Estimates false discovery rates by a Bonferroni-type procedure (A.I. Glaser).
|Controls printed output (
||Controls the method used for calculating π0 (
||Whether to take logs of π0 when
||Degrees of freedom for smoothing spline; default 3|
||Controls plots (
||Window for the graphs; default 1|
||Window for the key (zero for none); default 2|
||Significance values, must lie between 0 and 1|
||Values of tuning parameter λ, equivalent to significance levels at which to test the
||Saves the False Discovery Rates (i.e. q-values) at the sorted p-values in
||Saves the False Rejection Rates at the sorted p-values in
||Saves the power estimates as a function of the sorted p-values in
||Saves the value of π0, i.e. the maximum value of the FDR|
||Lower bound of q-values to use with
||Upper bound of q-values to to use with
When testing m multiple hypotheses there are various outcomes that can occur, summarized in the table below
|Decision on null hypothesis:||Accept||Reject||Total|
|Situation: null true||U||V||m0|
where R is the total number of rejected hypotheses and W = m – R. The proportion of tests that are truly null, π0, is m0 divided by m. The false discovery rate (FDR), also known as the q-value of a test, is a commonly used error measure in multiple-hypotheses, defined as
FDR = E(V/R | R > 0) × Pr(R > 0)
i.e. the expected proportion of false positives findings among all the rejected hypotheses multiplied by the probability of making at least one rejection; the FDR is zero when R = 0. Similarly the false rejection rate (FRR) is defined as
FDR = E(T/W | W > 0) × Pr(W > 0),
i.e. the expected proportion of false negatives findings among all the accepted hypotheses times the probability of accepting at least one test. We also define the power to be equal to E(S/m1 | m1 > 0) × Pr(m1 > 0).
The p-values from the multiple hypotheses are supplied, in a variate, using the
PROBABILITIES parameter. The analysis uses a Bonferroni-type multiple-testing procedure to calculate the corresponding q-values. The p-values are assumed independent, or may be weakly dependent if there are many of them. The parameter π0 is calculated using the method of Storey (2002). This involves a tuning parameter λ, which can be set using the
LAMBDA parameter; the default is a variate containing the numbers 0, 0.05, … 0.9. Λ can be thought of as the value beyond which the individual p-values are considered null. As λ gets larger the bias of π0 gets smaller, but its variance increases. If you set
LAMBDA to a scalar, π0 is estimated by dividing the number of null tests (i.e. the number of p-values greater than λ) by the expected number of null tests m × (1 – Λ). If you set
LAMBDA to a variate with several values, two methods are available, selected by the following settings of the
||fits a smoothing spline of λ onto initial estimates of π0 calculated as for a single λ value, and takes the estimate of π0 as the value corresponding to the largest value of λ;|
||estimates π0 by bootstrap sampling from the variate of p-values.|
The default is
smoother, as the
bootstrap method may be time-consuming when there are many p-values. The number of degrees of freedom to use in the smoothing is specified by the
DF option (default 3). Also, you can set option
LOGP=yes to do the smoothing on log-transformed π0 values.
||prints π0, the estimate of the expected proportion of null p-values corresponding to the largest q-value.|
Various graphs can be selected by the following settings of the
||histogram of p-values;|
||histogram of q-values (i.e. the FDR values);|
||λ against π0 (when only one value of λ is specified the default values of
||q-values against p-values;|
||plot of the sorted tests against q-values;|
||plot of the number of expected false positives against the sorted tests;|
||plot of the FDR, FRR and power statistics against the sorted p-values; and|
||plots the FDR, FRR and power statistics on log scales, against the sorted p-values restricted to p<0.5, with a background grid to enable estimates to be read for specific probability values. Due to the small numbers used in this plot the p-values and FDR, FRR & power statistics are displayed as percentages.|
By default all the plots are produced. The
WINDOW option specifies the window for the graphs, and the
KEYWINDOW option species the window for keys.
FDRBONFERRONI uses the method of Storey (2002), with the definitions of FRR and power given in Genovese & Wasserman (2002).
PROBABILITIES parameter can be restricted. All output estimates will then be based only on those unrestricted units.
Storey, J.D. (2002). A direct approach to false discovery rates. Journal of the Royal Statistical Society Series B, 64, 479-498.
Genovese, C. & Wasserman, L. (2002). Operating characteristics and extensions of the false discovery rate procedure. Journal of the Royal Statistical Society Series B, 64, 499-518.
Commands for: Microarray data.
CAPTION 'FDRBONFERRONI example'; STYLE=meta SCALAR ntests,seed,delta,phi,sampsize; VALUE=\ 20000,1231,0.3,0.90,50 " Create ntest data sets (rows), each of size sampsize (columns)." GRANDOM [NVALUES=1; SEED=seed] dum POINTER [NVALUES=sampsize] x CALCULATE x = GRNORMAL(ntests; 0; 1) " Add an offset, delta, to a fraction, (1-phi), of the samples. Derive test statistic, z, for Test Ho: mean=0 against Ha:mean0 and significance, p." CALCULATE nHa = ROUND(ntests * (1-phi)) & nHo = ntests - nHa VARIATE [VALUES=#nHo(0),#nHa(delta)] vdelta CALCULATE xbar = vmean(x) + vdelta & t = ABS(xbar * SQRT(sampsize)) & p = 2 * CUNORMAL(t; 0; 1) " Derive FDR " FDRBONFERRONI p; FDR=qvals; LAMBDA=!(0,0.01...0.99); UPPER=0.4