Replicability Adjusted p-Values for Two Independent Studies with Multiple Endpoints

Calculates adjusted p-values for the null hypothesis of no replicability across studies for two study designs: (i) a primary and follow-up study, where the features in the follow-up study are selected from the primary study, as described in Bogomolov and Heller (2013) and Heller, Bogomolov and Benjamini (2014) ; (ii) two independent studies, where the features for replicability are first selected in each study separately, as described in Bogomolov and Heller (2018) . The latter design is the one encountered in a typical meta-analysis of two studies, but the inference is for replicability rather than for identifying the features that are non-null in at least one study.


Given p-values from two independent studies with multiple endpoints (features), the functions in the package return the adjusted p-values for false discovery rate control on replicability claims.

In replicability analysis we seek to reject the null hypothesis of no replicability in favor of the alternative hypothesis of replicability: that the finding replicated across the studies.
We do so by testing for signal in both studies. This is in contrast to a typical meta-analysis, where the test can also reject when only a single study has signal.

The procedures implemented in the functions compute the adjusted p-values for FDR control on replicability claims. By declaring as replicability discoveries the features with adjusted p-values (termed r-values) below the desired nominal level (e.g., 0.05), the FDR on replicability claims is controlled at the nominal level. See Bogomolov and Heller (2013), Heller, Bogomolov and Benjamini (2014), and Bogomolov and Heller (2018) for details.

The function radjust_sym should be used for replicability analysis of two independent studies, each examining multiple features. The features for replicability are first selected in each study separately based on the results of that study.

The function radjust_pf should be used for replicability analysis of a primary study and an independent follow-up study, where the features in the follow-up study are selected from the primary study.


Using radjust_sym:

## transform the example two-sided p-values to one-sided in the same direction (left):
## (we use the direction of the test statistic to do so and assume that it is continuous)
pv1 <- ifelse(mice$dir_is_left1, mice$twosided_pv1/2, 1-mice$twosided_pv1/2)
pv2 <- ifelse(mice$dir_is_left2, mice$twosided_pv2/2, 1-mice$twosided_pv2/2)
radjust_sym(pv1, pv2, input_type = "all", directional_rep_claim = TRUE, variant = "adaptive", alpha=0.05)
>   -> matching features by location.

> Note: Directional replicability claim option is set to TRUE.
>   Make sure you have entered the *left* sided p-values.

>   Replicability Analysis
> Call:
> radjust_sym(pv1 = pv1, pv2 = pv2, input_type = "all", directional_rep_claim = TRUE, 
>     variant = "adaptive", alpha = 0.05)
> Selection (adaptive):
> 20 features selected in study 1.
> 19 features selected in study 2.
> 12 features selected in both studies.
> Estimates for fraction of nulls among the selected in the other study:
> 0.4432133 in study 1.
> 0.4736842 in study 2.
> Features selected in both studies:
>  name    p_value1    p_value2     r_value Direction Significant
>     2 1.18873e-03 1.61210e-06 0.004004153      Left           *
>     9 6.11236e-03 3.16097e-08 0.012868127      Left           *
>    14 4.34268e-05 4.77527e-03 0.012868127      Left           *
>    16 5.88782e-03 1.96218e-04 0.012868127      Left           *
>    17 1.75750e-02 3.26740e-04 0.026909119     Right           *
>    20 1.57223e-02 6.52192e-05 0.026479584      Left           *
>    21 2.64690e-06 2.34075e-02 0.036959205      Left           *
>    23 3.32734e-09 5.37832e-05 0.000509525      Left           *
>    24 6.65468e-09 7.59238e-03 0.015983952      Left           *
>    25 3.32734e-09 1.37186e-05 0.000259932      Left           *
>    26 6.65468e-09 3.15068e-04 0.001492426      Left           *
>    27 6.65468e-09 9.48060e-05 0.000598774      Left           *
> 12 features are discovered in the directional replicability analysis (alpha = 0.05).

Primary and follow-up studies (radjust_pf):

rv  <- radjust_pf(pv1 = crohn$pv1, pv2 = crohn$pv1, m = 635547)
> [1] 6.419025e-30 2.027395e-28 5.719923e-19 6.380892e-17 6.380892e-17
> [6] 2.711667e-16


You can install radjust from github with:

# install.packages("devtools")

How to cite

Use the citation() R function:

> To cite radjust in publications, please use:
>   Shay Yaacoby, Marina Bogomolov and Ruth Heller (2018). radjust:
>   Replicability Adjusted p-values for Two Independent Studies with
>   Multiple Endpoints. R package version 0.1.0.
> To cite radjust_sym(), add:
>   Bogomolov, M. and Heller, R. (2018). Assessing replicability of
>   findings across two studies of multiple features. Biometrika.
> To cite radjust_pf(), add:
>   Bogomolov, M. and Heller, R. (2013). Discovering findings that
>   replicate from a primary study of high dimension to a follow-up
>   study. Journal of the American Statistical Association, Vol.
>   108, No. 504, Pp. 1480-1492.
>   Heller, R., Bogomolov, M., & Benjamini, Y. (2014). Deciding
>   whether follow-up studies have replicated findings in a
>   preliminary large-scale omics study. Proceedings of the National
>   Academy of Sciences of the United States of America, Vol. 111,
>   No. 46, Pp. 16262–16267.
> To see these entries in BibTeX format, use 'print(<citation>,
> bibtex=TRUE)', 'toBibtex(.)', or set
> 'options(citation.bibtex.max=999)'.


Reference manual

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0.1.0 by Ruth Heller, a year ago

Browse source code at

Authors: Shay Yaacoby [aut] , Marina Bogomolov [aut] , Ruth Heller [aut, cre]

Documentation:   PDF Manual  

GPL-3 license

Suggests covr, testthat

See at CRAN