Continuous Independent Variables Research Articles

Ordered Polytomous Regression: An Example Relating Serum Biochemistry and Haematology to Alcohol Consumption

Ordered polytomous logistic regression is a useful technique for relating a dependent ordered categorical variable to both categorical and continuous independent variables. This paper demonstrates the practical implementation of this model to a large data set. The problem of selecting an appropriate subset of independent variables is considered. A method of checking the model's goodness of fit is described. Three methods of allocating subjects to categories are compared, one based on fixed proportions allocated to each category being most informative. A statistical policy for amalgamating adjacent categories is developed. These techniques are demonstrated by a study of the relationship of alcohol consumption to serum biochemistry and haematology in 7735 middle‐aged men. The ordered polytomous model is used here to estimate a summary measure of an individual's biochemical response to alcohol intake.

The logistic regression analysis of psychiatric data

Logistic regression is presented as the statistical method of choice for analyzing the effects of independent variables on a binary dependent variable in terms of the probability of being in one of its two categories vs the other. The method, which must be applied by computer, is illustrated on data from the DSM-III field trials. The dependent variable is treatment with behaviourally-oriented psychotherapy vs treatment with psychoanalytically-oriented psychotherapy, and the independent variables are several patient and clinician characteristics. Like ordinary multiple regression, the method is shown capable of analyzing categorical as well as continuous independent variables. Unlike ordinary multiple regression when applied to binary data, logistic regression analysis necessarily yields estimated probabilities that lie between 0 and 1. The measure of association derived from logistic regression analysis, the odds ratio, is defined. Methods for making inferences about it are presented and illustrated.

The effects of slaughter age on growth and carcass traits in an intensively managed crossbred beef herd

The purpose of this study was to determine and describe the effects of slaughter age (at constant weights) on pre- and post-weaning growth rate and carcass traits of unselected steer and heifer progeny from a crossbred beef herd. The data were obtained from 340 heifers and 377 steers weaned at 305 days of age and fed a high-energy post-weaning diet. Heifers and steers were slaughtered at individual unshrunk weights of 420 and 470 kg, respectively, and divided into slaughter age classes based on 25-day intervals. There were differences ( P < 0.01) between slaughter age classes for all growth traits including 305-day (weaning) weight, post-weaning daily gain, days on feed, and weight/day. Mean 305-day weight of steers in the youngest slaughter age class (347.8 kg; 376–400 days of age) was 28% greater than the mean of steers in the oldest age class (272.2 kg;>501 days of age). The youngest steer class averaged 63% more for daily gain than the oldest steer class (1.462 vs. 0.894 kg); the corresponding value for heifers was 64%. Post-weaning time in the feedlot ranged from 84.7 days for the youngest steer to 215.6 for the oldest steer class. Calculations based on mean differences between age classes in 305-day weight, daily gain and days on feed indicated that variability in weaning weight and in daily gain were of similar importance in determining slaughter age. Marbling score was greater ( P < 0.01) for older steer classes, but was not different in the heifer data. Fat thickness of the oldest steer class was 25% more than the youngest steer class (1.89 vs. 1.51 cm); the corresponding percent in heifers was 30%. With the exception of differences in untrimmed rib and chuck percents in steers ( P < 0.05), and rib specific gravity in heifers ( P < 0.05), carcass quantity traits were essentially unaffected by slaughter age class when marbling score and fat thickness were included as continuous independent variables. None of the organoleptic traits (tenderness, juiciness and flavour) were affected by slaughter age class.

Logistic Regression Models with Continuous Independent Variables

Logistic Regression Models with Continuous Independent Variables

"Battered data"--some clinical effects of the abuse of multiple regression methods: the NSD.

