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Mill's Methods are five methods of induction described by philosopher John Stuart Mill in his 1843 book A System of Logic . [1] They are intended to establish a causal relationship between two or more groups of data, analyzing their respective differences and similarities.
If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon.
For a property to be a necessary condition it must always be present if the effect is present. Since this is so, then we are interested in looking at cases where the effect is present and taking note of which properties, among those considered to be 'possible necessary conditions' are present and which are absent. Obviously, any properties which are absent when the effect is present cannot be necessary conditions for the effect. This method is also referred to more generally within comparative politics as the most different systems design. Symbolically, the method of agreement can be represented as:
To further illustrate this concept, consider two structurally different countries. Country A is a former colony, has a centre-left government, and has a federal system with two levels of government. Country B has never been a colony, has a centre-left government and is a unitary state. One factor that both countries have in common, the dependent variable in this case, is that they have a system of universal health care. Comparing the factors known about the countries above, a comparative political scientist would conclude that the government sitting on the centre-left of the spectrum would be the independent variable which causes a system of universal health care, since it is the only one of the factors examined which holds constant between the two countries, and the theoretical backing for that relationship is sound; social democratic (centre-left) policies often include universal health care.
If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance save one in common, that one occurring only in the former; the circumstance in which alone the two instances differ, is the effect, or cause, or an indispensable part of the cause, of the phenomenon.
This method is also known more generally as the most similar systems design within comparative politics.
As an example of the method of difference, consider two similar countries. Country A has a centre-right government, a unitary system and was a former colony. Country B has a centre-right government, a unitary system but was never a colony. The difference between the countries is that Country A readily supports anti-colonial initiatives, whereas Country B does not. The method of difference would identify the independent variable to be the status of each country as a former colony or not, with the dependant variable being supportive for anti-colonial initiatives. This is because, out of the two similar countries compared, the difference between the two is whether or not they were formerly a colony. This then explains the difference on the values of the dependent variable, with the former colony being more likely to support decolonization than the country with no history of being a colony.
If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance; the circumstance in which alone the two sets of instances differ, is the effect, or cause, or a necessary part of the cause, of the phenomenon.
Also called the "Joint Method of Agreement and Difference", this principle is a combination of two methods of agreement. Despite the name, it is weaker than the direct method of difference and does not include it.
Symbolically, the Joint method of agreement and difference can be represented as:
Subduct [2] from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents.
If a range of factors are believed to cause a range of phenomena, and we have matched all the factors, except one, with all the phenomena, except one, then the remaining phenomenon can be attributed to the remaining factor.
Symbolically, the Method of Residue can be represented as:
Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation.
If across a range of circumstances leading to a phenomenon, some property of the phenomenon varies in tandem with some factor existing in the circumstances, then the phenomenon can be associated with that factor. For instance, suppose that various samples of water, each containing both salt and lead, were found to be toxic. If the level of toxicity varied in tandem with the level of lead, one could attribute the toxicity to the presence of lead.
Symbolically, the method of concomitant variation can be represented as (with ± representing a shift):
Unlike the preceding four inductive methods, the method of concomitant variation doesn't involve the elimination of any circumstance. Changing the magnitude of one factor results in the change in the magnitude of another factor.
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
Causality is an influence by which one event, process, state, or object (acause) contributes to the production of another event, process, state, or object (an effect) where the cause is partly responsible for the effect, and the effect is partly dependent on the cause. In general, a process has many causes, which are also said to be causal factors for it, and all lie in its past. An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future. Some writers have held that causality is metaphysically prior to notions of time and space.
The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The idea that "correlation implies causation" is an example of a questionable-cause logical fallacy, in which two events occurring together are taken to have established a cause-and-effect relationship. This fallacy is also known by the Latin phrase cum hoc ergo propter hoc. This differs from the fallacy known as post hoc ergo propter hoc, in which an event following another is seen as a necessary consequence of the former event, and from conflation, the errant merging of two events, ideas, databases, etc., into one.
Simpson's paradox is a phenomenon in probability and statistics in which a trend appears in several groups of data but disappears or reverses when the groups are combined. This result is often encountered in social-science and medical-science statistics, and is particularly problematic when frequency data are unduly given causal interpretations. The paradox can be resolved when confounding variables and causal relations are appropriately addressed in the statistical modeling.
The Baconian method is the investigative method developed by Francis Bacon, one of the founders of modern science, and thus a first formulation of a modern scientific method. The method was put forward in Bacon's book Novum Organum (1620), or 'New Method', and was supposed to replace the methods put forward in Aristotle's Organon. This method was influential upon the development of the scientific method in modern science; but also more generally in the early modern rejection of medieval Aristotelianism.
