CORRELATIONS ~ PARTIAL REGRESSION COEFFICIENTS (GROWTH STUDY PAPER #29) and. Charles E. Werts

Size: px

Start display at page:

Download "CORRELATIONS ~ PARTIAL REGRESSION COEFFICIENTS (GROWTH STUDY PAPER #29) and. Charles E. Werts"

Morgan Curtis
5 years ago
Views:

1 RB-69-6 ASSUMPTIONS IN MAKING CAUSAL INFERENCES FROM PART CORRELATIONS ~ PARTIAL CORRELATIONS AND PARTIAL REGRESSION COEFFICIENTS (GROWTH STUDY PAPER #29) Robert L. Linn and Charles E. Werts This Bulletin is a draft for interoffice circulation. Corrections and suggestions for revision are solicited. The Bulletin should not be cited as a reference without the specific permission of the authors. It is automatically superseded upon fonnal publication of the material. Educational Testing Service Princeton~ New Jersey January 1969

2 ASSUMPrIONS IN MAKING CAUSAL INFERENCES FROM PART CORRELATIONS} PARTIAL CORRELATIONS AND PARTIAL REGRESSION COEFFICIENTS Robert L. Linn and Charles E. Werts Educational Testing Service Abstract Given a linear model and that X is antecedent to Y,a third variable, W which is antecedent to both X and Y,is o~ten used as a control variable to remove any spurious association between X and Y Four different methods that have been used as measures of the "influence" of X on Yare described. The implicit assumptions that are made in using each of these methods to make causal inferences are stated and compared.

3 ASSUMPTIONS IN MAKING CAUSAL INFERENCES FROM PART CORRELATIONS, PARTIAL CORRELATIONS AND PARTIAL REGRESSION COEFFICIENTS Robert L. Linn and Charles E. Werts Educational Testing Service It is frequently the case in quasi-experimental and naturalistic studies that the investigator wishes to make "causal" inferences about the "influence" of variable X on variable Y when it is clear from the logic or design of the analysis that X is antecedent to Y To rule out alternative explanations for the observed association between X and Y,variables believed to be antecedent to both X and Yare controlled to remove "spurious" association. For example, the input-output model employed in studies of schools employs this logic since inferences are made about the effect of the school environment on an output variable with input variables controlled to eliminate the alternate hypothesis that the observed association between school environment and output may be the spurious resultant of both being influenced by input. In these studies it is reasonable to suppose that the school atmosphere may be influenced by the kinds of students who enter that school and that the output will be influenced by the input. In the above situation four rather different measures of "influence" of X on Y have been employed, often interchangeably: Method 1: The partial correlation between X and Y with prior variables controlled, e.g., in the case of one common antecedent variable, W, r XY W Method 2; The pg.rt correlation of X and Y with antecedent variables removed from X,e.g., r(x.w)y This correlation squared is I:arlington's (1968) "usefulness," Le., the proportion of

4 -2- variance that X adds to the prediction of Y in a stepwise regression after the common antecedent variables are entered. Method 3: The part correlation of X and Y with antecedent variables removed from Y, e.g., rx(y.w) Tnis procedure is especially common in school effects studies, in which a "residual" output is obtained by partialing input out of output and correlating the school variables with residual output. Most users of Method 3 have unwisely ignored McNemar's (1962) statement that this part correlation will be most useful when it can be argued that the common antecedent variables do not influence X, e.g., that the kind of student a school receives does not influence the school environment. Method 4: The standardized partial regression (i.e., beta) weight of Y on X,e.g., bh.w,obtained from the regression equation in which the antecedent variables and X are independent variables. Even though these four procedures may give Widely divergent estimates of the magnitude of the presumed influence (Werts & Linn, 1968) of X on Y (antecedents controlled), the size of the respective coefficients is interpreted as indicating a "large" or a "small" effect. Although it is clear that the different methods ask somewhat different questions and make somewhat different assumptions about the nature of the underlying phenomena, few investigators are aware of these differences or justify which method is relevant to the question under study. The assumptions (Duncan, Featherman, & Duncan, 1968) implied by the use of these methods for causal interpretation will be detailed in this paper, so thatinvest1gators may better decide which statistic 1s most appropriate.

