The pandemic progression is a dynamic process, in which measures yield outcomes, and outcomes in turn influence subsequent measures and outcomes. Due to the dynamics of pandemic progression, it is challenging to analyse the long-term influence of an individual measure in the sequence on pandemic outcomes. To demonstrate the problem and find solutions, in this article, we study the first wave of the pandemic—probably the most dynamic period—in the Nordic countries and analyse the influences of the Swedish measures relative to the measures adopted by its neighbouring countries on COVID-19 mortality, general mortality, COVID-19 incidence, and unemployment. The design is a longitudinal observational study. The linear regressions based on the Poisson distribution or the binomial distribution are employed for the analysis. To show that analysis can be timely conducted, we use table data available during the first wave. We found that the early Swedish measure had a long-term and significant causal effect on public health outcomes and a certain degree of long-term mitigating causal effect on unemployment during the first wave, where the effect was measured by an increase of these outcomes under the Swedish measures relative to the measures adopted by the other Nordic countries. This information from the first wave has not been provided by available analyses but could have played an important role in combating the second wave. In conclusion, analysis based on table data may provide timely information about the dynamic progression of a pandemic and the long-term influence of an individual measure in the sequence on pandemic outcomes.

Plurisubharmonic functions of two groups of complex variables (x1,...,x(n)) and (a1,...,a(m)) are considered; their partial functions are defined by f(a)(x) = f(x, a). We discuss analyticity theorems for the level sets associated to Lelong numbers of the parameter-dependent currents dd(c)f(a).

A divergent integral can sometimes be handled by assigning to it as its value the finite part in the sense of Hadamard. This is done by expanding the integral over the complement of a symmetric neighborhood of a singularity in powers of the radius, and throwing away the negative powers. In this paper the finite part of a singular integral of Cauchy type is defined, and this is then used to describe the boundary behavior of derivatives of a Cauchy-type integral. The finite part of a singular integral of Bochner-Martinelli type is studied, and an extension of the Plemelj jump formulas is shown to hold.

A cancer diagnosis is part of a complex stochastic process, which involves patient's characteristics, diagnosing methods, an initial assessment of cancer progression, treatments and a certain outcome of interest. To evaluate the performance of diagnoses, one needs not only a consistent estimation of the causal effect under a specified regime of diagnoses and treatments but also reliable confidence interval, P-value and hypothesis testing of the causal effect. In this article, we identify causal effects under various regimes of diagnoses and treatments by the point effects of diagnoses and treatments and thus are able to estimate and test these causal effects by estimating and testing point effects in the familiar framework of single-point causal inference. Specifically, using data from a Swedish prognosis study of stomach cancer, we estimate and test the causal effects on cancer survival under various regimes of diagnosing and treating hospitals including the optimal regime. We also estimate and test the modification of the causal effect by age. With its simple setting, one can readily extend the example to a large variety of settings in the area of cancer diagnosis: different personal characteristics such as family history, different diagnosing procedures such as multistage screening, and different cancer outcomes such as cancer progression.

In different studies for treatment effect on dichotomous outcome of a certain population, one uses different regression models, leading to different measures of the treatment effect. In observational studies, the common measures of the treatment effect are the conditional risk ratio based on a log-linear model and the conditional odds ratio based on a logistic model; in randomized trials, the common measures are the marginal risk difference based on a linear model, the marginal risk ratio based on a log-linear model, and the marginal odds ratio based on a logistic model. In this paper we express these measures in terms of the risk of a dichotomous outcome conditional on covariates and treatment, where the risk is described by a regression model. Therefore these measures do not explicitly depend on the regression model. As a result, we are able to use one regression model in one study to estimate all these measures by their maximum likelihood estimates. We show that these measures have causal interpretations and reflect different aspects of the same underlying treatment effect under the assumption of no unmeasured confounding covariate given observed covariates. We construct approximate distributions of the maximum likelihood estimates of these measures and then by using the approximate distributions we get confidence intervals for these measures. As an illustration, we estimate these measures for the effect of a triple therapy on eradication of Helicobacter pylori among Vietnamese children and are able to compare the treatment effect in this study with those in other studies.

