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where x is known, α and β are unknown, ε is a normally distributed random variable with mean 0 and unknown variance σ 2, and Y is the outcome of interest. We want to test the null hypothesis that the slope β is equal to some specified value β 0 (often taken to be 0, in which case the null hypothesis is that x and y are uncorrelated).
It is possible to calculate the extent to which the two scales overlap by using the following formula where is correlation between x and y, is the reliability of x, and is the reliability of y: r x y r x x ⋅ r y y {\displaystyle {\cfrac {r_{xy}}{\sqrt {r_{xx}\cdot r_{yy}}}}}
Sample size determination or estimation is the act of choosing the number of observations or replicates to include in a statistical sample.The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample.
Psychological statistics is application of formulas, theorems, numbers and laws to psychology. Statistical methods for psychology include development and application statistical theory and methods for modeling psychological data. These methods include psychometrics, factor analysis, experimental designs, and Bayesian statistics. The article ...
In statistics, an effect size is a value measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the value of a parameter for a hypothetical population, or to the equation that operationalizes how statistics or parameters lead to the effect size ...
Naive application of classical formula, n − p, would lead to over-estimation of the residuals degree of freedom, as if each observation were independent. More realistically, though, the hat matrix H = X(X ' Σ −1 X) −1 X ' Σ −1 would involve an observation covariance matrix Σ indicating the non-zero correlation among observations.
This is not easy to calculate, and the biserial coefficient is not widely used in practice. A specific case of biserial correlation occurs where X is the sum of a number of dichotomous variables of which Y is one. An example of this is where X is a person's total score on a test composed of n dichotomously scored items. A statistic of interest ...
It is possible to have multiple independent variables or multiple dependent variables. For instance, in multivariable calculus, one often encounters functions of the form z = f(x,y), where z is a dependent variable and x and y are independent variables. [8] Functions with multiple outputs are often referred to as vector-valued functions.