Correlation does not imply causation in the context of Granger causality

In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the demand curve.

Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., correlation does not imply causation).

An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of event A taking place in the presence of B, and the odds of A in the absence of B. Due to symmetry, odds ratio reciprocally calculates the ratio of the odds of B occurring in the presence of A, and the odds of B in the absence of A. Two events are independent if and only if the OR equals 1, i.e., the odds of one event are the same in either the presence or absence of the other event. If the OR is greater than 1, then A and B are associated (correlated) in the sense that, compared to the absence of B, the presence of B raises the odds of A, and symmetrically the presence of A raises the odds of B. Conversely, if the OR is less than 1, then A and B are negatively correlated, and the presence of one event reduces the odds of the other event occurring.

Note that the odds ratio is symmetric in the two events, and no causal direction is implied (correlation does not imply causation): an OR greater than 1 does not establish that B causes A, or that A causes B.

In statistics, correlation is a kind of statistical relationship between two random variables or bivariate data. Usually it refers to the degree to which a pair of variables are linearly related. In statistics, more general relationships between variables are called an association, the degree to which some of the variability of one variable can be accounted for by the other.

The presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., correlation does not imply causation).Furthermore, the concept of correlation is not the same as dependence: if two variables are independent, then they are uncorrelated, but the opposite is not necessarily true: even if two variables are uncorrelated, they might be dependent on each other.

In causal inference, a confounder is a variable that affects both the dependent variable and the independent variable, creating a spurious relationship.

Confounding is a causal concept rather than a purely statistical one, and therefore cannot be fully described by correlations or associations alone. The presence of confounders helps explain why correlation does not imply causation, and why careful study design and analytical methods (such as randomization, statistical adjustment, or causal diagrams) are required to distinguish causal effects from spurious associations.

Fertility factors are determinants of the number of children that an individual is likely to have. Fertility factors are mostly positive or negative correlations without certain causations.

Factors associated with increased fertility include the intention to have children, remaining religiosity, general inter-generational transmission of values, high status of marriage and cohabitation, maternal and social support, rural residence, a small subset of pro-family social programs, low IQ, personality traits such as conscientiousness, and generally increased food production.

Several types of correlation coefficient exist, each with their own definition and own range of usability and characteristics. They all assume values in the range from −1 to +1, where ±1 indicates the strongest possible correlation and 0 indicates no correlation. As tools of analysis, correlation coefficients present certain problems, including the propensity of some types to be distorted by outliers and the possibility of incorrectly being used to infer a causal relationship between the variables (for more, see Correlation does not imply causation).

Post hoc ergo propter hoc (Latin: 'after this, therefore because of this') is an informal fallacy that states "Since event Y followed event X, event Y must have been caused by event X." It is a fallacy in which an event is presumed to have been caused by a closely preceding event merely on the grounds of temporal succession. This type of reasoning is fallacious because mere temporal succession does not establish a causal connection. It is often shortened simply to post hoc fallacy. A logical fallacy of the questionable cause variety, it is subtly different from the fallacy cum hoc ergo propter hoc ('with this, therefore because of this'), in which two events occur simultaneously or the chronological ordering is insignificant or unknown. Post hoc is a logical fallacy in which one event seems to be the cause of a later event because it occurred earlier.

Post hoc is a particularly tempting error because correlation sometimes appears to suggest causality. The fallacy lies in a conclusion based solely on the order of events, rather than taking into account other factors potentially responsible for the result that might rule out the connection.

Convergent cross mapping (CCM) is a statistical test for a cause-and-effect relationship between two variables that, like the Granger causality test, seeks to resolve the problem that correlation does not imply causation. While Granger causality is best suited for purely stochastic systems where the influences of the causal variables are separable (independent of each other), CCM is based on the theory of dynamical systems and can be applied to systems where causal variables have synergistic effects. As such, CCM is specifically aimed to identify linkage between variables that can appear uncorrelated with each other.