strong suggestion that the failure probabilities used at the design stage were too high. Since most of the methodological research involving sample size re-estimation has focused on normal or binary outcomes, we developed a method which preserves blinding to re-estimate sample size in our time to event trial.

PURPOSE

A mistakenly high estimate of the failure rate at the design stage may reduce the power unacceptably for a clinically important hazard ratio. We describe an ongoing trial and an application of a sample size re-estimation method that combines current trial data with prior trial data or assumes a parametric model to re-estimate failure probabilities in a blinded fashion.

METHODS

Using our current blinded trial data and additional information from prior studies, we re-estimate the failure probabilities to be used in sample size re-calculation. We employ bootstrap re-sampling to quantify uncertainty in the re-estimated sample sizes.

RESULTS

At the time of re-estimation data from 278 patients were available, averaging 1.2 years of follow up. Using either method, we estimated a sample size increase of zero for the hazard ratio because the estimated failure probabilities at the time of re-estimation differed little from what was expected. We show that our method of blinded sample size re-estimation preserves the type I error rate. We show that when the initial guess of the failure probabilities are correct, the median increase in sample size is zero.

LIMITATIONS

Either some prior knowledge of an appropriate survival distribution shape or prior data is needed for re-estimation.

CONCLUSIONS

In trials when the accrual period is lengthy, blinded sample size re-estimation near the end of the planned accrual period should be considered. In our examples, when assumptions about failure probabilities and HRs are correct the methods usually do not increase sample size or otherwise increase it by very little. Clinical Trials 2010; 7: 219. http://ctj.sagepub.com.

Six months of disuse during hibernation does not increase intracortical porosity or decrease cortical bone geometry, strength, or mineralization in black bear (Ursus americanus) femurs

Disuse typically uncouples bone formation from resorption, leading to bone loss which compromises bone mechanical properties and increases the risk of bone fracture. Previous studies suggest that bears can prevent bone loss during long periods of disuse (hibernation), but small sample sizes have limited the conclusions that can be drawn regarding the effects of hibernation on bone structure and strength in bears. Here we quantified the effects of hibernation on structural, mineral, and mechanical properties of black bear (Ursus americanus) cortical bone by studying femurs from large groups of male and female bears (with wide age ranges) killed during pre-hibernation (fall) and post-hibernation (spring) periods. Bone properties that are affected by body mass (e.g. bone geometrical properties) tended to be larger in male compared to female bears. There were no differences (p>0.226) in bone structure, mineral content, or mechanical properties between fall and spring bears. Bone geometrical properties differed by less than 5% and bone mechanical properties differed by less than 10% between fall and spring bears. Porosity (fall: 5.5+/-2.2%; spring: 4.8+/-1.6%) and ash fraction (fall: 0.694+/-0.011; spring: 0.696+/-0.010) also showed no change (p>0.304) between seasons. Statistical power was high (>72%) for these analyses. Furthermore, bone geometrical properties and ash fraction (a measure of mineral content) increased with age and porosity decreased with age. These results support the idea that bears possess a biological mechanism to prevent disuse and age-related osteoporoses.

POWERLIB: SAS/IML Software for Computing Power in Multivariate Linear Models

The POWERLIBSAS/IML software provides convenient power calculations for a wide range of multivariate linear models with Gaussian errors. The software includes the Box, Geisser-Greenhouse, Huynh-Feldt, and uncorrected tests in the "univariate" approach to repeated measures (UNIREP), the Hotelling Lawley Trace, Pillai-Bartlett Trace, and Wilks Lambda tests in "multivariate" approach (MULTIREP), as well as a limited but useful range of mixed models. The familiar univariate linear model with Gaussian errors is an important special case. For estimated covariance, the software provides confidence limits for the resulting estimated power. All power and confidence limits values can be output to a SAS dataset, which can be used to easily produce plots and tables for manuscripts.

BIAS IN LINEAR MODEL POWER AND SAMPLE SIZE CALCULATION DUE TO ESTIMATING NONCENTRALITY

Data analysts frequently calculate power and sample size for a planned study using mean and variance estimates from an initial trial. Hence power, or the sample size needed to achieve a fixed power, varies randomly. Such calculations can be very inaccurate in the General Linear Univariate Model (GLUM). Biased noncentrality estimators and censored power calculations create inaccuracy. Censoring occurs if only certain outcomes of an initial trial lead to a power calculation. For example, a confirmatory study may be planned (and a sample size estimated) only following a significant result in the initial trial. Computing accurate point estimates or confidence bounds of GLUM noncentrality, power, or sample size in the presence of censoring involves truncated noncentral F distributions. We recommend confidence bounds, whether or not censoring occurs. A power analysis of data from humans exposed to carbon monoxide demonstrates the substantial impact on sample size that may occur. The results highlight potential biases and should aid study planning and interpretation.

Catechol-O-methyl-transferase functional polymorphism and nicotine dependence: an evaluation of nonreplicated results

Review articles have focused attention on and cited possible reasons for the nonreplication of genetic association studies. Herein, we illustrate how one might work through these possible reasons to make a judgment about the most plausible reason(s) when faced with two or more studies which yield seemingly inconsistent results. In the first study, 342 treatment-seeking smokers were genotyped for the Val108Met polymorphism in the functional catechol-O-methyl-transferase (COMT) locus. Alleles coding Val at codon 108 are denoted as H and those coding Met are denoted as L. An association between presence of the "H" (high activity) allele and pretreatment level of nicotine dependence level using the Fagerstrom Test for Nicotine Dependence was detected (P = 0.0072), after controlling for baseline body mass index (BMI, kg/m2), depression symptoms, and age. To validate this initial finding, 443 treatment-seeking smokers from an independent smoking cessation clinical trial were genotyped for the COMT polymorphism. Within the second study, no association between presence of the "H" allele and nicotine dependence was detected (P = 0.6418) after controlling for baseline BMI, depression symptoms, and age. We critically reviewed both studies with regard to often cited reasons for nonreplication, including type I error, population stratification, low statistical power, and imprecise measures of phenotype. Although in our opinion the failure to replicate the initial association in the second study is likely either the result of low statistical power to detect a small effect or effect heterogeneity, thorough analyses failed to definitively identify the reason for nonreplication.

BIAS IN LINEAR MODEL POWER AND SAMPLE SIZE DUE TO ESTIMATING VARIANCE

Planning a study using the General Linear Univariate Model often involves sample size calculation based on a variance estimated in an earlier study. Noncentrality, power, and sample size inherit the randomness. Additional complexity arises if the estimate has been censored. Left censoring occurs when only significant tests lead to a power calculation, while right censoring occurs when only non-significant tests lead to a power calculation. We provide simple expressions for straightforward computation of the distribution function, moments, and quantiles of the censored variance estimate, estimated noncentrality, power, and sample size. We also provide convenient approximations and evaluate their accuracy. The results allow demonstrating that ignoring right censoring falsely widens confidence intervals for noncentrality and power, while ignoring left censoring falsely narrows the confidence intervals. The new results allow assessing and avoiding the potentially substantial bias that censoring may create.