Using Global Curve Fitting to Determine Dose Response Parallelism
Introduction
Dose response curves are parallel if they are
only shifted right or left on the concentration
(X) axis. So if you were to fit a 4 parameter
logistic function to multiple dose response curves
then, for curves which are parallel, only the EC50
parameters would be significantly different. If
the data is normalized from 0 to 100, say, then
the min, max and Hillslope parameters would not
be significantly different.
A simple procedure to determine parallelism is
to use global curve fitting to fit the dose response
curve data two ways - 1) with no shared parameters
and 2) with min, max and Hillslope shared for all
data sets. Then use an F test to see if the two
fits are significantly different. If there is
no difference between the fits then only differences
in the EC50 values caused the different data sets
and the dose response curves would be considered
to be parallel.
An Example
The data in the graph below has 3 replicates for
each concentration. In this example there are
two data sets to be compared – a standard and a
sample data set – though any number of data sets
may be compared. A global curve fit of both data
sets using a 4 parameter logistic function was
performed with no shared parameters. The fit lines
are shown in the left panel below. Visually the
curves are not very different from parallel.
To determine parallelism quantitatively you perform
a a second global curve fit but, in this case,
share the 3 parameters min, max and Hillslope. The
results are shown in the right panel below.
| Parameters Unshared |
Parameters min, max, Hillslope Shared |
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The two curves with shared parameters are clearly
more uniformly shifted horizontally but are these
fits significantly different from the previous? To
determine this use the F test
where the residual degrees of freedom are the
number of data points in the fit minus the number
of fitted parameters (this definition applies for
n data sets not just the two in this example). The
residual degrees of freedom (Residual DF), required
for the F test, are obtained from the Anova tables
in the curve fit reports. They are 76 and 79 for
the no-sharing and sharing cases, respectively.
The residual sum of squares for shared parameters
shown in the tables above are only slightly larger
than for unshared (0.0091 vs 0.0090).
A transform is used to compute the F and associated
P value to determine if there is a significant
difference. The curve fit option for placing the
fit residuals in the worksheet was set and the
column numbers for these residuals are specified
in the transform.
The results of running this transform are
Since P > 0.05 the conclusion is that there is no reason to suspect that the data is not parallel.
A Second Example
A second data set pair shows a significant difference
with P = 0.0039 indicating that these data sets
are not parallel.
Parameters Unshared |
Parameters min, max, Hillslope Shared |
|
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The Procedure
To repeat the second analysis shown above, do the following:
1. Download
the notebook containing the data in the examples
above , Dose Response Parallelism.jnb, and open
the third worksheet "Demonstration". This contains
the data for the second example.
2. Select
the Global Fit Wizard under Nonlinear Regression.
3. Select
the Ligand Binding Equation Category, and the equation "sigmoidal
dose-response (variable slope)", Click Next
4. In
the Shared Parameters panel do not select any parameters,
Click Next
5. In
the Variables Panel, in "Data From" select the
data format "X Many Y Replicates"
6. Select
the following worksheet columns and click Next
a. column
2 for x (x: 2-log Conc)
b. column
3 for the first of the standard group y replicates
(First y 1: 3-Standard 1)
c. column
5 for the last of the standard group y replicates
(Last y 1: 5-Standard 3)
d. column
6 for the first of the sample group y replicates
(First y 2: 6-Sample 1)
e. column
8 for the last of the sample group y replicates
(Last y 2: 8-Sample 3)
7. Scroll
down the Fit Results panel to see that both data
sets have been fit by the Global Fit Wizard (independently
in this case since no parameters were shared),
Click Next
8. Select the following options in the Numeric Output Options
panel and click Next
9. Select the "Create new graph" checkbox and click Finish
10. Repeat steps 2 through 8 in exactly the same way except
in the Shared Parameters Panel share the min, max and Hillslope parameters
11. Download
the "F test.xfm" transform.
12. Select Transform, User-Defined and open the "F test.xfm" transform
that you downloaded from this site.
13. If necessary enter values for the two degrees of freedom
and the column numbers for the four Residuals columns
in your worksheet (if you just downloaded the transform
then these values will be set for you) and run
the transform to produce the P value equal to 0.0039
shown above.
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