Simple ratios in which a measurement variable is divided by a size variable are commonly used but known to be inadequate for eliminating size correlations from morphometric data. Deficiencies in the simple ratio can be alleviated by incorporating regression coefficients describing the bivariate relationship between the measurement and size variables. Recommendations have included: 1) subtracting the regression intercept to force the bivariate relationship through the origin (intercept-adjusted ratios); 2) exponentiating either the measurement or the size variable using an allometry coefficient to achieve linearity (allometrically adjusted ratios); or 3) both subtracting the intercept and exponentiating (fully adjusted ratios). These three strategies for deriving size-adjusted ratios imply different data models for describing the bivariate relationship between the measurement and size variables (i.e., the linear, simple allometric, and full allometric models, respectively). Algebraic rearrangement of the equation associated with each data model leads to a correctly formulated adjusted ratio whose expected value is constant (i.e., size correlation is eliminated). Alternatively, simple algebra can be used to derive an expected value function for assessing whether any proposed ratio formula is effective in eliminating size correlations. Some published ratio adjustments were incorrectly formulated as indicated by expected values that remain a function of size after ratio transformation. Regression coefficients incorporated into adjusted ratios must be estimated using least-squares regression of the measurement variable on the size variable. Use of parameters estimated by any other regression technique (e.g., major axis or reduced major axis) results in residual correlations between size and the adjusted measurement variable. Correctly formulated adjusted ratios, whose parameters are estimated by least-squares methods, do control for size correlations. The size-adjusted results are similar to those based on analysis of least-squares residuals from the regression of the measurement on the size variable. However, adjusted ratios introduce size-related changes in distributional characteristics (variances) that differentially alter relationships among animals in different size classes.

No abstract

Background: According to the original model of cranial osteopathy, intrinsic rhythmic movements of the human brain cause rhythmic fluctuations of cerebrospinal fluid and specific relational changes among dural membranes, cranial bones, and the sacrum. Practitioners believe they can palpably modify parameters of this mechanism to a patient's health advantage.

After any therapy, when symptoms improve, healthcare providers (and patients) are tempted to award credit to treatment. Over time, a particular treatment can seem so undeniably helpful that scientific verification of efficacy is judged an inconvenient waste of time and resources. Unfortunately, practitioners' accumulated, day-to-day, informal impressions of diagnostic reliability and clinical efficacy are of limited value. To help clarify why even treatments entirely lacking in direct effect can seem helpful, I will explain why real signs and symptoms often improve, independent of treatment. Then, I will detail quirks of human perception, interpretation, and memory that often make symptoms seem improved, when they are not. I conclude that healthcare will grow to full potential only when judgments of clinical efficacy routinely are based in properly scientific, placebo-controlled, outcome analysis.