Interaction Effects in Statistical Analysis
Explore the concept of interaction effects in statistical analysis, how exposures can be modified, challenges in interpreting regression coefficients, considerations for studying systolic blood pressure data, and more in this series of lectures.
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Presentation Transcript
Interaction Part 1 Nina Gunnes April 8, 2021 04/08/2021 Spring 2021 - Lecture 6 1
Interaction effect Effect of an exposure depending on the level of other exposure(s) Also known as effect modification Effect of an exposure changed or modified by another exposure Can exist between different types of variables Continuous variables Categorical variables Continuous and categorical variables Included in the model as the product between the variables 04/08/2021 Spring 2021 - Lecture 6 2
Interaction effect, cont. Regression coefficients no longer interpreted directly Only plausible interactions considered in practice Difficult to test all possible interactions with many explanatory variables High correlation between variables for main and interaction effects Involving the same variables Non-significant interactions often resulting in non-significant main effects Large sample size often required to demonstrate interaction effects 04/08/2021 Spring 2021 - Lecture 6 3
Data on systolic blood pressure Considering data on systolic blood pressure (SBP) Sample of 32 subjects Five variables in the data set ??: ID number ???: SBP (in mmHg) ?????: age (in years) ???: body mass index (BMI) (in kg/m2) ? ?????: smoking (0 no, 1 yes) 04/08/2021 Spring 2021 - Lecture 6 4
Data on systolic blood pressure, cont. Described in Chapter 3 in the book by Laake et al. (2007) Data and variable description available for download https://www.med.uio.no/imb/personer/vit/veierod/boker/epidemiologiske- og-kliniske-forskningsmetoder/datafiler/ Inspecting the data Summary of data set and values of variables Frequency counts and summary statistics 04/08/2021 Spring 2021 - Lecture 6 5
Data on systolic blood pressure, cont. 04/08/2021 Spring 2021 - Lecture 6 6
Data on systolic blood pressure, cont. 04/08/2021 Spring 2021 - Lecture 6 7
Data on systolic blood pressure, cont. 04/08/2021 Spring 2021 - Lecture 6 8
Interaction in linear regression: Categorical and continuous variables Using the SBP data Defining the response variable ??? Defining the exposure variables ??? ? ????? Fitting a multiple linear regression model without interaction ??? = ?0+ ?1 ??? + ?2 ? ????? 04/08/2021 Spring 2021 - Lecture 6 9
Interaction in linear regression: Categorical and continuous variables, cont. 04/08/2021 Spring 2021 - Lecture 6 10
Interaction in linear regression: Categorical and continuous variables, cont. Calculating mean SBP in non-smokers ??? = 61.18 + 3.58 ??? + 8.72 0 = 61.18 + 3.58 ??? Calculating mean SBP in smokers ??? = 61.18 + 3.58 ??? + 8.72 1 = 69.90 + 3.58 ??? Effect of BMI independent on smoking Equations expressing two parallel straight lines Different intersections: 61.18 (non-smokers) and 69.90 (smokers) Same slope: 3.58 04/08/2021 Spring 2021 - Lecture 6 11
Interaction in linear regression: Categorical and continuous variables, cont. 04/08/2021 Spring 2021 - Lecture 6 12
Interaction in linear regression: Categorical and continuous variables, cont. Effect of BMI depending on smoking and vice versa? Using the same variables as before Response variable: ??? Exposure variables: ??? and ? ????? Considering interaction between BMI and smoking Product of ??? and ? ????? Fitting a multiple linear regression model with interaction ??? = ?0+ ?1 ??? + ?2 ? ????? + ?3 ??? ? ????? 04/08/2021 Spring 2021 - Lecture 6 13
