Adjustment of p-Values

adjust(PValues, <:PValueAdjustment)
adjust(PValues, Int, <:PValueAdjustment)

Adjustment of p-values

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, BenjaminiHochberg())
4-element Array{Float64,1}:
 0.004
 0.02
 0.039999999999999994
 0.5

julia> adjust(pvals, 6, BenjaminiHochberg()) # 4 out of 6 p-values
4-element Array{Float64,1}:
 0.006
 0.03
 0.06
 0.75

julia> adjust(pvals, BarberCandes())
4-element Array{Float64,1}:
 0.3333333333333333
 0.3333333333333333
 0.3333333333333333
 1.0

See also

PValueAdjustments:

Bonferroni BenjaminiHochberg BenjaminiHochbergAdaptive BenjaminiYekutieli BenjaminiLiu Hochberg Holm Hommel Sidak ForwardStop BarberCandes

Bonferroni adjustment

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, Bonferroni())
4-element Array{Float64,1}:
 0.004
 0.04
 0.12
 1.0

julia> adjust(pvals, 6, Bonferroni())
4-element Array{Float64,1}:
 0.006
 0.06
 0.18
 1.0

References

Bonferroni, C.E. (1936). Teoria statistica delle classi e calcolo delle probabilita (Libreria internazionale Seeber).

Benjamini-Hochberg adjustment

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, BenjaminiHochberg())
4-element Array{Float64,1}:
 0.004
 0.02
 0.039999999999999994
 0.5

julia> adjust(pvals, 6, BenjaminiHochberg())
4-element Array{Float64,1}:
 0.006
 0.03
 0.06
 0.75

References

Benjamini, Y., and Hochberg, Y. (1995). Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing. Journal of the Royal Statistical Society. Series B (Methodological) 57, 289–300.

Adaptive Benjamini-Hochberg adjustment

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, BenjaminiHochbergAdaptive(Oracle(0.5))) # known π₀ of 0.5
4-element Array{Float64,1}:
 0.002
 0.01
 0.019999999999999997
 0.25

julia> adjust(pvals, BenjaminiHochbergAdaptive(StoreyBootstrap())) # π₀ estimator
4-element Array{Float64,1}:
 0.0
 0.0
 0.0
 0.0

julia> adjust(pvals, 6, BenjaminiHochbergAdaptive(StoreyBootstrap()))
4-element Array{Float64,1}:
 0.0
 0.0
 0.0
 0.0

References

Benjamini, Y., Krieger, A. M. & Yekutieli, D. (2006). Adaptive linear step-up procedures that control the false discovery rate. Biometrika 93, 491–507.

Benjamini-Yekutieli adjustment

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, BenjaminiYekutieli())
4-element Array{Float64,1}:
 0.00833333333333334
 0.0416666666666667
 0.0833333333333334
 1.0

julia> adjust(pvals, 6, BenjaminiYekutieli())
4-element Array{Float64,1}:
 0.01470000000000001
 0.07350000000000005
 0.14700000000000008
 1.0

References

Benjamini, Y., and Yekutieli, D. (2001). The Control of the False Discovery Rate in Multiple Testing under Dependency. The Annals of Statistics 29, 1165–1188.

Benjamini-Liu adjustment

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, BenjaminiLiu())
4-element Array{Float64,1}:
 0.003994003998999962
 0.022275750000000066
 0.02955000000000002
 0.125

julia> adjust(pvals, 6, BenjaminiLiu())
4-element Array{Float64,1}:
 0.0059850199850060015
 0.04084162508333341
 0.07647146000000005
 0.4375

References

Benjamini, Y., and Liu, W. (1999). A step-down multiple hypotheses testing procedure that controls the false discovery rate under independence. Journal of Statistical Planning and Inference 82, 163–170.

Hochberg adjustment

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, Hochberg())
4-element Array{Float64,1}:
 0.004
 0.03
 0.06
 0.5

julia> adjust(pvals, 6, Hochberg())
4-element Array{Float64,1}:
 0.006
 0.05
 0.12
 1.0

References

Hochberg, Y. (1988). A sharper Bonferroni procedure for multiple tests of significance. Biometrika 75, 800–802.

Holm adjustment

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, Holm())
4-element Array{Float64,1}:
 0.004
 0.03
 0.06
 0.5

julia> adjust(pvals, 6, Holm())
4-element Array{Float64,1}:
 0.006
 0.05
 0.12
 1.0

References

Holm, S. (1979). A Simple Sequentially Rejective Multiple Test Procedure. Scandinavian Journal of Statistics 6, 65–70.

Hommel adjustment

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, Hommel())
4-element Array{Float64,1}:
 0.004
 0.03
 0.06
 0.5

julia> adjust(pvals, 6, Hommel())
4-element Array{Float64,1}:
 0.006
 0.05
 0.12
 1.0

References

Hommel, G. (1988). A stagewise rejective multiple test procedure based on a modified Bonferroni test. Biometrika 75, 383–386.

Šidák adjustment

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, Sidak())
4-element Array{Float64,1}:
 0.003994003998999962
 0.039403990000000055
 0.11470719000000007
 0.9375

julia> adjust(pvals, 6, Sidak())
4-element Array{Float64,1}:
 0.0059850199850060015
 0.058519850599
 0.1670279950710002
 0.984375

References

Šidák, Z. (1967). Rectangular Confidence Regions for the Means of Multivariate Normal Distributions. Journal of the American Statistical Association 62, 626–633.

Forward-Stop adjustment

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, ForwardStop())
4-element Array{Float64,1}:
 0.0010005003335835344
 0.005525418093542492
 0.013836681223931188
 0.1836643060579347

julia> adjust(pvals, 6, ForwardStop())
4-element Array{Float64,1}:
 0.0010005003335835344
 0.005525418093542492
 0.013836681223931188
 0.1836643060579347

References

G’Sell, M.G., Wager, S., Chouldechova, A., and Tibshirani, R. (2016). Sequential selection procedures and false discovery rate control. J. R. Stat. Soc. B 78, 423–444.

Barber-Candès adjustment

Examples

julia> pvals = PValues([0.001, 0.01, 0.03, 0.5]);

julia> adjust(pvals, BarberCandes())
4-element Array{Float64,1}:
 0.3333333333333333
 0.3333333333333333
 0.3333333333333333
 1.0

References

Barber, R.F., and Candès, E.J. (2015). Controlling the false discovery rate via knockoffs. Ann. Statist. 43, 2055–2085.

Arias-Castro, E., and Chen, S. (2017). Distribution-free multiple testing. Electron. J. Statist. 11, 1983–2001.