If appels and peaches tell you the very same about the response, then it does not matter which one you chose for prediction. Similarily, a variable which is half "appleness" and haf "peacheness" will do the same job. This is exactely the case where the information from both variables can be used in combination/alternatively/simultaneously. This is the case when the correlation is really high and deviations from a perfect correlation is considered caused by (random) effects that are also not related to the response. Using the average will average out the noise (same as if you average pairs of measurements of the same predictor). For multidemensional data, PCA provides composite variables that are still useful predictors in a model, but the interpretation of the meaning of the predictors may be next to impossible.

Proportional odds models are a better bet for ordinal data. With ordinal data, especially with so few levels, you can't be sure whether the intervals between levels are consistent across your scale. Proportional odds models allow you to model the likelihood of moving up or down a level, in an analogous way that logistic regression allows you to model the likelihood of presence or absence of a binary outcome.

You prbably may find this helpful: http://www.biostat.sdu.dk/courses/f12/Biostatistics/misc/ordinalResponseModels.pdf

An SPSS solution is described here: http://www.norusis.com/pdf/ASPC_v13.pdf

Further:

http://www.stat.ufl.edu/~aa/sta6127/ch15.pdf

http://people.csail.mit.edu/jrennie/papers/other/mccullagh-ordinal-80.pdf

I'm not clear about your problem. If u are facing a problem of dichotomous variable in ur left side of ur equation & with so many variables for your explanatory variables, even they are binary, continuous, or scale variable; better go for a logit regression. U can do it in SPSS.

First of all it is not clear that what type of variable (continuous or Categorical) you are using as dependent. If you’re using a continuous variable in your right hand side, as you know, first step in multiple regression analysis is to produce scatter plots taking dependent variable and an independent variable at a time and this need to repeat for all pairs of variables. It will help you to decide whether independent variables are linearly related with dependent variable. Categorical variables can be treated as dummies but need to avoid dummy variable trap. If this is the case I propose that linearity of dummy variables is not an issue. Alternatively, you can estimate regressions with different slops and can compare. SPSS allows different modelling strategies. Final warning is that multiple regressions may not be best modelling technique for your data. Try with some multivariate techniques.

Thanks a lot for your valuable advices, Damian, Jochen, Mahesh and Nisantha. I'll try to do my best on the basis of your instruccions. Very kind of you!

In my opinion another tool of analysis is bootstrap regression. When nonparametric can adapt to their data.

http://investigacion.us.es/sisius/sis_showpub.php?idpers=1060

Prof. Rafael Pino Mejias has experience in this method.

You may want to see if you can use logarithmic variables.

Gerard, (1) what's the fun of transforming varianbles when there are models available properly working on the original scale? (2) What would be the meaning of a log(Likert scale)?

The model may be most accurately or correctly captured using logarithmic variables. That's the nature of some data. We often simply don't choose the nature of the data. There could be underlying non-linear relationships that cannot be captured in any other way.

Gerard, you are right. It's just that I cannot see how a transformation of Likert-scaled values should help.

Thanks again Jochen: I agree with you. Best wishes!

I would go with Dummy Variables

Go for transformation of data may be log values can fit with linearity assumption otherwise try non parametric tests

first ask yourself how many of world researchers go for linearity tests before doing regression. I haven't seen a paper yet with normality & linearity tests. Nothing is linear, all statistical tests are mere approximation.

why don't u try SEM

Wouldn't the linearity assumption violation point to the need for using Generalized Linear Model approach?

The other option would be to try and fit a higher order polynomial.

There seems to be a considerable confusion about what was ment by "assumption of linearity". Most respondents seem to think that the functional relationship between predictor and response is not linear, i.e. curved. However, the originally posted problem was that the response variable is not on a continuous scale, not even on an interval scale. In fact the Likert scale is just ordinal. Such responses are modeled by ordinal regression, an extension to generalized linear models (which, btw, are linear in their coefficients - and not(!) neccesarily linear in the functional relationship(s) between response and predictor(s)).

Yes, I agree with Jochen's assessment here. While it's true that "assumptions of linearity" failing unfortunately covers a wide swath of territory, Jaume's questions was quite specific, and so, Jochen highlights the specific use of GLM appropriate to the question. Perhaps Jaume will get back in touch once he has tried that and let us know how it went.

