Non-Linear Regression Analysis By Chanaka Kaluarachchi
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Presentation outline
• Linear regression
• Checking linear Assumptions
• Linear vs non-linear
• Non linear regression analysis
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Linear regression (reminder)
• Linear regression is an approach for modelling dependent variable(푦) and one or more explanatory variables (푥).
푦 = 훽0 + 훽1푥 + 휀 Assumptions: 휀~푁(0, 휎2 ) – iid ( independently identically distributed)
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Checking linear Assumptions
iid- residual plot (휀 푣푠 푦 ) can be inspect to check that assumptions are met.
• Constant variance- Scattering is a constant magnitude
• Normal data- few outliers, systematic spared above and below the axis
• Liner relationship- No curve in the residual plot
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Residual plot in SPSS
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Residual plots in SPSS
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Linear vs non linear
Linear
• Linear scatter plot • No curves in residual plot • Correlation between variable is significant
Non-linear
• Curves in scatter plot • Curves in residual plot • No significant correlation between variables
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Non linear regression
• Non linear regression arises when predictors and response follows particular function form.
푦 = 푓 훽, 푥 + 휀
Examples
푦 = 훽2푥 + 휀 - non linear 푦 = 훽푥2 + 휀 - linear
1 1 푦 = 푥 + 휀 - non linear 푦 = 훽 + 휀 - linear 훽 푥 푦 = 푒훽푥 + 휀 - non linear 푦 = 훽 ln 푥 + 휀 - linear
1 푦 = + 휀 - non linear 1+훽푥
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Transformation
• Some nonlinear regression problems can be moved to a linear domain by a suitable transformation of the model formulation.
• Four common transformations to induce linearity are: logarithmic transformation, square root transformation, inverse transformation and the square transformation
Examples
• 푦 = 푒훽푥 ln 푦 = 훽푥 if 푦 ≥ 0
1 1 • 푦 = − 1 = 훽푥 if 푦 ≠ 0 1+훽푥 푦
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Curve Estimation
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points.
Example – Viral growth model
• An internet service provider (ISP) is determining the effects of a virus on its networks. As part of this effort, they have tracked the (approximate) percentage of e-mail traffic on its networks over time, from the moment of discovery until the threat was contained.
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Curve Estimation- Cont.
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Output
Scatter plot P value< 0.05 means model is significant Higher the 푅2 better the model fit
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Segmentation
We can split the graph in to segments and fit a segmented model. Example – Viral growth model
We can fit a logistic equation for the first 19 hours and an asymptotic regression for the remaining hours should provide a good fit and interpretability over the entire time period.
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Logistic model and choosing starting values
훽1 푦 = −훽 푥 1 + 훽2푒 3
Starting values
• 훽1- upper value of growth (0.65)
• 훽2- ratio upper value and lowest value (0.65/0.13=5)
• 훽3- estimated slop between points in plot. (0.6-0.12/19-3)=0.03
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Asymptotic regression model
휃3푥 푦 = 휃1 + 휃2푒
Starting values
• 휃1- lowest value (0)
• 휃2- difference upper value and lowest value (0.6)
• 훽3- estimated slop between points in plot. (0.6-0.1/20-40)=-0.025
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Output
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Output
Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago Thanks for listening
ResearchResearch in Pharmacoepidemiology in Pharmacoepidemiology (RIPE) @ National (RIPE) School @ of Pharmacy,National UniversitySchool of of Pharmacy,Otago University of Otago