9780195171808

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

by ;
  • ISBN13:

    9780195171808

  • ISBN10:

    0195171802

  • Format: Paperback
  • Copyright: 2004-05-27
  • Publisher: Oxford University Press

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Supplemental Materials

What is included with this book?

Summary

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.

Author Biography

Harvey Motulsky has been using nonlinear regression since 1981, while performing pharmacological research at the University of California, San Diego. He is now president of GraphPad Software Arthur Christopoulos, NHMRC Senior Research Fellow at the University of Melbourne

Table of Contents

Preface 12(1)
A. Fitting data with nonlinear regression 13(34)
1. An example of nonlinear regression
13(6)
Example data
13(1)
Step 1: Clarify your goal. Is nonlinear regression the appropriate analysis?
14(1)
Step 2: Prepare your data and enter it into the program
15(1)
Step 3: Choose your model
15(1)
Step 4: Decide which model parameters to fit and which to constrain
16(1)
Step 5: Choose a weighting scheme
17(1)
Step 6: Choose initial values
17(1)
Step 7: Perform the curve fit and interpret the best-fit parameter values
17(2)
2. Preparing data for nonlinear regression
19(6)
Avoid Scatchard, Lineweaver-Burk, and similar transforms whose goal is to create a straight line
19(1)
Transforming X values
20(1)
Don't smooth your data
20(1)
Transforming Y values
21(1)
Change units to avoid tiny or huge values
22(1)
Normalizing
22(1)
Averaging replicates
23(1)
Consider removing outliers
23(2)
3. Nonlinear regression choices
25(4)
Choose a model for how Y varies with X
25(1)
Fix parameters to a constant value?
25(2)
Initial values
27(1)
Weighting
27(1)
Other choices
28(1)
4. The first five questions to ask about nonlinear regression results
29(3)
Does the curve go near your data?
29(1)
Are the best-fit parameter values plausible?
29(1)
How precise are the best-fit parameter values?
29(1)
Would another model be more appropriate?
30(1)
Have you violated any of the assumptions of nonlinear regression?
30(2)
5. The results of nonlinear regression
32(6)
Confidence and prediction bands
32(1)
Correlation matrix
33(1)
Sum-of-squares
33(1)
R2 (coefficient of determination)
34(1)
Does the curve systematically deviate from the data?
35(2)
Could the fit be a local minimum?
37(1)
6. Troubleshooting "bad" fits
38(9)
Poorly defined parameters
38(1)
Model too complicated
39(2)
The model is ambiguous unless you share a parameter
41(2)
Bad initial values
43(2)
Redundant parameters
45(1)
Tips for troubleshooting nonlinear regression
46(1)
B. Fitting data with linear regression 47(11)
7. Choosing linear regression
47(4)
The linear regression model
47(1)
Don't choose linear regression when you really want to compute a correlation coefficient
47(1)
Analysis choices in linear regression
48(1)
X and Y are not interchangeable in linear regression
49(1)
Regression with equal error in X and Y
49(1)
Regression with unequal error in X and Y
50(1)
8. Interpreting the results of linear regression
51(7)
What is the best-fit line?
51(2)
How good is the fit?
53(2)
Is the slope significantly different from zero?
55(1)
Is the relationship really linear?
55(1)
Comparing slopes and intercepts
56(1)
How to think about the results of linear regression
56(1)
Checklist: Is linear regression the right analysis for these data?
57(1)
C. Models 58(22)
9. Introducing models
58(4)
What is a model?
58(1)
Terminology
58(2)
Examples of simple models
60(2)
10. Tips on choosing a model
62(5)
Overview
62(1)
Don't choose a linear model just because linear regression seems simpler than nonlinear regression
62(1)
Don't go out of your way to choose a polynomial model
62(1)
Consider global models
63(1)
Graph a model to understand its parameters
63(1)
Don't hesitate to adapt a standard model to fit your needs
64(2)
Be cautious about letting a computer pick a model for you
66(1)
Choose which parameters, if any, should be constrained to a constant value
66(1)
11. Global models
67(5)
What are global models?
67(1)
Example 1. Fitting incomplete data sets
67(1)
Example 2. The parameters you care about cannot be determined from one data set
68(1)
Assumptions of global models
69(1)
How to specify a global model
70(2)
12. Compartmental models and defining a model with a differential equation
72(8)
What is a compartmental model? What is a differential equation?
72(1)
Integrating a differential equation
73(1)
The idea of numerical integration
74(3)
More complicated compartmental models
77(3)
D. How nonlinear regression works 80(17)
13. Modeling experimental error
80(4)
Why the distribution of experimental error matters when fitting curves
80(1)
Origin of the Gaussian distribution
80(2)
From Gaussian distributions to minimizing sums-of-squares
82(1)
Regression based on nongaussian scatter
83(1)
14. Unequal weighting of data points
84(7)
Standard weighting
84(1)
Relative weighting (weighting by 1/Y2)
84(2)
Poisson weighting (weighting by 1/Y)
86(1)
Weighting by observed variability
86(1)
Error in both X and Y
87(1)
Weighting for unequal number of replicates
87(2)
Giving outliers less weight
89(2)
15. How nonlinear regression minimizes the sum-of-squares
91(6)
Nonlinear regression requires an iterative approach
91(1)
How the nonlinear regression method works
91(5)
Independent scatter
96(1)
E. Confidence intervals of the parameters 97(37)
16. Asymptotic standard errors and confidence intervals
97(7)
Interpreting standard errors and confidence intervals
97(1)
How asymptotic standard errors are computed
98(1)
An example
99(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 AICe
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 -- AICe 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 AIC~ 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 AICe
158(1)
Compare with t test
159(1)
G. How does a treatment change the curve? 160
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 logEC5o? 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 Ka 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 logEC5o 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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