Test of hypotheses for linear regression models with non-sample prior information

Test of hypotheses for linear regression models with
non-sample prior information

Classical inferences about population parameters are usually drawn from the sample data alone. This applies to methods used in parameter estimation and hypothesis
testing. Inferences about population parameters could be improved using non-sample prior information (NSPI) on the value of another related parameter. However, any NSPI on the value of any parameter is likely to be uncertain (or unsure). The NSPI can be classified as (i) unknown (unspecified), (ii) known (specified), and (iii) uncertain
if the suspected value is unsure. For the three different scenarios, three different statistical tests: (i) unrestricted test (UT), (ii) restricted test (RT) and (iii) preliminary test test (PTT) are defined. The current research is to test the intercept parameter(s) when NSPI is available on the slope parameter(s). The test statistics,
their sampling distributions, and power functions of the tests are derived. Comparison of power functions of the tests are used to recommend a best test. In this thesis, we test (1) the intercept of the simple regression model (SRM) when there is NSPI on the slope, (2) the intercept vector of the multivariate simple regression model (MSRM) when there is NSPI on the slope vector, (3) a subset of regression parameters of the multiple regression model (MRM) when NSPI is available on another subset of the
regression parameters, and (4) the equality of the intercepts for p (≥ 2) lines of the parallel regression model (PRM) when there is NSPI on the slopes.
For each of the above four regression models, the following steps are carried out: (1) derived the test statistics of the UT, RT and PTT for both known and unknown variance, (2) derived the sampling distributions of the test statistics of the UT, RT and PTT, (3) derived and compared the power function and the size of the UT, RT and PTT. For known variance, under a sequence of an alternative hypothesis, the sampling distributions of the UT and RT of the simple regression model follows a normal distribution. However, the PTT follows a bivariate normal distribution. For unknown
variance, the sampling distribution of the UT and RT of the simple regression model follows a Student’s t distribution but the PTT follows a correlated bivariate Student’s
t distribution. For the multivariate simple regression, multiple regression and parallel regression models, the sampling distribution of the UT and RT follows a univariate
noncentral F distribution under the alternative hypothesis. However, the PTT follows a correlated bivariate noncentral F distribution. For the four regression models above, there is a correlation between the UT and PT but there is no such correlation between the RT and PT. To evaluate the power function of the PTT the probability integral of the bivariate normal, bivariate Student’s t and bivariate noncentral F distributions are used. For the computations of the power function of the PTT of the MSRM, MRM and PRM require the cumulative distribution function (cdf) of a correlated bivariate noncentral F (BNCF) distribution. But the correlated BNCF distribution is not available in the literature, and hence we derive the probability density function (pdf) and cdf of the BNCF distribution. The R package is used for all computations and graphical analyses.
The statistical criteria that are used to compare the performance of the UT, RT and PTT are the size and power of the tests. A test that minimizes the size and maximizes the power is preferred over any other tests. In reality, the size of a test is fixed, and then the choice of the best test is based on its maximum power. The study shows that the power of the RT is always higher than that of the UT
and PTT, and the power of the PTT lies between the power of the RT and UT. The size of the UT is smaller than that of the RT and PTT. Among the three tests, the UT has the lowest power and lowest size. In terms of power it is the worst and in terms of size it is the best. The RT has maximum power and size. The PTT has smaller size than the RT and the RT has larger power than the PTT. The PTT protects against maximum size of the RT and minimum power of the UT. Thus the the PTT attains a reasonable dominance over the UT and RT for all regression models when the suspected value of the slope parameter(s) suggested by the NSPI is not too far away from that under the null hypothesis.