Empirical Bayes Estimates

In a microarray experiment, as hundreds and often thousands of probes are being processed in parallel there is a loss of power if you consider the variation of each probe in isolation. If this parallelism is used between the genes to gain extra information on the variation of an individual probe, then more powerful tests of the level of differential expression of a probe can be obtained. To do this, a prior distribution of the standard deviations is assumed or equivalently the variances over the probes. In this approach it is assumed that the reciprocal of the variance is distributed with a multiple of a chi-square distribution with d₀ degrees of freedom, i.e.

1/s_p² ~ 1/(d₀s₀²) _d₀. (Equation 1)

If the parameters of this distribution, the prior degrees of freedom and standard deviation, d₀ and s₀ are estimated, more information can be gained on an individual probe, by shrinking it towards the prior, depending on the amount of information in the standard deviation of an individual probe, s_p, (in this case its degrees of freedom, d_p). The modified standard deviation s^~_p is then given by:
(Equation 2).

A modified t-test can then be performed using the modified standard deviation with d₀ + d_p degrees of freedom. The method can also produce the two-sided p values from a test of the differential expression being different from 0.

Available Data

This lists data structures appropriate to the current input field. Double-click on a name to copy it to the current input field; alternatively, you can type the name directly into the input field.

Data Type

The data can be supplied in either of the following formats:

Means	A variate containing means for each probe. These are normally the estimates of differential expression for the effect or a contrast from the trial.
t-values	A variate containing the t-values for each probe, i.e. an effect or contrast divided by its standard error.
Pointer	A pointer to the columns of data, one variate per slide, with the probes in the same position within each variate. The means and standard deviations will be calculated from the raw data.

Means/t-values/Pointer

Specifies the data for the analysis. The label displayed and required data structure type for this option depends on the current selection in the Data Type list.

Standard Deviations

The standard deviations for each probe when the data are supplied as means or t-values.

Degrees of Freedom

The degrees of freedom when the data are supplied as means or t-values.

Save

Allows you to save results from the analysis. After checking the appropriate boxes, you need to type the names for the identifiers of the data structures into the corresponding In: fields.

Modified t-values	variate	the adjusted t-values, using the modified standard deviations calculated as below
Modified SDs	variate	the modified standard deviations for each probe as specified in equation 2. Note that the values for d₀ and s₀ can be displayed in the output window, if the options to display these are selected.
Modified Probabilities	variate	the modified probabilities of the t-values being significantly different from zero when the true mean is 0. This is a two sided probability, for example using the GenStat command language, CALC Mod_Pr = 2*CLT(ABS(T);D0+DG)

Display in Spreadsheet

When selected, the saved results will also be displayed within spreadsheet windows. If a spreadsheet containing the Probability variate is currently open within GenStat, the saved results will be added to this spreadsheet.

Action Buttons

Run	Run the analysis.
Cancel	Close the menu without further changes.
Options	Opens a dialog where additional options and settings can be specified for the analysis.
Defaults	Set the menu settings back to the default settings. Clicking the right mouse on this button produces a shortcut menu where you can choose to set the menu using the currently stored defaults or the GenStat default settings.

Example

The following menu shows the estimation of the modified t-values and probabilities for the differential expression in a knock out mouse experiment. The menu to do this analysis is shown below:

The options controlling the output and plots was:

This produces the output:

Empirical Bayes estimation of modified t-values

	Data	CTVal[1]	(variate of t-values)
	Number of tests 	 	 6384
	Mean standard deviation	 	 0.2958
	Prior standard deviation	 0.06875
	Prior degrees of freedom	 5.973

This produces the plot of the modified t-values vs the raw t-values:

and the plot of the modified versus the raw t-values which shows a very strong linear relationship between the two as the degrees of freedom only range between 11 and 14.