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| Preface | |
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| Fitting data with nonlinear regression | |
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| An example of nonlinear regression | |
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| Example data | |
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| Clarify your goal. Is nonlinear regression the appropriate analysis? | |
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| Prepare your data and enter it into the program | |
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| Choose your model | |
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| Decide which model parameters to fit and which to constrain | |
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| Choose a weighting scheme | |
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| Choose initial values | |
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| Perform the curve fit and interpret the best-fit parameter values | |
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| Preparing data for nonlinear regression | |
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| Avoid Scatchard, Lineweaver-Burk, and similar transforms whose goal is to create a straight line | |
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| Transforming X values | |
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| Don't smooth your data | |
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| Transforming Y values | |
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| Change units to avoid tiny or huge values | |
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| Normalizing | |
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| Averaging replicates | |
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| Consider removing outliers | |
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| Nonlinear regression choices | |
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| Choose a model for how Y varies with X | |
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| Fix parameters to a constant value? | |
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| Initial values | |
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| Weighting | |
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| Other choices | |
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| The first five questions to ask about nonlinear regression results | |
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| Does the curve go near your data? | |
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| Are the best-fit parameter values plausible? | |
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| How precise are the best-fit parameter values? | |
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| Would another model be more appropriate? | |
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| Have you violated any of the assumptions of nonlinear regression? | |
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| The results of nonlinear regression | |
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| Confidence and prediction bands | |
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| Correlation matrix | |
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| Sum-of-squares | |
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| R[superscript 2] (coefficient of determination) | |
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| Does the curve systematically deviate from the data? | |
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| Could the fit be a local minimum? | |
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| Troubleshooting "bad" fits | |
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| Poorly defined parameters | |
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| Model too complicated | |
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| The model is ambiguous unless you share a parameter | |
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| Bad initial values | |
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| Redundant parameters | |
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| Tips for troubleshooting nonlinear regression | |
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| Fitting data with linear regression | |
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| Choosing linear regression | |
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| The linear regression model | |
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| Don't choose linear regression when you really want to compute a correlation coefficient | |
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| Analysis choices in linear regression | |
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| X and Y are not interchangeable in linear regression | |
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| Regression with equal error in X and Y | |
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| Regression with unequal error in X and Y | |
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| Interpreting the results of linear regression | |
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| What is the best-fit line? | |
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| How good is the fit? | |
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| Is the slope significantly different from zero? | |
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| Is the relationship really linear? | |
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| Comparing slopes and intercepts | |
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| How to think about the results of linear regression | |
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| Checklist: Is linear regression the right analysis for these data? | |
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| Models | |
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| Introducing models | |
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| What is a model? | |
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| Terminology | |
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| Examples of simple models | |
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| Tips on choosing a model | |
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| Overview | |
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| Don't choose a linear model just because linear regression seems simpler than nonlinear regression | |
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| Don't go out of your way to choose a polynomial model | |
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| Consider global models | |
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| Graph a model to understand its parameters | |
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| Don't hesitate to adapt a standard model to fit your needs | |
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| Be cautious about letting a computer pick a model for you | |
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| Choose which parameters, if any, should be constrained to a constant value | |
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| Global models | |
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| What are global models? | |
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| Fitting incomplete data sets | |
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| The parameters you care about cannot be determined from one data set | |
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| Assumptions of global models | |
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| How to specify a global model | |
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| Compartmental models and defining a model with a differential equation | |
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| What is a compartmental model? What is a differential equation? | |
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| Integrating a differential equation | |
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| The idea of numerical integration | |
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| More complicated compartmental models | |
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| How nonlinear regression works | |
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| Modeling experimental error | |
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| Why the distribution of experimental error matters when fitting curves | |
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| Origin of the Gaussian distribution | |
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| From Gaussian distributions to minimizing sums-of-squares | |
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| Regression based on nongaussian scatter | |
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| Unequal weighting of data points | |
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| Standard weighting | |
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| Relative weighting (weighting by 1/Y[superscript 2]) | |
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| Poisson weighting (weighting by 1/Y) | |
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| Weighting by observed variability | |
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| Error in both X and Y | |
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| Weighting for unequal number of replicates | |
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| Giving outliers less weight | |
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| How nonlinear regression minimizes the sum-of-squares | |
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| Nonlinear regression requires an iterative approach | |
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| How the nonlinear regression method works | |
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| Independent scatter | |
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| Confidence intervals of the parameters | |
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| Asymptotic standard errors and confidence intervals | |
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| Interpreting standard errors and confidence intervals | |
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| How asymptotic standard errors are computed | |
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| An example | |
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| Because asymptotic confidence intervals are always symmetrical, it matters how you express your model | |
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| Problems with asymptotic standard errors and confidence intervals | |
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| What if your program reports "standard deviations" instead of "standard errors"? | |
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| How to compute confidence intervals from standard errors | |
