1. Regression models in bio-medical research RNDr. Karel Hrach, Ph.D.
Biomedicínský výzkum s podporou evropských zdrojů v nemocnicích ( )
Regression models in bio-medical research
RNDr. Karel Hrach, Ph.D.

2. Multiple regression („classical“)

3. Multiple regression (in Excel)

4. Multiple regression (in Excel)

5. 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.

6. 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

7. 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)

8. 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 (???)

9. 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)

10. 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

11. 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.)

12. 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

13. Logistic Regression Application: Data from the project
„The use of diffusion tensor imaging in preoperative planning and intraoperative neuronavigation“
(Masaryk Hospital, dpt. of neurosurgery)

14. 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

15. Probabilities of positive stimulation for each value of TTD+TR:

16. „Time-to-event“ variable and censoring
SURVIVAL ANALYSIS
„Time-to-event“ variable and censoring

17. 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

22. 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)

23. The difference seems to be clear …

24. FW „R-project“
coef e(coef) p
group

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