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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 ...
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.
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 ...
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:
Violations to this assumption result in a large reduction in power. Suggested solutions to this violation are: delete a variable, combine levels of one variable (e.g., put males and females together), or collect more data. 3. The logarithm of the expected value of the response variable is a linear combination of the explanatory variables.
If the response data used to estimate the model contain values that change sign, or if the lowest response value is far from zero (for example, when data are left-truncated), a location parameter, L, may be added to the response so that the expressions for the quantile function and for the median become, respectively:
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).
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 ...