Fitting Models to Biological Data Using Linear and Nonlinear Regression A Practical Guide to Curve Fitting

, by Motulsky, Harvey; Christopoulos, Arthur

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ISBN: 9780195171792 | 0195171799
Cover: Hardcover
Copyright: 5/27/2004

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Most biologists use nonlinear regression more than any other statistical technique, but there are very few places to learn about curve-fitting. This book, by the author of the very successful Intuitive Biostatistics, addresses this relatively focused need of an extraordinarily broad range ofscientists.

Preface

(1)

A. Fitting data with nonlinear regression

(34)

1. An example of nonlinear regression

(6)

Example data

(1)

Step is Clarify your goal. Is nonlinear regression the appropriate analysis?

(1)

Step 2: Prepare your data and enter it into the program

(1)

Step 3: Choose your model

(1)

Step 4: Decide which model parameters to fit and which to constrain

(1)

Step 5: Choose a weighting scheme

(1)

Step 6: Choose initial values

(1)

Step 7: Perform the curve fit and interpret the best-fit parameter values

(2)

2. Preparing data for nonlinear regression

(6)

Avoid Scatchard, Lineweaver-Burk, and similar transforms whose goal is to create a straight line

(1)

Transforming X values

(1)

Don't smooth your data

(1)

Transforming Y values

(1)

Change units to avoid tiny or huge values

(1)

Normalizing

(1)

Averaging replicates

(1)

Consider removing outliers

(2)

3. Nonlinear regression choices

(4)

Choose a model for how Y varies with X

(1)

Fix parameters to a constant value?

(2)

Initial values

(1)

Weighting

(1)

Other choices

(1)

4. The first five questions to ask about nonlinear regression results

(3)

Does the curve go near your data?

(1)

Are the best-fit parameter values plausible?

(1)

How precise are the best-fit parameter values?

(1)

Would another model be more appropriate?

(1)

Have you violated any of the assumptions of nonlinear regression?

(2)

5. The results of nonlinear regression

(6)

Confidence and prediction bands

(1)

Correlation matrix

(1)

Sum-of-squares

(1)

R² (coefficient of determination)

(1)

Does the curve systematically deviate from the data?

(2)

Could the fit be a local minimum?

(1)

6. Troubleshooting "bad" fits

(9)

Poorly defined parameters

(1)

Model too complicated

(2)

The model is ambiguous unless you share a parameter

(2)

Bad initial values

(2)

Redundant parameters

(1)

Tips for troubleshooting nonlinear regression

(1)

B. Fitting data with linear regression

(11)

7. Choosing linear regression

(4)

The linear regression model

(1)

Don't choose linear regression when you really want to compute a correlation coefficient

(1)

Analysis choices in linear regression

(1)

X and Y are not interchangeable in linear regression

(1)

Regression with equal error in X and Y

(1)

Regression with unequal error in X and Y

(1)

8. Interpreting the results of linear regression

(7)

What is the best-fit line?

(2)

How good is the fit?

(2)

Is the slope significantly different from zero?

(1)

Is the relationship really linear?

(1)

Comparing slopes and intercepts

(1)

How to think about the results of linear regression

(1)

Checklist: Is linear regression the right analysis for these data?

(1)

C. Models

(22)

9. Introducing models

(4)

What is a model?

(1)

Terminology

(2)

Examples of simple models

(2)

10. Tips on choosing a model

(5)

Overview

(1)

Don't choose a linear model just because linear regression seems simpler than nonlinear regression

(1)

Don't go out of your way to choose a polynomial model

(1)

Consider global models

(1)

Graph a model to understand its parameters

(1)

Don't hesitate to adapt a standard model to fit your needs

(2)

Be cautious about letting a computer pick a model for you

(1)

Choose which parameters, if any, should be constrained to a constant value

(1)

11. Global models

(5)

What are global models?

(1)

Example 1. Fitting incomplete data sets

(1)

Example 2. The parameters you care about cannot be determined from one data set

(1)

Assumptions of global models

(1)

How to specify a global model

(2)

12. Compartmental models and defining a model with a differential equation

(8)

What is a compartmental model? What is a differential equation?