5 -3- In discussing the interpretation of regression weights in a causal sense, Larlington (1968) notes the assumption that "all variables which might affect the dependent variable are included in the regression equation or are uncorrelated with the variables which are included." It follows that when a variable is not included as an independent variable the implicit causal assumption has been made that there are no other causes of the dependent variables which are correla.ted with the independent variables (linearity of all relationships is assumed throughout this paper). This principle derives from the necessity, when making causal interpretations, of assuming a closed theoretical system, i.e., no variables omitted. Since it is always the case that some variables are omitted the researcher must make the assumption that the variables omitted (Le., the "implicit" variables) do not affect the associations among observed variables. Thus if X 2 = b Xl l + e it is necessary to assume that e is 2 2 uncorrelated with the independent variable Xl vlliile many authors call e an error term, in a causal schema it represents the implicit (i.e., unmeasured) factors that act on the explicitly measured dependent variable (Blalock & Blalock, 1968, p. 200). X 3 = b l Xl + b 2 X 2 + e 3 If the causal system includes other variables, e.g., then it is necessary to assume that the implicit factors represented by e 2 and e 3 are uncorrelated in order for the causal model to be equivalent to a closed system. In actual research, if causal interpretations are to be made, at some point the simplifying assumption must be made that all variables which affect the dependent variables under study are either included in the system or are uncorrelated with the relevant independent variables and the implicit factors (Blalock, 1967). Because of the necessary assumptions about linearity, measurement error, and the effect of unmeasured variables, it will always be the case that causal inferences will be speculative.

6 ~- Since in a causal schema, factors influencing a particular dependent variable must be either included as independent variables or uncorrelated with the independent variables that are included, the implicit causal models for the four methods mentioned above can be specified since each procedure can be studied in terms of the sequence of regression equations used to obtain the respective coefficients. Thus the procedure for a partial correlation is to: (a) Use a regression equation with X as the dependent variable and the common antecedent variables as the independent variables to obtain the residual of X with antecedents controlled, i.e., herein labelled X'. (b) Use a regression equation with Y as the dependent variable and the common antecedent variables as independent variables to obtain the residuals of Y with antecedents controlled, which is herein labelled y t (c) In the regression equation with Y' as the dependent variable and XI X' the independent variable, the standardized regression weight for is the partial correlation coefficient which represents the influence of X' on y t It follows that when the partial correlation is interpreted in a causal sense, it is assumed that: (1) there are no other causes of X correlated with the common antecedent variables; (2) there are no other causes of Y correlated with the common antecedent variables, in particular X cannot be a cause of Y unless it 1s uncorrelated with the common antecedent variables; and (3) there are no other causes of y t which are correlated with X' The resulting implicit causal model for the partial correlation is shown in Figure 1 for the case of a single common antecedent variable, W It can be seen from Figure 1 that the

7 -5- zero order correlation between X and Y is considered to be totally spurious due to the common antecedent variables Wand X' (X' influences Y through the intervening variable Y'), the correlation r is the influence of W on WX X,the correlation r wy is the influence of W on Y,the correlation r X 'X( =,f I - rex).; I - rey) is the is the influence of X' on X the correlation ry,y( influence of y' on Y, the correlation rx'y' ( = r XY W ) is the influence of X' on y', the correlation r XY ' ( = r XY W v" I - rex) is completely spurious because of the common antecedent variable XI and the correlation results from X' influencing y' which in turn influences Y Insert Figure I about here In the case of method 2, the steps are: (a) Use a regression equation with X as the dependent variable and the common antecedent variables as independent variables to obtain the X residuals. (b) In the regression equation with X' as the independent variable and Y as the dependent variable, the standardized regression weight for Xl is the part correlation which measures the influence of XI on Y. It follows that method 2, when used in a causal sense, implies that all other influences on X are uncorrelated with the antecedent variables and that it is the hypothetical variable X' which influences Y as depicted in Figure 2. In Figure 2, r XY is the spurious resultant of the common antecedent variables Wand X' Note that both X' and W can be influences on Y since they are uncorrelated.

