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The response values are placed in a vector, (), and the predictor values are placed in the design matrix, (), where each row is a vector of the predictor variables (including a constant) for the th data point.
x m,i (also called independent variables, explanatory variables, predictor variables, features, or attributes), and a binary outcome variable Y i (also known as a dependent variable, response variable, output variable, or class), i.e. it can assume only the two possible values 0 (often meaning "no" or "failure") or 1 (often meaning "yes" or ...
If this assumption is violated (i.e. if the data is heteroscedastic), it may be possible to find a transformation of Y alone, or transformations of both X (the predictor variables) and Y, such that the homoscedasticity assumption (in addition to the linearity assumption) holds true on the transformed variables [5] and linear regression may ...
The vector contains information about the variable of interest (in this case, minutes spent exercising) for individual in stratum . The value α i {\displaystyle \alpha _{i}} is the impact of demographics on cardiovascular disease incidence Y i ℓ {\displaystyle Y_{i\ell }} , which is assumed to be the same for all people in the stratum.
GAMLSS assumes the response variable follows an arbitrary parametric distribution, which might be heavy or light-tailed, and positively or negatively skewed. In addition, all the parameters of the distribution – location (e.g., mean), scale (e.g., variance) and shape (skewness and kurtosis) – can be modeled as linear, nonlinear or smooth ...
where , is a regression coefficient associated with the mth explanatory variable and the kth outcome. As explained in the logistic regression article, the regression coefficients and explanatory variables are normally grouped into vectors of size M + 1, so that the predictor function can be written more compactly:
Ordinary linear regression predicts the expected value of a given unknown quantity (the response variable, a random variable) as a linear combination of a set of observed values (predictors). This implies that a constant change in a predictor leads to a constant change in the response variable (i.e. a linear-response model). This is appropriate ...
It is also possible in some cases to fix the problem by applying a transformation to the response variable (e.g., fitting the logarithm of the response variable using a linear regression model, which implies that the response variable itself has a log-normal distribution rather than a normal distribution).