(1)

Integrating a differential equation

(1)

The idea of numerical integration

(3)

More complicated compartmental models

(3)

D. How nonlinear regression works

(17)

13. Modeling experimental error

(4)

Why the distribution of experimental error matters when fitting curves

(1)

Origin of the Gaussian distribution

(2)

From Gaussian distributions to minimizing sums-of-squares

(1)

Regression based on nongaussian scatter

(1)

14. Unequal weighting of data points

(7)

Standard weighting

(1)

Relative weighting (weighting by 1/Y²)

(2)

Poisson weighting (weighting by 1/Y)

(1)

Weighting by observed variability

(1)

Error in both X and Y

(1)

Weighting for unequal number of replicates

(2)

Giving outliers less weight

(2)

15. How nonlinear regression minimizes the sum-of-squares

(6)

Nonlinear regression requires an iterative approach

(1)

How the nonlinear regression method works

(5)

Independent scatter

(1)

E. Confidence intervals of the parameters

(37)

16. Asymptotic standard errors and confidence intervals

(7)

Interpreting standard errors and confidence intervals

(1)

How asymptotic standard errors are computed

(1)

An example

(1)

Because asymptotic confidence intervals are always symmetrical, it matters how you express your model

100

(2)

Problems with asymptotic standard errors and confidence intervals

102

(1)

What if your program reports "standard deviations" instead of "standard errors"?

102

(1)

How to compute confidence intervals from standard errors

103

(1)

17. Generating confidence intervals by Monte Carlo simulations

104

(5)

An overview of confidence intervals via Monte Carlo simulations

104

(1)

Monte Carlo confidence intervals

104

(3)

Perspective on Monte Carlo methods

107

(1)

How to perform Monte Carlo simulations with Prism

107

(1)

Variations of the Monte Carlo method

108

(1)

18. Generating confidence intervals via model comparison

109

(9)

Overview on using model comparison to generate confidence intervals

109

(1)

A simple example with one parameter

109

(3)

Confidence interval for the sample data with two parameters

112

(1)

Using model comparison to generate a confidence contour for the example data

112

(3)

Converting the confidence contour into confidence intervals for the parameters

115

(1)

How to use Excel's solver to adjust the value of a parameter to get the desired sum-of-squares

116

(1)

More than two parameters

117

(1)

19. Comparing the three methods for creating confidence intervals

118

(10)

Comparing the three methods for our first example

118

(1)

A second example. Enzyme kinetics

119

(4)

A third example

123

(4)

Conclusions

127

(1)

20. Using simulations to understand confidence intervals and plan experiments

128

(6)

Example 1. Should we express the middle of a dose-response curve as EC50 or log(EC50)?

128

(1)

Example simulation 2. Exponential decay

129

(2)

How to generate a parameter distribution with Prism

131

(3)

F. Comparing models

134

(26)

21. Approach to comparing models

134

(4)

Why compare models?

134

(1)

Before you use a statistical approach to comparing models

134

(1)

Statistical approaches to comparing models

135

(3)

22. Comparing models using the extra sum-of-squares F test

138

(5)

Introducing the extra sum-of-squares F test

138

(1)

The F test is for comparing nested models only

138

(1)

How the extra sum-of-squares F test works

139

(3)

How to determine a P value from F

142

(1)

23. Comparing models using Akaike's Information Criterion (AIC)

143

(6)

Introducing Akaike's Information Criterion (AIC)

143

(1)

How AIC compares models

143

(1)

A second-order (corrected) AIC

144

(1)

The change in AICc tells you the likelihood that a model is correct

145

(1)

The relative likelihood or evidence ratio

146

(1)

Terminology to avoid when using AICc

147

(1)

How to compare models with AICc by hand

147

(1)

One-way ANOVA by AICc

148

(1)

24. How should you compare models -- AICc or F test?

149

(3)

A review of the approaches to comparing models

149

(1)

Pros and cons of using the F test to compare models

149

(1)

Pros and cons of using AICc to compare models

150

(1)

Which method should you use?

151

(1)

25. Examples of comparing the fit of two models to one data set

152

(5)

Example 1. Two-site competitive binding model clearly better

152

(2)

Example 2: Two-site binding model doesn't fit better

154

(2)

Example 3. Can't get a two-site binding model to fit at all

156

(1)

26. Testing whether a parameter differs from a hypothetical value

157

(3)

Example. Is the Hill slope factor statistically different from 1.0?

