Albert J. Getson

Crossover Comparison of Rizatriptan 5 mg and 10 mg Versus Sumatriptan 25 mg and 50 mg in Migraine

Rizatriptan is a selective 5‐HT1B/1D receptor agonist with rapid oral absorption and early onset of action in the acute treatment of migraine. This double‐blind, placebo‐controlled, crossover study compared rizatriptan 5 mg versus sumatriptan 25 mg, and rizatriptan 10 mg versus sumatriptan 50 mg. A total of 1329 patients were allocated to one of five groups for treatment of two attacks: rizatriptan 5 mg/sumatriptan 25 mg; sumatriptan 25 mg/rizatriptan 5 mg; rizatriptan 10 mg/sumatriptan 50 mg; sumatriptan 50 mg/rizatriptan 10 mg; placebo/placebo. For each attack, patients rated headache severity, presence of associated symptoms, and functional disability prior to dosing and at intervals through 4 hours thereafter. Patients also rated their satisfaction with medication. Rizatriptan 5 mg and 10 mg provided faster relief of headache pain and greater relief of migraine symptoms than the 25‐mg and 50‐mg doses of sumatriptan, respectively. The response to rizatriptan was better than sumatriptan on additional measures including functional disability and satisfaction with medication. All active treatments were highly effective compared to placebo and acted as early as 30 minutes after dosing. All active treatments were well‐tolerated and showed comparable safety profiles.

{2}-Inverses and Their Statistical Application

I Introduction.- II Time for {2}-Inverses.- 2.0 Introduction.- 2.1 The Three Phase Inversion Procedure.- 2.2 Constrained Inverses.- 2.3 {2}- and {1,2}-Inverses: The Null Augmented Mappings.- 2.4 {1,5}-Inverses: The Nonnull Augmented Mappings.- 2.5 Construction of Moore-Penrose Type Generalized Inverses.- 2.6 A Geometric Representation of {2}-Inverses.- 2.7 {1,5}-Inverses and Projections.- 2.8 {1,5}-Inverses and Solutions to Linear Equations.- 2.9 Decomposition of {2}-Inverses.- 2.10 Spectral Decomposition in Terms of {2}Inverses.- 2.11 Computation of {2}-Inverses.- III {2}-Inverses, Quadratic Forms and Second Degree Polynomials.- 3.0 Introduction.- 3.1 x2 Distribution and Independence of Quadratic Forms and Second Degree Polynomials.- 3.2 Generalized Inverses and Quadratic Forms.- 3.3 {2}-Inverses and x2 Distributed Quadratic Forms.- 3.4 On The Uniqueness of the {2}-Inverse Representation of x2 Distributed Quadratic Forms.- 3.5 A Minimal Sufficient Set of Coefficient Matrices for All X2 Distributed Quadratic Forms.- 3.6 Independence of X2 Distributed Quadratic Forms.- 3.7 A Canonical Representation of Second Degree Polynomials.- 3.8 X2 Distributed Second Degree Polynomials.- 3.9 {2}-Inverses and the Distribution and Independence of Second Degree Polynomials.- IV {2}-Inverses and Least Squares Solutions.- 4.0 Introduction.- 4.1 The Least Squares Problem.- 4.1 Strategies For Obtaining Least Squares Solutions.- 4.3 Symmetric {1,2}-Inverses and Sets of Nonestimable Constraints.- 4.4 Bott-Duffin Inverses and Constrained LSSs.- 4.5 {1,5}-Inverses and LSSs.- 4.6 Relationships Among LSSs.- 4.7 Minimum Norm LSSs.- 4.8 A General Theorem on Constrained LSSs.- 4.9 Residual Sum of Squares and Their Difference.- 4.10 Computing Constrained LSSs and Residual Sum of Squares.- V {2}-Inverses in Linear Models.- 5.0 Introduction.- 5.1 The Models.- 5.2 The Distribution and Relationships Among the LSSs For the Prameters in Various Models.- 5.3 Hypothesis Testing in Linear Models.- 5.4 Equivalent Numerator Sum of Squares for a Test of Hypothesis.- 5.5 Hypotheses Invariant to Cell Sizes.- 5.6 The R Approach and SAS Type I and Type II Sums of Squares.- 5.7 The R* Approach and the SAS Type III Sum of Squares.- References.