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| Generating confidence intervals by Monte Carlo simulations | |
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| An overview of confidence intervals via Monte Carlo simulations | |
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| Monte Carlo confidence intervals | |
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| Perspective on Monte Carlo methods | |
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| How to perform Monte Carlo simulations with Prism | |
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| Variations of the Monte Carlo method | |
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| Generating confidence intervals via model comparison | |
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| Overview on using model comparison to generate confidence intervals | |
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| A simple example with one parameter | |
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| Confidence interval for the sample data with two parameters | |
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| Using model comparison to generate a confidence contour for the example data | |
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| Converting the confidence contour into confidence intervals for the parameters | |
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| How to use Excel's solver to adjust the value of a parameter to get the desired sum-of-squares | |
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| More than two parameters | |
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| Comparing the three methods for creating confidence intervals | |
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| Comparing the three methods for our first example | |
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| A second example. Enzyme kinetics | |
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| A third example | |
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| Conclusions | |
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| Using simulations to understand confidence intervals and plan experiments | |
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| Should we express the middle of a dose-response curve as EC[subscript 50] or log(EC[subscript 50])? | |
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| Exponential decay | |
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| How to generate a parameter distribution with Prism | |
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| Comparing models | |
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| Approach to comparing models | |
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| Why compare models? | |
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| Before you use a statistical approach to comparing models | |
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| Statistical approaches to comparing models | |
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| Comparing models using the extra sum-of-squares F test | |
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| Introducing the extra sum-of-squares F test | |
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| The F test is for comparing nested models only | |
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| How the extra sum-of-squares F test works | |
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| How to determine a P value from F | |
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| Comparing models using Akaike's Information Criterion (AIC) | |
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| Introducing Akaike's Information Criterion (AIC) | |
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| How AIC compares models | |
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| A second-order (corrected) AIC | |
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| The change in AICc tells you the likelihood that a model is correct | |
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| The relative likelihood or evidence ratio | |
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| Terminology to avoid when using AIC[subscript c] | |
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| How to compare models with AICc by hand | |
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| One-way ANOVA by AICc | |
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| How should you compare models--AIC[subscript c] or F test? | |
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| A review of the approaches to comparing models | |
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| Pros and cons of using the F test to compare models | |
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| Pros and cons of using AIC[subscript c] to compare models | |
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| Which method should you use? | |
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| Examples of comparing the fit of two models to one data set | |
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| Two-site competitive binding model clearly better | |
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| Two-site binding model doesn't fit better | |
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| Can't get a two-site binding model to fit at all | |
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| Testing whether a parameter differs from a hypothetical value | |
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| Example. Is the Hill slope factor statistically different from 1.0? | |
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| Compare models with the F test | |
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| Compare models with AIC[subscript c] | |
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| Compare with t test | |
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| How does a treatment change the curve? | |
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| Using global fitting to test a treatment effect in one experiment | |
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| Does a treatment change the EC[subscript 50]? | |
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| Does a treatment change the dose-response curve? | |
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| Using two-way ANOVA to compare curves | |
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| Situations where curve fitting isn't helpful | |
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| Introduction to two-way ANOVA | |
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| How ANOVA can compare "curves" | |
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| Post-tests following two-way ANOVA | |
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| The problem with using two-way ANOVA to compare curves | |
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| Using a paired t test to test for a treatment effect in a series of matched experiments | |
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| The advantage of pooling data from several experiments | |
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| An example. Does a treatment change logEC[subscript 50]? Pooling data from three experiments | |
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| Comparing via paired t test | |
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| Why the paired t test results don't agree with the individual comparisons | |
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| Using global fitting to test for a treatment effect in a series of matched experiments | |
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| Why global fitting? | |
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| Setting up the global model | |
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| Fitting the model to our sample data | |
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| Was the treatment effective? Fitting the null hypothesis model | |
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| Using an unpaired t test to test for a treatment effect in a series of unmatched experiments | |
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| An example | |
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| Using the unpaired t test to compare best-fit values of V[subscript max] | |
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| Using global fitting to test for a treatment effect in a series of unmatched experiments | |
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| Setting up a global fitting to analyze unpaired experiments | |
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| Fitting our sample data to the global model | |
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| Comparing models with an F test | |
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| Comparing models with AIC[subscript c] | |
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| Reality check | |
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| Fitting radioligand and enzyme kinetics data | |
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| The law of mass action | |
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| What is the law of mass action? | |
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| The law of mass action applied to receptor binding | |
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| Mass action model at equilibrium | |
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| Fractional occupancy predicted by the law of mass action at equilibrium | |
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| Assumptions of the law of mass action | |
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| Hyperbolas, isotherms, and sigmoidal curves | |
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| Analyzing radioligand binding data | |
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| Introduction to radioligand binding | |
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| Nonspecific binding | |
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| Ligand depletion | |
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| Calculations with radioactivity | |
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| Efficiency of detecting radioactivity | |
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| Specific radioactivity | |
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| Calculating the concentration of the radioligand | |
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| Radioactive decay | |
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| Counting errors and the Poisson distribution | |
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| The GraphPad radioactivity web calculator | |
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| Analyzing saturation radioligand binding data | |
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| Introduction to saturation binding experiments | |
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| Fitting saturation binding data | |
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| Checklist for saturation binding | |
