Entering Multidimensional Space: Multiple Regression

1. Statistics for Health Research Entering Multidimensional Space: Multiple Regression Peter T. Donnan Professor of Epidemiology and Biostatistics

2. Objectives of session Recognise the need for multiple regression Understand methods of selecting variables Understand strengths and weakness of selection methods Carry out Multiple Regression in SPSS and interpret the output

3. Why do we need multiple regression? Research is not as simple as effect of one variable on one outcome, Especially with observational data Need to assess many factors simultaneously; more realistic models

4. Consider Fitted line of y = a + b1x1 + b2x2 Dependent (y) Explanatory (x2) Explanatory (x1)

5. 3-dimensional scatterplot from SPSS of Min LDL in relation to baseline LDL and age

6. When to use multiple regression modelling (1) Assess relationship between two variables while adjusting or allowing for another variable Sometimes the second variable is considered a ‘nuisance’ factor Example: Physical Activity allowing for age and medications

7. When to use multiple regression modelling (2) In RCT whenever there is imbalance between arms of the trial at baseline in characteristics of subjects e.g. survival in colorectal cancer on two different randomised therapies adjusted for age, gender, stage, and co-morbidity at baseline

8. When to use multiple regression modelling (2) A special case of this is when adjusting for baseline level of the primary outcome in an RCT Baseline level added as a factor in regression model This will be covered in Trials part of the course

9. When to use multiple regression modelling (3) With observational data in order to produce a prognostic equation for future prediction of risk of mortality e.g. Predicting future risk of CHD used 10-year data from the Framingham cohort

10. When to use multiple regression modelling (4) With observational designs in order to adjust for possible confounders e.g. survival in colorectal cancer in those with hypertension adjusted for age, gender, social deprivation and co-morbidity

11. Definition of Confounding A confounder is a factor which is related to both the variable of interest (explanatory) and the outcome, but is not an intermediary in a causal pathway

12. Example of Confounding Lung Cancer Deprivation Smoking

13. But, also worth adjusting for factors only related to outcome Lung Cancer Deprivation Exercise

14. Not worth adjusting for intermediate factor in a causal pathway Exercise Blood viscosity Stroke In a causal pathway each factor is merely a marker of the other factors i.e correlated - collinearity

15. SPSS: Add both baseline LDL and age in the independent box in linear regression

16. Output from SPSS linear regression on Age at baseline

17. Output from SPSS linear regression on Baseline LDL

18. Output: Multiple regression R2 now improved to 13% Both variables still significant INDEPENDENTLY of each other

19. How do you select which variables to enter the model? • Usually consider what hypotheses are you testing? • If main ‘exposure’ variable, enter first and assess confounders one at a time • For derivation of CPR you want powerful predictors • Also clinically important factors e.g. cholesterol in CHD prediction • Significance is important but • It is acceptable to have an ‘important’ variable without statistical significance

20. How do you decide what variables to enter in model? Correlations? With great difficulty!

21. 3-dimensional scatterplot from SPSS of Time from Surgery in relation to Duke’s staging and age

22. Approaches to model building 1. Let Scientific or Clinical factors guide selection 2. Use automatic selection algorithms 3. A mixture of above

23. 1) Let Science or Clinical factors guide selection Baseline LDL cholesterol is an important factor determining LDL outcome so enter first Next allow for age and gender Add adherence as important? Add BMI and smoking?

24. 1) Let Science or Clinical factors guide selection Results in model of: Baseline LDL age and gender Adherence BMI and smoking Is this a ‘good’ model?

25. 1) Let Science or Clinical factors guide selection: Final Model Note three variables entered but not statistically significant

26. 1) Let Science or Clinical factors guide selection Is this the ‘best’ model? Should I leave out the non-significant factors (Model 2)? Adj R2 lower, F has increased and number of parameters is less in 2nd model. Is this better?