Interaction in linear regression: Categorical and continuous variables, cont. 04/08/2021 Spring 2021 - Lecture 6 14
Interaction in linear regression: Categorical and continuous variables, cont. Estimated effect of BMI for non-smokers: ?1 Estimated effect of BMI for smokers: ?1+ ?3 Calculating mean SBP in non-smokers ??? = 49.76 + 4.10 ??? + 26.25 0 0.80 ??? 0 = 49.76 + 4.10 ??? Calculating mean SBP in smokers ??? = 49.76 + 4.10 ??? + 26.25 1 0.80 ??? 1 = 76.01 + 3.30 ??? 04/08/2021 Spring 2021 - Lecture 6 15
Interaction in linear regression: Categorical and continuous variables, cont. Effect of BMI dependent on smoking Equations expressing two non-parallel straight lines Different intersections: 49.76 (non-smokers) and 76.01 (smokers) Different slopes: 4.10 (non-smokers) and 3.30 (smokers) Larger effect of BMI among non-smokers Interaction term statistically non-significant ? = 0.414 Too low statistical power? 04/08/2021 Spring 2021 - Lecture 6 16
Interaction in linear regression: Categorical and continuous variables, cont. 04/08/2021 Spring 2021 - Lecture 6 17
Interaction in linear regression: Categorical and continuous variables, cont. 04/08/2021 Spring 2021 - Lecture 6 18
Interaction in linear regression: Categorical and continuous variables, cont. 04/08/2021 Spring 2021 - Lecture 6 19
Interaction in linear regression: Categorical and continuous variables, cont. 04/08/2021 Spring 2021 - Lecture 6 20
Interaction in linear regression: Continuous variables Using the SBP data Considering interaction between age and BMI Fitting a multiple linear regression model with interaction ??? = ?0+ ?1 ????? + ?2 ??? + ?3 ? ????? + ?4 ????? ??? Effect of age on mean SBP dependent on BMI Effect of BMI on mean SBP dependent on age Effect of smoking independent of both age and BMI 04/08/2021 Spring 2021 - Lecture 6 21
Interaction in linear regression: Continuous variables, cont. 04/08/2021 Spring 2021 - Lecture 6 22
Interaction in linear regression: Continuous variables, cont. Calculating effect of BMI for 45-year-olds 0.62 + 0.06 45 = 2.08 Mean SBP increased by 2.08 mmHg for every 1 kg/m2increase of BMI Calculating effect of BMI for 65-year-olds 0.62 + 0.06 65 = 3.28 Mean SBP increased by 3.28 mmHg for every 1 kg/m2increase of BMI Lager effect of BMI among 65-year-olds Calculating effect of age for different BMI values in a similar manner 04/08/2021 Spring 2021 - Lecture 6 23
References Laake P, Hjart ker A, Thelle DS, Veier d MB. Epidemiologiske og kliniske forskningsmetoder. Oslo: Gyldendal akademisk; 2007. https://www.med.uio.no/imb/forskning/publikasjoner/boker/2007/epidemiolgiske- kliniske-forskningsmetoder.html. StataCorp. 2017. Stata 15 Base Reference Manual. College Station, TX: Stata Press. Corraini P, Olsen M, Pedersen L, Dekkers OM, Vandenbroucke JP. Effect modification, interaction and mediation: an overview of theoretical insights for clinical investigators [published correction appears in Clin Epidemiol. 2018 Mar 01;10 :223] [published correction appears in Clin Epidemiol. 2019 Apr 09;11:245]. Clin Epidemiol. 2017;9:331-338. Published 2017 Jun 8. doi:10.2147/CLEP.S129728 Erratum: Effect modification, interaction and mediation: an overview of theoretical insights for clinical investigators [Corrigendum]. Clin Epidemiol. 2018;10:223. Published 2018 Mar 1. doi:10.2147/CLEP.S162236 Erratum: Effect modification, interaction and mediation: an overview of theoretical insights for clinical investigators [Corrigendum]. Clin Epidemiol. 2019;11:245. Published 2019 Apr 9. doi:10.2147/CLEP.S198519 04/08/2021 Spring 2021 - Lecture 6 24
