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<br />study to: I) identify imporlant dependent variables, <br />and 2) develop equalions for the sociological compon- <br />enl of Ihe model. Weightings for the Iheorelical rela. <br />tionships described later in lhis report were also ob- <br />tained using this methodology. <br /> <br />Standardization of Measurements <br /> <br />Combining unlike measurements inlo Ihe same <br />equalion requires a standardization or weighting pro- <br />cedure. Therefore, the malhematical equations in <br />lhis study are expressed in two forms. The firsl form <br />usesnonslandardized coefficienls based on the num- <br />bers directly measured. The second or slandardized <br />form is derived from Ihe nonslandardized form by <br />multiplying Ihe coefficient by its slandard devialion <br />and by dividing by the slandard devialion of the de- <br />pendenl variable in the equation. The standardized <br />form thus compensates for differences in the measure- <br />ment scales used and for variations in the distributions <br />of variable values. The standard deviation is used as a <br />measure of variability and not for statistical inference. <br />No particular underlying distribution is assumed, but <br />the values of each of the variables should be reason- <br />ably well distributed, (Blalock, 1961; Coleman, 1966; <br />and Duncan, 1966). <br /> <br />The slandardized form permits an evaluation of <br />the relative sensitivity of Ihe dependen t variable to <br />changes in the various independent variables in the <br />relationship under consideration. The sign of the co- <br />efficient indicates the type of relationship, direct or <br />inverse, between the respective independent variable <br />and Ihe dependent variable. The larger the coefficient <br />associated with an independent variable, the greater is <br />the sensitivity of the dependent variable to variations <br />in that variable alone. However. the variable with the <br />largest coefficienl in the standardized form is not nec- <br />essarily the IImost important" because that variable it. <br />self may vary considerably less than does a variable <br />wilh a relatively low coefficient. Also, a variable with <br />low coefficient may have concomitant variation with <br />other varaibles in the equation (Gordon, 1968) and <br />thereby be capable of introducing considerable varia- <br />tion in the dependent variable (Blalock, 1964). <br /> <br />Statistical Assumptions <br /> <br />The standardized relationships are valid for mod- <br />el building provided the equalion is accurate and re- <br />cursive (Blalock, 1964). While accuracy and recursive- <br />ness are not entirely attained with sociological data, <br />these limitations do not mean the equation are inap- <br />propriate or inapplicable to social science work pro- <br />viding Ihe user is aware oflhe consequent degree of <br />approximation (Coleman, 1964). In addition, as an <br />increased understanding of Ihe sociological syslem is <br />reflected in improved dala and relationships, the two <br />conditions are expected 10 be met more closely. <br /> <br />A further problem associaled with statistical re- <br />lationships is explained by Coleman (1964: 101) as <br />follows: <br /> <br />Other variables which affect the dependent variable <br />are assumed to be uncorrelated to the independent <br />variable, and this assumption is not normally en- <br />tirely true. , .if this assumption is not true, as often <br />it is not, then the observed relation may be a spuri- <br />ous one because of the variables not taken into <br />account. It is to reduce this difficulty that more <br />variables are added and multiple regression is used. <br /> <br />However, too many variables cause redundancy and <br />lead to serious problems (see Gordon, 1968; Schoen- <br />berg, 1971). <br /> <br />Two other assumptions (Coleman, 1964) are: <br />I) Ihe independent variables are theoretically causally <br />related as described by the equation to the dependent <br />variable; and 2) the parameters of the equations are <br />alike or nearly so for all unils in which observations <br />are made. The second assumption is often met in soci- <br />ological samples drawn from Ihe same population. <br />Meeting Ihe flrst assumption requires knowledge of <br />the system being sludied. <br /> <br />For the initial model, the relalionships within <br />Ihe sociological component of the system are assumed <br />10 be linear in order 10 simplify the analysis and be- <br />cause the syslem was not sufficiently well defmed to <br />develop more complex relationships.! The linear hy- <br />polhesis is a first approximation. Since Ihe relation- <br />ships of some important social variables in the hydro- <br />logic-sociologic system are nOllinear, efforts should <br />be made in the future to develop more accurate non- <br />linear equations. <br /> <br />One frequently stated requirement for linear re- <br />gression analysis and related stalistical techniques is <br />that variables should be measured on a continuous <br />scale even though multiple regression can be run with <br />variables classified by discrete calegories. Recent <br />investigations have shown thai powerful paramelric <br />statistics are useful even when scales do not meet all <br />of the assumptions for the statistics. Labovilz (1967, <br />1970) and Baker et al. (1971) demonstrated thai even <br />radically differen t numbering systems for ordinal data <br />do not greatly change Ihe resulls when slatistica! tech- <br />niques normally requiring interval measurements are <br />applied to ordinal scales. He wrole: <br /> <br />Empirical evidence supports the treatment of <br />ordinal variables as if they conformed to inter- <br />val scales. Although some small error may ac- <br /> <br />1 Not all relationships resulting from this analysis are <br />linear, although they are monotonic. This is because of the <br />presence of interaction terms and of multiple indirect affects <br />of variables in the equations of the model as well as because <br />of categorization of one equation. See Chapter VII. <br /> <br />13 <br />