The NSD equation, D = 1850 x T0.11 x N0.24, is a celebrated transmogrification of Cohen's two well-known collations of data on response to clinical irradiations (3 degrees erythema and 0.90 ablation of skin cancer) in which the relation of D and T is fixed by the data selected by Cohen and the additional constraint that N correspond to a schedule of five treatments per week is imposed subsequently by Ellis. The present paper shows that the equation, if correct, would have little clinical significance because the proportion, P, in which the dose, D, elicits the 3 degree erythema is unspecified: D = D(P) = D(?). Since the two Cohen collations each summarize the measurements on a different set of observational units, it is questionable whether the equation can be correct. This paper further shows that the appropriate (Least Squares) estimates of the three-parameter equation derived for the Cohen data under the Ellis constraint (five treatments per week) is in fact: D(1.0) = 1710 x T0.54 x N-0.26. The paper shows that the NSD equation is also incorrect because the ad hoc method by which Ellis estimates the exponents is inconsistent with the constraints imposed by Cohen and Ellis upon the parameters of the multivariate frequency distribution of the data set. The paper shows that the method by which the correct LS estimates of the exponents were obtained from the Cohen-Ellis data is consistent with these constraints and, therefore, this equation is a correct graduation of any other set of treatment regimens which is also consistent with the Cohen and Ellis constraints. The paper further shows that for such data sets there are, in fact, only two independent continuous variables, either D and T or D and N, since the Ellis constraint requires that N and T be collinear. Thus, the best linear graduation has the typical form: D congruent to 1900T0.32. This is "best" in the usual sense: both prediction and confidence intervals are provided for the estimates of the conditional "tolerance dose" D; these are not inflated by the presence of a collinear variable. This equation is biased, however, by the absence of the collinear variable. The TDF and the CRE concepts are derived from the NSD and, therefore, the deficiencies of the latter concept which we discuss may be expected to encumber these progeny as well. The two characteristic features of the Cohen (Ellis)-type data which impede the construction of useful estimates of the putative separate effects of N and T upon the response of tissues to irradiation are that (1) these data do not include specifications of either a tissue defect or its incidence, and (2) the variables N and T are collinear. Appendices I and II describe methods by which the effects of these features may be eliminated (I) or reduced (II).

King Crab Meal in Concentrates for Lactating Cows1,2

King crab (Paralithodes camtschatica) meal was compared with soybean meal at two compared with soybean meal at two percents of concentrate supplementation for lactating cows in a 22 factorial feeding experiment with an unsupplemented negative control. All cows were fed the high soybean concentrate wk 5 through 7. Milk production and live weight from this positive control were independent continuous variables for covariance analysis of wk 8 through 16. Silage and concentrates were fed independently. Cows rejected .5 and .2 kg/day of high and low crab concentrates. Milk production was lowest for cows receiving the unsupplemented control and next low for those receiving concentrates supplemented with the smaller amounts of either soybean or crab meal. The linear regression of milk production on time was significant for all concentrates and of weight on time for soybean meal. With certain qualifications king crab meal can be a potential source of supplemental protein in concentrates for lactating cows.

Periparturient Endocrine Changes of Conceptus and Maternal Units in Jersey Cows Bred for Milk Yield

Control cows, sired by and bred to bulls with assumed zero estimated breeding values, and selected cows, sired by and bred to bulls of high predicted differences for milk yield, were used. Blood samples were collected via jugular venipuncture on alternate days from 35 to 14 days prepartum, from 14 to 28 days postpartum, and daily from 14 days prepartum to 14 days postpartum. We examined blood hematocrit, and concentrations in plasma of protein, estrone, estradiol, estrone sulfate, progestins, glucocorticoids, luteinizing hormone, and prolactin. Total plasma volume was measured on days −21, −7, and +11. Weekly body weights, prepartum and postpartum, were recorded. All data were analyzed by least squares analysis of variance with day and hematocrit as continuous independent variables. Hematocrit was higher for selected cows throughout the sampling period. Prepartum concentrations of progesterone were higher in selected cows, but concentrations of estrone, luteinizing hormone, and prolactin were lower than in control cows. Hormonal and physiological responses indicated that selection for milk yield influenced both the conceptus and maternal units as measured by prepartum endocrine function.

A Case for Using Multiple Regression Instead of ANOVA in Educational Research

Researchers in the behavioral sciences have utilized “ANOVA theory” to test research hypotheses ever since R. A. Fisher first derived the solution. There are, however, limitations to classical ANOVA, particularly with respect to the handling of unbalanced design, continuous independent variables, and multiple covariates. Further, classical ANOVA fails to provide a strength of relationship index. The authors provide evidence which demonstrates that ANOVA is a special case of a multivariate procedure termed “multiple regression.” Because multiple regression is the more general method, it is capable of handling data which have traditionally been a nemesis for ANOVA researchers. Since multiple regression is a more powerful method of statistical analysis, and since it also provides a strength of relationship index, the authors argue that it be used in place of ANOVA in educational research. Also discussed is the coding of categorical variables as well as available computer programs for multiple regression.

Catastrophe Theory and the Study of War

The methodologies and conceptual frameworks employed by contemporary scholars do not satisfactorily account for the outbreak of international violence. Particularly frustrating to the analyst are the facts: (1) Some wars start and escalate suddenly while others begin and gradually build in intensity; yet in either case assumed causal factors are seen to vary smoothly and continuously through time. (2) The international system may be either at war or at peace at two separate time points even though the causal factors exhibit similar configurations. What is required is a framework that can deal with continuous and discontinuous dependent variables and continuous independent variables. Catastrophe theory provides the necessary classifactory structure in which to construct a systemic-level theory of the occurrence of international violence. Changes in the level of violence in World War I and World War II and the differences between these two conflicts are interpreted in the context of a particular catastrophe model—the butterfly. Catastrophe theory is discussed in light of the problems of description, classification, and empirical generalization in the construction of social science theory.