27. Kullback-Leibler Information f Kullback and Leibler (1951) quantified the meaning of ‘information’ – related to Fisher’s ‘sufficient statistics’ Basically we have reality f And a model g to approximate f So K-L information is I(f,g) g

28. Kullback-Leibler Information We want to minimise I (f,g) to obtain the best model over other models I (f,g) is the information lost or ‘distance’ between reality and a model so need to minimise:

29. Akaike’s Information Criterion It turns out that the function I(f,g) is related to a very simple measure of goodness-of-fit: Akaike’s Information Criterion or AIC

30. Selection Criteria • With a large number of factors type 1 error large, likely to have model with many variables • Two standard criteria: • 1) Akaike’s Information Criterion (AIC) • 2) Schwartz’s Bayesian Information Criterion (BIC) • Both penalise models with large number of variables if sample size is large

31. Akaike’s Information Criterion • Where p = number of parameters and -2*log likelihood is in the output • Hence AIC penalises models with large number of variables • Select model that minimises (-2LL+2p)

32. Generalized linear models • Unfortunately the standard REGRESSION in SPSS does not give these statistics • Need to use • Analyze • Generalized Linear Models…..

33. Generalized linear models. Default is linear • Add Min LDL achieved as dependent as in REGRESSION in SPSS • Next go to predictors…..

34. Generalized linear models: Predictors • WARNING! • Make sure you add the predictors in the correct box • Categorical in FACTORS box • Continuous in COVARIATES box

35. Generalized linear models: Model • Add all factors and covariates in the model as main effects

36. Generalized Linear Models Parameter Estimates Note identical to REGRESSION output

37. Generalized Linear Models Goodness-of-fit Note output gives log likelihood and AIC = 2835 (AIC = -2x-1409.6 +2x7= 2835) Footnote explains smaller AIC is ‘better’

38. Let Science or Clinical factors guide selection: ‘Optimal’ model • The log likelihood is a measure of GOODNESS-OF-FIT • Seek ‘optimal’ model that maximises the log likelihood or minimises the AIC Change is 1.6

39. 1) Let Science or Clinical factors guide selection Key points: Results demonstrate a significant association with baseline LDL, Age and Adherence Difficult choices with Gender, smoking and BMI AIC only changes by 1.6 when removed Generally changes of 4 or more in AIC are considered important

40. 1) Let Science or Clinical factors guide selection Key points: Conclude little to chose between models AIC actually lower with larger model and consider Gender, and BMI important factors so keep larger model but have to justify Model building manual, logical, transparent and under your control

41. 2) Use automatic selection procedures These are based on automatic mechanical algorithms usually related to statistical significance Common ones are stepwise, forward or backward elimination Can be selected in SPSS using ‘Method’ in dialogue box

42. 2) Use automatic selection procedures (e.g Stepwise) Select Method = Stepwise

43. 2) Use automatic selection procedures (e.g Stepwise) 1st step 2nd step Final Model

44. 2) Change in AIC with Stepwise selection Note: Only available from Generalized Linear Models

45. 2) Advantages and disadvantages of stepwise • Advantages • Simple to implement • Gives a parsimonious model • Selection is certainly objective • Disadvantages • Non stable selection – stepwise considers many models that are very similar • P-value on entry may be smaller once procedure is finished so exaggeration of p-value • Predictions in external dataset usually worse for stepwise procedures – tends to add bias

46. 2) Automatic procedures: Backward elimination • Backward starts by eliminating the least significant factor form the full model and has a few advantages over forward: • Modeller has to consider the ‘full’ model and sees results for all factors simultaneously • Correlated factors can remain in the model (in forward methods they may not even enter) • Criteria for removal tend to be more lax in backward so end up with more parameters

47. 2) Use automatic selection procedures (e.g Backward) Select Method = Backward

48. 2) Backward elimination in SPSS 1st step Gender removed 2nd step BMI removed Final Model

49. Summary of automatic selection • Automatic selection may not give ‘optimal’ model (may leave out important factors) • Different methods may give different results (forward vs. backward elimination) • Backward elimination preferred as less stringent • Too easily fitted in SPSS! • Model assessment still requires some thought

50. 3) A mixture of automatic procedures and self selection Use automatic procedures as a guide Think about what factors are important Add ‘important’ factors Do not blindly follow statistical significance Consider AIC