Viewing By Month : July 2008 / Main
July 25, 2008
Graphing tip: Creating a graph with only a right Y axis, and no left Y axis. 

When Prism makes a graph with one Y axis, it calls it the left Y axis. But you can place that axis wherever you want, including the right edge of the graph:

  1. Double click on the left Y axis to bring up the Format Axis dialog.
  2. Set the origin to the lower right corner.
  3. Go to the Left Y axis tab, and change the location of the numbering to the right side of the  axis. (Yes, use the 'Left' Y axis tab, as Prism calls the first axis 'left' even if you move it to the right.)
  4. Remember that Prism still thinks of the axis as being the left Y axis, so leave data sets (in Format Graph) assigned to the left Y axis. 

July 16, 2008

Tips to make Prism 5 files much smaller!

Prism 5 files can get large, but we offer two ways to make them smaller. 

The best  method is to save the projects in the new PZFX file format, instead of the PZF format. This is a choice in the File... Save As dialog. Change the default in the File&Printer tab of Prism's Preferences dialog.

With this format, the data and info tables are in plain text XML, but the rest of the file is greatly compressed. Overall, PZFX files tend to be MUCH smaller than PZF files. Typically, a PZFX file is less than 10% the size of the corresponding PZF file.

If you only use Prism 5,  there is no reason to avoid PZFX files. Not only are they smaller, but also are a bit more secure. All the data are stored in the first part of the file in plain text that could be recovered with a text editor, even if the file is damaged. The only drawback of choosing the PZFX format, is that these files can only be opened by Prism 5, and not by Prism 4. In contrast, PZF files can be opened by either Prism 4 or 5.

If you want to use PZF files (for compatibility with Prism 4), here is a trick to reduce the file size a bit. Go to the Preferences dialog, File tab, and check the option to "Save compact". With this option checked, Prism won't save the results of analyses, but rather recomputes the results when it opens the file. The difference in file size depends on how many analyses your file contains. 

July 15, 2008
How to compare two (or more) means when the groups have different standard deviations.

The unpaired t test depends on the assumption that the samples come from populations that have identical standard deviations (and thus identical variances). One-way ANOVA makes the same assumption.

Prism tests this assumption using an F test (to compare the variance of two groups) and Barlett’s test (three or more groups).

The P value answers the question:

If the populations really had identical standard deviations, what is the chance of observing as large a discrepancy among sample standard deviations as occurred in the data (or an even larger discrepancy)?

Note: Don’t mix up the P value testing for equality of the standard deviations of the groups with the P value testing for equality of the means. That latter P value is the one that answers the question you most likely were thinking about when you chose the t test or one-way ANOVA.

If the P value is large (>0.05) you conclude that there is no evidence that the standard deviations differ. If the P value is small, you conclude that the data are likely to be sampled from populations with different standard deviations.

Then what? There are four answers.

  • Conclude that the populations are different, that the treatment had an effect. In many experimental contexts, the finding of different variances is as important as the finding of different means. If the standard deviations are truly different, then the populations are different regardless of what the t test or ANOVA concludes about differences between the means. This may be the most important conclusion from the experiment.
  • Transform your data in an attempt to equalize the standard deviations, and then run the t test or ANOVA on the transformed results. Logs are especially useful. If the data are sampled from a lognormal distribution, then the logarithms of the data will follow a Gaussian distribution.
  • Ignore the result. With equal, or nearly equal, sample size (and moderately large samples), the assumption of equal standard deviations is not a crucial assumption. The t test and one-way ANOVA work pretty well even with unequal standard deviations. In other words, the t test and one-way ANOVA are remarkably robust to violations of that assumption so long as the sample size isn’t tiny and the sample sizes aren’t far apart.
  • Go back and rerun the t test, checking the option to do the modified Welch t test that allows for unequal variance. While this sounds sensible, Moser and Stevens (Amer. Statist. 46:19-21, 1992) have shown that it is not a good idea to first look at the F test to compare variances, and then switch to the modified (Welch modification to allow for different variances) t test when the P value is less than 0.05. The Welch test must be specified as part of the experimental design. (Modifications of one-way ANOVA are also available that do not assume equal standard deviations.)

Note that switching to a nonparametric test is not an appropriate approach. If the groups are sampled from populations with distinct standard deviations, then the nonparametric tests simply test whether the distributions are different. They do not test whether the medians differ

July 8, 2008

Asymmetrical (five-parameter) logistic dose-response curves

The standard log(dose) vs. response curve is defined by the bottom, top, EC50, and slope. In this curve, the top and bottom parts are mirror images of each other -- the curve is symmetrical.

Some log(dose) vs. response curves are not symmetrical. This can be modeled by including a fifth parameter that describes the asymmetry of the curve. The standard curve is sometimes called a four-parameter logistic model, so the asymmetrical curve is called a five parameter logistical model. 

Of course, an equation should not be referred to by its number of parameters. Some authors assume that any nonspecific signal is already subtracted off, so present the equations in a form where Bottom is defined to be zero and doesn't appear in the equation. Then the symmetrical variable slope equation has three (not four) parameters, and the asymmetrical form has four (not five) parameters.

Asymmetrical dose-response curves can be described by several equations. Prism uses the RIchards version (from Giraldo et. al.), which  is built-in to the 'Dose-response -- Special' group of equations in Prism 5. This is also called the generalized Hill equation.

LogXb = LogEC50 + (1/HillSlope)*Log((2^(1/S))-1)
Numerator = Top - Bottom
Denominator = (1+10^((LogXb-X)*HillSlope))^S
Y = Bottom + (Numerator/Denominator)

This equation assumes that X has been entered as (or transformed to) the logarithm of concentration, and that Y is response in any convenient units. 

S is the asymmetry parameter. If s=1.0, then this is the same as the four parameter equation. When s is not 1.0, the curve will be asymmetrical. S must be greater than zero, but can be less than, or greater than, 1.0. 

Top and Bottom are the Y values at the top and bottom plateaus of the curve. If you have normalized the data, you may want to constrain these values to 100 and 0. 

The equation above fits the logEC50, which is the X value when Y is half-way between the Top and Bottom plateaus. It is in the same units as the X values -- the logarithm of concentration.  Note that the logEC50 is not the same as the inflection point. The first line in the equation computes the inflection point from the logEC50, HillSlope, and S. Like the logEC50, this inflection point (called logXb) is in the same scale as the X values (the logarithm of concentration). 

 

If your goal is to obtain meaningful best-fit parameters, then you'll need lots of high quality data. It is very hard to fit both slope and asymmetry with tight confidence intervals. If your goal is just to interpolate unknowns from a standard curve, the width of the confidence intervals of the parameters doesn't really matter. What you want is a curve that follows the data, and in some cases an asymmetrical five parameter model does so better than a four parameter model. 

 

Other formulations of asymmetrical dose-response curves have been developed. For example, Ricketts and Head developed a model for use in baroreflex studies. 

Bindslev has written a lengthy on-line text, Drug-Acceptor Interactions.  Chapter 10,  Hill in Hell discusses many models of dose-response curves, including asymmetrical ones.