Osittaisjäännösplotit

Osittaisjäännösplotit, known in English as partial residual plots, are graphical tools used in statistical modeling to visualize the relationship between a specific predictor variable and the response variable, after accounting for the effects of all other predictors in the model. They are particularly useful in regression analysis, where they help to identify non-linear relationships and assess the influence of individual predictors.

The construction of a partial residual plot involves a series of steps. First, a linear regression model is fitted to the data. Then, for the predictor variable of interest, say $X_i$, a new variable is created by taking the residuals from a regression of $X_i$ on all other predictors in the model. Subsequently, another regression is performed, this time regressing the original response variable $Y$ on all predictor variables except $X_i$. The residuals from this second regression represent the part of $Y$ that is not explained by the other predictors. Finally, a scatterplot is created with the residuals from the regression of $X_i$ on the x-axis and the residuals from the regression of $Y$ on the y-axis.

A straight line on the partial residual plot indicates a linear relationship between the predictor variable and the response, holding other variables constant. A curved pattern suggests a non-linear relationship, which might imply that a transformation of the predictor or a non-linear term should be included in the model. Deviations from any fitted line can also highlight outliers or influential observations related to that specific predictor. Partial residual plots are valuable for model diagnostics, helping researchers to understand the individual contributions of predictor variables and to improve the overall fit of the statistical model.