Degrees of necessity and of sufficiency: Further results and extensions, with an application to covid-19 mortality in Austria

Visualizing a marker's degrees of necessity and of sufficiency in the predictiveness curve

BACKGROUND

The degrees to which a factor is necessary or sufficient for an event have been proposed as generalizations of attributable risk based on simple functions of unconditional and conditional event probabilities. Predictiveness curves show the risk for an event, as derived by a model with one or more predictors, depending on risk percentiles that represent the predictors' distribution in the underlying population.

METHODS

Connections between the degrees of necessity and of sufficiency and explained variation on the one hand and the predictiveness curve on the other hand are mathematically proved and exemplified using data of in-hospital death of Covid- 19 patients.

RESULTS

We show that the degrees of necessity and of sufficiency can be represented as proportions of areas easily identifiable in the plot of the predictiveness curve. In addition, we show that the proportion of explained variation, a common measure of predictiveness and relative importance of prognostic factors, is also closely connected to these areas.

CONCLUSION

Our investigations demonstrate that the predictiveness curve extended by these new interpretations of areas provides a comprehensive evaluation of markers or sets of markers for prediction.

TRIAL REGISTRATION

Austrian Coronavirus Adaptive Clinical Trial (ACOVACT); ClinicalTrials.gov, identifier NCT04351724.

Explained variation and degrees of necessity and of sufficiency for competing risks survival data

In this contribution, the Schemper-Henderson measure of explained variation for survival outcomes is extended to accommodate competing events (CEs) in addition to events of interest. The extension is achieved by moving from the unconditional and conditional survival functions of the original measure to unconditional and conditional cumulative incidence functions, the latter obtained, for example, from Fine and Gray models. In the absence of CEs, the original measure is obtained as a special case. We define explained variation on the population level and provide two different types of estimates. Recently, the authors have achieved a multiplicative decomposition of explained variation into degrees of necessity and degrees of sufficiency. These measures are also extended to the case of competing risks survival data. A SAS macro and an R function are provided to facilitate application. Interesting empirical properties of the measures are explored on the population level and by an extensive simulation study. Advantages of the approach are exemplified by an Austrian study of breast cancer with a high proportion of CEs.

An international comparison of age and sex dependency of COVID-19 deaths in 2020: a descriptive analysis

The number of reported coronavirus disease (COVID-19) deaths per 100,000 persons observed so far in 2020 is described in 15 European countries and the USA as dependent on age groups and sex. It is compared with the corresponding historic all-cause mortality per year depending on age and sex observed in these countries. Some common features exist although substantial differences in age and sex dependency of COVID-19 mortality were noted between countries. An exponential increase with age is a good model to describe and analyze both COVID-19 and all-cause mortality above 40 years old, where almost all COVID-19 deaths occur. Moreover, age dependency is stronger for COVID-19 mortality than for all-cause mortality, and males have an excess risk compared with women, which is less pronounced in the higher age groups. Additionally, concerning calendar time, differences in the age and sex dependency between countries were noted with the common tendency that male excess risk for COVID-19 mortality was smaller in the second half of the year.

Degrees of necessity and of sufficiency: Further results and extensions, with an application to covid-19 mortality in Austria

The purpose of this paper is to extend to ordinal and nominal outcomes the measures of degree of necessity and of sufficiency defined by the authors for dichotomous and survival outcomes in a previous paper. A cause, represented by certain values of prognostic factors, is considered necessary for an event if, without the cause, the event cannot develop. It is considered sufficient for an event if the event is unavoidable in the presence of the cause. The degrees of necessity and sufficiency, ranging from zero to one, are simple, intuitive functions of unconditional and conditional probabilities of an event such as disease or death. These probabilities often will be derived from logistic regression models; the measures, however, do not require any particular model. In addition, we study in detail the relationship between the proposed measures and the related explained variation summary for dichotomous outcomes, which are the common root for the developments for ordinal, nominal, and survival outcomes. We introduce and analyze the Austrian covid-19 data, with the aim of quantifying effects of age and other potentially prognostic factors on covid-19 mortality. This is achieved by standard regression methods but also in terms of the newly proposed measures. It is shown how they complement the toolbox of prognostic factor studies, in particular when comparing the importance of prognostic factors of different types. While the full model's degree of necessity is extremely high (0.933), its low degree of sufficiency (0.179) is responsible for the low proportion of explained variation (0.193).