1. Non-linear and Multiple Regression
Scientific Practice
Non-linear and Multiple Regression

2. Where We Are/Where We Are Going
Simple linear regression calculates the ‘straight line of best fit’
y = mx + c
The slope (m) can be tested statistically against zero (the Null Hypothesis)
There are two ways we can extend this…
‘curved’ relationships (non-linear regression)
eg y = ax2 + bx + c
note it’s still just y versus x
more than one variable (multiple regression)
eg y = ax + bz + c
one variable, y, influenced by two things, x and z

3. Non-linear Regression
Below is some data, plotted with a linear line of best fit
significant, but dots look like they fall on a curve rather than a straight line

4. Non-linear Regression
Below is some data, plotted with a 2nd order polynomial line of best fit
note a straight line is a 1st order polynomial
even better fit

5. Non-linear Regression
But don’t get carried away…
here’s a 6th order polynomial through the first seven points
perfect (but useless) fit!

6. Non-linear Regression in Minitab
Using the 10 pairs of data used earlier…
what does simple linear regression look like?
MTB > regress c1 1 c2
The regression equation is
Y = X
Predictor Coef SE Coef T P
Constant X
S = R-Sq = 95.0% R-Sq(adj) = 94.4%
Both intercept and slope significant

7. Non-linear Regression in Minitab
For a 2nd order polynomial, we include x2
MTB > let c3 = c2 * c2
MTB > regress c1 2 c2 c3
Regression Analysis: Y versus X, C3
The regression equation is
Y = X C3
Predictor Coef SE Coef T P
Constant X C
S = R-Sq = 99.4% R-Sq(adj) = 99.3%
Only the square term is significant! y=1.06x2

8. Non-linear Regression in Minitab
As the ‘x’ term is not sig, repeat without
MTB > regress c1 1 c3
Regression Analysis: Y versus C3
The regression equation is
Y = C3
Predictor Coef SE Coef T P
Constant C
S = R-Sq = 99.4% R-Sq(adj) = 99.3%
Note intercept not sig diff to 0, and the slope term has changed; y=1.13x2

9. Multiple Regression
This is where we have different things affecting the response variable
not just one thing represented in different ways
eg ventilation determined by concentrations of oxygen and carbon dioxide and body size
Here are some data (128 lines)
Minute Vol (L/min)
Oxygen conc (%)
Carbon dioxide (%)
Body Surface Area (m2)

10. Multiple Regression in Minitab
Same approach, really…
MTB > regress c1 3 c2 c3 c4
Regression Analysis: MV versus O2, CO2, BSA
The regression equation is
MV = O CO BSA
Predictor Coef SE Coef T P
Constant O CO
BSA
S = R-Sq = 30.8% R-Sq(adj) = 29.1%
Oxygen not sig, so leave it out and repeat the regression…

11. Multiple Regression in Minitab
MTB > regress c1 2 c3 c4
Regression Analysis: MV versus CO2, BSA
The regression equation is
MV = CO BSA
Predictor Coef SE Coef T P
Constant CO
BSA
S = R-Sq = 30.6% R-Sq(adj) = 29.5%
All terms now significant, so stop
Called Stepwise Multiple Regression

12. Multiple Regression Interpretation
The regression equation is
MV = CO BSA
Predictor Coef SE Coef T P
Constant CO
BSA
S = R-Sq = 30.6% R-Sq(adj) = 29.5%
Oxygen has no sig effect on Min Vol
Carbon dioxide does
1% increase increased MV by 2.87 L/min
BSA does
1 m2 increase increases MV by 20.1 L/min
About 30% of variation in MV explained

13. Summary
Non-linear and Multiple Regression are extensions of simple linear regression
In non-linear you can add extra terms that cover the same independent variable
allow ‘curved’ relationships to be explored
In multiple regression, additional terms represent other variables
eg MV determined by carbon dioxide and BSA
Stepwise regression is a process by which you eliminate non-sig terms until only the sig ones left
NB if intercept is non-sig, it is dropped at the end