8 -6- Insert Figure 2 about here Method 3, the ~rt correlation of X with Y, common antecedent variables removed only from Y, proceeds as follows: (a) Use a regression equation with Y as the dependent variable and common antecedent variables as independent variables to obtain the Y residuals. (b) In the regression equation with X as the independent variable and y 1 as the dependent variable obtain the measure of the influence of X on Y' It follows from (a) that this ~rt correlation (e.g., rx(y.w» assumes that there are no other influences on Y correlated with the antecedent variables. However, if X is influenced by the antecedent variable(s) and X influences Y', then the antecedent variable(s) would be correlated with y r, which would. create a logical contradiction in the model since Y' is by definition uncorrelated with the antecedent variables. l Since we wish to measure the influence of X on y t we must assume that the antecedent variables do not influence X (1.e., r = 0) to avoid having contradictory assumptions in the model. WX Only when X is uncorrelated with (i.e., is not influenced by) the common antecedent variables, the standardized regression coefficient of Y' on X will equal the part correlation (e.g., rx(y.w» The causal model with this assumption is shown in Figure 3, for the case of a single common antecedent variable; the correlation r XY ( = rx(y.w) '/1 - r~ ) results from the influence of X on Y' which in turn influences Y The use of method 3 in the study of the influence of school environments on some output variable is not generally applicable since it 1s unusual for school variables to be uncorrelated with input variables.

9 -7- Insert Figure 3 about here Method 4, in contrast to the other methods, involves only the regression equation with X and the antecedent variables (e.g., W ) as independent variables and Y as the dependent variable. Although X and W may be correlated it is assumed that all other causes of Yare uncorrelated with both X and the common antecedent variables. As depicted in Figure 4, for the case of a single common antecedent variable (w), the correlation r XY has a nonspurious component bfx w representing the influence of X on Y and a spurious component (r XY - bh w) representing association due to the common antecedent variable W Algebraically r XY which can be recognized as a "normal" equation. The correlation r represents the total influence of W on Y,but this correlatio:::l is divided lnto WY two parts, bfw X the direct influence of W on Y and (ryw - bfw.x) which represents the indirect influence of W on Y via X Algebraically r yw b* which can be recognized as the other "normal" equation - b~.x = r vlx YXAI used to solve for the regression coefficients. An interesting property of the partial regression coefficients (Blalock & Blalock, 1968, p ) in these circumstances is that regardless of whether X is influenced by or influences the other independent variables in the equation (e.g., W ) the same regression coefficient for X will be obtained. It is the interpretation of (r yw - b~.x) which changes depending on the relationships of X to the other independent variables. Insert Figure 4 about here

10 -8- Algebraic Correspondences It will be noticed that the coefficient obtained in methods 1, 2, and 4 and in method 3 when X is uncorrelated with the antecedent variables (e.g., r wx = 0) is a standardized regression coefficient for a specified pair of independent and dependent variables. It may be shown further that the W'lstandardized regression coefficient between each of these pairs of variables will be the same for all four methods, i.e., each of the four procedures (for method 3, r wx = 0 ) is merely the W'lstandardized partial regression coefficient of Y on X ( W controlled) standardized by a different set of variances depending on what the independent and what the dependent variables are considered to be. The unstandardized coefficient for the case of a single common variable W is ) where 0y = standard deviation of y Ox = standard deviation of X Multiplying B yx W by the standard deviation of the independent variable and dividing by the standard deviation of the dependent variable we obtain: (1) r xy w = B yx W ox' 0y' 1/1 - r~, Ox r yx - r yw r XW = Byx W = 0y -V 1- r~ (2) r(x.w)y = Byx W ax, r WX r yx - ryyfj r - XW = B yx = 0y W 0y.yl- r 2 wx Ox 11-2

11 (3) rx(y.w) = Byx -W -9- Ox Ox r yx = Byx when 0, 0y' W ~l _ r 2 {I - r 2 r WX 0y WY WY (4) bh.w = Byx w Since all four methods when properly used are in fact simply variations of i~he unstandardized partial regression coefficient, why not simply study unstandardized regression coefficients in the manner of the econometricians? The standardized methods are highly subject to change due to changes in the variances of the variables from one sample to another. The effects of group heterogeneity on correlation coefficients are well known (see Lord & Novick, 1968, Chapter 6, for an excellent treatment of this topic). The standardized regression coefficients are also subject to this same influence. For this reason, the use of any of the four standardized methods must limit generalizations to very specifically defined populations since anything that restricted the range of the sample on one or more of the variables could lead to serious bias in the estimates and therefore the conclusions_ Unstandardized regression weights have the distinct advantage of not being affected by changes in the group heterogeneity. For this reason, Blalock (1967) argues that the unstandardized coefficients are more useful for purposes of stating general laws. A similar argument for the use of unstandardized weights rather than standardized weights is made by Tukey (1954). Tukey also argues that if the scales of the variables have any meaning, then the interpretation of the unstandardized weights is more straightforward. Due to the arbitrary nature of the scale units for most variables in the social sciences, however, the standardized coefficients will undoubtedly continue to be more generally used. Standardized measures

