An in-depth yet elementary exposition of regression and its power in statistical inference. Suitable for researchers in diverse fields of intellectual endeavor interested in exploring relationships between variables.
Standard Assumptions of Multiple-Linear
Regression: A Gentle Introduction to Residual
Analysis – Checking for Violations and Remedies
(Compiled by Emmanuel Makotsi-BSc Mathematics & Statistics)
(emmanuek5050@gmail.com ; +27621456161)
Assumption 1: Correct functional Form Assumption: Linear relationship
The model specified from the given data may be correct or incorrect.
In linear regression, the mean of the response yi is a linear combination of the
predictors x1i , · · · , xk1.
In simple words, the relationship between the predictor and
dependent(outcome/response) variables must be linear.
The linearity assumption can be tested with scatter plots or the Multiple R but these
are not sufficient as a quadratic relationship may appear linear on a scatter plot with
a high correlation. A more robust way to check for any violations is to use residuals.
If the functional form of a regression model is incorrect, the residual plots
constructed by using the model often display a pattern suggesting the form of a
more appropriate model. For instance, if we use a simple linear regression model
when the true relationship between y and x is curved, the residual plot will have a
curved appearance.
We illustrate the above with the output below.
The table below shows a bivariate sample data, that is ordered pairs of two variables
(x and y).
Note that y is the response/outcome variable and x is the predictor.
, The outputs below show the scatter plot indicating the relationship between the two
variables and the Pearson’s correlation coefficient is also shown with other sample
statistics.
Model Fit Measures
Model R R²
1 0.981 0.963
Scatterplot
At first glance, both the scatter plot and the Pearson’s correlation coefficient suggest a strong linear
relationship, but looking at the residuals plot below shows that a quadratic model will fit the data more
appropriately than a linear model.
Residuals plot
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