Self-directed Treatment Of Intermittent Heartburn: A Randomized, Multicenter, Double-blind, Placebo-controlled Evaluation Of Antacid And Low Doses Of An H2-receptor Antagonist (famotidine)

BackgroundHeartburn, a common symptom, is self-treated with oral antacids. Efficacy of antacids has not been demonstrated for individual, spontaneous heartburn episodes. MethodsWe conducted a double-blind, randomized, placebo-controlled, parallel-group study of self-directed treatment for episodic heartburn comparing famotidine (FAM) 5, 10, or 20 mg and antacid (11 mEq ANC) to placebo (PBO) during a 4-week period. Twenty-nine US investigators enrolled a total of 565 outpatients, ages 18–81 years (mean 44.1 years) with heartburn but not seeking care for heartburn. Treatment of spontaneous heartburn episodes was permitted as needed, up to twice daily, with self-administered test drug. An open-label, backup antacid was provided to use if test drug did not provide adequate relief. Patients assessed heartburn relief hourly and recorded use of backup antacid. Relief was defined as complete relief of symptoms without the use of backup antacid. ResultsThe median proportion of episodes relieved was: PBO, 41%; FAM 5 mg, 59%, 0.05 ≤ p < 0.10; FAM 10 mg, 70%, p < 0.001; FAM 20 mg, 69%, p < 0.001; antacid, 62%, p < 0.05 (p-values versus PBO). Supplemental analyses incorporating time to relief confirmed that famotidine and antacid provided more rapid and more frequent relief than placebo (odds ratio for relief relative to PBO: FAM 5 mg, 1.55, p = 0.003; FAM 10 mg, 1.94, p < 0.001; FAM 20 mg, 2.13, p < 0.001; antacid 1.57, p = 0.003). The tolerability profile was similar with famotidine, antacid, and placebo. ConclusionsThe positive results with antacid demonstrated for the first time the efficacy of antacid in self-treatment of individual heartburn episodes and provided internal validation of this study paradigm. Patients in this study self-medicated effectively using low doses of famotidine on an as needed basis for spontaneous episodes of heartburn.

{2}-Inverses and Least Square Solutions

The concept of least squares is the cornerstone of regression and linear model analysis. Many statistical texts consider at length the associated theory and propose various strategies for obtaining least squares solutions (LSS’s), e.g. Searle (1971), Graybill (1976), and Draper and Smith (1981).

{2}-Inverses in Linear Models

For an experiment designed with an equal number of replications in each cell, there is little controversy concerning the sum of squares to be used in testing the various effects. However, in reality most data are imbalanced and there is no unanimity of opinion as to the sum of squares appropriate for testing the various effects.

{2}-Inverses, Quadratic forms and Second Degree Polynomials

Due to the importance of quadratic forms, for example in the analysis of variance as established by Fisher (1926) and Cochran (1934), the theory of these statistics has been well explored in the statistical literature. Beginning with quadratic forms in normally and independently distributed random variables, Craig (1943) and Hotelling (1944) established the elegant, easily implemented and well known matrix conditions for their independence and X2 distribution. Subsequently, in a series of papers by Craig (1947), Matern (1949), Aitken (1950), Carpenter (1950), and Graybill and Marsaglia (1957), these conditions were extended to quadratic forms in correlated normal variables with a positive definite covariance matrix. In this case the conditions still retain the simplicity established by Craig and Hotelling. In turn, these conditions were further extended to quadratic forms in correlated normal variables with only nonnegative definite and possibly singular covariance by Good (1963, 1969, 1970), Khatri (1963, 1977) and Shanbhag (1966, 1968). Unfortunately in this case the conditions become more complicated and lose their previous simplicity.

Time For {2}-Inverses

As noted in the previous chapter, various classes of generalized inverses have been proposed in the literature. Geometric characterizations of generalized inverses were presented by Kruskal (1975) and, more recently, by Rao and Yanai (1985). The principal aim of this chapter is to unify and expand upon these diverse approaches in a consistent way.