I have checked whether my ordinal scale-dependent variable has a normal distribution. The Kolmogorov-Smirnov test says it is acceptable, but the Shapiro-Wilk test says it is not. My ranking has a scale of 1 to 150. Please see attached files for tests of normal distributions.
Hello Jason, if you go to the favourite links page of my website then go to the electronic stats textbook, this is a good resource. This is what I found when I searched for ordinal.
Deborah Hilton Statistics Online
http://sites.google.com/site/deborahhilton/
Specialized methods for analyzing multinomial and ordinal multinomial response variables can be found in Generalized Linear Models.
Deborah- Thank you for referring me to the electronic stats textbook. It has some useful information.
Hello Jason, I have some important points for you:
1-when the ordinal variable has more than 10 level, it can be considered as a quantitative variable.
2- your sample size is very important to judjment about normality test and adequacy of regression model.
3- checking the normality of response variable for fitting a regression model is not correct, you should fitted the regression model and then assess the normallity of residuals.
2- normality tests are very conservative when the sample size is small. some of normality tests are sensetive to skewness and some of them are sensetive to kurtosis, so please chech these to statistics and their standard errors to assesing the normality. also, histogram and pp plot are useful.
3- regression model is robust against majore departure from normallity assumption when the sample size is sufficiently large.
Azadeh- This is extremely helpful. If it is not too much to ask, would it please be possible for you to send me a reference or source concerning the rule that any ordinal variable that has a scale of greater than 10 can be considered quantitative? Again, thank you for this extremely helpful and sound advice.
I am wondering how you would consider your results. Since the measure of your dependent variable is from rank to 150, the possibility of getting regressed values of 2.333, 2.435, 5.222, 6.783 and predicted values of 151.111, 151.234, 189.999 is very likely, depending on the measure of your independent variable.
Hi Jason- when ranked are based on a pattern (such as Likert scale) or judgmaents (such as VAS) the behavior of ranks are likes as scores. Also, you can use of latent variable (random effect) models to accounting the discretness of ranks. A main reference is: Categorical data Analysis; Author: Agresti,(page 277-278)
Raid- Thank you for that advice. I will try that! Azadeh's advice above about using assessing the normallity of residuals was also very helpful..
Normal quantile transformation will give you (perfect) normal data. It also will make the interpretation of the results easier as each value is expressed as how far it is from the center in standard deviations.
In SAS: I use proc rank with the BLOM option. This can be googled.
It will give you a robust statistical test.
Raid- Thank you for your advice. If it is not too much to ask, do you please know what the procedure is called in SPSS? That is the statistical package that I am currently using.
Hi Pedro- Thank you for your response. My sample size is 110. I was operating under the assumption that the Kolmogorov-Smirnov was generally valid for sample sizes greater than 200. It is useful to know that it can be effective for samples as small as 50. Again, thanks! - Jason
Rank score analyses can certainly be justified if the raw score distribution is asymetrical enough, or the variances heterogeneous enough, so analysis of the raw scores misleads you. If I could simplify your question, and address it by analogy: is the Spearman correlation coefficient a justifiable index of association? If you think that it is, then the multiple regression generalization is also justified, because computing a Pearson correlation on ranked data is equivalent to computing the Spearman correlation on the unranked source of those data. Conover and Iman (American Statistician, 1981) provided me a brilliant analysis of this problem.
By truncating the original scale, ranked data often contain less information than the source data, and this can be an undesirable side effect of using the ranked information. In practice, I compare the raw score and ranked analyses, and use the untransformed analysis only when they are comparable (and say so, in the report). If they differ, however, and the additional 'information' in the crude measure reflects, eg, the influence of outliers, then this can be an instance where I would find that 'less is more'.
But perhaps you are asking a different question. That is, after you have rank transformed your dependent variable, it appears non-normal. It must, of course, as rank transformations produce rectangular distributions. I would check that this is true of your data, and if so, if it appears that the S-W test result then reflects the deviation from normal of this shape. The question for me, then, would focus on whether this distribution is an improvement over that provided by the raw scores.
The asymptotic relative efficiency of rank based inferences often exceeds 85% and it can reach infinity. This is a sign to me to always try out the parametric test in addition to the corresponding nonparametric or rank based test. Like Malvin said above, I will use the nonparametric test if the parametric test differs greatly from it, and I will use the parametric test otherwise, while inserting a line to state that a nonparametric test was also performed and that the results agree with each other.
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