157

(1)

Compare models with the F test

157

(1)

Compare models with AICc

158

(1)

Compare with t test

159

(1)

G. How does a treatment change the curve?

160

(27)

27. Using global fitting to test a treatment effect in one experiment

160

(6)

Does a treatment change the EC50?

160

(3)

Does a treatment change the dose-response curve?

163

(3)

28. Using two-way ANOVA to compare curves

166

(5)

Situations where curve fitting isn't helpful

166

(1)

Introduction to two-way ANOVA

166

(1)

How ANOVA can compare "curves"

167

(1)

Post-tests following two-way ANOVA

168

(2)

The problem with using two-way ANOVA to compare curves

170

(1)

29. Using a paired t test to test for a treatment effect in a series of matched experiments

171

(3)

The advantage of pooling data from several experiments

171

(1)

An example. Does a treatment change logEC50? Pooling data from three experiments

171

(1)

Comparing via paired t test

172

(1)

Why the paired t test results don't agree with the individual comparisons

173

(1)

30. Using global fitting to test for a treatment effect in a series of matched experiments

174

(7)

Why global fitting?

174

(1)

Setting up the global model

174

(1)

Fitting the model to our sample data

175

(2)

Was the treatment effective? Fitting the null hypothesis model

177

(4)

31. Using an unpaired t test to test for a treatment effect in a series of unmatched experiments

181

(2)

An example

181

(1)

Using the unpaired t test to compare best-fit values of Vmax

181

(2)

32. Using global fitting to test for a treatment effect in a series of unmatched experiments

183

(4)

Setting up a global fitting to analyze unpaired experiments

183

(1)

Fitting our sample data to the global model

184

(1)

Comparing models with an F test

185

(1)

Comparing models with AICc

186

(1)

Reality check

186

(1)

H. Fitting radioligand and enzyme kinetics data

187

(69)

33. The law of mass action

187

(5)

What is the law of mass action?

187

(1)

The law of mass action applied to receptor binding

187

(1)

Mass action model at equilibrium

188

(1)

Fractional occupancy predicted by the law of mass action at equilibrium

189

(1)

Assumptions of the law of mass action

190

(1)

Hyperbolas, isotherms, and sigmoidal curves

191

(1)

34. Analyzing radioligand binding data

192

(2)

Introduction to radioligand binding

192

(1)

Nonspecific binding

192

(1)

Ligand depletion

193

(1)

35. Calculations with radioactivity

194

(5)

Efficiency of detecting radioactivity

194

(1)

Specific radioactivity

194

(1)

Calculating the concentration of the radioligand

195

(1)

Radioactive decay

195

(1)

Counting errors and the Poisson distribution

196

(1)

The GraphPad radioactivity web calculator

197

(2)

36. Analyzing saturation radioligand binding data

199

(12)

Introduction to saturation binding experiments

199

(1)

Fitting saturation binding data

199

(5)

Checklist for saturation binding

204

(1)

Scatchard plots

205

(3)

Analyzing saturation binding with ligand depletion

208

(3)

37. Analyzing competitive binding data

211

(11)

What is a competitive binding curve?

211

(2)

Competitive binding data with one class of receptors

213

(2)

Shallow competitive binding curves

215

(4)

Competitive binding with two receptor types (different Kd for hot ligand)

219

(1)

Heterologous competitive binding with ligand depletion

220

(2)

38. Homologous competitive binding curves

222

(11)

Introducing homologous competition

222

(1)

Theory of homologous competition binding

223

(1)

Why homologous binding data can be ambiguous

223

(1)

Using global curve fitting to analyze homologous (one site) competition data

224

(2)

Analyzing homologous (one site) competition data without global curve fitting

226

(3)

Homologous competitive binding with ligand depletion

229

(2)

Fitting homologous competition data (two sites)

231

(2)

39. Analyzing kinetic binding data

233

(12)

Dissociation ("off rate") experiments

233

(1)

Association binding experiments

234

(2)

Fitting a family of association kinetic curves

236

(2)

Globally fitting an association curve together with a dissociation curve

238

(2)

Analysis checklist for kinetic binding experiments

240

(1)

Using kinetic data to test the law of mass action

241

(2)

Kinetics of competitive binding

243

(2)

40. Analyzing enzyme kinetic data

245

(11)

Introduction to enzyme kinetics

245

(3)

How to determine V. and Km

248

(1)

Comparison of enzyme kinetics with radioligand binding

249

(1)

Displaying enzyme kinetic data on a Lineweaver- Burk plot

250

(1)

Allosteric enzymes

251

(1)

Enzyme kinetics in the presence of an inhibitor

251

(5)

I. Fitting dose-response curves

256

(40)

41. Introduction to dose-response curves

256

(10)

What is a dose-response curve?

