Multivariable_Regression_Dec_2025 about reg
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about multivariate regression advances concepts
Multivariable_Regression_Dec_2025 about reg
- 2. Why do we need multivariable analysis? Reminder of confounders and adjusting for confounders The idea of a risk prediction (or risk factor) model Two of the main regression models linear and logistic Strategies of adjusting for confounders Strategies of building a risk prediction model Opportunities to see how this is used in Sports Medicine publications and examples Outline of Learning Objectives
- 3. Scene setting example Researchers looked at the association between active commuting and obesity (BMI) in mid-life Men who walk or cycle to work have a BMI (body mass index) of 1.4Kg/m2 less than those who drive For women it was 0.9 Kg/m2 less Are there any alternative explanations for this association?
- 4. active commuting BMI Confounders? A confounder is: • associated with the exposure of interest • independently associated with the disease outcome Age? Healthy lifestyle? Confounding… the issue of observational data
- 5. Multivariable analysis allows us to adjust for confounders Analysis depends on the type of outcome variable: Continuous multiple linear regression Binary multiple logistic regression Survival time Cox regression Multivariable (multivariate) analysis
- 6. y = β0 + β1x1 Remember the simple linear regression model: where y = continuous outcome, x = exposure variable β0 = constant (intercept) β1 = regression coefficient (slope) NB: if x1 is continuous or binary, β1 is amount y increases for a unit increase in x1 BMI = β0 + β1 x active commuting (yes = 1 versus no = 0) Extend this model quite naturally: y = β0 + β1x1 + β2x2 + … BMI =β0 + β1 x active commuting + β2 x age β1 is the effect of active commuting adjusted for age
- 7. Effect of active commuting with adjustment for confounders.. After adjustment for age and other things, men who use active transport have a weight 0.97Kg less than men who dont
- 8. Multiple linear regression in SPSS Other variables can be added to the independent box (note that for this option in SPSS they must be continuous or binary)
- 9. To predict blood pressure given waist to hip ratio and age: Blood pressure = 70.748 + 30.725 x whr + 0.594 x age Blood pressure increases by 30.725 mmHg per 1 unit increase in whr, after adjusting for age The effect of whr remains significant (p < 0.001) so is independent of age
- 10. Logistic regression For binary outcomes … Based on odds ratios …
- 11. Odds & Odds Ratios Odds of wheeze in exposed = 0.045 / 0.955 = 0.048 Odds of wheeze unexposed = 0.032 / 0.968 =0.033 OR = 0.048 = 1.45 0.033 wheeze no wheeze Total exposed 61 (4.5%) 1282 (95.5%) 1343 unexposed 185 (3.2%) 5627 (96.8%) 5812 Total 246 6909 7155 Association between exposure to parental tobacco smoke and wheezing in children:
- 12. In the logistic regression model, we model log odds: where log odds = log odds of outcome (or disease) x = binary exposure variable β0 = constant (log odds of outcome in unexposed group) β1 = measure of effect (log odds ratio) log odds = β0 + β1 x
- 13. Does the risk of wheeze differ between those exposed and unexposed to parental tobacco smoke? ie want to estimate odds ratio for wheeze comparing exposed with unexposed and obtain p-value Outcome: Must be coded 1 for event (+ve outcome), 0 for no event no wheeze = 0 wheeze = 1 Exposure: parent’s don’t smoke = 0 (unexposed/baseline gp) at least one parent smokes = 1 Example in SPSS: one binary exposure
- 14. Logistic regression in SPSS
- 15. Click on options and tick ‘CI for exp(β)’ Move outcome into ‘dependent’ box and binary exposure into ‘covariates’ box
- 16. Variables in the Equation .370 .151 6.005 1 .014 1.447 1.077 1.945 -3.415 .075 2088.807 1 .000 .033 anysmoke Constant Step 1 a B S.E. Wald df Sig. Exp(B) Lower Upper 95.0% C.I.for EXP(B) Variable(s) entered on step 1: anysmoke. a. Odds ratio of wheeze for exposed vs unexposed Odds of wheeze in unexposed 95% CI for OR P value for smoking effect (Wald test) As no other explanatory variables (exposures) in the model, the odds ratio above is the ‘unadjusted OR’ and the same as that computed from the cross-tab. The p-value above will be very similar to that obtained from chi-squared test earlier β0, ie log odds of wheeze in unexposed β1, ie log OR SPSS output and interpretation
