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 Analysis checklist: Fitting a model

In many cases, your goal is to create a standard curve from which to interpolate unknown values. We've created a different checklist for this purpose.

More often, your goal is to determine the best-fit values of the model. If that is your goal, here are some questions to ask yourself as you evaluate the fit:

## Curve Does the graph look sensible?

Your first step should be to inspect a graph of the data with superimposed curve. Most problems can be spotted that way. Does the runs or replicate test tell you that the curve deviates systematically from the data?

The runs and replicates tests are used to determine whether the curve follows the trend of your data. The runs test is used when you have single Y values at each X. It asks if data points are clustered on either side of the curve rather than being randomly scattered above and below the curve. The replicates test is used when you have replicate Y values at each X. It asks if the points are 'too far' from the curve compared to the scatter among replicates.

If either the runs test or the replicates test yields a low P value, you can conclude that the curve doesn't really describe the data very well. You may have picked the wrong model, or applied invalid constraints.

## Parameters Are the best-fit parameter values plausible?

When evaluating the parameter values reported by nonlinear regression, check that the results are scientifically plausible. Prism doesn't 'know' what the parameters mean, so can report best-fit values of the parameters that make no scientific sense. For example, make sure that parameters don't have impossible values (rate constants simply cannot be negative). Check that EC50 values are within the range of your data. Check that maximum plateaus aren't too much higher than your highest data point.

If the best-fit values are not scientifically sensible, then the results won't be useful. Consider constraining the parameters to a sensible range, and trying again. How precise are the best-fit parameter values?

You don't just want to know what the best-fit value is for each parameter. You also want to know how certain that value is. Therefore an essential part of evaluating results from nonlinear regression is to inspect the 95% confidence intervals for each parameter.

If all the assumptions of nonlinear regression are true, there is a 95% chance that the interval contains the true value of the parameter. If the confidence interval is reasonably narrow, you've accomplished what you wanted to do – found the best fit value of the parameter with reasonable certainty. If the confidence interval is really wide, then you've got a problem. The parameter could have a wide range of values. You haven't nailed it down. How wide is 'too wide' depends on the scientific context of your work. Are the confidence bands 'too wide'?

Confidence bands visually show you how precisely the parameters have been determined. Choose to plot confidence bands by checking an option on the Fit tab of the nonlinear regression dialog. If all the assumptions of nonlinear regression have been met, then there is a 95% chance that the true curve falls between these bands. This gives you a visual sense of how well your data define the model.

## Residuals Does the residual plot look good?

A residual plot shows the relationship between the X values of your data and the distance of the point from the curve (the residuals). If the assumptions of the regression are met, the residual plot should look bland, with no trends apparent. Does the scatter of points around the best-fit curve follow a Gaussian distribution?

Least squares regression is based on the assumption that the scatter of points around the curve follows a Gaussian distribution. Prism offers three normality tests (in the Diagnostics tab) that can test this assumption (we recommend the D'Agostino test). If the P value for a normality test is low, you conclude that the scatter is not Gaussian. Could outliers be impacting your results?

The presence of one or a few outliers (points much further from the curve than the rest) can overwhelm the least-squares calculations and lead to misleading results.

You can spot outliers by examining a graph (so long as you plot individual replicates, and not mean and error bar). But outliers can also be detected automatically. GraphPad has developed a new method for identifying outliers we call the ROUT method. Check the option on the diagnostics tab to count the outliers, but leave them in the calculations. Or check the option on the Fit tab to exclude outliers from the calculations.

## Models Would another model be more appropriate?

Nonlinear regression finds parameters that make a model fit the data as closely as possible (given some assumptions). It does not automatically ask whether another model might work better.

Even though a model fits your data well, it may not be the best, or most correct, model. You should always be alert to the possibility that a different model might work better. In some cases, you can't distinguish between models without collecting data over a wider range of X. In other cases, you would need to collect data under different experimental conditions. This is how science moves forward. You consider alternative explanations (models) for your data, and then design experiments to distinguish between them. If you chose to share parameters among data sets, are those datasets expressed in the same units?

Global nonlinear regression (any fit where one or more parameter is shared among data sets) minimizes the sum (over all datasets) of the sum (over all data points) of the squared distance between data point and curve. This only makes sense if the Y values for all the datasets are expressed in the same units.

## Goodness of fit Is the R2 'too low' compared to prior runs of this experiment?

While many people look at R2 first, it really doesn't help you understand the results very well. It only helps if you are repeating an experiment you have run many times before. If so, then you know what value of R2 to expect. If the R2 is much lower than expected, something went wrong. One possibility is the presence of outliers. Are the values of sum-of-squares and sy.x 'too low' compared to prior runs of this experiment?

These values are related to the R2, and inspecting the results can only be useful when you have done similar experiments in the past so know what values to expect.