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The bias of an estimator is the difference between an estimator's expected value and the true value of the parameter being estimated.
Omitted-variable bias is the bias that appears in estimates of parameters in a regression analysis when the assumed specification omits an independent variable that should be in the model.
In statistical hypothesis testing, a test is said to be unbiased if for some alpha level (between 0 and 1), the probability the null is rejected is less than or equal to the alpha level for the entire parameter space defined by the null hypothesis, whilst the probability the null is rejected is greater than or equal to the alpha level for the entire parameter space defined by the alternate hypothesis.
Detection bias occurs when a phenomenon is more likely to be observed for a particular set of study subjects. For instance, the syndemic involving obesity and diabetes may mean doctors are more likely to look for diabetes in obese patients than in thinner patients, leading to an inflation in diabetes among obese patients because of skewed detection efforts.
In educational measurement, bias is defined as "Systematic errors in test content, test administration, and/or scoring procedures that can cause some test takers to get either lower or higher scores than their true ability would merit. The source of the bias is irrelevant to the trait the test is intended to measure." 
Funding bias may lead to selection of outcomes, test samples, or test procedures that favor a study's financial sponsor.
Reporting bias involves a skew in the availability of data, such that observations of a certain kind are more likely to be reported.
Exclusion bias arise due to the systematic exclusion of certain individuals from the study.
Attrition bias arises due to a loss of participants e.g. loss to follow up during a study.
Recall bias arises due to differences in the accuracy or completeness of participant recollections of past events. e.g. a patient cannot recall how many cigarettes they smoked last week exactly, leading to over-estimation or under-estimation.
Observer bias arises when the researcher subconsciously influences the experiment due to cognitive bias where judgement may alter how an experiment is carried out / how results are recorded.