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Register a new userConfidence intervals in regression utilizing prior information
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Added by
Paul Kabaila
Publication date
2009
fields
Mathematical Statistics
and research's language is
English
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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.
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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.
We consider a general regression model, without a scale parameter. Our aim is to construct a confidence interval for a scalar parameter of interest $theta$ that utilizes the uncertain prior information that a distinct scalar parameter $tau$ takes the specified value $t$. This confidence interval should have good coverage properties. It should also have scaled expected length, where the scaling is with respect to the usual confidence interval, that (a) is substantially less than 1 when the prior information is correct, (b) has a maximum value that is not too large and (c) is close to 1 when the data and prior information are highly discordant. The asymptotic joint distribution of the maximum likelihood estimators $theta$ and $tau$ is similar to the joint distributions of these estimators in the particular case of a linear regression with normally distributed errors having known variance. This similarity is used to construct a confidence interval with the desired properties by using the confidence interval, computed using the R package ciuupi, that utilizes the uncertain prior information in this particular linear regression case. An important practical application of this confidence interval is to a quantal bioassay carried out to compare two similar compounds. In this context, the uncertain prior information is that the hypothesis of parallelism holds. We provide extensive numerical results that illustrate the properties of this confidence interval in this context.
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.
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?
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.
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