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Interpreting the Results of Weighted Least-Squares Regression: Caveats for the Statistical Consumer DOCUMENT RESUME ED 282 906 TM 870 317 AUTHOR Willett, John B.; Singer, Judith D. TITLE Interpreting the Results of Weighted Least-Squares Regression: Caveats for the Statistical Consumer. PUB DATE Apr 87 NOTE 29p.; Paper presented at the Annual Meeting of the American Educational Research Association (Washington, DC, April 20-24, 1987). PUB TYPE Speeches/Conference Papers (150) -- Reports Research/Technical (143) EDRS PRICE MF01/PCO2 Plus Postage. DESCRIPTORS Error of Measurement; Estimation (Mathematics); *Goodness of Fit; *Least Squares Statistics; *Mathematical Models; *Predictor Variables; *Regression (Statistics); Scaling; Statistical Studies IDENTIFIERS *SPSS Computer Program; *Weighted Data ABSTRACT In research, data sets often occur in which the variance of the distribution of the dependent variable at given levels of the predictors is a function of the values of the predictors. In this situation, the use of weighted least-squares (WLS) or techniques is required. Weights suitable for use ina WLS regression analysis must be estimated. A variety of techniques have been proposed for the empirical selection of weights with the ultimate objective being a better "fit." The outcomes of the analysis must be interpreted once the fitting is complete. Problems can arise in the interpretation of some of the statistics when using a computer package. In this paper, such problems in the application and interpretation of WLS regression using the SPSS statistical package are demonstrated, both algebraically and by example. For the purposes of the example, an artificial data set (whose underlying parametric structure is known) has been created. Each of the statistics commonly reported in the WLS regression analysis of such a data set are isolated and their interpretation discussed. Where necessary, adjusted statistics that more reasonably represent the outcomes of the analysis are proposed and their use illustrated. (BAE) *********************************************************************** Reproductions supplied by EARS are the best that can be made from the original document. *********************************************************************** e Working paper -- not for quotation Comments welcomed INTERPRETING THE RESULTS OF WEIGHTED LEAST-SQUARES REGRESSION: CAVEATS FOR THE STATISTICAL CONSUMER U.S. DEPARTMENT OF EDUCATION "PERMISSION TO REPRODUCE THIS Office of Educational Research and Improvement MATERIAL HAS BEEN GRANTED BY EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) #skthis document has been reproduced as j. Bk);iI received from the person or organization originating it. 0 Minor changes have been made to improve reproduction quality. PoIntsof view or opinionsstated inthisdocu- TO THE EDUCATIONAL RESOURCES ment do not necessarily represent official OERI position or pohcy. INFORMATION CENTER (ERIC)." * John B. Willett and Judith D. Singer Harvard University Graduate School of Education N) March, 1987 N. Do The order of the authors has been determined by randomization. f: Paper presented at the American Educational Research Association Conference, Washingion, D.C., April 1987. 2 BEST COPY AVAILABLE INTERPRETING THE RESULTS OF WEIGHTED LEAST-SQUARES REGRESSIONi CAVEATS FOR THE STATISTICAL CONSUMER Quite frequently in educational, psychological and sociological research, datasets occur in which the variance of the distribution of the dependent variable at given levels of the predictors is a function of the values of the predictors. In this situation, the use of weighted, rather than ordinlry, least-squares techniques is required in the fitting of regression models (Draper & Smith, 1981). Typically, weights suitable for use in a weighted least- squares (WLS) regression analysis are not known in advance and must be estimated ill situ by "a combination of prior knowledge, intuition and evidence" (Chatterjee & Price, 1977, p.101). A variety of techniques have been proposed the statistical literature for the empirical selection of the weights, ranging from strategies that incorporate substantive knowledge of the form of the residual variance as a function of the predictors (Miller, 1986) to two-stage strategies in which an initial unweighted (OLS) analysis is used to inform the selection of weights (for instance, biweighting in Mosteller & Tukey, 1977). Whatever the selected approach, the ultimate objective is to achieve a "better" +it in that "while the [ordinary] least-squares estimates and fit may be satisfactory, the precision of the [ordinary] least-squares estimates may be different from that indicated under standard 3 assumptions" (Cox & Snell, 1981, p. 83). Of course, regardless of the manner in which the empirical weights have been selected, there remains the question of interpreting the outcomes of the analysis once the fitting is complete. It is during this interpretation that the consumer can be lead wildly astray by the output from a computer package such as SPSSx. For some of the computed statistics (such as the estimated slopes) there is no problem. However, pitfalls can arise