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In a data set I have, all of the results of the residual vs. predicted values appear as parallel lines as pictured below. I interpret this according to my training as a violation of homoscedasticity which makes running a multiple regression untenable.

enter image description here

Here is some context about the data sets, type of analysis used, the design of the study, etc.

  1. The type of analysis I conducted was multiple regression. In SPSS - Analyze > Regression > Linear.

  2. The independent (predictor) and dependent (outcome) variables are all ordinal data (though I'm treating them as "scale" in the analysis). Both the independent and dependent variables represent satisfaction scores on a 7-point scale. These satisfaction scores were collected from the same participants at the same point in time (one survey). There are 4 independent variables that represent participants' satisfaction with several more granular dimensions or aspects of a service (e.g., easy to use, effective in meeting needs, etc.) The dependent variable is satisfaction overall with the service.

  3. More about the other results from testing the multiple regression assumptions: (1) The data set passed the "independence of observations" test as measured by a Durbin-Watson statistics which was approximately 2; (2) the independent and dependent variables are approximately linearly related which was based on an inspection of the partial plots produced by the regression analysis.

  4. There are around 20 highly influential data points out of approximately 450 participants as measured by studentized deleted residuals of over +/- 3.

I've tried transforming the data using a log transformation and this did not at all help the apparent heteroscedasticity problem. Plotting the log transformed residuals vs. predicted values yielded the same general pattern in the chart.

The other technique I tried was outlined in this paper http://afhayes.com/public/BRM2007.pdf which described providing a regression model with heteroscedasticity-corrected error residuals. The beta coefficients for each predictor variable were essentially the same as the non-heteroscedasticity corrected model, as well as the overall R^2 value for the model.

Some of the predictor variables are highly correlated with one another, but around .6 - .8 - so there may be a slight multicollinearity problem - however, the tolerance and VIF values are fine. More intuitively, a lot of the participants provided similar satisfaction ratings for all of the predictor variables and the dependent variable. Quite a few on the 7-point scale answered things like " 7 7 7" for the predictors and "7" for the outcome or "1 1 1" and "1". However, this was not always the case.

Looking for some guidance around potential reasons for this, alternative ways to analyze the model, etc. Thanks in advance.

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    $\begingroup$ There was already many questions more or less similar. Search parallel lines residuals diagonal lines residuals straight lines residuals $\endgroup$ – ttnphns Jun 28 '17 at 8:19
  • $\begingroup$ The definition residual = observed $-$ fitted necessarily implies that each distinct observed value lies on a line of negative slope when you plot residual versus fitted. Lo and behold! You have 7 lines for 7 distinct values. (Standardization as linear scaling doesn't affect this, clearly.) $\endgroup$ – Nick Cox Jun 28 '17 at 11:28
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    $\begingroup$ Backing up, the fundamental issue is arguably not homoscedasticity at all but fitting a linear model to a outcome that is bounded (and the bounds are attained). Even if you choose to treat the outcome as measured, it is still bounded! You are clearly aware of some of the issues here but I'd underline that many researchers would regard this model as fundamentally wrong as applied to this data. As an important detail, it's hard to see that a Durbin-Watson test is pertinent here outside a time framework. Otherwise put, what order defines which observations are compared with which? $\endgroup$ – Nick Cox Jun 28 '17 at 11:36
  • $\begingroup$ Thanks for the comments. I do recognize this violates some assumptions of multiple regression and homoscedasticity is not the problem here, as the nature of the measurement instrument itself may cause this to be produced. In my experience (industrial / commercial), I'm surprised that many researchers in applied settings treat ordinal data as scale data if this is seemingly untenable in the view of the statistics community. Perhaps this is a different question, but I'm wondering if there are any resources / papers you would recommend for how to conduct regression with ordinal data? $\endgroup$ – Appendix Jul 8 '17 at 2:53
  • $\begingroup$ To clarify further -- and to step away from the measurement scientist perspective for a moment -- I don't think most individuals would see reporting, for example, an "average" satisfaction score based on data from an ordinal scale as offensive. Having said that, I'm not trained enough at this time to understand how this treatment of ordinal data affects / translates into linear regression analyses and the procedure for that analysis. The interpretive convenience of R^2 / beta coefficients this analysis produces is helpful in comparison to ordinal regression. $\endgroup$ – Appendix Jul 8 '17 at 3:01
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The parallel lines are a logical consequence of the fact that your dependent variable has only a few possible values. Try and plot your dependent variable against your one of your independent variables and overlay a regression line. You will see a couple of horizontal line, and a sloping regression line. Now look at a particular value of your independent variable and see what the residuals (deviation from the regression line) is. Look at a value of your independent variable to the right and see what happens to those residuals.

Transforming your variable cannot change the fact that you have only a few possible values, so no transformation can change this pattern.

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    $\begingroup$ This calls for ordinal regression $\endgroup$ – Frank Harrell Jun 28 '17 at 11:41
  • $\begingroup$ Agreed with answer + comment. The fundamental problem here is not heteroskedacity; it is the level of measurement of the data, which is ordinal, rather than interval, violating the assumptions of the linear regression analysis. $\endgroup$ – Ruben van Bergen Jun 28 '17 at 11:45

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