Nathan Mantel

Why Stepdown Procedures in Variable Selection

Recent reviews have dealt with the subject of which variables to select and which to discard in multiple regression problems. Lindley (1968) emphasized that the method to be employed in any analysis should be related to the use intended for the finally fitted regression. In the report by Beale et al. (1967), the emphasis is on selecting the best subset for any specified number of retained independent variables. Here we will be concerned with pointing out the advantages of the variable selection scheme in which independent variables are successively discarded one at a time from the original full set. While these advantages are not unknown to workers in this field, they are however not appreciated by the statistical community in general. For the purposes of this demonstration it is assumed that we are in the nonsingular case so that the number of observations exceeds the number of regressor variables. Let us begin by considering economy of effort. Suppose that we were using a step-up regression procedure, ignoring for the while its theoretical deficiencies (to be discussed later). We should then first fit k simple regressions, one for each of the k regressor variables considered, selecting the single most significant individual regressor variable. Having made this selection we would proceed with k - 1 additional fits to determine which of the remaining variables in conjunction with the first selected yielded the greatest reduction in residual variation. This process is continued on so as to provide a successive selection and ordering of variables. We may even require the ordering of all k variables, leaving for later decision what critical juncture is to be employed in determining which of the k variables to retain, which to reject-if we do so we shall have made a total of k(k + 1)/2 fits, albeit they may have differed greatly in their degree of complexity. A complete stepdown regression procedure however requires but k fits, as will now be indicated. Suppose we have done a multiple regression on all k variables and wish to consider the k possible multiple regressions on all sets of k - 1 variables, that is where 1 variable has been deleted. The results for these k possible multiple regressions are implicit in the initial k-variable regression, provided we have secured the inverse matrix, or at least its diagonal, necessary for testing the significance of the fitted partial regression coefficients. The case

A case‐control study of twin pairs discordant for Parkinson's disease A search for environmental risk factors

A previous study of twins with Parkinsons disease (PD) revealed low concordance, suggesting that genetic factors play a minor role in the etiology of PD. To identify possible environmental determinants of PD while maximally controlling for hereditary factors, 31 monozygotic twin pairs discordant for PD were interviewed by telephone. Information about possible risk factors was obtained from systematic and uniform interviews with cases and controls. The only statistically significant result was less cigarette smoking by PD patients (p < 0.05). Thirteen dizygotic discordant twin pairs were evaluated with the same techniques, but there were no statistically significant differences between affected and unaffected twins.

Motor neuron disease on Guam: Geographic and familial occurrence, 1956-85

We investigated the geographic and familial occurrence of motor neuron disease (MND) on Guam, and then considered etiologic hypotheses related to cycad use and metal intoxication. The research was based on 303 Chamorros from Guam and 3 Chamorros from other Mariana Islands, all with MND onset on Guam during 1956–85. Inarajan and Umatac, two southern districts, each had, for both sexes combined, an average incidence rate significantly higher than the corresponding overall rate for Guam. Also, for each sex, geographic patterns of incidence were significantly related to 1)socioeconomic level (men only), 2)cycasin concentrations in cycad flour samples (men and women), 3)iron concentrations in water samples (men and women), 4)silicon concentrations in water samples (men only), and 5)cobalt and nickel concentrations in soil samples (men and women). The MND risk in susceptible sibships was about 7–28 times greater than that in the general population. The cycad hypothesis conforms somewhat better than the metal intoxication hypothesis with the data presented.

Restricted Least Squares Regression and Convex Quadratic Programming

A parsimonious stepwise procedure for obtaining least squares solutions of multiple regression eqllntions when the regression coefficients are subject to arbitrary but consistent linear restraints is presented. The method is also applicable to the minimization of positive definite quadratic functions. Key to the method is the use of the elements of the appropriate inverse matrix for determining the standardized distance from any unrestricted, or conditionally unrestricted, solution to any boundary or boundary intersection of the permissible region for the regression coefficients. Various other aspects of the problem are discussed.

Understanding Wald's Test for Exponential Families

Abstract Anomalous behavior of Wards test for exponential families is explained on the basis of the use of parameter estimates rather than null values in the variance part of the Wald expression. Illustration of the difficulty is made through use of single-parameter situations, albeit multiparameter situations were first cited to bring out the difficulty. Situations had been claimed in which statistical significance disappeared as data became more inconsistent with the null hypothesis. It is shown here that with use of the Wald test, according to the parameterization employed, the same data can be both consistent with all possible null values for the parameter and inconsistent with all possible values.

