Fitting Models to Biological Data using Linear and ...
Fitting Models
to Biological Data
using Linear and
Nonlinear
Regression
A practical guide to
curve fitting
Harvey Motulsky &
Arthur Christopoulos
Copyright ? 2003 GraphPad Software, Inc. All rights reserved.
GraphPad Prism and Prism are registered trademarks of GraphPad Software, Inc.
GraphPad is a trademark of GraphPad Software, Inc.
Citation: H.J. Motulsky and A Christopoulos, Fitting models to biological data using
linear and nonlinear regression. A practical guide to curve fitting. 2003, GraphPad
Software Inc., San Diego CA, .
To contact GraphPad Software, email support@ or sales@.
Contents at a Glance
A. Fitting data with nonlinear regression.................................... 13
B. Fitting data with linear regression .......................................... 47
C. Models ....................................................................................58
D. How nonlinear regression works........................................... 80
E. Confidence intervals of the parameters .................................. 97
F. Comparing models ................................................................ 134
G. How does a treatment change the curve?.............................. 160
H. Fitting radioligand and enzyme kinetics data ....................... 187
I. Fitting dose-response curves .................................................256
J. Fitting curves with GraphPad Prism......................................296
3
Contents
Preface ........................................................................................................12
A. Fitting data with nonlinear regression.................................... 13
1.
An example of nonlinear regression ......................................................13
Example data ............................................................................................................................13
Step 1: Clarify your goal. Is nonlinear regression the appropriate analysis? .........................14
Step 2: Prepare your data and enter it into the program. ....................................................... 15
Step 3: Choose your model....................................................................................................... 15
Step 4: Decide which model parameters to fit and which to constrain..................................16
Step 5: Choose a weighting scheme ......................................................................................... 17
Step 6: Choose initial values..................................................................................................... 17
Step 7: Perform the curve fit and interpret the best-fit parameter values ............................. 17
2. Preparing data for nonlinear regression................................................19
Avoid Scatchard, Lineweaver-Burk and similar transforms whose goal is to create a
straight line ............................................................................................................................19
Transforming X values ............................................................................................................ 20
Don¡¯t smooth your data........................................................................................................... 20
Transforming Y values..............................................................................................................21
Change units to avoid tiny or huge values .............................................................................. 22
Normalizing ............................................................................................................................. 22
Averaging replicates ................................................................................................................ 23
Consider removing outliers ..................................................................................................... 23
3. Nonlinear regression choices ............................................................... 25
Choose a model for how Y varies with X................................................................................. 25
Fix parameters to a constant value? ....................................................................................... 25
Initial values..............................................................................................................................27
Weighting..................................................................................................................................27
Other choices ........................................................................................................................... 28
4. The first five questions to ask about nonlinear regression results ........ 29
Does the curve go near your data? .......................................................................................... 29
Are the best-fit parameter values plausible? .......................................................................... 29
How precise are the best-fit parameter values? ..................................................................... 29
Would another model be more appropriate? ......................................................................... 30
Have you violated any of the assumptions of nonlinear regression? .................................... 30
5. The results of nonlinear regression ...................................................... 32
Confidence and prediction bands ........................................................................................... 32
Correlation matrix ................................................................................................................... 33
Sum-of-squares........................................................................................................................ 33
R2 (Coefficient of Determination) ........................................................................................... 34
Does the curve systematically deviate from the data? ........................................................... 35
Could the fit be a local minimum? ...........................................................................................37
6. Troubleshooting ¡°bad¡± fits.................................................................... 38
Poorly-defined parameters...................................................................................................... 38
Model too complicated ............................................................................................................ 39
4
The model is ambiguous unless you share a parameter .........................................................41
Bad initial values...................................................................................................................... 43
Redundant parameters............................................................................................................ 45
Tips for troubleshooting nonlinear regression....................................................................... 46
B. Fitting data with linear regression .......................................... 47
7.
Choosing linear regression ................................................................... 47
The linear regression model.....................................................................................................47
Don¡¯t choose linear regression when you really want to compute a correlation coefficient .47
Analysis choices in linear regression ...................................................................................... 48
X and Y are not interchangeable in linear regression ............................................................ 49
Regression with equal error in X and Y .................................................................................. 49
Regression with unequal error in X and Y.............................................................................. 50
8. Interpreting the results of linear regression ......................................... 51
What is the best-fit line?........................................................................................................... 51
How good is the fit? ................................................................................................................. 53
Is the slope significantly different from zero? .........................................................................55
Is the relationship really linear? ..............................................................................................55
Comparing slopes and intercepts............................................................................................ 56
How to think about the results of linear regression............................................................... 56
Checklist: Is linear regression the right analysis for these data?............................................57
C. Models ....................................................................................58
9. Introducing models...............................................................................58
What is a model?...................................................................................................................... 58
Terminology............................................................................................................................. 58
Examples of simple models..................................................................................................... 60
10. Tips on choosing a model ......................................................................62
Overview .................................................................................................................................. 62
Don¡¯t choose a linear model just because linear regression seems simpler than nonlinear
regression.............................................................................................................................. 62
Don¡¯t go out of your way to choose a polynomial model ....................................................... 62
Consider global models ........................................................................................................... 63
Graph a model to understand its parameters......................................................................... 63
Don¡¯t hesitate to adapt a standard model to fit your needs ................................................... 64
Be cautious about letting a computer pick a model for you................................................... 66
Choose which parameters, if any, should be constrained to a constant value ...................... 66
11. Global models ....................................................................................... 67
What are global models? ..........................................................................................................67
Example 1. Fitting incomplete data sets. .................................................................................67
Example 2. The parameters you care about cannot be determined from one data set. ....... 68
Assumptions of global models ................................................................................................ 69
How to specify a global model................................................................................................. 70
12. Compartmental models and defining a model with a differential
equation ............................................................................................ 72
What is a compartmental model? What is a differential equation? .......................................72
Integrating a differential equation...........................................................................................73
The idea of numerical integration............................................................................................74
More complicated compartmental models..............................................................................77
5
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