Biomedicínský výzkum s podporou evropských zdrojů v nemocnicích (12. 4. 2012) Regression models in biomedical research RNDr. Karel Hrach, Ph. D.
Multiple regression („classical“)
Multiple regression (in Excel)
Multiple regression (in Excel)
Reduction of the model? Question: What if we omit the variable VĚK (Age)? Is the resulting model really BETTER than the previous one? There exists a model-building approach.
Stepwise regression (NOT in Excel) • type forward: adds the best candidating regressor so, that the new model is significantly better than the sub-model • type backward: removes the weekest regressor so, that the sub-model remains significant
Other types of regressors? • e. g. Y=blood pressure decrease (BPD) • X (X 1, …) might be nominal, e. g. „treatment“: X=1 … standard medication X=2 … experimantal medication X=3 … no medication (life-style change)
Other types of regressors? Y=α+βX+ε interpretation of β? • It should express the change of BPD, corresponding to the unit change of X (? ? ? )
Other types of regressors? • Solution = dummy regressors • e. g. (BPD example): let’s define X 1=1 (if X=1), X 1=0 otherwise (it is an indicator of the treatment n. 1) X 2=1 (if X=2), X 2=0 otherwise (it is an indicator of the treatment n. 2)
Other types of regressors? • Re-definition of the model Y=α+βX+ε : Y=β 0+β 1 X 1+β 2 X 2+ε with only these possible situations: X=1 … Y=β 0+β 1∙ 1+β 2∙ 0+ε =β 0+β 1+ε X=2 … Y=β 0+β 1∙ 0+β 2∙ 1+ε =β 0+β 2+ε X=3 … Y=β 0+β 1∙ 0+β 2∙ 0+ε =β 0+ε
Other types of regression? • „classical“=model for dependency of continuous variable(s) Y (Y 1, …) • dependent variable Y = binary (outcome yes/no) … logistic regr. • dependent variable Y = „survival“ … Cox regression (prop. hazards) • and other types (e. g. Poisson regr. )
Logistic (Logit) Regression • Simple case (i. e. one regressor X) model: =β 0+β 1 X+ε (i. e. as before) BUT Logit = β 0+β 1 X+ε Logit = ln(π/(1 - π)) π=probability of Y=1 (event) 1 -π =probability of Y=0
Logistic Regression Application: Data from the project „The use of diffusion tensor imaging in preoperative planning and intraoperative neuronavigation“ (Masaryk Hospital, dpt. of neurosurgery)
Logistic Regression The model found: logit = 4, 05– 0, 68∙(TTD+TR) logit… motor response stimulated? TTD… tumor-to-tract distance TR… thickness of the remnant
Probabilities of positive stimulation for each value of TTD+TR:
SURVIVAL ANALYSIS „Time-to-event“ variable and censoring
• SURVIVAL FUNCTION S: S (t ) = P (T ≥ t) • KAPLAN-MEIER ESTIMATE: SKM (t ) … cumulative relative frequency of surviving at the time t = time of event • LIFE-TABLE ESTIMATE : SLT (t ) … cumulative relative frequency of surviving inside given time-interval
Data (FZS UJEP, dpt. of physioth. ): group … =0 (mamma ablation) =1 (tumorectomy) 70 surv … time (months 1 -6) until the angle returns to the value of 70° 70 event … censoring (did it happen within 6 months or not? … 1/0)
The difference seems to be clear …
FW „R-project“ group coef 1. 33 e(coef) p 3. 79 0. 00068
The offer for co-operation with clinicians: Application of statistical methods (even „less-traditional“) SW available (Excel, R, STATISTICA)
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