The function rankPvalue calculates the p-value for observing that an object (corresponding to a row of the input
data frame datS) has a consistently high ranking (or low ranking) according to multiple ordinal scores
(corresponding to the columns of the input data frame datS).
rankPvalue(datS, columnweights = NULL,
na.last = "keep", ties.method = "average",
calculateQvalue = TRUE, pValueMethod = "all")a data frame whose rows represent objects that will be ranked. Each column of datS represents an
ordinal variable (which can take on negative values). The columns correspond to (possibly signed) object
significance measures, e.g., statistics (such as Z statistics), ranks, or correlations.
allows the user to input a vector of non-negative numbers reflecting weights for the different columns of
datZ.
If it is set to NULL then all weights are equal.
controls the treatment of missing values (NAs) in the rank function. If TRUE, missing values in the data are
put last (i.e. they get the highest rank values). If FALSE, they are put first;
if NA, they are removed; if
"keep" they are kept with rank NA. See rank for more details.
represents the ties method used in the rank function for the percentile rank method. See rank for
more details.
logical: should q-values be calculated? If set to TRUE then the function calculates corresponding q-values (local false discovery rates) using the qvalue package, see Storey JD and Tibshirani R. (2003). This option assumes that qvalue package has been installed.
determines which method is used for calculating p-values. By default it is set to "all", i.e. both methods are used. If it is set to "rank" then only the percentile rank method is used. If it set to "scale" then only the scale method will be used.
A list whose actual content depends on which p-value methods is selected, and whether q0values are calculated.
The following inner components are calculated, organized in outer components datoutrank and
datoutscale,:
This is the minimum between pValueLowRank and pValueHighRank, i.e. min(pValueLow, pValueHigh)
Asymptotic p-value for observing a consistently low value across the columns of datS based on the rank method.
Asymptotic p-value for observing a consistently low value across the columns of datS based on the rank method.
This is the minimum between pValueLowScale and pValueHighScale, i.e. min(pValueLow, pValueHigh)
Asymptotic p-value for observing a consistently low value across the columns of datS based on the Scale method.
Asymptotic p-value for observing a consistently low value across the columns of datS based on the Scale method.
local false discovery rate (q-value) corresponding to the p-value pValueExtremeRank
local false discovery rate (q-value) corresponding to the p-value pValueLowRank
local false discovery rate (q-value) corresponding to the p-value pValueHighRank
local false discovery rate (q-value) corresponding to the p-value pValueExtremeScale
local false discovery rate (q-value) corresponding to the p-value pValueLowScale
local false discovery rate (q-value) corresponding to the p-value pValueHighScale
The function calculates asymptotic p-values (and optionally q-values) for testing the null hypothesis that the values in the columns of datS are independent. This allows us to find objects (rows) with consistently high (or low) values across the columns.
Example: Imagine you have 5 vectors of Z statistics corresponding to the columns of datS. Further assume that a gene has ranks 1,1,1,1,20 in the 5 lists. It seems very significant that the gene ranks number 1 in 4 out of the 5 lists. The function rankPvalue can be used to calculate a p-value for this occurrence.
The function uses the central limit theorem to calculate asymptotic p-values for two types of test statistics that measure consistently high or low ordinal values. The first method (referred to as percentile rank method) leads to accurate estimates of p-values if datS has at least 4 columns but it can be overly conservative. The percentile rank method replaces each column datS by the ranked version rank(datS[,i]) (referred to ask low ranking) and by rank(-datS[,i]) (referred to as high ranking). Low ranking and high ranking allow one to find consistently small values or consistently large values of datS, respectively. All ranks are divided by the maximum rank so that the result lies in the unit interval [0,1]. In the following, we refer to rank/max(rank) as percentile rank. For a given object (corresponding to a row of datS) the observed percentile rank follows approximately a uniform distribution under the null hypothesis. The test statistic is defined as the sum of the percentile ranks (across the columns of datS). Under the null hypothesis that there is no relationship between the rankings of the columns of datS, this (row sum) test statistic follows a distribution that is given by the convolution of random uniform distributions. Under the null hypothesis, the individual percentile ranks are independent and one can invoke the central limit theorem to argue that the row sum test statistic follows asymptotically a normal distribution. It is well-known that the speed of convergence to the normal distribution is extremely fast in case of identically distributed uniform distributions. Even when datS has only 4 columns, the difference between the normal approximation and the exact distribution is negligible in practice (Killmann et al 2001). In summary, we use the central limit theorem to argue that the sum of the percentile ranks follows a normal distribution whose mean and variance can be calculated using the fact that the mean value of a uniform random variable (on the unit interval) equals 0.5 and its variance equals 1/12.
The second method for calculating p-values is referred to as scale method. It is often more powerful but its asymptotic p-value can only be trusted if either datS has a lot of columns or if the ordinal scores (columns of datS) follow an approximate normal distribution. The scale method scales (or standardizes) each ordinal variable (column of datS) so that it has mean 0 and variance 1. Under the null hypothesis of independence, the row sum follows approximately a normal distribution if the assumptions of the central limit theorem are met. In practice, we find that the second approach is often more powerful but it makes more distributional assumptions (if datS has few columns).
Killmann F, VonCollani E (2001) A Note on the Convolution of the Uniform and Related Distributions and Their Use in Quality Control. Economic Quality Control Vol 16 (2001), No. 1, 17-41.ISSN 0940-5151
Storey JD and Tibshirani R. (2003) Statistical significance for genome-wide experiments. Proceedings of the National Academy of Sciences, 100: 9440-9445.