Chi-squared distribution

Abstract: A frequently used statistic for testing homogeneity in a meta-analysis of K independent studies is Cochran’s Q For a standard test of homogeneity the Q statistic is referred to a chi-square distribution with K−1 degrees of freedom For the situation in which the effects of the studies are logarithms of odds ratios, the chi-square distribution is much too conservative for moderate size studies, although it may be asymptotically correct as the individual studies become large Using a mixture of theoretical results and simulations, we provide formulas to estimate the shape and scale parameters of a gamma distribution to fit the distribution of Q Simulation studies show that the gamma distribution is a good approximation to the distribution for Q Use of the gamma distribution instead of the chi-square distribution for Q should eliminate inaccurate inferences in assessing homogeneity in a meta-analysis (A computer program for implementing this test is provided) This hypothesis test is competitive with the Breslow-Day test both in accuracy of level and in power

Abstract: Background and Purpose: The purpose of this prospective cohort study was to identify whether variables from the baseline examination or physical therapy interventions received could predict clinical outcomes for people with cervical radiculopathy. Subjects and Methods: A total of 96 consecutive patients referred for physical therapy for cervical radiculopathy were the sources of data for this study. All subjects underwent a standardized examination and completed the Neck Disability Index (NDI), the Patient-Specific Functional Scale (PSFS), and the Numeric Pain Rating Scale (NPRS) at baseline and at discharge. The subjects were treated according to the discretion of the individual therapists. At the time of discharge, the subjects completed the Global Rating of Change as well. Subjects surpassing the minimal clinically important change for all 4 outcome tools were categorized as achieving short-term success. Individual variables from the examination and interventions provided were tested for univariate relationships with outcomes. Variables with a significance level of less than .10 were retained as potential predictor variables and were entered into a stepwise logistic regression model to determine the most accurate set of variables for predicting outcomes. Results: The pretest probability for the likelihood of short-term (28-day follow-up) success was 53%. A 4-variable model optimally identified subjects who were most likely to achieve success with physical therapy interventions (age of <54 years; dominant arm is not affected; looking down does not worsen symptoms; and multimodal treatment including manual therapy, cervical traction, and deep neck flexor muscle strengthening for at least 50% of visits). When 3 of these 4 variables were present, the positive likelihood ratio (+LR) was 5.2 (95% confidence interval [CI]=2.4, 11.3), and the posttest probability of success was 85%. When all 4 variables were present, the +LR was 8.3 (95% CI=1.9, 63.9), and the posttest probability of success was 90%. Discussion and Conclusion: These results suggest that a subset of predictor variables can accurately identify which people with cervical radiculopathy are likely to experience short-term successful outcomes. The study design did not allow for the identification of a cause-and-effect relationship, but it appears that intermittent cervical traction, manual therapy, and deep neck flexor muscle strengthening may be beneficial in the management of cervical radiculopathy. Future research is needed to substantiate these findings.