Simultaneous confidence intervals for the population cell means, for two-by-two factorial data, that utilize uncertain prior information

Consider a two-by-two factorial experiment with more than 1 replicate. Suppose that we have uncertain prior information that the two-factor interaction is zero. We describe new simultaneous frequentist confidence intervals for the 4 population cell means, with simultaneous confidence coefficient 1-alpha, that utilize this prior information in the following sense. These simultaneous confidence intervals define a cube with expected volume that (a) is relatively small when the two-factor interaction is zero and (b) has maximum value that is not too large. Also, these intervals coincide with the standard simultaneous confidence intervals obtained by Tukeys method, with simultaneous confidence coefficient 1-alpha, when the data strongly contradict the prior information that the two-factor interaction is zero. We illustrate the application of these new simultaneous confidence intervals to a real data set.

Further properties of frequentist confidence intervals in regression that utilize uncertain prior information

Consider a linear regression model with n-dimensional response vector, regression parameter beta = (beta_1, ..., beta_p) and independent and identically N(0, sigma^2) distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau = c^T beta - t where c and t are specified. Also suppose that we have uncertain prior information that tau = 0. Part of our evaluation of a frequentist confidence interval for theta is the ratio (expected length of this confidence interval)/(expected length of standard 1-alpha confidence interval), which we call the scaled expected length of this interval. We say that a 1-alpha confidence interval for theta utilizes this uncertain prior information if (a) the scaled expected length of this interval is significantly less than 1 when tau = 0, (b) the maximum value of the scaled expected length is not too much larger than 1 and (c) this confidence interval reverts to the standard 1-alpha confidence interval when the data happen to strongly contradict the prior information. Kabaila and Giri, 2009, JSPI present a new method for finding such a confidence interval. Let hatbeta denote the least squares estimator of beta. Also let hatTheta = a^T hatbeta and hattau = c^T hatbeta - t. Using computations and new theoretical results, we show that the performance of this confidence interval improves as |Corr(hatTheta, hattau)| increases and n-p decreases.

Admissibility of the usual confidence interval in linear regression

Consider a linear regression model with independent and identically normally distributed random errors. Suppose that the parameter of interest is a specified linear combination of the regression parameters. We prove that the usual confidence interval for this parameter is admissible within a broad class of confidence intervals.

Computation of confidence intervals in regression utilizing uncertain prior information

We consider a linear regression model with regression parameter beta =(beta_1, ..., beta_p) and independent and identically N(0, sigma^2)distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau = c^T beta - t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that tau = 0. Kabaila and Giri (2009c) present a new frequentist 1-alpha confidence interval for theta that utilizes this prior information. This interval has expected length that (a) is relatively small when the prior information about tau is correct and (b) has a maximum value that is not too large. It coincides with the standard 1-alpha confidence interval (obtained by fitting the full model to the data) when the data strongly contradicts the prior information. At first sight, the computation of this new confidence interval seems to be infeasible. However, by the use of the various computational devices that are presented in detail in the present paper, this computation becomes feasible and practicable.

Confidence intervals in regression utilizing prior information

We consider a linear regression model with regression parameter beta=(beta_1,...,beta_p) and independent and identically N(0,sigma^2) distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau=c^T beta-t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that tau = 0. We present a new frequentist 1-alpha confidence interval for theta that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-alpha confidence interval when the data strongly contradicts this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when tau=0. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about tau is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2-by 2 factorial experiment with 20 replicates. Suppose that the parameter of interest theta is a specified simple effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for theta that utilizes this prior information.

Large-Sample Confidence Intervals for the Treatment Difference in a Two-Period Crossover Trial, Utilizing Prior Information

Consider a two-treatment, two-period crossover trial, with responses that are continuous random variables. We find a large-sample frequentist 1-alpha confidence interval for the treatment difference that utilizes the uncertain prior information that there is no differential carryover effect.

Confidence intervals for the normal mean utilizing prior information

Consider X_1,X_2,...,X_n that are independent and identically N(mu,sigma^2) distributed. Suppose that we have uncertain prior information that mu = 0. We answer the question: to what extent can a frequentist 1-alpha confidence interval for mu utilize this prior information?

Upper bounds on the minimum coverage probability of confidence intervals in regression after variable selection

We consider a linear regression model, with the parameter of interest a specified linear combination of the regression parameter vector. We suppose that, as a first step, a data-based model selection (e.g. by preliminary hypothesis tests or minimizing AIC) is used to select a model. It is common statistical practice to then construct a confidence interval for the parameter of interest based on the assumption that the selected model had been given to us a priori. This assumption is false and it can lead to a confidence interval with poor coverage properties. We provide an easily-computed finite sample upper bound (calculated by repeated numerical evaluation of a double integral) to the minimum coverage probability of this confidence interval. This bound applies for model selection by any of the following methods: minimum AIC, minimum BIC, maximum adjusted R-squared, minimum Mallows Cp and t-tests. The importance of this upper bound is that it delineates general categories of design matrices and model selection procedures for which this confidence interval has poor coverage properties. This upper bound is shown to be a finite sample analogue of an earlier large sample upper bound due to Kabaila and Leeb.