Practical Biostatistics

Transcription

1 Practical Biostatistics Clinical Epidemiology, Biostatistics and Bioinformatics AMC Multivariable regression

2 Day 5 Recap Describing association: Correlation Parametric technique: Pearson (PMCC) Non-parametric: Spearman rank Predicting data: Linear regression Y=constant + b*x

3 Day 6 Multiple regression ASSOCIATIONS CROSSSECTIONAL LONGITUDINAL BIVARIATE MULTIVARIATE measurement level distribution DATA TYPE numerical continuous normal not-normal multiple regression binary balanced categorical nominal balanced ordinal balanced

4 Day 6 Multiple regression Contents Why? When? How? Key concepts SPSS menu SPSS output

5 Why perform Linear regression? Identify explanatory variables that are associated with an Outcome Variable: Is length related to lung function (FVC)? Is treatment related to outcome? Predict: Which weight do I expect for which length?

6 Why perform multiple linear regression? To what extent are multiple explanatory variables related to the outcome variable? Effect of treatment on outcome, adjusted for prognostic factors (e.g. age, sex)

7 Simple linear regression FVC at visit Length

8 Simple linear regression Y = a + bx Lung function = (0.098 * length) Lung function = (0.098*165) = y is the dependent, outcome or response variable - x is the independent, predictor or explanatory variable - a is the intercept, or constant: it is the value of y when x = 0 - b is the slope: it presents the amount by which y increases on average if we increase x by one unit

9 Simple linear regression FVC female: 2.24 FVC male: FVC at visit Male man Female vrouw Length

10 Simple linear regression 7,00 6,00 FVC female: 2.24 FVC male: 3.69 FVC at visit 0 5,00 4,00 3,00 Female Male 2,00 1,00 0, Sex

11 Simple linear regression Y = a + bx Lung function = *sex) Sex: value label: 0 = female 1 = male Lung function female: 2.24 Lung function male: 3.69

12 Linear regression FVC female: 2.24 FVC male: FVC at visit Male man Female vrouw Length

13 Linear regression FVC female: 2.24 FVC male: FVC at visit man Male vrouw Female Length

14 Why perform Multiple linear regression? Identify explanatory variables that are associated with an outcome variable: Which variables (length x1, sex x2, ) are related to lung function (y), after adjusting for the other xs? To what extent are multiple explanatory variables related to the outcome variable: Effect of treatment on outcome, adjusted for prognostic factors (e.g. age, sex) Predict: Predict lung function of males/females, adjusted for length

15 Multiple linear regression Y=a+b1x1+b2x2 Lung func. = intercept + (0.72*sex) + (0.064*length) Lung func. = (0.72*sex) + (0.064*length) Lung func. female = (0.064* length) Lung func. male = (0.064* length) Male, 165 cm => lung function = 3.25 Female, 165 cm => lung function = 2.53 Female, 175 cm => lung function = 2.85

16 Multiple linear regression Lung func. = (0.064*length) + (0.72*sex) Taking into account the length of a person (adjusted for length) the difference in predicted lung function between males and females is Without correction for length this difference was 1.45 Length is a confounder for the relationship between sex and lung function

17 Confounding Confounder: The association between the increase in length and the increase in lung function (b or slope) is similar for males and female, yet the intercept is NOT similar (a).

18 Confounding FVC female: 2.2 FVC male: Y = a + bx 4 Slope (b): similar Intercept (a): different FVC at visit Male man Female vrouw Length

19 Interaction: Interferences The association between the increase in length and the increase in lung function (b or slope) is NOT similar for males and females

20 Interaction

21 interaction Y = a + b1 X1 + b2 X2 + b3 X1*X2 X1 = length X2 = sex X12 = length * sex (interaction)

22 interaction Y = a + b1x1 + b2x2 + b3x1*x2 interaction X1 = length X2 = sex X1*2 = length * sex main effect Mind: interaction is only allowed in case BOTH main effects are included into the model as well

23 SPSS menu

24 multiple linear regression age & sex % fat SPSS-menu (1)

25 multiple linear regression age & sex % fat SPSS-menu (2)

26 multiple linear regression age and sex % fat SPSS-output (1) Model 1 2 Regression Residual Total Regression Residual Total a. Predictors: (Constant), AGE b. ANOVA c Sum of Squares df Mean Square F Sig a b Predictors: (Constant), AGE, SEX c. Dependent Variable: % FAT test H o : b 1 = b 2 = 0

27 Age and sex % fat SPSS-output (2) Model Summary Model 1 2 Std. Error Change Statistics Adjusted of the R Square Sig. F R R SquareR Square Estimate Change F Change df1 df2 Change,812 a,659,645 6,3898,659 44, ,000,828 b,686,657 6,2782,026 1, ,190 Predictors: a. (Constant), AGE Predictors: b. (Constant), AGE, SEX total R 2 = 69% age explains 66% of the variance in % fat. sex adds another 3% (non significant contribution)

28 Age and sex % fat SPSS-output (3) b coefficients Compare among themselves Model 1 2 (Constant) AGE (Constant) AGE SEX a. Dependent Variable: % FAT Coefficients a Unstandardized Coefficients Standardi zed Coefficien ts B Std. Error Beta t Sig prediction: y = 0,761 + (0.629 * age) + (-4,233 * sex) Interpretation Dependent of value codes

29 Categorical variables Sex, dichotomous 0,1 or 1,2 just put it into the model Site of onset (nominal) bulbar code 1 arm code 2 trunk code 3 leg code 4 SPSS takes leg (code 4) as 2 times arm (code 2), so you can NOT put it just into the model.

30 bulbar arm trunk leg Categorical variables site1 site2 site3 site4 Four categories: Create three dummy variables, code 0 of 1 Site1 => code 1 in case bulbar, code 0 in case other Site2 => code 1 in case arm, code 0 in case other Site3 => code 1 in case trunk, code 0 in case other Site4 => unnecessary, n-1 categories necessary Example: bulbar arm trunk leg Site Site Site

31 Categorical variables Model 1 Regression Residual Total ANOVA b Sum of Squares df Mean Square F Sig a a. Predictors: (Constant), arm, bulbar b. Dependent Variable: FVC at visit 0 Test complete model Arm, bulbar and leg

32 Site1 Categorical variables Coefficients a Model 1 (Constant) bulbar arm Unstandardized Coefficients a. Dependent Variable: FVC at visit 0 Standardi zed Coefficien ts B Std. Error Beta t Sig Onset bulbar: Lung function = => Onset arm: Lung function = => Onset leg: Lung function = => Onset trunk: (no patients with trunk) In case onset is bulbar lung function is different compared to a patient with onset leg, lung function for onset arm does not differ from onset leg

33 When is multiple linear regression allowed? Same restrictions as for univariate linear regression analyses linear relationship normal distribution constant variance Test assumptions: Analyses of the residuals: scatterplots

34 Before you build a model? Consider clinical/biological plausibility knowledge of confounders Knowledge of (patho)physiology Beware! generalizability? # Predictors vs. sample size - clinical validity/plausibility? - association =/= causation

35 Recap Why? Identify Explanatory Variables associated with Outcome Adjust effect for (other) prognostic factors Predict outcome with independent variables When? continuous outcome variable continuous and/or categorical explanatory variables How? SPSS linear regression Key concepts regression coefficients (intercept, slope), R 2, interaction, residuals, model building Assumptions linearity, distribution of residuals