in the interpretation of other statistics for three reasons. First, by virtue of the manner inwhich the empirical weighting must be X applied in the WLS regression by SPPS , several important statistics (i.e., the standard errors associated with the slope estimates, elements of the regression ANOVA table, the root mean- square error and related statistics) are likely to be incorrect. Second, because the regression statistics created in a WLS analysis are expressed in the metric of the weighted variates, it is not immediately obvious how even those statistics which have been computed correctly (i.e., the coefficient of determination) should be interpreted. Third, even though an optimal set of weights may have been selected, many of the important statistics and diagnostics (i.e., the standard errors and associated t-statistics, the regression ANOVA sums-of-squares, mean-squares, F-statistic and other related statistics) may not be invariant under multiplication of the weights by a constant -- a process which modifies the measurement metric in the weighted world and raises questions about how the empirical weights might optimally be scaled. In this paper, such pitfalls in the application and 4 interpretation of WLS regression using the Sne statistical package are demonstrated, both algebraically and by example. For the purposes of the example, an artifical dataset (whose underl),ing parametric structure is known) has been created. Each of the statistics commonly reported in the WLS regression analysis of such a dataset are isolated and their interpretation discussed. Where necessary, adjusted statistics that more reasonably represent the outcomes cf the analysis are proposed and their use illustrated. WEIGHTED LEAST-SQUARES As Mosteller & Tukey (1977, p.346) suggest, the action of assigning "different weights to different observations, either for objective reasons or as a matter of judgement" in order to recognize "some observations as "better" or,"stronger" than others" has an extensive history. Whether the investigator wishes to downplay the importance of datapoints that are intrinsically more variable at specific levels of the predictor variables, or simply to decrease the effect on the fit of remote datapoints, the strategy is the same. Although the results of this paper are easily generalizable to the multiple predictor case, the discussion presented here deals with the estimation of the relationship between a dependent variable and a single predictor. We will assume that observations or measures on two related variables, Y and X, have been obtained from a random sample of n independent subjects and that the 5 5" relationship between these two variables in the population is given by: y. 1= p p x. E . [1] 0 1 1 where p and p are the unknown intercept and slope parameters to 0 i be estimated. Furthermore, we will assume that the Ei are unobserved random errors which are normally distributed with zero 22 mean and varianceaEX . Thus, the random errorsare heteroscedastic and the typical OLS strategy for estimating po, pi and (TE is inefficient (Neter et al., 1965). Typically, an empirical response to the inefficiency of the OLS estimation involves the creation of a set of weights, wi, which 4r@ inv@rtoly proportional to the squared magnitudes of the observed X.. These IN.are then applied in the re-fitting of the 1 1 regression model by weighted least-squares. Of course. in practice, it is unlikely that the functional dependence of the heteroscedasticerrorvarianceontheX.will be known exactly. However, in an empirical analysis, the error structure is usually inferred from a "combination of prior knowledge, intuition, and evidence" (Chatterjee & Price, 1977, p.101). Often. the required evidence is obtained by inspection of residuals created in an initial unweighted (OLS) regression analysis (Neter et al., 1985. p. 170). 6 Equations for the WLS estimates Providing the w are known, the more efficient WLS estimates i of po and Pil their sampling variances, and (YE can be estimated by direct minimization of the sum of the squared weighted residuals (for instance, see Neter et al., 1985, pp. 167-170). Equivalent results can also be obtained by transformation, in which the original variates are multiplied by the square-roots of the wi (Neter et al., 1985, pp. 171-172). Whatever the method of estimation, of particular interest are the estimates of po and pi(all summations taken over the index i=1, n): (EW.)(EW.X.Y.) (EW.X.)(2W.Y.) 1 1 1 1 1 3. 1 3. P 1:23 aw.)(sw.xt) - 1. 1. awiYi) PiawiXi) Po - [3] ) and their estimated sampling variances (standard errors): ) =0E1 L'43 1 2. (Ew..1('Ev). X. ,) (Ew. X. ) 1 1 1 7 7 (N.V") 1 S.8.(t50) = aE 1 C53 2 (Ew )(Ew.X.) (Ew X.) " 3. 3. i '2 where the mean-s,11,,re erroraE, an unb4ased estimate of the variance of the q is estimated from the sum of the weighted squared residuals: Ew. (Y.- Y. )4- 3. 3. 3. ccE n - 2 C63 and Y is the predicted value of Yobtained in the WLS analysis. i i 2 The coefficient of determination, R , is also estimated in the transformed world. It is a measure of the proportion of the variation in weighteq Y that can be accounted for by weighted X.
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