Regression models in bio-medical research RNDr. Karel Hrach, Ph.D. Biomedicínský výzkum s podporou evropských zdrojů v nemocnicích (12.4.2012) Regression models in bio-medical 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 (X1,…) 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 X1=1 (if X=1), X1=0 otherwise (it is an indicator of the treatment n.1) X2=1 (if X=2), X2=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=β0+β1X1+β2X2+ε with only these possible situations: X=1 … Y=β0+β1∙1+β2∙0+ε =β0+β1+ε X=2 … Y=β0+β1∙0+β2∙1+ε =β0+β2+ε X=3 … Y=β0+β1∙0+β2∙0+ε =β0+ε

Other types of regression? „classical“=model for dependency of continuous variable(s) Y (Y1,…) 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: =β0+β1X+ε (i.e.as before) BUT Logit = β0+β1X+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:

„Time-to-event“ variable and censoring 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) 70surv … time (months 1-6) until the angle returns to the value of 70° 70event … censoring (did it happen within 6 months or not? … 1/0)

The difference seems to be clear …

FW „R-project“ coef e(coef) p group 1.33 3.79 0.00068

The offer for co-operation with clinicians: Application of statistical methods (even „less-traditional“) SW available (Excel, R, STATISTICA)