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The correlation coefficient, often represented as $r$, is a numerical measure of the strength and direction of a linear relationship between two variables. It can range from -1 to 1 where -1 indicates a perfect negative linear correlation, 0 indicates no linear correlation, and 1 indicates a perfect positive linear correlation.

An RCT is an experiment that randomly assigns participants into an experimental group or a control group to determine the causality between an intervention and an outcome. An example is testing a new medication's effectiveness against a placebo.

A scatter plot is a graphical representation of the relationship between two quantitative variables. It shows the values of one variable along the x-axis and the other variable along the y-axis. Placement of each dot on the scatter plot indicates an individual data point's values. Correlation can be visually assessed with scatter plots.

A spurious correlation is a connection between two variables that appears causal or related due to either coincidence or the presence of a third, unseen factor (confounder), but without a direct causal link. An example would be the correlation between the number of pirates and global warming.

A longitudinal study is an observational research method in which data is gathered from the same subjects repeatedly over a period of time. It can provide evidence of temporal precedence but cannot conclusively prove causation on its own. For instance, a study that measures the same individuals’ dietary patterns and health outcomes over decades.

An experimental study is designed to test whether a specific variable causes an effect on another variable by manipulating one variable (the independent variable) and observing changes in another (the dependent variable), while controlling for confounding factors.

The Pearson correlation coefficient is a measure of the linear correlation between two variables $X$ and $Y$, given by the formula:

Regression analysis is a statistical method used to estimate the relationships among variables. It often includes fitting a line or curve to a scatter plot. Linear regression, for example, estimates the relationship between a dependent variable and one or more independent variables, assuming the relationship is linear.

A confounding variable is an external variable that affects both variables of interest, potentially giving a false impression of a relationship. For example, ice cream sales and shark attacks are correlated, but the confounding variable is the summer season, during which both increase.

Causation indicates a relationship where one event is the result of the occurrence of the other event; there is a causal relationship between the two events. For instance, smoking is causally linked to lung cancer.

Causal inference is the process of using data analysis and statistical methods to draw conclusions about a causal relationship between variables. It's often challenging due to the potential presence of confounding variables, and thus, robust methods like randomized experiments are preferable.

A cross-sectional study observes data at a single point in time or over a short period, examining the relations between variables at that snapshot. While useful for identifying correlations, it is not well-suited for establishing causation due to the lack of temporal data. An example is a survey of smoking habits and lung cancer prevalence in a population at a given time.

Correlation is a statistical measure that describes the extent to which two variables fluctuate together. A positive correlation indicates that both variables tend to increase together, while a negative correlation indicates that as one variable increases, the other decreases. For example, height and weight in humans often show a positive correlation.

Temporal precedence is the principle that the cause must occur before the effect. This criterion is essential in establishing causation; if two variables are correlated, the one that comes first in time is more likely the cause. For example, learning (cause) must precede improvement in test scores (effect).