Here, we extend the concept of cointegration from single-equation analysis to systems containing unit root data. We begin with a discussion of the intuition of vector cointegration and introduce the VECM, which is the basis for estimation of cointegrated VARs. We discuss the method for estimation of the VECM. In parallel to Chapter 8, we describe how to build the model that will be used to test for cointegration. We then introduce tests for the existence of multiple cointegrating relationships. If there are one or more cointegrating relationships, the model is reestimated imposing that number of cointegrating relationships. This model is referred to as the unrestricted reduced rank model. We discuss the identification problems inherent in the unrestricted reduced rank model, as well as what can be learned from this model. Next, we present a number of hypothesis tests that can be conducted to aid in identifying reasonable restrictions to impose on the model and explain how to impose restrictions in practice. We cover two different interpretation strategies as well and then offer a practical guide to cointegrated vector autoregression and an application.

Survivors are not a random sample of patients with disease, they are biased. How they differ from non-survivors must be understood before survival can be attributed to a disease process, or therapeutic intervention. Live-birth bias is a particular example; many conceptions fail before term birth and this influences the live-birth population. The importance of collider bias is reviewed. Workers are generally healthier than those not working introducing bias into occupational health studies.

In this chapter we examine multi-equation models for stationary data – vector autoregression. We describe a general system of equations in which each variable is allowed to be a function of contemporaneous values of all other variables, as well as lags of itself, and all other variables in the system. We explain the estimation problem inherent in a system with contemporaneous relationships and show how this estimation problem is solved by rewriting the system in reduced form. While this system can be estimated, writing the system in reduced form trades the estimation problem for an identification problem. We then describe how to build a reduced form VAR and discuss the interpretational tools widely used in VAR analysis. We cover some extensions of the VAR model and discuss the role of Granger causality in VAR analysis. We offer a practical guide to VAR analysis, including the basic steps in the framework, and provide an example using a five-variable system of attention allocation using Twitter data.

The propensity to interpret data according to prior beliefs, confirmation bias is one of the most insidious forms of bias in research: old and modern examples are offered. Misinterpretation of study results is commonplace in the courtroom, often described under the rubric of “junk science.” The association of a rare exposure with a rare outcome is increasingly the focus of biomedical research, this incurs increased opportunity for bias to influence study results. Absolute rather than relative risks are an important form of interpreting rare study data. Reverse causality is a profound source of error: Is the disease responsible for increasing exposure to the putative risk factor? Various biases are linked with time: In the context of public health screening, there is lead time bias and length time bias; and for survival studies, immortal time bias. Stein’s paradox offers a caution that the results of a larger sample may actually be more predictive of the subgroup experience within that sample than the study result observed for that subgroup.

The statistical foundations of ordinary least squares (OLS) regression are built on independent random variables. But time series observations are seldom, if ever, independent. Observations ordered sequentially in time exhibit time dependence. Time series can also exhibit trends, structural breaks, and repeated patterns that induce seasonality. All of these data features must be accounted for in OLS regression for statistical inference to be meaningful. This chapter discusses the key features of time series data. We explain the difference between stationary and nonstationary data and provide guidelines for how the two types behave differently.

In this chapter we consider single-equation regression analysis involving unit root processes. Unit root time series processes present additional problems for estimation and inference. In particular, hypothesis tests in regressions involving unit root processes will tend to have nonstandard distributions. We begin with a discussion of the intuition underlying cointegrating relationships and provide a formal definition of the concept. Two approaches are widely used to test for, and estimate, cointegrating relationships in a single-equation framework. We walk through both the Engle–Granger approach and the generalized error correction model. We discuss the different roles deterministic terms play in each of these approaches and provide guidelines on how to specify deterministic terms hypothesized to affect both the short- and long-run relationships. In parallel to Chapter 5, we discuss the importance of diagnostic tests to ensure consistent estimates from either approach. We then delve deeper into issues of interpretation of quantities of interest in the cointegrated case and provide a practical guide to working with unit root time series followed by an example.

Among the key constructs of biomedical research (random error [chance], risk, and bias in the search for causation), bias (or systemic error) is the most formidable source of inefficient and wasteful research, leading to incorrect or exaggerated results. The cause of most disease is complex, owed to many inherent (genetic) and environmental risk factors. It is in studying the interplay of these, each incurring modest risk, that many biases come into play.

Much biomedical research is funded by governments. The examples of abortion, gun safety, and some breast cancer research are offered to demonstrate political bias in prioritizing research. Limited evidence for bias during funding review and bias in citation are discussed.

Bias in reporting study outcomes, so as to favor statistically significant results, is a relatively recently documented phenomenon, which conflates hypothesis testing with hypothesis generation. The biased publication of significant results is enhanced by the preference for authors to submit publications with statistically “positive” results, and journal editors to publish them. Citation bias has been shown to occur throughout the scientific process, from grant writing to final manuscripts. Peer review, from grant submission to manuscripts, is also subject to bias, favoring the status of the authors. All of this contributes to the “winners curse,” the propensity for the first studies declaring an association to be wrong.

Tuskegee, the early twentieth-century eugenics movement, and Roma studies, are examples of major research efforts that were biased and misdirected. More recent examples from the study of electro-magnetic field exposure and childhood cancer, as influenced by interviewer bias, are provided.

Here, we discuss more formal approaches for identifying the influence of possible bias on a study design, or in the conduct of a study, what is called quantitative bias assessment. Then we consider the even more difficult problem of managing systemic bias in researchers themselves, and in research teams. Investigators find it particularly difficult to recognize cognitive and psychological bias in themselves and suggestions for facilitating this are made. The importance of creating a “bias calculation” in grant proposals is emphasized. A hierarchical structure to research bias is recognized.

Here we sail into an analysis of what is random error (chance) and what is systemic error (bias), the focus of this present work. Bias produces research whose results are the most destructive; the result appears to be precise (chance does not exert a large effect), but it is wrong. Early descriptions of bias list 35 types but later catalogues describe well over 200. Documenting an association between risk factor and disease is the sine qua non in a causal analysis, and this is where bias most fully operates.

Building on our discussion of stationary and nonstationary processes in Chapter 2, this chapter focuses on how to determine whether an observed time series is consistent with a (trend) stationary process or a nonstationary process containing a unit root. The presence or absence of a unit root fundamentally shapes our understanding of the behavior of a time series, influences our choice of analytical tools, and carries profound implications for theories of social, economic, and political change. This chapter presents the theory behind unit root testing and provides practical guidance for applied researchers.

Control and comparison groups are essential for most biomedical research, but who is employed fundamentally influences risk estimation. Historical and hospital control groups are especially susceptible to bias. Natural experiments can prove useful if conducted carefully. Pharmaco-epidemiology has to manage the, not quite, intractable problem of indication bias: Are treatment effects a result of the drug or the disease being treated? Some control groups may be appropriate for detecting and ameliorating bias: respectively, negative and sibling controls.