Alan B. Forsythe

Robust Tests for the Equality of Variances

Abstract Alternative formulations of Levenes test statistic for equality of variances are found to be robust under nonnormality. These statistics use more robust estimators of central location in place of the mean. They are compared with the unmodified Levenes statistic, a jackknife procedure, and a χ2 test suggested by Layard which are all found to be less robust under nonnormality.

The Small Sample Behavior of Some Statistics Which Test the Equality of Several Means

Four statistics which may be used to test the equality of population means are com-pared with respect to their robustness under heteroscedasticity, their power, and the overlap of their critical regions. The four are: the ANOVA F-statistic; a modified F which has the same numerator as the ANOVA but an altered denominator; and two similar statistics proposed by Welch and James which differ primarily in their approximations for their critical values. The critical values proposed by Welch are a better approximation for small sample sizes than that proposed by James. Both Welchs statistic and the modified F are robust under the inequality of variances. The choice between them depends upon the magnitude of the means and their standard errors. When the population variances are equal, the critical region of the modified F more closely approximates that of the ANOVA than does Welchs.

372: The Anova and Multiple Comparisons for Data with Heterogeneous Variances

The lack of robustness of the one-way ANOVA to the heterogeneity of the within group variances is well-known (Scheffe [1959]). For two samples a robust version of the t-test has been developed (Welch [1947], Wang [1971]). For more than two samples Welch [1951] proposed a statistic based on an asymptotic confidence interval approach. An alternative robust method is to replace the denominator of the usual ANOVA F-statistic by an estimate of the variance of the numerator (Brown and Forsythe [1974]). Here we show how this latter statistic can be derived by combining individual orthonormal contrasts. This result is extended to the analysis of a two-way ANOVA and to test the lack of fit of contrasts partitioned from the mode]. A multiple comparisons test procedure is proposed which is similar to that of Scheffe [1959].

Robust Estimation of Straight Line Regression Coefficients by Minimizing pth Power Deviations

It is well known that the mean is very sensitive to deviations from normality, especially due to outliers or long tails. We propose the use of an estimator which has been demonstrated to be more robust than least squares for estimating the simple mean, that is, the estimator which minimizes the pth power of the deviations for a power of p between one and two. We also show that a reasonably fast and widely available computer subroutine is available to solve the problem.

Rapid eye movement (REM) activity in normal and autistic children during REM sleep.

Thirty normal children (aged 3–68 months) and 16 autistic children (aged 36–62 months) were recorded during nonmedicated sleep and data pertaining to rapid eye movements (REM) were measured during the first three REM periods of the night. When time of night from which data were gathered was held constant, normal children showed a significant relationship between age and the organization of eye movements into discrete bursts. When autistic children were compared to age-matched normal controls, they showed an immaturity in this phenomena, their results being similar to those found in children less than 18 months of age. Such an immaturity could result from dysfunction at a number of diverse levels and sites in the central nervous system.

Prenatal and intrapartum high-risk screening: II. Risk factors reassessed☆

A method of identifying the high-risk pregnancy by a quantitative assessment of prenatal, intrapartum, and neonatal factors is presented. Calculating the probabilities of neonatal risk can be done with a hand-held calculator. The technique described provides a method of assessing the importance of perinatal variables and determining the effect of the process of health care on outcome.

A THEORETICAL ANALYSIS OF ANTECEDENTS OF YOUNG COUPLES' FERTILITY DECISIONS AND OUTCOMES

Longitudinal survey data from 509 couples who at Time 1 interview had recently married or had their first child did not support the hypothesis that demographic factors influence fertility intentions, decisions, and outcomes only indirectly through their effects on attitudes and motivations. Husbands’ and wives’ attitudes exerted reciprocal influence on one another. However, while husbands’ sex-role traditionalism and motivation for parenthood strongly influenced wives’ traditionalism and motivation in the case of recently married couples, this pattern was reversed for riew parents. Birth control use was directly affected by wives’ fertility intentions, but not by husbands’ intentions. Difficulties in examining couple interaction variables such as relative power and the possible limitations of fitting these data to a complex theoretical model using LISREL are discussed.

Epilepsy and Mental Retardation Following Febrile Seizures In Childhood

ABSTRACT. In an unselected group of children who were seen following an initial febrile convulsion, the frequency of subsequent afebrile seizures was 3.5% and of mental retardation 1%. The most common afebrile seizure type was generalized major (86%). About 3/4 of the children who developed afebrile seizures did so by three years and all by five years following the initial febrile seizure. The children with afebrile seizures differed from those without afebrile seizures in the frequency of neonatal abnormality, family history of mental retardation, focal initial febrile convulsions, and delay in psychomotor milestones before the initial febrile seizure. Only about 1/3 of the children who developed afebrile seizures ever had a recurrent febrile convulsion and none had complex recurrent febrile seizures. Half the children with mental retardation had histories of delay in psychomotor milestones prior to the initial febrile seizure, and no child with mental retardation had any seizure longer than five minutes. The administration of daily phenobarbital did not reduce the frequency of epilepsy, in spite of a significant reduction in the incidence of recurrent febrile seizures. There remains no evidence that the prevention of recurrent febrile convulsions significantly decreases the frequency of afebrile seizures or mental retardation.

Radiotherapy for Prostatic Carcinoma: Post-Irradiation Prostatic Biopsy and Recurrence Patterns With Long-term Followup

From 1968 through 1975, 159 patients with stages A, B and C adenocarcinoma of the prostate were treated with supervoltage radiation therapy. A median dose of 7,200 rad was given. The influence upon survival of grade, stage and a positive biopsy result after irradiation is analyzed. Over-all survival is a patently imprecise test of local treatment because patients die of intercurrent disease and are alive with metastases for significant intervals. The classification of survival with no evidence of recurrent disease excluded those patients dying of intercurrent disease. Therefore, the rate of survival free of disease always will be greater than the over-all survival rate in these patients, and a certain percentage of the patients without evidence of cancer will harbor subclinical disease. The 5 and 10-year survival rates free of disease were 80 and 69 per cent, respectively, in 51 patients with stages A and B cancer, and 66 and 47 per cent, respectively, in 108 with stage C disease. The 5 and 10-year over-all survival rates were 72 and 56 per cent, respectively, in the former and 68 and 39 per cent, respectively, in the latter patients. Survival free of disease and death of prostatic carcinoma were influenced adversely by advancing grade and stage (p less than 0.05), while over-all survival was influenced adversely by grade (p equals 0.02) but not by stage (p greater than 0.05). A positive biopsy result after irradiation did not predict survival free of disease, over-all survival or death of prostatic cancer in patients followed for 10 years (p greater than 0.05).

A Stopping Rule for Variable Selection in Multiple Regression

Abstract Stepwise regression is a member of a class of statistical methods which may be called opportunistic. The statistician is naturally concerned that an artifact should not be reported so he needs a stopping rule that is geared to the way in which variables are selected. A sampled permutation test is outlined which offers such a stopping rule for forward stepping.