Graphpad Prism 7

Odds ratio

Here are the sample data for a case-control study (the first study to link smoking to lung cancer). The investigators chose to study a group of cases with lung cancer and a group of controls without lung cancer. They then asked whether each person had smoked or not (Doll and Hill, British Med. J, 1950, 739-748). The results were:

	Cases (lung cancer)	Control
Smoked	688	650
Never smoked	21	59

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With a retrospective case-control data, direct calculations of the relative risk or the difference between proportions should not be performed, as the results are not meaningful. When designing this kind of study, you decide how many cases and controls to study. Those numbers don't have to be equal. Changing the ratio of cases to controls would also change the computed values for the relative risk and difference between proportions. For that reason, it makes no sense to compute or try to interpret these values from case-control data.

In contrast, changing the ratio of cases to controls does not change the expected value of the odds ratio. If the disease or condition you are studying is rare, you can interpret the Odds ratio as an approximation of the relative risk

For the sample data above, the odds of a case being a smoker is 688/21 or 32.8. The odds of a control being a smoker is 650/59 or 11.0. The odds ratio is 32.8/11.0, which is 3.0. Prism reports the value more precisely as 2.974 with a 95% confidence interval ranging from 1.787 to 4.950. You can interpret this odds ratio as a relative risk. The risk of a smoker getting lung cancer is about three times the risk of a nonsmoker getting lung cancer.

How Prism computes the confidence interval of the odds ratio

Prism computes the confidence interval of the odds ratio using computed either using the Woolf logit method (reference 1; the only method used by Prism 6 and earlier) or the Baptista-Pike method (2) which we recommend. Choose on the Options tab of the Contingency table dialog. Fagerland (3) reviews the various methods available to compute this confidence interval.

If any cell has a zero and you choose the Woolf method, Prism adds 0.5 to all cells before calculating the odds ratio and its confidence interval. In this case, we suggest you switch to the Baptista-Pike method.

References

1.Woolf B. On estimating the relation between blood group and disease. Ann Human Gene 1955; 19: 251–253.

2. Baptista J and Pike MC. Exact two-sided confidence limits for the odds ratio in a 2 2 table. J R Stat Soc C Appl Stat 1977; 26: 214–220.

3.Fagerland MW, Lydersen S, Laake P. Recommended confidence intervals for two independent binomial proportions. Stat Methods Med Res. SAGE Publications; 2011 Oct 13.

Introduction

In a saturation binding experiment, you vary the concentration of radioligand and measure binding. The goal is to determine the Kd (ligand concentration that binds to half the receptor sites at equilibrium) and Bmax (maximum number of binding sites).

The ligand binds not only to receptors sites, but also to nonspecific sites. There are three approaches to dealing with nonspecific binding.

•Subtract off the nonspecific, and analyze only the specific binding. Read on for this approach.

•Analyze the total binding only, inferring the amount of nonspecific binding from the shape of the total binding curve. Learn more.

•Globally analyze the total and nonspecific binding at one time. Learn more.

Step by step

Create an XY data table. Enter radioligand concentration into X, and specific binding into Y. If you have several experimental conditions, place the first into column A, the second into column B, etc.

An alternative approach would be to enter total binding into column A, and nonspecific into column B. Then use the Remove Baseline analysis to subtract column B from column A, creating a new results table with the specific binding.

From the table of specific binding, click Analyze, choose nonlinear regression, choose the panel of Saturation Binding equations, and choose One site specific binding.

Model

Y = Bmax*X/(Kd + X)

Interpret the parameters

Bmax is the maximum specific binding in the same units as Y. It is the specific binding extrapolated to very high concentrations of radioligand, and so its value is almost always higher than any specific binding measured in your experiment.

Kd is the equilibrium dissociation constant, in the same units as X. It is the radioligand concentration needed to achieve a half-maximum binding at equilibrium.

Create a Scatchard plot

Before nonlinear regression was available, investigators had to transform curved data into straight lines, so they could analyze with linear regression. One way to do this is with a Scatchard plot, which plots specific binding vs. the ratio of specific binding to the concentration of free radioligand.

If you create a Scatcahrd plot, use it only to display your data. The human retina and visual cortex evolved to detect edges (straight lines), not rectangular hyperbolas, and so it can help to display data this way. Scatchard plots are often shown as insets to the saturation binding curves. They are especially useful when you want to show a change in Bmax or Kd.

Don't use the slope and intercept of a linear regression line to determine values for Bmax and Kd. If you do this, you won't get the most accurate values for Bmax and Kd. The problem is that the transformation distorts the experimental error, so the data on the Scatchard plot do not obey the assumptions of linear regression. Use nonlinear regression to obtain the most accurate values of Kd and Bmax.

To create a Scatchard plot from your specific binding data, use Prism's Transform analysis, and choose the Scatchard transform from the panel of biochemistry and pharmacology transforms.

To create a Scatchard line corresponding to the nonlinear regression fit, follow these steps:

1.Create a new XY data table, with no subcolumns.

2.Into row 1 enter X=0, Y=Bmax/Kd (previously determined by nonlinear regression).You need to do the calculation manually, and enter a number.

3.Into row 2 enter X=Bmax and Y=0. Again enter the number into the X column, not the text 'Bmax'.

4.Note the name of this data table. Perhaps rename it to something appropriate.

5.Go to the Scatchard graph.

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6.Drag the new table from the navigator and drop onto the graph.

7.Double-click on one of the new symbols for that data set to bring up the Format Graph dialog.

8.Choose to plot no symbols, but to connect with a line.

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Notes

•This is not the best way to determine Bmax and Kd. It is better to globally fit total and nonspecific binding, without subtracting to compute specific binding.

•When making a Scatchard plot, you have to choose what units you want to use for the Y-axis. Some investigators express both free ligand and specific binding in cpm so the ratio bound/free is a unitless fraction. While this is easy to interpret (it is the fraction of radioligand bound to receptors), an alternative is to express specific binding in sites/cell or fmol/mg protein, and to express the free radioligand concentration in nM. While this makes the Y-axis hard to interpret visually, it provides correct units for the slope (which equals -1/Kd).

•This equation is equivalent to the Langmuir isotherm for the absorption of a gas onto a surface.