GraphPad Curve Fitting Guide
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PRINCIPLES OF CURVE FITTING
Understanding mathematical models
What is a model?
Three example models
The problem with choosing models automatically
Advice: How to understand a model
Principles of linear regression
The goal of linear regression
How linear regression works
Comparing linear regression to correlation
Comparing linear regression to nonlinear regression
Advice: Look at the graph
Advice: Avoid Scatchard, Lineweaver-Burke and similar transforms
Getting started with nonlinear regression
Distinguishing nonlinear regression from other kinds of regression
The goal of nonlinear regression
The six steps of nonlinear regression
Preparing data for nonlinear regression
Don't fit a model to smoothed data
Reparameterizing an equation can help
Weighted nonlinear regression
The need for unequal weighting in nonlinear regression
Math theory of weighting
Don't use weighted regression with normalized data
What are the consequences of choosing the wrong weighting method
The many uses of global nonlinear regression
What is global nonlinear regression?
The uses of global nonlinear regression
Using global regression to fit incomplete datasets
Fitting models where the parameters are defined by multiple data sets
Advice: Don't use global regression if datasets use different units
Comparing fits of nonlinear models
Questions that can be answered by comparing models
Approaches to comparing models
How the F test works to compare models
How the AICc computations work
Outlier elimination and robust nonlinear regression
When to use automatic outlier removal
When to avoid automatic outlier removal
Outliers aren't always 'bad' points
The ROUT method of identifying outliers
Robust nonlinear regression
How nonlinear regression works
Why minimize the sum-of-squares?
How nonlinear regression works
Nonlinear regression with unequal weights
How standard errors are computed
How confidence intervals are computed
How confidence and prediction bands are computed
Replicates
How dependency is calculated
Who developed nonlinear regression?
CURVE FITTING WITH PRISM 7
Linear regression with Prism
How to: Linear regression
Finding the best-fit slope and intercept
Interpolating from a linear standard curve
Advice: When to fit a line with nonlinear regression
Confidence and prediction bands (linear regression)
Graphing tips: Linear regression
Results of linear regression
Slope and intercept
r2, a measure of goodness-of-fit of linear regression
Is the slope significantly different than zero?
Comparing slopes and intercepts
Runs test following linear regression
Analysis checklist: Linear regression
Questions and answers
Deming regression
Key concepts: Deming regression
How to: Deming regression
Q&A: Deming Regression
Analysis checklist: Deming regression
Interpolating from a standard curve
Key concept: Interpolating
How to interpolate
How to graph the interpolated values
Example: Interpolating from a sigmoidal standard curve
Equations used for interpolating
The results of interpolation
Interpolating with replicates in side-by-side subcolumns
Interpolating several data sets at once
When X values are logarithms
Analysis checklist: Interpolating
Reasons for blank (missing) interpolated results
Q&A: Interpolating
How Prism interpolates
Standard Addition Method
Nonlinear regression tutorials
Example: Fitting an enzyme kinetics curve
Example: Comparing two enzyme kinetics models
Example: Automatic outlier elimination (exponential decay)
Example: Global nonlinear regression (dose-response curves)
Example: Ambiguous fit (dose-response)
Nonlinear regression with Prism
How to fit a model with Prism
Which choices are essential?
Nonlinear regression choices
Fit tab
Compare tab
Constrain tab
Weights tab
Initial values tab
Range tab
Output tab
Confidence tab
Diagnostics tab
Flags tab
Graphing tips: Nonlinear regression
Graphing best-fit curves
Graphing confidence and prediction bands
Adding the equation to the graph
Graphing outliers
Residual plot
Interpreting nonlinear regression results
Interpreting results: Nonlinear regression
Best-fit parameter values
Standarad error of parameters
Confidence intervals of parameters
Normality tests of residuals
R squared
Sum-of-squares
Standard deviation of the residuals
Why Prism doesn't report the chi-square of the fit
Runs test
Replicates test
Dependency of each parameter
Covariance matrix
Confidence and prediction bands
Hougaard's measure of skewness
Test for appropriate weighting/homoscedasticity
Could the fit be a local minimum?
Outliers
Troubleshooting nonlinear regression
Why results can differ in various Prism versions
Interpreting results: Comparing models
Interpreting comparison of models
Interpreting the extra sum-of-squares F test
Interpreting AIC model comparison
How Prism compares models when outliers are eliminated
Interpreting the adjusted R2
Comparing fits with ANOVA
Analysis checklists: Nonlinear regression
Analysis checklist: Fitting a model
Analysis checklist: Comparing nonlinear fits
Analysis checklist: Interpolating from a standard curve
Error messages from nonlinear regression
"Bad initial values"
"Interrupted"
"Not converged"
ambiguous
"Ambiguous"
"Hit constraint"
"Don't fit"
"Too few points"
"Perfect fit"
"Impossible weights"
"Equation not defined"
"Can't calculate"
Models (equations) built-in to Prism
Dose-response - Key concepts
What are dose-response curves?