A controversy about the Swedish strategy of dealing with COVID-19 during the early period is how decision-making was based on evidence, which refers to data and data analysis. During the earliest period of the pandemic, the Swedish decision-making was based on subjective perspective. However, when more data became available, the decision-making stood on mathematical and descriptive analyses. The mathematical analysis aimed to model the condition for herd immunity while the descriptive analysis compared different measures without adjustment of population differences and updating pandemic situations. Due to the dubious interpretations of these analyses, a mild measure was adopted in Sweden upon the arrival of the second wave, leading to a surge of poor public health outcomes compared to the other Nordic countries (Denmark, Norway, and Finland). In this article, using data available during the first wave, we conduct longitudinal analysis to investigate the consequence of the shred of evidence in the Swedish decision-making for the first wave, where the study period is between January 2020 and August 2020. The design is longitudinal observational study. The linear regressions based on the Poisson distribution and the binomial distribution are employed for the analysis. We found that the early Swedish measure had a long-term and significant effect on general mortality and COVID-19 mortality and a certain mitigating effect on unemployment in Sweden during the first wave; here, the effect was measured by an increase of general deaths, COVID-19 deaths or unemployed persons under Swedish measure relative to the measures adopted by the other Nordic countries. These pieces of statistical evidence were not studied in the mathematical and descriptive analyses but could play an important role in the decision-making at the second wave. In conclusion, a timely longitudinal analysis should be part of the decision-making process for containing the current pandemic or a future one.

In observational studies for the interaction between exposures on a dichotomous outcome of a certain population, usually one parameter of a regression model is used to describe the interaction, leading to one measure of the interaction. In this article we use the conditional risk of an outcome given exposures and covariates to describe the interaction and obtain five different measures of the interaction, that is, difference between the marginal risk differences, ratio of the marginal risk ratios, ratio of the marginal odds ratios, ratio of the conditional risk ratios, and ratio of the conditional odds ratios. These measures reflect different aspects of the interaction. By using only one regression model for the conditional risk, we obtain the maximum-likelihood (ML)-based point and interval estimates of these measures, which are most efficient due to the nature of ML. We use the ML estimates of the model parameters to obtain the ML estimates of these measures. We use the approximate normal distribution of the ML estimates of the model parameters to obtain approximate non-normal distributions of the ML estimates of these measures and then confidence intervals of these measures. The method can be easily implemented and is presented via a medical example.

We use logistic model to get point and interval estimates of the marginal risk difference in observational studies and randomized trials with dichotomous outcome. We prove that the maximum likelihood estimate of the marginal risk difference is unbiased for finite sample and highly robust to the effects of dispersing covariates. We use approximate normal distribution of the maximum likelihood estimates of the logistic model parameters to get approximate distribution of the maximum likelihood estimate of the marginal risk difference and then the interval estimate of the marginal risk difference. We illustrate application of the method by a real medical example. 

Suppose that a sequence of treatments are assigned to influence an outcome of interest that occurs after the last treatment. Between treatments, there are time-dependent covariates that may be post-treatment variables of the earlier treatments and confounders of the subsequent treatments. In this article, we study identification and estimation of the net effect of each treatment in the treatment sequence. We construct a point parametrization for the joint distribution of treatments, time-dependent covariates and the outcome, in which the point parameters of interest are the point effects of treatments considered as single-point treatments. We identify net effects of treatments by their expressions in terms of point effects of treatments and express patterns of net effects of treatments by constraints on point effects of treatments. We estimate net effects of treatments through their point effects under the constraint by maximum likelihood and reduce the number of point parameters in the estimation by the treatment assignment condition. As a result, we obtain an unbiased consistent maximum-likelihood estimate for the net effect of treatment even in a long treatment sequence. We also show by simulation that the interval estimation of the net effect of treatment achieves the nominal coverage probability.

In sequential causal inference, two types of causal effects are of practical interest, namely, the causal effect of the treatment regime (called the sequential causal effect) and the blip effect of treatmenton on the potential outcome after the last treatment. The well-known G-formula expresses these causal effects in terms of the standard paramaters. In this article, we obtain a new G-formula that expresses these causal effects in terms of the point observable effects of treatments similar to treatment in the framework of single-point causal inference. Based on the new G-formula, we estimate these causal effects by maximum likelihood via point observable effects with methods extended from single-point causal inference. We are able to increase precision of the estimation without introducing biases by an unsaturated model imposing constraints on the point observable effects. We are also able to reduce the number of point observable effects in the estimation by treatment assignment conditions.

