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x" i i i i 4 4 4 $ X $ 4 4 4 X 4 4 4 Assumptions in Correlation and Regression Models
Both correlation and regression models are based on the general linear model, EMBED Equation.3 , but they differ with respect to whether the X variables are considered random or fixed. In the correlation model they are considered random that is, the values of the X variables obtained in the sample and the number of cases obtained at each level of the X variables is random another sample from the same population would yield a different set of values of X and different probability distributions of X. In the (fixed) regression model the values of X and their distributions are assumed to be, in the sample, identical to that in the population.
Some have argued that the correlation coefficient is meaningless in a regression analysis, since it depends, in large part, on the fixed particular values of X obtained in the sample and the probability distribution of X in the sample (see Cohen & Cohen, 1975, p. 5). While this relationship between r and the distribution of X in the sample is certainly true, it does not, IMHO, necessarily follow that R and R2 are not useful statistics in a regression analysis, as long as the reader understands that their values depend, in part, on the fixed values and distribution of X.
The fixed regression model fits best with experimental research where the researcher has arbitrarily chosen particular values of the X variables and particular numbers of cases at each value of each X. In this context, the fixed regression model is most often called the Analysis of Variance model. It is, however, true, that it is common practice to apply the regression model to data where the X variables are clearly not fixed.
When you use t or F to get a p value or a confidence interval involving , y o u a s s u m e t h a t t h e j o i n t d i s t r i b u t i o n o f X a n d t h e Y ( o r Y s ) i s b i v a r i a t e ( o r m u l t i v a r i a t e ) n o r m a l . W h e n t h e d i s t r i b u t i o n i s b i v a r i a t e n o r m a l t h e n i t i s a l s o t r u e t h a t t h e m a r g i n a l d i s t r i b u t i o n s o f X a n d Y a r e b o t h n o r m a l , t h e c o n d i t i o n a l d i s t r i b u t i o n s of X given Y and of Y given X are all normal, the variance in X does not vary with Y and the variance in Y does not change with X (see Winkler and Hays, 1975, p. 644-652).
When you use t or F to get a p value or a confidence interval involving regression coefficients you make no assumptions about the X variables. You do assume that the distribution of Y is normal at every value of X and that the variance in Y does not change with X. This assumption can be restated in terms of the error term: the distribution of the error term (the residuals) is normal at every value of X and constant in variance across values of X.
Pedhazur (1982) stated the assumptions of regression analysis as follows:
X is measured without error. Now there is an assumption that will always be violated.
The population means of Y|X are on a straight line. I consider this to be part of the tested null hypothesis that is, we are testing a linear model.
X is fixed, Y is random.
For each observed Y, the mean error is, over very many replications, zero.
Errors associated with any one observed Y are independent of errors associated with any other observed Y. The Durbin-Watson statistic can be used to evaluate one form of non-independence the situation where the value of a given case is related to the values of cases which are adjacent in the data stream. This is likely when cases are entered in temporal or geographical order. The Durbin-Watson statistic is available under Statistics in SPSS Regression, and its value is included in the Model Summary table in the output. It is often recommended that one need not worry if the Durbin-Watson statistic is with the range of 1.5 to 2.5.
The error variance is constant across values of X. If you encounter heteroscedasticity, you may be able to resolve it by transforming Y. You may also consider HYPERLINK "http://www2.chass.ncsu.edu/garson/pa765/wls.htm" weighted least squares regression instead of Ordinary Least Squares regression.
The errors are independent of X. This sounds like homogeneity of variance to me.
Pedhazur states that these assumptions are necessary for the obtained estimators to be best linear unbiased estimators. If t or F will be employed for tests of significance it is also assumed that the errors are normally distributed.
References
Cohen, J., & Cohen, P. (1975) Applied multiple regression/correlation analysis for the behavioral sciences. Hillsdale, NJ: Lawrence Erlbaum.
Pedhazur, E. J. (1982). Multiple regression in behavioral research. (2nd ed.). New York: CBS College Publishing. [especially Chapter 2]
Winkler, R. L., & Hays, W. L. (1975). Statistics: Probability, inference, and decision. (2nd ed.). New York: Holt, Rinehart, & Winston.
Links
HYPERLINK "http://core.ecu.edu/psyc/wuenschk/docs30/corr6430.doc" Bivariate Linear Correlation
HYPERLINK "http://core.ecu.edu/psyc/wuenschk/docs30/regr6430.doc" Bivariate Linear Regression
HYPERLINK "http://www2.chass.ncsu.edu/garson/pa765/assumpt.htm" \l "normal" Testing of Assumptions 0 1
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