256

(3)

The equation for a dose-response curve

259

(1)

Other measures of potency

260

(1)

Dose-response curves where X is concentration, not log of concentration

261

(2)

Why you should fit the logEC50 rather than EC50

263

(1)

Decisions when fitting sigmoid dose-response curves

264

(1)

Checklist: Interpreting a dose-response curve

265

(1)

42. The operational model of agonist action

266

(10)

Limitations of dose-response curves

266

(1)

Derivation of the operational model

266

(2)

Shallower and steeper dose-response curves

268

(1)

Designing experiments to fit to the operational model

269

(1)

Fitting the operational model to find the affinity and efficacy of a full agonist

270

(3)

Fitting the operational model to find the affinity and efficacy of a partial agonist

273

(3)

43. Dose-response curves in the presence of antagonists

276

(14)

Competitive antagonists

276

(4)

Using global fitting to fit a family of dose-response curves to the competitive interaction model

280

(3)

Fitting agonist EC50 values to the competitive interaction model

283

(3)

Antagonist inhibition curves

286

(4)

44. Complex dose-response curves

290

(6)

Asymmetric dose-response curves

290

(1)

Bell-shaped dose-response curves

291

(4)

Biphasic dose-response curves

295

(1)

J. Fitting curves with GraphPad Prism

296

(52)

45. Nonlinear regression with Prism

296

(2)

Using Prism to fit a curve

296

(1)

Which choices are most fundamental when fitting curves?

296

(1)

Prism's nonlinear regression error messages

297

(1)

46. Constraining and sharing parameters

298

(4)

The constraints tab of the nonlinear regression parameters dialog

298

(1)

Constraining to a constant value

298

(2)

Data set constants

300

(1)

Constrain to a range of values

301

(1)

Shared parameters (global fitting)

301

(1)

47. Prism's nonlinear regression dialog

302

(10)

The equation tab

302

(1)

Comparison tab

303

(2)

Initial values tab

305

(1)

Constraints for nonlinear regression

306

(1)

Weighting tab

306

(3)

Output tab

309

(1)

Range tab

310

(1)

Default preferences for nonlinear regression

311

(1)

48. Classic nonlinear models built into Prism

312

(10)

Equilibrium binding

312

(3)

Dose-response

315

(2)

Exponential

317

(2)

Other classic equations

319

(3)

49. Importing equations and equation libraries

322

(2)

Selecting from the equation library

322

(1)

Adding equations to the equation library

322

(1)

Importing equations

323

(1)

50. Writing user-defined models in Prism

324

(10)

What kinds of equations can you enter?

324

(1)

Equation syntax

324

(1)

Available functions for user-defined equations

325

(3)

Using the IF function

328

(1)

How to fit different portions of the data to different equations

328

(2)

How to define different models for different data sets

330

(1)

Defining rules for initial values and constraints

331

(1)

Managing your list of equations

332

(1)

Modifying equations

332

(2)

51. Linear regression with Prism

334

(5)

Entering data for linear regression

334

(1)

Choosing a linear regression analysis

334

(2)

Default preferences for linear regression

336

(1)

Using nonlinear regression to fit linear data

336

(1)

Deming (Model II) linear regression

337

(1)

Inverse linear regression with Prism

338

(1)

52. Reading unknowns from standard curves

339

(5)

Introduction to standard curves

339

(1)

Determining unknown concentrations from standard curves

340

(1)

Standard curves with replicate unknown values

341

(1)

Potential problems with standard curves

342

(2)

53. Graphing a family of theoretical curves

344

(2)

Creating a family of theoretical curves

344

(2)

54. Fitting curves without regression

346

(2)

Introducing spline and lowess

346

(1)

Spline and lowess with Prism

346

(2)

Annotated bibliography

348

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Fitting Models to Biological Data Using Linear and Nonlinear Regression A Practical Guide to Curve Fitting

Summary

Author Biography

Table of Contents

Supplemental Materials