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

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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

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