Chance can be expressed either as a probability or as odds. In most contexts, there is no particular reason to prefer one over the other. Most scientists tend to feel more comfortable thinking about probabilities than odds, but that is a matter of training and custom, not logic.

The distinction is simple:

• The probability that an event will occur is the fraction of times you expect to see that event in many trials. Probabilities always range between 0 and 1.
• The odds are defined as the probability that the event will occur divided by the probability that the event will not occur.

A probability of 0 is the same as odds of 0. Probabilities between 0 and 0.5 equal odds less than 1.0. A probability of 0.5 is the same as odds of 1.0. Think of it this way: The probability of flipping a coin to heads is 50%. The odds are “fifty: fifty,” which equals 1.0.

As the probability goes up from 0.5 to 1.0, the odds increase from 1.0 to approach infinity. For example, if the probability is 0.75, then the odds are 75:25, three to one, or 3.0.

Converting between odds and probability is straightforward:

• To convert from a probability to odds, divide the probability by one minus that probability. So if the probability is 10% or 0.10 , then the odds are 0.1/0.9 or ‘1 to 9’ or 0.111.
• To convert from odds to a probability, divide the odds by one plus the odds. So to convert odds of 1/9 to a probability, divide 1/9 by 10/9 to obtain the probability of 0.10.

The example shows data fit to a two-component binding curve, with dotted and dashed lines showing the two components. This page explains how those curves were generated. Download the Prism file.  It shows the versatility of hooking analysis constants. This example assumes you are using  GraphPad Prism 5. It will not work with earlier versions.

The data were fit with nonlinear regression to the built-in model "Two sites -Specific binding" from the section "Binding - Saturation" . Nonlinear regression fit five parameters: Bmaxi, KdHi, BmaxLo, KdLo, and PercentHiAffinity. 

To generate the dotted and dashed curves:

1. Click Analyze and choose to Create a Family of Theoretical curves. 
2. Enter a range of X values to match the graph (0 to 60). 
3. On the Equation tab, choose the equation, "One site - Specific binding".
4. On the Parameter Value tab, choose to simulate a family of two curves.
5. Click on Curve A to define its parameters. 
6. The first parameter is the Bmax. Don't enter a value. Instead, click on the fish hook icon to hook an analysis constant. 

7. On the Hook Constant dialog, choose best-fit value of BmaxHi, and click ok.

8. Click the other fish hook icon to hook Kd.
9. On the hook constant dialog, choose the best fit value of KdHi. 
10. Repeat for data set B, hooking BmaxLo and KdLo. 

By hooking the icons, the link is live. If you enter new data and so get a different best-fit curve, the two component curves will automatically be recomputed.

 Problems with this fit

When you first look at this  log(dose) vs. response curve, it looks fine. The curve comes close to the points, and so has a high R² value ( 0.997).

But viewing the results table reveals three problems:

• Prism labels the fit ‘ambiguous’.
• Instead of reporting the confidence interval of some parameters, it simply reports "very wide".
• The best-fit value of the EC50 (1200 nM) can’t be correct. It is higher than any of the X values.

To understand the problem, you need to learn about dependency.

The concept of dependency

The curve is defined by an equation that uses the best-fit values of the parameters. What happens if you change the value of one parameter away from its best-fit value, while holding the other parameters constant? This will always move the curve further from the data and increase the sum-of-squares and decrease the R². Now what happens if you adjust the value of other parameters? The dependency quantifies the degree to which it is possible to move the curve back to be near the points again. If changing the value of other parameters cannot move the curve at all closer to the points, then the dependency is defined to equal 0.0. If  changing the value of the other parameters can move the curve back so it is just as close to the points as the original curve, then the dependency equals 1.0. With actual data, of course, the dependency is between these two extremes. Read details about how dependency is computed.

To ask Prism to report this value, check an option on the Parameters tab of the nonlinear regression dialog. Each parameter has a distinct dependency (unless there are only two parameters, in which case both dependencies are the same). The value of dependency is always between 0.0 and 1.0.

The term "ambiguous"

When the dependency is greater than 0.9999, Prism labels the fit "ambiguous" (a term coined by GraphPad). As you can see in the results above, when a parameter that is 'ambiguous' (there may be one or several), Prism puts '~' before the best fit values and standard errors, and reports "very wide" for the corresponding confidence intervals.

When a fit is 'ambiguous', the best-fit values of some parameters are meaningless. If your goal is to interpret or compare the best-fit parameter values, an ambiguous fit is essentially useless. On the other hand, the curve may be quite useful if your goal is to interpolate values from a standard curve.

When to suspect high dependency

By default, Prism does not report the dependency. If all the dependencies are less than 0.9999, Prism won't alert you to any problem. But that threshold is arbitrary. When the dependency is high, the results may not be helpful. Suspect his problem when best-fit values of the parameters seem wrong,  when the confidence intervals of the parameters are wider than you expectm or when the standard errors are larger than you expect. If you suspect this problem, go to the Parameters tab and choose to report the Dependency. If the dependency of any parameter is high, you know that you are asking Prism to fit more parameters than the data and model really define. How high is high? Any rule of thumb is arbitrary, but be suspicious of any dependency greater than 0.95, and very suspicious of any dependency greater than 0.99.

 How to salvage this fit

In this example, Prism was asked to fit all four parameters that define the curve: the bottom plateau, the top plateau, the EC50 (middle) and the Hill Slope (steepness). But the data don’t even give a hint of a top plateau. The best-fit curve shown has an EC50 of 1.2 μM (far to the right of the concentrations used) and a Top plateau of 416 (much higher than any observed value).

Since these Y values represent responses normalized to run from 0% to 100%, the results shown above make no sense. But Prism can’t know that. All Prism can do is report that the dependency is high and the confidence intervals are wide.

It is easy to fix the problem. Go to the Constraints tab and force Bottom to equal 0 and Top to equal 100. The resulting curve looks nearly identical, but the EC50 drops to 49 nM. The dependency of the log(EC50) parameter drops from 1.000 down to 0.05.

Wilcoxon's name is used to describe several statistical tests.

Two tests are related:

• The Wilcoxon matched-pairs signed-rank test is a nonparametric method to compare before-after, or matched subjects. How to run this test with Prism 5.
• The Wilcoxon signed rank test is a nonparametric test that compares the median of a set of numbers against a hypothetical median. How to run this test with Prism 5.

The two tests are related. The matched-pairs signed-rank test works by first computing the difference between each set of matched pairs, and then using the Wilcoxon signed rank test to ask if the median of these differences differs from zero. Often the term "Wilcoxon signed rank" test is used to refer to either test. This is not really confusing as it is usually obvious whether the test is comparing one set of numbers against a hypothetical median, or comparing a set of matched values.

Wilcoxon's name is also associated with two other tests:

• The Wilcoxon rank sum test is a nonparametric test to compare two unmatched groups, more commonly known as the Mann-Whitney test.
• The Gehan-Wilcoxon test is a method to compare survival curves.