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`

.

Suppose you want to get FAB *p*-values for the predictors *x*_{i, 1}, …, *x*_{i, p} in the linear model

*y*_{i} = *α*_{0} + *α*_{1}*w*_{i, 1} + *α*_{2}*w*_{i, 2} + *β*_{1}*x*_{i, 1} + ⋯ + *β*_{p}*x*_{i, p} + *ϵ*_{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

column-bind the

*x*-variables into an*n*×*p*matrix`X`

, e.g.`X<-cbind(x1,x2,x3)`

;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 *β*_{1}, …, *β*_{p} correspond to *p* objects about which you have additional covariate information (say attributes {(*v*_{j, 1}, *v*_{j, 2}), *j* = 1, …, *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 *β*_{1}, …, *β*_{p}. The additional term specifies a *linking model* for *β*_{1}, …, *β*_{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.

In the simplest case of a normally distributed estimator *θ̂* of *θ* such that *θ̂* ∼ *N*(*θ*, *σ*^{2}), a standard *p*-value and confidence interval are based on the test statistic |*θ̂*|. A FAB *p*-value and confidence interval is based on the statistic |*θ̂* + *a*|, where *a* is determined from indirect information about the sign and magnitude of *θ*. The functional form of the FAB *p*-value is extremely simple:

*p*_{FAB}(*θ̂*, *a*) = 1 − |*Φ*(*θ̂* + 2*a*) − *Φ*( − *θ̂*)|,

where *Φ* 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.

```
# Release version on CRAN
install.packages("FABInference")
# Development version on GitHub
devtools::install_github("pdhoff/FABInference")
```

“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.

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)