- 17. Have put site in the model as well, and have chosen simple(first) contract for each so SPSS gives odds ratio for levels of site and smoking relative to the lowest level. Multiple logistic regression: other “confounders” can be added to the model for “adjustment” eg site
- 18. Deciding on confounding Variables in the Equation .560 .154 13.174 1 .000 1.751 1.294 2.369 -.894 .155 33.170 1 .000 .409 .302 .555 -3.339 .082 1639.188 1 .000 .035 anysmoke(1) site(1) Constant Step 1 a B S.E. Wald df Sig. Exp(B) Lower Upper 95.0% C.I.for EXP(B) Variable(s) entered on step 1: anysmoke, site. a. Interpretation: After adjusting for site, • the OR for smoking has increased from 1.45 (unadjusted) to 1.75 suggesting site is a confounder in this relation • We sometimes use a 10% change in the effect of interest to decide if there is important confounding. • In this case, the change is more than 10% so site is an important confounder. We should adjust for it. OR and 95% CI for being exposed to smoke vs unexposed, adjusted for site
- 19. Modelling strategies Strategy depends on study aim: Strategy 1: If study aim is to describe a single exposure- outcome relation, adjusted for potential confounding variables Strategy 2: If study aim is to establish which of a list of exposure variables are associated with outcome, ie to find the 'best' model to predict outcome
- 20. STRATEGY 1 Examples: Study to investigate whether physical activity reduces blood pressure Study of the relation between physical inactivity and depression. i.e. one exposure, one outcome
- 21. Want to get an ‘adjusted’ regression coefficient, and associated CI and p-value, for exposure of interest 1.Start by looking at unadjusted regression coefficient for exposure of interest 2.Add all a priori confounders (age, sex, …) to the model. What is the regression coefficient for the exposure of interest now? Strategy 1: Steps to follow
- 22. Strategy 1: Steps to follow (continued) 3. Does adding any other potential confounder to the model make a difference to the magnitude of the regression coefficient of interest (>10% change)? NO - present coefficient from step 2, adjusted for a priori confounders only. YES - include confounder in model and present coefficient for exposure of interest arising from this model.
- 23. Example: What are the risk factors for predicting inactivity? i.e. multiple exposures, one outcome Want to find the ‘best’ model: best at predicting dependent variable best at explaining variation in dependent variable Should aim to have a simple a model as possible STRATEGY 2
- 24. Multiple logistic regression: Used to look at predictors o f physical inactivity
- 25. Strategy 2: Model building No perfect answer Formal selection procedures (forward, backward and stepwise), particularly the automatic procedures in SPSS, should be used with caution • Can lead to very different results depending on procedure used • Avoids thinking about the specific research question and can hence lead to models that include ‘implausible’ variables and exclude ‘known risk factors’
- 26. Strategy 2: Model building 1. Use a systematic approach where you think about the particular problem 2. If the literature points to any established risk factors, these should probably be included. 3. Look at the univariate (unadjusted) significance of each predictor 4. Fit the significant ones together into a multivariate model 5. Remove the non-significant ones 6. Try adding back any that are omitted. 7. And repeat …..!
- 27. 1. Don’t include any explanatory variable whose relation with the outcome is implausible 2. Don’t include two explanatory variables that are closely related (collinear), eg both Townsend and Carstairs social deprivation score 3. Don’t try modelling large numbers of variables to small datasets. There should be at least 10 times as many observations as variables in the model 4. If you are unsure which of a number of models is best, see how alternative models effect the conclusions. General principles of model building