The terms "agonist", "antagonist", and "normalized" in equation names
Converting concentration to log(concentration)
The EC50
Confidence intervals of the EC50
Hill slope
Choosing a dose-response equation
Pros and cons of normalizing the data
The term "logistic"
50% of what? Relative vs absolute IC50
Fitting the absolute IC50
Incomplete dose-respone curves
Troubleshooting fits of dose-response curves
Dose-response - Stimulation
Equation: log(agonist) vs. response
Equation: log(agonist) vs. response -- Variable slope
Equation: log(agonist) vs. normalized response
Equation: log(agonist) vs. normalized response -- Variable slope
Equation: [Agonist] vs. response
Equation: [Agonist] vs. response -- Variable slope
Equation: [Agonist] vs. normalized response
Equation: [Agonist] vs. normalized response -- Variable slope
Dose-response - Inhibition
Equation: log(inhibitor) vs. response
Equation: log(inhibitor) vs. response -- Variable slope
Equation: log(inhibitor) vs. normalized response
Equation: log(inhibitor) vs. normalized response -- Variable slope
Equation: [Inhibitor] vs. response
Equation: [Inhibitor] vs. response -- Variable slope
Equation: [Inhibitor] vs. normalized response
Equation: [Inhibitor] vs. normalized response -- Variable slope
Dose-response -- Special
Asymmetrical (five parameter)
Equation: Biphasic dose-response
Equation: Bell-shaped dose-response
Equation: Operational model - Depletion
Equation: Operational model - Partial agonist
Equation: Gaddum/Schild EC50 shift
Equation: EC50 shift
Equation: Allosteric EC50 shift
Equation: ECanything
Receptor binding - Key concepts
Law of mass action
Nonspecific binding
Ligand depletion
The radioactivity web calculator
Receptor binding - Saturation binding
Key concepts: Saturation binding
Equation: One site -- Total binding
Equation: One site -- Fit total and nonspecific binding
Equation: One site -- Total, accounting for ligand depletion
Equation: One site -- Specific binding
Binding potential
Equation: Two sites -- Specific binding only
Equation: Two sites -- Fit total and nonspecific binding
Equation: One site with allosteric modulator
Equation: Specific binding with Hill slope
Receptor binding - Competitive binding
Key concepts: Competitive binding
Equation: One site - Fit Ki
Equation: One site - Fit logIC50
Equation: Two sites - Fit Ki
Equation: Two sites - Fit logIC50
Equation: One site - Ligand depletion
Equation: One site - Homologous
Equation: Allosteric modulator
Receptor binding - Kinetics
Key concepts: Kinetics of binding
Equation: Dissociation kinetics
Equation: Association kinetics (one ligand concentration)
Equation: Association kinetics (two ligand concentrations)
Equation: Association then dissociation
Equation: Kinetics of competitive binding
Enzyme kinetics -- Key concepts
Key concepts: Terminology
Key concepts: Assumptions
Enzyme kinetics - Subtrate vs. velocity
Key concepts: Substrate vs. velocity
Equation: Michaelis-Menten model
Equation: Determine kcat
Equation: Allosteric sigmoidal
Enzyme kinetics -- Inhibition
Key concepts: Enzyme inhibition
Equation: Competitive inhibition
Equation: Noncompetitive inhibition
Equation: Uncompetitive inhibition
Equation: Mixed-model inhibition
Equation: Substrate inhibition
Equation: Tight inhibition (Morrison equation)
Comparing models of enzyme inhibition
Exponential
Key concepts: Exponential equations
Key concepts: Derivation of exponential decay
Equation: One phase decay
Equation: Plateau followed by one phase decay
Equation: Two phase decay
Equation: Three phase decay
Equation: One phase association
Equation: Plateau followed by one phase association
Equation: Two phase association
Equation: Exponential growth
Lines
Key concepts: Fitting lines
Equation: Fitting a straight line with nonlinear regression
Equation: Line through origin
Equation: Segmental linear regression
Equation: Fitting a straight line on a semi-log or log-log graph
Equation: Fitting a straight line on a graph with a probability axis
Polynomial
Key concepts: Polynomial
Centered polynomial equations
Equations: Polynomial models
Gaussian
Key concepts: Gaussian
Equation: Gaussian distribution
Equation: Log Gaussian distribution
Equation: Cumulative Gaussian distribution
Equation: Lorentzian
Sine waves
Standard sine wave
Damped sine wave
Sinc wave
Sine wave with nonzero baseline
Classic equations from prior versions of Prism
Equation: One site binding (hyperbola)
Equation:Two site binding
Equation: Sigmoidal dose-response
Equation: Sigmoidal dose-response (variable slope)
Equation: One site competition
Equation: Two site competition
Equation: Boltzmann sigmoid
Equation: One phase exponential decay
Equation: Two phase exponential decay
Equation: One phase exponential association
Equation: Two phase exponential association
Equation: Exponential growth
Equation: Power series
Equation: Sine wave
Equation: Gaussian distribution
Entering a user-defined model into Prism
Overview: User-defined equations
How to: Enter a new equation
How to: Clone an equation
How to: Manage your list of equations
Syntax of user-defined equations
Multiline models
Limitations when entering equations
Entering a differential equation
Entering an implicit equation
Available functions for user-defined equations
Fitting different segments of the data to different models
Fitting different models to different data sets
Column constants
Defining equation with two (or more) independent variables
Reparameterizing an equation
Rules for initial values
Default constraints
Reporting transforms of parameters
Plotting a function
How to: Plot a function
Plotting t, z, F or chi-square distributions
Plotting a binomial or Poisson distribution
Fitting a curve without a model
Spline and Lowess curves
Using nonlinear regression with an empirical model
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