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I would like to build a model where:

1) My dependent variable is an ordinal variable with 7 or 8 levels

(sadly, I know for a fact, that the intervals between them are not equal, because it is a likert response-scale with numerical values which grow 'almost exponentially')

2) I want to include multiple predictors:

a) categorical --> one variable with three categories

b) contineous --> four variables - results on four scales

I have a total of 1464 participants in two groups (a total of 1464 observations in two groups). I would like to analyse the groups separately, in a way described above.

As far as I am aware, for linear regression the following assumptions need to be met:

  1. both independent and dependent variables need to be normally distributed
  2. the model is linear
  3. there is more observations than variables
  4. homoascedasticity (variance is equal for all observations)
  5. The residuals have normal distribution
  6. None of the predictors are correlated

How does my data set look in terms of these assumptions?

1.None of the variables is normally distributed (even If I check it separately for each group x category). The p-values for Kolmogorov-Smirnov and Shapiro-Wilk are < 0.001. But I hear that according to central limit theorem, this is not important, cause the sample is big enough.

note: When I look at histograms, the distribution looks either skewed to one side, or roughly normal - it is not absolutely terrible and bare-eyed visibly not normal (an if skewness exists, it is actually in direction that is logical for a given group).

exapmples of a histogram and a q-q plot below: histogram q-q plot for the same variable

  1. The scatterplots don't look like there is much linearity there. an example of a scatter plot of one of my dependent variables on y axis and independent (contineous) variables on the y axis

  2. There sure is more observations than variables.

  3. Homoascedascity

the z residual /z predicted scatterplot looks almost sort of ok, but I guess not really.... (how am I actually supposed to make an executive decision based on looking at a scatterplot?) an example of scatterplot used to assess homoascedascity for one of my variables

  1. Normality of residuals (isn't this tested in the exact same way as homoascedascity?)

  2. I hear this can be remedied by mean-centering anyway, so that should not be an issue?

Questions: 1. Does central limit theory also free one from the homoascedascity and normal distribution of residuals assumptions?

  1. Would it ever be ok to perform linear regression on this dataset, if the dependent variable is ordinal (some say, it has enough levels to do it)? Or should I give up and settle on ordinal regression (odds ratio)? In this case, what type of ordinal regression would you recommend? Can I test the above described model using ordinal regression? How?

  2. Will I ever be able to perform an odds ratio analysis that would give me the same amount of information as a multiple regression?

NOTE: I am using SPSS, so answers taking this into account are very welcome.

I am beyond frustrated with this dataset, going back and forth between deciding to

a) combine my dependent variables into two scales, in order to be able to consider my dependent variables continuous (I have 6 dependent variables total, they actually are separate items on two scales - one has 3 items, the other has 2 items). But.... If I do this, my other assumptions are still violated - hence the question whether I am waived from them by central limit theory?

b) do an odds ratio analysis for each item separately - can this actually be done in SPSS, without having to modify my model?

From what I gather now, a) would mean losing information that could be gained from differentiating between separate items b) would mean losing information by means of having to simplify my model

but... I really have no experience in odds ratio analysis and maybe there is a way to actually do the same thing as I would do with multiple linear regression?

c) what about the lack of linearity? what does this mean for my endevour?

How would you go about analysing this data?

any help would be REALLY welcome, thank you in advance!

P.S. let me know if the description of my problem is too convoluted, and I will try to clarify

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  • $\begingroup$ When you share your graphics with statisticians, it's effectively useless to not label the axes. $\endgroup$ – AdamO Sep 30 '19 at 16:16
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I'm not sure where you got that list of assumptions, but it is seriously wrong.

1. both independent and dependent variables need to be normally distributed
2. the model is linear
3. there is more observations than variables
4. homoascedasticity (variance is equal for all observations)
5. The residuals have normal distribution
6. None of the predictors are correlated
  1. is incorrect, neither the DV nor the IVs need to be normal. 3. Is correct but too limited; there should be a lot more observations (although exactly how many more is debated, some say 10 observations for every IV). 6. Is sort of right, but modest correlations are OK and what is problematic is not correlations but colinearity, which is not quite the same. You have also misinterpreted "linear" in this context it means linear in the parameters.