Interesting point. But doesn't the assumption of linearity specifically pertain to the response-predictor relationship? I think that GLM is a way to go, but I'm not sure I follow how linearity assumption violation may refer to anything other than response -predictor relationship. There are more than one item in a Likert scale, which should be distinguished from the 5-point item rating scale used in the Likert scale. Scores are then added and represent a score on a factor. I guess you could argue that it is still ordinal data, but wouldn't it be meaningful to look for a linear relationship between such factor scores and factor scores on some other measure using a rating scale (e.g. conscientiousness will have a U-shape curve if plotted against some measures of performance and linear shape if plotted against others). Are you saying that this is a just a function of the fact that the data is not interval?

Bogdan, thank you for mentioning the scores as being sums of different 5-point-items. I think this procedure can introduce more problems than it solves, especially when the individual items are not very comparable. This ends up in adding 3 apples and 2 peaches. This give 5 ... what? Strangely, on the final score scale, 5 apples would mean exactely the same as 5 peaches. The results will only be interpretable -somehow- if different cultivars of apples are summed and different cultivars of peaches are summed up (without mixing apples and peaches). Still, the addition of ordinal values is actually not allowed, but one may be able to do some sensible inference from such sum-scores anyway. In fact, when really many items are added, the influence of each individual item becomes negligible and our knowledge about the characteristics of the (ordinal) distribution vanishes. I could imagine that the remaining knowledge about such scores could be described using the beta distribution, hence a beta regression model might be appropriate. I'm not an expert here - this is just an idea.

Jochem, interesting points. So are Jan's. I recently participated in a stats conference at my university, in which one of the presenters exposed many weaknesses of the Likert scales and suggested that they cannot be analyzed using traditional methods including LR. This leaves most of psychology in sort of a quandary. SEM is one option as Jan suggests and I guess GLM is another but few use these so far. Thanks for suggestions.

I think Jochen's point of view (so are Damian's and Jan's) has been crucial in this fruitful discussion. Your clarification, Jochen, exactly fits the problem I would like to solve. I'll try with SEM for ordinal data, as Jan suggests. Thanks to everybody!

Before trying to use SEM, I would like first to standardized the responses corresponding to each predictor (ordinal data) the check the ccolinearilty. If the found independent of each other then I'll try to utilize Multiple regression analysis ( linear as well as non-linear or Logistic ,if the data is dichotomized ) and compare their residuals.

Thanks a lot for your suggestion, Pankaj. We've already checked for multicolinearity. There are high correlations (p < .001) between some predictors, so, if we apply the multiple regression analysis, the model would be inestable.

Best wishes,

Jaume

Jaume, you can try to use only the most promising predictor of the coliniear ones, or you can use a summary statistic (the average, for instance) of the highly correlated predictors. A similar strategy is to perform a PCA and use the first few principal companents as predictors.

Yes, I know Jochen. But, from my point of view, the problem is that, according to our own data there are several "promising predictors" that are not conceptually related (again, "apples" and "peaches"...). Nevertheless, thanks a lot for your valuable suggestions.

All the best

Jaume

If appels and peaches tell you the very same about the response, then it does not matter which one you chose for prediction. Similarily, a variable which is half "appleness" and haf "peacheness" will do the same job. This is exactely the case where the information from both variables can be used in combination/alternatively/simultaneously. This is the case when the correlation is really high and deviations from a perfect correlation is considered caused by (random) effects that are also not related to the response. Using the average will average out the noise (same as if you average pairs of measurements of the same predictor). For multidemensional data, PCA provides composite variables that are still useful predictors in a model, but the interpretation of the meaning of the predictors may be next to impossible.

I think the difference between apples and peaches matters very much if the issue is trying to understand what gave rise to a response variable. This game is not as much fun if we just get to "choose" which predictor we're going to report--or if we just average across conceptually disparate variables. Options worth considering might be a) that the variables have a yet unrecognized conceptual relationship or b) there are yet-unrecognized global variables whose drift is tipping the whole boat of currently collected predictors over one way or another, leading to similar nonstationarities in all of them. Said nonstationarities might then be overwhelming any independent information they might otherwise offer for predicting the response variable.

Damian, I agree and thank you for making this point. One has to make one's mind up about the aim: just prediction or "true" modeling. In some cases, being able to make good predictions is also a reputable aim.

I agree with the responses of Damian and Jochen.

Thanks for your feed-back, Pankaj.

There is a vast literature on smoothers, which are designed to deal with curvature. There are inferential methods based on them as well. Suggest looking at Wilcox

(2012, INTRODUCTION TO ROBUST ESTIMATION AND HYPOTHESIS TESTING. ACADEMIC PRESS, 3RD EDITION). See in particular, section 11.5.

I've got the book, Rand. Great recommendation!