In observational studies with dichotomous outcome of a population, researchers usually report treatment effect alone, although both baseline risk and treatment effect are needed to evaluate the significance of the treatment effect to the population. In this article, we study point and interval estimates including confidence region of baseline risk and treatment effect based on logistic model, where baseline risk is the risk of outcome of the population under control treatment while treatment effect is measured by the risk difference between outcomes of the population under active versus control treatments. Using approximate normal distribution of the maximum-likelihood (ML) estimate of the model parameters, we obtain an approximate joint distribution of the ML estimate of the baseline risk and the treatment effect. Using the approximate joint distribution, we obtain point estimate and confidence region of the baseline risk and the treatment effect as well as point estimate and confidence interval of the treatment effect when the ML estimate of the baseline risk falls into specified range. These interval estimates reflect nonnormality of the joint distribution of the ML estimate of the baseline risk and the treatment effect. The method can be easily implemented by using any software that generates normal distribution. The method can also be used to obtain point and interval estimates of baseline risk and any other measure of treatment effect such as risk ratio and the number needed to treat. The method can also be extended from logistic model to other models such as log-linear model.

The control of confounding is essential in many statistical problems, especially in those that attempt to estimate exposure effects. In some cases, in addition to the 'primary' sample, there is another 'secondary' sample which, though having no direct information about the exposure effect, contains information about the confounding factors. The purpose of this article is to study the influence of the secondary sample on likelihood inference for the exposure effect. In particular, we investigate the interplay between the efficiency improvement and the possible bias introduced by the secondary sample as a function of the degree of confounding in the primary sample and the sizes of the primary and secondary samples. In the case of weak confounding, the secondary sample can only little improve estimation of the exposure effect, whereas with strong confounding the secondary sample can be much more useful. On the other hand, it will be more important to consider possible biasing effects in the latter case. For illustration, we use a formal example of a generalized linear model and a real example with sparse data from a case-control study of the association between gastric cancer and HM-CAP/Band 120. Copyright (c) 2006 John Wiley & Sons, Ltd.

Cancer diagnosis is part of a complex stochastic process, in which patients' personal and social characteristics influence the choice of diagnosing methods, diagnosing methods in turn influence the initial assessment of cancer stage, cancer stage in turn influences the choice of treating methods, and treating methods in turn influence cancer outcomes such as cancer survival. To evaluate the performance of diagnoses, one needs to estimate and test the sequential causal effect (SCE) under a specified regime of diagnoses and treatments in such a complex observational study, where the data-generating mechanism is unknown and modeling is needed for statistical inference. In this article, we introduce a method of statistical modeling to estimate and test SCEs under regimes of treatments (diagnoses and treatments in cancer diagnosis) in complex observational studies. By applying the alternative G-formula, we express the SCE in terms of the point effects of treatments in the sequence, so that the modeling can be conducted via the point effects in the framework of single-point causal inference. We illustrate our method by a medical example of cancer diagnosis with data from a Swedish prognosis study of cardia cancer.

In this article we estimate confidence regions of the common measures of (baseline, treatment effect) in observational studies, where the measure of a baseline is baseline risk or baseline odds while the measure of a treatment effect is odds ratio, risk difference, risk ratio or attributable fraction, and where confounding is controlled in estimation of both the baseline and treatment effect. We use only one logistic model to generate approximate distributions of the maximum-likelihood estimates of these measures and thus obtain the maximum-likelihood-based confidence regions for these measures. The method is presented via a real medical example.

When estimating treatment effect on count outcome of given population, one uses different models in different studies, resulting in non-comparable measures of treatment effect. Here we show that the marginal rate differences in these studies are comparable measures of treatment effect. We estimate the marginal rate differences by log-linear models and show that their finite-sample maximum-likelihood estimates are unbiased and highly robust with respect to effects of dispersing covariates on outcome. We get approximate finite-sample distributions of these estimates by using the asymptotic normal distribution of estimates of the log-linear model parameters. This method can be easily applied to practice.

In observational studies for the interaction between treatments, one needs to estimate and present both the treatment effects and the interaction to learn the significance of the interaction to the treatment effects. In this article, we estimate the treatment effects and the interaction jointly by using only one logistic model and based on maximum-likelihood. We present the interaction by (1) point estimate and confidence interval of the interaction, (2) point estimate and confidence region of (treatment effect, interaction) and (3) point estimate and confidence interval of the interaction when the maximum-likelihood estimate of one treatment effect falls into specified range.