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| Scatchard plots | |
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| Analyzing saturation binding with ligand depletion | |
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| Analyzing competitive binding data | |
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| What is a competitive binding curve? | |
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| Competitive binding data with one class of receptors | |
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| Shallow competitive binding curves | |
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| Competitive binding with two receptor types (different K[subscript d] for hot ligand) | |
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| Heterologous competitive binding with ligand depletion | |
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| Homologous competitive binding curves | |
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| Introducing homologous competition | |
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| Theory of homologous competition binding | |
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| Why homologous binding data can be ambiguous | |
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| Using global curve fitting to analyze homologous (one site) competition data | |
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| Analyzing homologous (one site) competition data without global curve fitting | |
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| Homologous competitive binding with ligand depletion | |
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| Fitting homologous competition data (two sites) | |
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| Analyzing kinetic binding data | |
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| Dissociation ("off rate") experiments | |
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| Association binding experiments | |
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| Fitting a family of association kinetic curves | |
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| Globally fitting an association curve together with a dissociation curve | |
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| Analysis checklist for kinetic binding experiments | |
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| Using kinetic data to test the law of mass action | |
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| Kinetics of competitive binding | |
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| Analyzing enzyme kinetic data | |
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| Introduction to enzyme kinetics | |
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| How to determine V[subscript max] and K[subscript m] | |
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| Comparison of enzyme kinetics with radioligand binding | |
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| Displaying enzyme kinetic data on a Lineweaver- Burk plot | |
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| Allosteric enzymes | |
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| Enzyme kinetics in the presence of an inhibitor | |
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| Fitting dose-response curves | |
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| Introduction to dose-response curves | |
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| What is a dose-response curve? | |
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| The equation for a dose-response curve | |
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| Other measures of potency | |
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| Dose-response curves where X is concentration, not log of concentration | |
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| Why you should fit the logEC[subscript 50] rather than EC[subscript 50] | |
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| Decisions when fitting sigmoid dose-response curves | |
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| Checklist: Interpreting a dose-response curve | |
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| The operational model of agonist action | |
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| Limitations of dose-response curves | |
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| Derivation of the operational model | |
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| Shallower and steeper dose-response curves | |
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| Designing experiments to fit to the operational model | |
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| Fitting the operational model to find the affinity and efficacy of a full agonist | |
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| Fitting the operational model to find the affinity and efficacy of a partial agonist | |
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| Dose-response curves in the presence of antagonists | |
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| Competitive antagonists | |
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| Using global fitting to fit a family of dose-response curves to the competitive interaction model | |
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| Fitting agonist EC[subscript 50] values to the competitive interaction model | |
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| Antagonist inhibition curves | |
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| Complex dose-response curves | |
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| Asymmetric dose-response curves | |
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| Bell-shaped dose-response curves | |
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| Biphasic dose-response curves | |
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| Fitting curves with GraphPad Prism | |
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| Nonlinear regression with Prism | |
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| Using Prism to fit a curve | |
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| Which choices are most fundamental when fitting curves? | |
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| Prism's nonlinear regression error messages | |
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| Constraining and sharing parameters | |
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| The constraints tab of the nonlinear regression parameters dialog | |
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| Constraining to a constant value | |
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| Data set constants | |
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| Constrain to a range of values | |
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| Shared parameters (global fitting) | |
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| Prism's nonlinear regression dialog | |
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| The equation tab | |
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| Comparison tab | |
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| Initial values tab | |
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| Constraints for nonlinear regression | |
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| Weighting tab | |
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| Output tab | |
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| Range tab | |
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| Default preferences for nonlinear regression | |
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| Classic nonlinear models built into Prism | |
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| Equilibrium binding | |
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| Dose-response | |
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| Exponential | |
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| Other classic equations | |
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| Importing equations and equation libraries | |
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| Selecting from the equation library | |
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| Adding equations to the equation library | |
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| Importing equations | |
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| Writing user-defined models in Prism | |
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| What kinds of equations can you enter? | |
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| Equation syntax | |
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| Available functions for user-defined equations | |
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| Using the IF function | |
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| How to fit different portions of the data to different equations | |
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| How to define different models for different data sets | |
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| Defining rules for initial values and constraints | |
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| Managing your list of equations | |
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| Modifying equations | |
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| Linear regression with Prism | |
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| Entering data for linear regression | |
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| Choosing a linear regression analysis | |
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| Default preferences for linear regression | |
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| Using nonlinear regression to fit linear data | |
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| Deming (Model II) linear regression | |
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| Inverse linear regression with Prism | |
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| Reading unknowns from standard curves | |
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| Introduction to standard curves | |
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| Determining unknown concentrations from standard curves | |
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| Standard curves with replicate unknown values | |
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| Potential problems with standard curves | |
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| Graphing a family of theoretical curves | |
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| Creating a family of theoretical curves | |
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| Fitting curves without regression | |
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| Introducing spline and lowess | |
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| Spline and lowess with Prism | |
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| Annotated bibliography | |