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 Using global regression to fit incomplete datasets

The graph below shows two dose-response curves. The goal of the experiment is to determine the two EC50 values. The EC50 is the concentration (dose) that gives a response half-way between the minimum and maximum responses. Each curve in the graph below was fit individually to one of the data sets. The horizontal lines show the 95% confidence interval of the EC50.

While the curves nicely fit the data points, the confidence intervals are quite wide. We really haven't determined the EC50 with sufficient precision to make useful conclusions. The problem is that the control data (squares) don’t really define the bottom plateau of the curve, and the treated data (circles) don’t really define the top plateau of the curve. Since the data don’t define the minimum and maximum responses very well, the data also don’t define very clearly the point half-way between the minimum and maximum responses. Accordingly, the confidence intervals for each EC50 extend over more than an order of magnitude. The whole point of the experiment was to determine the two EC50 values, but there is an unacceptable amount of uncertainty in the value of the best-fit values of the EC50.

The problem is solved by sharing parameters. For this example, share the parameters that define the top and bottom plateaus and the slope. But don’t share the EC50 value, since the EC50 values for control and treated data are clearly distinct.

Here are the results.

The graph of the curves looks only slightly different. But now the program finds the best-fit parameters with great confidence. The 95% confidence intervals for the EC50 values span about a factor of two (compared to a factor of ten or more when no parameters were shared).

The control data define the top of the curve pretty well, but not the bottom. The treated data define the bottom of the curve pretty well, but not the top. By fitting both data sets at once, sharing some parameters, both EC50 values were determined with reasonable certainty.