The Reductive Hourglass Test aims to statistically evaluate the
existence of a phylotranscriptomic hourglass pattern based on
The corresponding p-value quantifies the probability that a given TAI or TDI pattern (or any phylotranscriptomics pattern)
does not follow an hourglass like shape. A p-value < 0.05 indicates that the corresponding phylotranscriptomics pattern does
indeed follow an hourglass (high-low-high) shape.
ReductiveHourglassTest( ExpressionSet, modules = NULL, permutations = 1000, lillie.test = FALSE, plotHistogram = FALSE, runs = 10, parallel = FALSE, gof.warning = FALSE, custom.perm.matrix = NULL )
a standard PhyloExpressionSet or DivergenceExpressionSet object.
a list storing three elements: early, mid, and late. Each element expects a numeric
vector specifying the developmental stages or experiments that correspond to each module.
a numeric value specifying the number of permutations to be performed for the
a boolean value specifying whether the Lilliefors Kolmogorov-Smirnov Test shall be performed to quantify the goodness of fit.
a boolean value specifying whether a Lillifor's Kolmogorov-Smirnov-Test shall be performed to test the goodness of fit of the approximated distribution, as well as additional plots quantifying the significance of the observed phylotranscriptomic pattern.
specify the number of runs to be performed for goodness of fit computations, in case
a logical value indicating whether non significant goodness of fit results should be printed as warning. Default is
a list object containing the list elements:
p.value : the p-value quantifying the statistical significance (high-low-high pattern) of the given phylotranscriptomics pattern.
std.dev : the standard deviation of the N sampled phylotranscriptomics patterns for each developmental stage S.
lillie.test : a boolean value specifying whether the Lillifors KS-Test returned a p-value > 0.05,
which indicates that fitting the permuted scores with a normal distribution seems plausible.
The reductive hourglass test is a permutation test based on the following test statistic.
(1) A set of developmental stages is partitioned into three modules - early, mid, and late - based on prior biological knowledge.
(3) The two differences D1 = T_early - T_mid and D2 = T_late - T_mid are calculated.
(4) The minimum D_min of D1 and D2 is computed as final test statistic of the reductive hourglass test.
In order to determine the statistical significance of an observed minimum difference D_min
the following permutation test was performed. Based on the
is calculated from each of the permuted
approximated by a Gaussian distribution with method of moments estimated parameters returned by
and the corresponding p-value is computed by
pnorm given the estimated parameters of the Gaussian distribution.
The goodness of fit for the random vector D_min is statistically quantified by an Lilliefors (Kolmogorov-Smirnov) test
In case the parameter plotHistogram = TRUE, a multi-plot is generated showing:
(1) A Cullen and Frey skewness-kurtosis plot generated by
This plot illustrates which distributions seem plausible to fit the resulting permutation vector D_min.
In the case of the Reductive Hourglass Test a normal distribution seemed plausible.
(2) A histogram of D_min combined with the density plot is plotted. D_min is then fitted by a normal distribution.
The corresponding parameters are estimated by moment matching estimation using the
(3) A plot showing the p-values for N independent runs to verify that a specific p-value is biased by a specific permutation order.
(4) A barplot showing the number of cases in which the underlying goodness of fit (returned by Lilliefors (Kolmogorov-Smirnov) test
for normality) has shown to be significant (
TRUE) or not significant (
This allows to quantify the permutation bias and their implications on the goodness of fit.
Drost HG et al. (2015) Mol Biol Evol. 32 (5): 1221-1231 doi:10.1093/molbev/msv012
Quint M et al. (2012). A transcriptomic hourglass in plant embryogenesis. Nature (490): 98-101.
M. L. Delignette-Muller, R. Pouillot, J.-B. Denis and C. Dutang (2014), fitdistrplus: help to fit of a parametric distribution to non-censored or censored data.
Cullen AC and Frey HC (1999) Probabilistic techniques in exposure assessment. Plenum Press, USA, pp. 81-159.
Evans M, Hastings N and Peacock B (2000) Statistical distributions. John Wiley and Sons Inc.
Sokal RR and Rohlf FJ (1995) Biometry. W.H. Freeman and Company, USA, pp. 111-115.
Juergen Gross and bug fixes by Uwe Ligges (2012). nortest: Tests for Normality. R package version 1.0-2.
Dallal, G.E. and Wilkinson, L. (1986): An analytic approximation to the distribution of Lilliefors' test for normality. The American Statistician, 40, 294-296.
Stephens, M.A. (1974): EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69, 730-737.
data(PhyloExpressionSetExample) # perform the reductive hourglass test for a PhyloExpressionSet # here the prior biological knowledge is that stages 1-2 correspond to module 1 = early, # stages 3-5 to module 2 = mid (phylotypic module), and stages 6-7 correspond to # module 3 = late ReductiveHourglassTest(PhyloExpressionSetExample, modules = list(early = 1:2, mid = 3:5, late = 6:7), permutations = 1000) #> $p.value #>  4.850444e-09 #> #> $std.dev #>  0.05577317 0.05521729 0.05297510 0.05147066 0.05055055 0.05156745 0.05461829 #> #> $lillie.test #>  NA #> # use your own permutation matrix based on which p-values (ReductiveHourglassTest) # shall be computed custom_perm_matrix <- bootMatrix(PhyloExpressionSetExample,100) ReductiveHourglassTest(PhyloExpressionSetExample, modules = list(early = 1:2, mid = 3:5, late = 6:7), custom.perm.matrix = custom_perm_matrix) #> $p.value #>  1.788496e-07 #> #> $std.dev #> Zygote Quadrant Globular Heart Torpedo Bent Mature #> 0.05203162 0.05028922 0.05146610 0.04911402 0.04760213 0.04831819 0.05337730 #> #> $lillie.test #>  NA #>