In any case, with 8 levels I would do an ordinal logistic regression. It gives you different information that linear regression, but it's hard to say if it is more information or less information.

As to the rest of your questions, I think they get too broad for this forum. A good book on ordinal logistic would help.

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    $\begingroup$ Thank you very much for your quick reply. I got that list from a book about statistical analysis, so I guess I need to look for a good book now, with the emphasis on good. $\endgroup$ – Mandarc Mar 19 '18 at 12:36
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The issue with statistical model assumptions in general is that they are never fulfilled precisely, so we more or less always do analyses in which assumptions are violated. The important question is whether assumptions are violated in ways that will cause misleading results, which is not exactly the question that misspecification tests such as Kolmogorov-Smirnov address (although not totally unrelated either).

The Central Limit Theorem implies that many analyses based on methods derived from a normal distribution assumption are approximately valid for non-normal error distributions. However, if you were to analyse your data using standard linear regression, there are problems that the CLT cannot amend:

  • Your dependent variable (y) has a fixed value range, and chances are that observations, at least in some regions of the explanatory variables (X, "X-space"), are not strongly concentrated in a safe distance from the limits of the value range. In this case a linear regression can give you predictions outside the value range for realistic values of X, which don't make sense.

  • If your regression coefficients are not very close to zero, in some parts of X-space the values for y will be close to the maximum, in others close to the minimum, with potentially strong "local" skewness (possibly in both directions, but maybe far stronger in one direction than in the other). I'd expect that the overall regression slopes will produce some kind of compromise between areas in which y-values are rather central and symmetric and linearity may be locally fine, and areas in which there is skewness and linearity cannot hold because of closeness to the borders of the value range, so chances are regression parameters will be too low in absolute value for fitting the "good central" area while attempting to fit the more extreme areas linearly where this doesn't work.

  • Homoscedasticity (equal variances) will also be violated because y-values have more space to vary in the central areas, i.e., where they are mostly in the middle of your ordinal scale. This means that different observations have different information content for fitting the regression and should be reweighted, which is ignored by standard linear regression (although the previous item may stop reweighting from helping here). The same issue can be caused by local skewness.

Note by the way that the y-histogram and the qq-plot don't help much diagnosing normality, because the normal assumption in regression requires normality to hold in every area of X-space whereas these plots ignore the location in X-space. In your situation it seems (from your other plots) that any illusion of "near normality" is caused by putting together quite non-normal distributions from different regions in X-space with some kind of non-normality (e.g., skewness in different directions) that cancel each other out to some extent.

Overall this is a bunch of good reasons to rather go for ordinal logistic regression as already recommended, rather than to use a standard linear (standard linear can be OKish if your regression slopes are low enough in absolute value that y-predictions are safely away from the borders of the y-value range in all realistic areas of X-space).

PS: Combining scales resulting in scales that have a bigger value range as you suggest in your item (a) could actually help with making borders of value ranges matter less in standard regression. It depends on the data how well that will work and of course the scales still need to be meaningful (which depends on the subject matter).

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A more useful subject is how to visualize the nature of ordinal data. A scatter plot needs to be jittered or aggregated and shown as a hexbin to be useful when there's a large number of tied responses.

A histogram of the unconditional response won't tell us much about it's distribution or whether it meets any useful assumptions. The paper below https://www.ncbi.nlm.nih.gov/pubmed/11910059 is a useful discussion about why normality is useless with large $N$. The results as you allude to come from the CLT. In fact, later in the paper, they discuss the advantage of linear model over proportional odds.

The linear model reports a mean difference which is easy to interpret whereas a proportional odds model reports an odds ratio whih is hard to interpret. We cannot say in general if the proportional odds model might be more efficient, but when $N>400$ who cares? Use a robust error estimator (also called Huber White or Sandwich).

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