pdhoff / FABInference

FAB p-values and confidence intervals

FABInference

Construction of FAB p-values and confidence intervals

Package website

About

In multiparameter problems, information sharing across parameters can be used to improve the power of statistical hypothesis tests, thereby providing smaller $p$-values and narrower confidence intervals, on average across parameters. The FABInference package provides information sharing in linear and generalized linear regression models using a syntax similar to the built-in R functions lm and glm.

Usage

Suppose you want to get FAB $p$-values for the predictors $x_{i,1},\ldots, x_{i,p}$ in the linear model

[ y_i = \alpha_0 + \alpha_1 w_{i,1} + \alpha_2 w_{i,2} + \beta_1 x_{i,1} + \cdots + \beta_p x_{i,p} + \epsilon_i,
]

where $w_{i,1}$ and $w_{i,2}$ (and potentially other $w_{i,j}$'s) are additional control variables you'd like to have in the model. Then you need to

1. column-bind the $x$-variables into an $n\times p$ matrix X, e.g. X<-cbind(x1,x2,x3);

2. run the command fit<-lmFAB(y~w1+w2,X).

The output is similar to the output of the lm command, so you can type summary(fit) to see the FAB $p$-values. The FAB $p$-values and confidence intervals are stored in fit$FABpv and fit$FABci.

If $\beta_1,\ldots, \beta_p$ correspond to $p$ objects about which you have additional covariate information (say attributes ${(v_{j,1},v_{j,2}), j =1,\ldots, p}$ you might be interested in fitting the model fit<-lmFAB(y~w1+w2,X,~v1+v2), where v1 and v2 are $p$-dimensional vectors giving the attributes associated with $\beta_1,\ldots, \beta_p$. The additional term specifies a linking model for $\beta_1,\ldots, \beta_p$. Importantly, the linking model doesn't have to be correct in any way for the FAB $p$-values of confidence intervals to be valid. However, the better the linking model, the smaller the $p$-values and the narrower the intervals.

FAB inference for generalized linear models can be obtained similarly using the command glmFAB. In this case, the $p$-values and confidence intervals are valid asymptotically (just like the standard $p$-values and intervals). Fitting a normal linear regression with glmFAB is much faster than using lmFAB because the former uses an asymptotic approximation.

Theoretical details

In the simplest case of a normally distributed estimator $\hat\theta$ of $\theta$ such that $\hat \theta \sim N(\theta,\sigma^2)$, a standard $p$-value and confidence interval are based on the test statistic $|\hat\theta|$. A FAB $p$-value and confidence interval is based on the statistic $|\hat\theta + a|$, where $a$ is determined from indirect information about the sign and magnitude of $\theta$. The functional form of the FAB $p$-value is extremely simple:

[ p_{FAB}(\hat\theta,a) = 1- | \Phi(\hat\theta+2a) - \Phi(-\hat\theta) |, ]

where $\Phi$ is the standard normal CDF. The FAB confidence interval is a bit more complicated. In multiparameter settings, the optimal choice for $a$ for one parameter may be estimated from data on the other parameters, using a linking model that relates the parameters to each other. Importantly, the FAB confidence intervals and $p$-values have correct frequentist error rates, even if the linking model is incorrect.

Installation

# Release version on CRAN
install.packages("FABInference")


# Development version on GitHub 
devtools::install_github("pdhoff/FABInference")  

Documentation and citation

"Smaller p-values via indirect information". P.D. Hoff. arXiv:1907.12589 Journal of the American Statistical Association, to appear.

"Smaller p-values in genomics studies using distilled historical information". arXiv:2004.07887 J.G. Bryan and P.D. Hoff. Biostatistics, to appear.

"Exact adaptive confidence intervals for small areas". K. Burris and P.D. Hoff. arXiv:1809.09159 Journal of Survey Statistics and Methodology, 8(2):206–230, 2020.

"Exact adaptive confidence intervals for linear regression coefficients". P.D. Hoff and C. Yu. arXiv:1705.08331 Electronic Journal of Statistics, 13(1):94–119, 2019.

"Adaptive multigroup confidence intervals with constant coverage". arXiv:1612.08287 C. Yu and P.D. Hoff. Biometrika, 105(2):319–335, 2018.

Some examples

• Demo and simulation study

• Small area estimation Replication file for Hoff(2019)

• Hidden Markov model Replication file for Hoff(2019)

• Linear model interactions Replication file for Hoff(2019)

• Logistic regression Replication file for Hoff(2019)