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.