12 -10- lend themselves to the familiar language of "percentage of variance accounted forb which 1s often used to suggest that a given effect is "large ll or "small. tt Such language ascribes a meaning to an arbitrary scale, ignoring the crucial question of how this scale in any sense is a measure of the underlying phenomena. The language of prediction equations cannot be arbitrarily assumed to have meaning when used in the interpretation of causal systems. Summary Examination of Figures 1 through 4 shows that each of the four methods when used for causal interpretation accounts for the observed pattern of correlations with a different explanation or "theory" about the nature of the phenomena under study. In the case of a single common antecedent variable (w): Method 1 the partial correlation r XY W asserts that it is only Xl (the implicit factor(s) inf'luencing X ) which influences y' (the implicit factor(s) influencing Y ). Method 2 the part correlation r(x w)y asserts that it is only XI (the implicit factor(s) influencing X ) which ini'luences Y Method 3 the part correlation rx(y.w) asserts (when r WX = 0) that X inf'luences only Y' (the implicit factor(s) inf'luencing Y ). Method 4 the beta weight b~.w asserts that X influences Y ~.W is the (standardized) rate of change in Y when X varies, W having a fixed value (i.e., the inf'luence of X on Y, W con- trolled) Method. 4 differs from the others in that no hypothetical unmeasured variables, i.e., X' or Y',are invoked to explain the observed associations. The challenge to investigators who invoke implicit, unmeasured factors to explain observed data is obviously to measure these factors and then to

13 -11- demonstrate that they in fact account for the observed associations. Investigators who do not invoke unmeasured variables should be conscientious in searching for and incorporating new variables which would affect the variables under study.

14 -12- References Blalock, H. M. Causal inference, closed populations, and measures of association. American Political Science Review, 1967, 61, Blalock, H. M., & Blalock, A. B. Methodology in social research. New York: McGraw-Hill, Darlington, R. B. Multiple regression in research and practice. Psychological Bulletin, 1968, 69, Duncan, O. D., Featherman, D. L., & Duncan, B. Partials, partitions and paths. Socioeconomic background and occupational achievement. Final Report, Project No (EO-191), Contract No. OE , U. S. Office of Education. Ann Arbor: University of Michigan, Lord, F. M., & Novick, M. R. Statistical theories of mental test scores. New York: Addison-Wesley, McNemar, Q. Psychological statistics. (3rd ed.) New York: Wiley, P Tukey, J. W. Causation, regression, and path analyses. In O. Kempthorne, T. A. Bancroft, J. W. Gowen, & J. L. Lush (Eds.), Statistics and mathematics in biology. Ames, Iowa: Iowa State College Press, Werts, C. E., & Linn, R. L. Analyzing school effects: How to use the same data to support different hypotheses. Research Bulletin Princeton, N. J.: Educational Testing Service, 1968.

15 -13- Footnote lin Figure 3, the dilemma could be resolved by assuming that W influences Y' negatively, but this would contradict the intent of partialing W out of y to obtain a measure uninfluenced by W,e.g., in school studies the intent of partialing input out of output is to obtain a residual output uninfluenced by input. Furthermore if VI influenced YI then both Wand X would have to be included in the regression equation with Y' as the dependent variable and the part correlation rx(y.w) would not be the standardized weight for X in that equation.

16 -14- Figure Captions Fig. 1. The causal diagram corresponding to the causal interpretation of a partial correlation. Fig. 2. The causal diagram corresponding to method 2, the part correlation (r(x.w)y)' Fig. 3. The causal diagram corresponding to method 3, the part correlation rx(y'w). Fig. 4. The causal diagram corresponding to method 4, the standardized partial regression weight b~x.w'

17 e

18 e'

19 e" / " " " r wx ;: 0 ",,-,, "" /

20 e* el

Do not copy, post, or distribute

Do not copy, post, or distribute 14 CORRELATION ANALYSIS AND LINEAR REGRESSION Assessing the Covariability of Two Quantitative Properties 14.0 LEARNING OBJECTIVES In this chapter, we discuss two related techniques for assessing a possible