Presentation Dubious evidence of heart and cancer deaths due to passive smoking

Glantz and Parmley [l] published a report, apparently commissioned by the U.S. Environmental Protection Agency, in which it was suggested that passive smoking or environmental tobacco smoke caused 53,000 heart and cancer deaths annually in the U.S. That estimated death toll is not independently developed by Glantz and Parmley but is essentially borrowed from Wells [2], with modifications by Kawachi et al. [3]. In their own area of expertise, Glantz and Parmley claim certain physiological effects, particularly on platelets, of exposure to tobacco smoke but do not say whether those effects are unique to such exposure or that they necessarily lead to increased mortality. In the event, that work provides them with an introduction to writing about the death toll from passive smoking. That there is such a toll is taken as unquestionably a given, so attention need be paid to only the magnitude of the toll. But away from their area of expertise, Glantz and Parmley seem to be naive. Thus they take as worthy of note that the confidence intervals on relative risk in their Table 1 are skewed towards higher risks-this is essentially a necessity, since a relative risk can range upwards to

Motor neuron disease on Guam: Temporal occurrence, 1941-85

Using a case registry, we investigated the temporal occurrence of motor neuron disease (MND) on Guam. MND with onset during 1941–85 was documented in 434 Chamorros and 9 non‐Chamorro migrants who had lived on Guam before onset. Increased median age at onset and decreased age‐adjusted incidence rates (since the early 1960s) were observed for Chamorros of both sexes. Our evidence about MND on Guam is consistent with: 1) The latent period duration has varied from years to decades; 2) With time, the exposure period or latent period, or both, have lengthened; 3) The high risk of acquiring the condition has been reduced since, at least, the early 1950s, and the most recent years of meaningful risk were the early to middle 1960s; 4) The critical age for acquiring the condition is in adolescence and adulthood; 5) Change of environment from Guam to overseas during childhood resulted in decreased risk of acquiring the condition.

Doubt and Certainty in Statistics

13 Lathers C, Schraeder P, Autonomic dysfunction in epilepsy: characterisation of autonomic cardiac neural discharge associated with pentylenetetrazol-induced epileptogenic activity. Epilepsia 1982;23:633-47 14 Lathers C, Schraeder P, Weiner F. Synchronization of cardiac autonomic neural discharge with epileptogenic activity: the lockstep phenomenon. Electroencephalogr Clin Neurophysiol1987;67:247-59 15 Dasheiff R, Dickinson L. Sudden unexpected death of epileptic patient due to cardiac arrhythmia after seizure. Arch Neurol 1986;43:194-6

An analysis of two recent epidemilogic report in the New England Journal of Medicine associating breast cancer in women with moderate alcohol consumption

Two cohort studies showed an association between moderate alcohol consumption and breast cancer, with an apparent relative risk of 1.50. Drinkers and nondrinkers already differed in important ways at the initiation of these two studies, precluding a causal interpretation of the associations found. Women in need of the relaxing effect of alcohol may indeed be the group at higher risk for breast cancer. Inasmuch as the studies did not inquire into other cancers, morbidities, mortalities, or total mortality, they leave open the possibility that, on balance, moderate alcohol consumption is beneficial.

Ordered Alternatives and the 1 1/2-Tail Test

Abstract Under proper conditions, two independent tests of the null hypothesis of homogeneity of means are provided by a set of sample averages. One test, with tail probability P 1, relates to the variation between the sample averages, while the other, with tail probability P 2, relates to the concordance of the rankings of the sample averages with the anticipated rankings under an alternative hypothesis. The quantity G = P 1 P 2 is considered as the combined test statistic and, except for the discreteness in the null distribution of P 2, would correspond to the Fisher statistic for combining probabilities. Illustration is made, for the case of four means, on how to get critical values of G or critical values of P 1 for each possible value of P 2, taking discreteness into account. Alternative measures of concordance considered are Spearmans ρ and Kendalls τ. The concept results, in the case of two averages, in assigning two-thirds of the test size to the concordant tail, one-third to the discordant tail.