In statistics, a spurious relationship or spurious correlation is a mathematical relationship in which two or more events or variables are associated but not causally related, due to either coincidence or the presence of a certain third, unseen factor.
In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment. Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term (endogenous), in which case ordinary least squares and ANOVA give biased results. A valid instrument induces changes in the explanatory variable but has no independent effect on the dependent variable and is not correlated with the error term, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable.
External validity is the validity of applying the conclusions of a scientific study outside the context of that study. In other words, it is the extent to which the results of a study can generalize or transport to other situations, people, stimuli, and times. Generalizability refers to the applicability of a predefined sample to a broader population while transportability refers to the applicability of one sample to another target population. In contrast, internal validity is the validity of conclusions drawn within the context of a particular study.
In econometrics, endogeneity broadly refers to situations in which an explanatory variable is correlated with the error term. The distinction between endogenous and exogenous variables originated in simultaneous equations models, where one separates variables whose values are determined by the model from variables which are predetermined. Ignoring simultaneity in the estimation leads to biased estimates as it violates the exogeneity assumption of the Gauss–Markov theorem. The problem of endogeneity is often ignored by researchers conducting non-experimental research and doing so precludes making policy recommendations. Instrumental variable techniques are commonly used to mitigate this problem.
This glossary of statistics and probability is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics and Glossary of experimental design.
In causal inference, a confounder is a variable that influences both the dependent variable and independent variable, causing a spurious association. Confounding is a causal concept, and as such, cannot be described in terms of correlations or associations. The existence of confounders is an important quantitative explanation why correlation does not imply causation. Some notations are explicitly designed to identify the existence, possible existence, or non-existence of confounders in causal relationships between elements of a system.
In the philosophy of science, a causal model is a conceptual model that describes the causal mechanisms of a system. Several types of causal notation may be used in the development of a causal model. Causal models can improve study designs by providing clear rules for deciding which independent variables need to be included/controlled for.
In statistics, a mediation model seeks to identify and explain the mechanism or process that underlies an observed relationship between an independent variable and a dependent variable via the inclusion of a third hypothetical variable, known as a mediator variable. Rather than a direct causal relationship between the independent variable and the dependent variable, which is often false, a mediation model proposes that the independent variable influences the mediator variable, which in turn influences the dependent variable. Thus, the mediator variable serves to clarify the nature of the relationship between the independent and dependent variables.
In the statistical analysis of observational data, propensity score matching (PSM) is a statistical matching technique that attempts to estimate the effect of a treatment, policy, or other intervention by accounting for the covariates that predict receiving the treatment. PSM attempts to reduce the bias due to confounding variables that could be found in an estimate of the treatment effect obtained from simply comparing outcomes among units that received the treatment versus those that did not. Paul R. Rosenbaum and Donald Rubin introduced the technique in 1983.
In statistics and regression analysis, moderation occurs when the relationship between two variables depends on a third variable. The third variable is referred to as the moderator variable or simply the moderator. The effect of a moderating variable is characterized statistically as an interaction; that is, a categorical or continuous variable that is associated with the direction and/or magnitude of the relation between dependent and independent variables. Specifically within a correlational analysis framework, a moderator is a third variable that affects the zero-order correlation between two other variables, or the value of the slope of the dependent variable on the independent variable. In analysis of variance (ANOVA) terms, a basic moderator effect can be represented as an interaction between a focal independent variable and a factor that specifies the appropriate conditions for its operation.
In statistics, dispersion is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. For instance, when the variance of data in a set is large, the data is widely scattered. On the other hand, when the variance is small, the data in the set is clustered.
Causal inference is the process of determining the independent, actual effect of a particular phenomenon that is a component of a larger system. The main difference between causal inference and inference of association is that causal inference analyzes the response of an effect variable when a cause of the effect variable is changed. The study of why things occur is called etiology, and can be described using the language of scientific causal notation. Causal inference is said to provide the evidence of causality theorized by causal reasoning.
Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc.
Universal causation is the proposition that everything in the universe has a cause and is thus an effect of that cause. This means that if a given event occurs, then this is the result of a previous, related event. If an object is in a certain state, then it is in that state as a result of another object interacting with it previously.
Biological tests of necessity and sufficiency refer to experimental methods and techniques that seek to test or provide evidence for specific kinds of causal relationships in biological systems. A necessary cause is one without which it would be impossible for an effect to occur, while a sufficient cause is one whose presence guarantees the occurrence of an effect. These concepts are largely based on but distinct from ideas of necessity and sufficiency in logic.
In his book A System of Logic (1843), Mill proposed four methods for testing causal hypotheses: the method of agreement, the method of difference, the joint method of agreement and difference, and the method of concomitant variation.7 (footnote 7: Mill also proposed a fifth method, which he called the method of residues.)