Synthesis of heat exchange networks by mixed integer optimization

AbstractProcess synthesis involves manipulation of the process arrangement, while studying the variables of each arrangement, to arrive at the optimal process. If each process arrangement is treated as a discrete variable, process synthesis becomes a mixed integer optimization problem. This paper examines the synthesis of heat exchanger networks using the adaptive random search procedure, which can be used to search continuous and discrete independent variables simultaneously. The means of handling the heat exchanger arrangement as a discrete variable is discussed, and the incorporation of various synthesis heuristics is presented. The results of synthesis of 2 × 2, 2 × 3, and 3 × 3 networks are presented and compared with other methods of synthesis.

Log-Linear Techniques and the Regression Analysis of Dummy Dependent Variables

Although previous comparisons of log-linear techniques with the regression analysis of dummy dependent variables have focused on the statistical superiority of the log-linear techniques, this paper presents three advantages of dummy dependent-variable regressions. First, dummy dependent-variable regression is able to accommodate both categorical and continuous independent variables. Second, the calculation of indirect effects and reciprocal effects is possible only with dummy dependent-variable regression. Third, the slopes yielded by dummy dependent-variable regression possess the properties of fundamental parameters; the effect parameters of log-linear models do not. In particular, controlling for a variable that is irrelevant to the causal system under investigation can yield a partial relationship whose expected value differs from that of the zero-order relationship when using log-linear techniques.

A Forum for Researchers: Is the (Psychological) World Flat?

The purpose of this note is to question the almost exclusive use of linear models in mathematics education research and to urge thoughtful exploration of alternate models when a linear model fits the data of a study poorly. (Here “linear model” refers to a model linear in one or more continuous independent variables, not to a model linear in the parameters to be estimated.)

Dynamic programming, quasilinearization and the dimensionality difficulty

One of the most powerful techniques for solving optimization problems is Bellman’s dynamic programming. This technique is very versatile and it can solve, at least in principle, many important optimization problems. However, because of the dimensionality difficulties, only problems with a relatively small number of state variables have been solved, Various techniques have been proposed to reduce the dimensionality difficulties. Some of these techniques are the use of Lagrange multipliers, the use of orthogonal polynomials, the use of successive approximations, and the state increment dynamic programming [l, 21. Except the last technique which has been applied to optimal control problems with continuous independent variables, all the other techniques have proven to be satisfactory in only a few cases. It should be noted that most of these techniques are trading computation time for computer memory. Thus, the use of these techniques will increase the required computation time. The quasilinearization technique has been shown to be a useful tool for overcoming the boundary value difficulties in the classical variational methods [3]. In this paper this technique is used to overcome the dimensionality difficulties in dynamic programming. Bellman and Kalaba [4,5] have shown that if the equations governing the transformation of the process are linear, a reduction in the dimensionality of the problem can be obtained. Since nonlinear equations can be linearized by the quasilinearization technique, a combined linearization and iteration scheme can be obtained. By the use of this scheme, the dimensionality of the functional equation of dynamic programming can be reduced to one in most cases. The advantage of this combined technique is that both the computer memory requirement and the computation time are reduced. In one example a three-dimensional problem is solved as a one-dimensional problem by this combined technique in one minute. If this problem were solved as a three-

Limits in terms of order, with example of limiting element not approachable by a sequence

The importance of the role of the order relation in the theory of the linear continuum has been recognized since the appearance, in 1872, of Dedekind's Stetigkeit und irrationale Zahlen,t and it received added importance at the hands of Cantor: in 1895. A later contribution to the theory was made by O. Veblen§ in 1905, and a systematic account of the theory of the continuum as a type of order was given by E. V. Huntington 1[ the same year. The work of these writers is directed toward a complete characterization of the linear continuum in terms of order alone. A set or class of elements, otherwise undefined, is assumed to fulfil conditions, stated in terms of order, sufficiently restrictive to admit of effective use of the class in the r61e of range of the continuous independent variable. Meanwhile certain classes have been recognized as being, in effect, the range of an independent variable, while not fulfilling all the conditions of the linear continuum. The desirability of utilizing analogies that exist between theories that pertain to these classes and the theory of the continuum is obvious. To this end it is important to generalize, as far as may be, the notions of point sets so as to be applicable to such classes. The thesis of M. FrechetS was a notable effort toward this form of generalization. He presupposes a definition of limit of a sequence of elements, the definition fulfilling certain mild conditions, and defines in terms of this the notion of limiting element of a subelass, and develops a theory applicable to any class for which a limit relation can be defined fulfilling the prescribed