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

by Motulsky, Harvey; Christopoulos, Arthur

• ISBN13: 9780195171808
• ISBN10: 0195171802
• Format: Paperback
• Copyright: 2004-05-27
• Publisher: Oxford University Press

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