![]() ![]() In other words, the higher the BMI the weaker its association with systolic blood pressure. The first thing to notice is that systolic blood pressure does not change as much with BMI for overweight people compared to underweight people. Mathematically, β 1 is the derivative of Y with respect to X (denoted \(\frac = 8.9 – 0.28 × 28 = 1.06\). In this case: β 1 is the change in Y associated with a 1 unit change in X. In a quadratic model, the variable X is associated with 2 coefficients β 1 and β 2 (\(Y = β_0 + β_1 X + β_2 X^2\)), so its effect will no longer have a straightforward interpretation.īut before going into the details of this interpretation, let’s first review how to interpret the effect of X on Y in a linear model \(Y = β_0 + β_1 X\). How to interpret a model with a quadratic term? (For more information, I recommend an article I wrote on using variable transformations to improve your regression model). Other options to correct a non-linear relationship between X and Y is to use a logarithmic or a square root transformation of X. If the pattern disappears (see right side of the figure below), then conclude that the quadratic model is a better fit to the data.īesides looking at the residuals vs fitted values, we can also assess the fit of the quadratic model by comparing the adjusted R-squared between the linear and the quadratic model, or by checking the statistical significance of the quadratic term’s coefficient (i.e. If this plot shows some pattern (for example, the U-shaped pattern in the left side of the figure below), try adding a quadratic term to the model (\(Y = β_0 + β_1 X + β_2 X^2\)). Start by fitting a linear regression model to the data (\(Y = β_0 + β_1 X\)), and plot the residuals versus the fitted values. Other curves can also be fitted using just a part of the parabola, as we see below: When to add a quadratic term? Note that the quadratic model does not require the data to be U-shaped. ![]() In this case, adding a quadratic term to the regression equation may help model the relationship between X and Y. If this assumption is not met, linear regression will be a poor fit to the data (as shown in the figure below). Linear regression assumes that the relationship between the predictor X and the outcome Y is linear. ![]()
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