The problem is not that you do hqave kurtosis, the problem is that you want to remove it. You should rather think about what this kurtosis tries to tell you. You should seek a reasonable model explaining the data - not trying to torture the data until it fits to a (obviousely suboptimal) model.

The problem is not that you do hqave kurtosis, the problem is that you want to remove it. You should rather think about what this kurtosis tries to tell you. You should seek a reasonable model explaining the data - not trying to torture the data until it fits to a (obviousely suboptimal) model.

Behzad,

In addition to the considerations Jochen offers, many modeling strategies have available estimators ("sandwich" estimators are the most common for modest to moderate sample sizes; bootstrapping is another option) for the standard errors and confidence intervals which are less sensitive to normality violations.

Pat

I would recommend bootstrapping as Patrick suggests. Bootstrapping is easy to implement, can provide estimates of pretty much any statistic you might be interested in. Also bootstrapping from the empirical distribution will (almost) certainly be more accurate than using an incorrectly specified parametric model.

The following provide a relatively easy to understand over view:

http://www.r-bloggers.com/bootstrap-example/

http://spark.rstudio.com/ahmed/bootstrap/

If you think that the bootstrap is not appropriate for your data or the questions you are posing, you'll need to provide a more detailed description of your data and research questions.

Dear Colleagues,

thank you very much for your great responses.

I was wondering if you could tell me more about using the bootstrapping method

(for example: the result of kurtosis for the first question is 4.091 and skewness: -2.061), how I can solve this with bootstrapping?

Another question, for measuring kurtosis, must I use the results of measuring all questions or the average of them? !

I have an addition problem that my data is not normally distributed!

Any suggestion will be appreciated.

Thank you very much in advance.

I just saw: you are talking about "questions" from which your data comes from. So I assume you have some kind of "scores", for instance Liekert data or something similar. In this case, the data is ordinal and not metric. This means that sums and differences of the values are not defined, and thus measures for the moments of a distribution of a metric variable are neither defined nor sensible.

Bootstrapping is relatively simple to perform. I suggest to start reading the Wikipedia article, to google a bit, and (optionally and eventually) read some of Efron's contributions. The major hurdle with bootstrapping is to really understand why it works and where and why it is more or less efficient than other methods - but since the t-test for instance is also not really understood by most "users", the lack of a deeper understanding should not be a knock-out criterion for using bootstrap instead.

Jochen is correct. By examining your data more closely you will likely find that you have mixed two or more populations.

Dear Colleagues,

thank you for your answers.

Dr. Donald, I don`t know what you mean Mixed Two Or More Populations!

I have used bootstrapping, but I could not take acceptable results!

Dr. Jochen, could you please give me more about, What This Kurtosis Tries To Tell You! what you mean?

I have used also the Winsorizing more than removing outliers (base on David C Hollew`s (2013) interpretations) for extreme values, but the data still are not normally distributed!

For solving this, I have used the robust and doing CFA by SmartPLS (for using PLS), due to non-normally data distribution.

Any suggestion will be appreciated.

Please, see this paper... https://www.statmodel.com/download/Skew.pdf

This reminds me of a quote I once read in a textbook preface: "If you discover a use for kurtosis, be sure to let me know." I'm not saying that kurtosis is completely useless, just that it has very low practical value. Kurtosis provides some sort of qualitative description of what your data look like compared to normally distributed data. But for any quantitative question that I can conceive, there are better tools.

It is still unclear (at least to me) what your data look like, what question(s) you're asking, and what methods you're applying. It would be helpful if you could supply a sample of your data, and a clear statement of the question(s) you're trying to address. Otherwise, the only further advice I can provide would be to consult a local statistician.

Hi,

@Behzad Behzadnia: You say "what you mean Mixed Two Or More Populations?" and I will try to answer this question, it will hopefully shed some light on your question that initiated this thread.

So, to you it seems that the problem you have is the fact that the distribution [of I don't know what] has too low a kurtosis.

Many contributors to this thread pointed out to you that a low kurtosis in not necessarily a problem per se, mainly because there is no a priori reason why you should get a normal distribution to start with.

As I understand your "problem", it is that you expected a distribution with a kurtosis value compatible with a Gaussian distribution. It is not what you get, your data behave in a different way.

In my experience (and in that of Donald A. Singer I guess), you may get that because there is a variable that you ought to have taken into account and did not. Let me exemplify this.

Imagine you measure the height of all Iranians and look at the data. You will get a low kurtosis value. Why ? Because you have not separated men from women and women tend to have a lesser height than men. Confounding women and men will make your confounded measure (i.e., one distribution of height including the height of both women and men) a mixture of 2 normal distributions: the normal distribution of height in men and the normal distribution of height in women. As the peaks of these 2 normal distributions do not overlap, the probability density value "at the center" of your height distribution (I am referring now to that one distribution that includes both men and women's height) will be less than expected under a normal law, so a kurtosis index will return a low kurtosis value. Note that if I am right you may (or may not, but most probably you will) also get a skewness value that is not compatible with a normal distribution.

This is an easy example, but reality is more complicated. Even my first year students come up with the fact that gender (i.e., mixing 2 distributions, one by gender) would explain why the distribution of people's heights (irrespective of gender) is not a normal distribution. But *you* will have to find out what is the confounding variable in your study without my help, because I have no suggestion to make since finding out what is the confounding variable is a theory-guided work (and I know very little about the domain you are referring to).

Maybe there is a (or some) variable(s) that once taken into account would allow you to find normal distributions to work with. But that does not necessarily have to be so. It all depends on your dependent variable. And from what you have said we know not much about that, and certainly not enough to help.

I hope this makes it clearer to you what "Mixed Two Or More Populations" means, and I wish you good luck with your data analysis.

Cheers,

Serban

Hello Colleagues,

Thank you for your great comments.

I have used the data Winsorizing instead of using outliers, base on some studies.

In this method, the results showed that all of items had + - 2 skewness and kutosis, base on Tabachnick and Fidell (2013).

Any suggestion about winsorizing will be appreciated.

Kind regards,

Data, and by this I mean *all* data you have is telling a story. Eliminating some of this data or winsorizing may lead you to tell a different story...

My 2 cents.

Cheers,

CM

you have 350 values. Ever thought that the central limit theorem might ensure a sufficiently good approximation of the sampling distribution to the normal distribution?

Dear Serban and Jochen,

Thank you for your responses.

Serban, Maybe by using data winsorizing, the data give me a different story, but what would you prefer to do instead of this?

I have gather more than 350 data that all of them are base on 5 or 7 point-Likert  and ther are not height or sth like this.

As well as by these data (without winsorizing), I could not take an acceptable CFA.

Jochen, I have thought about the centeral limit theorem, but I selected my samples base on their interested in participating in this research and not selecting from the whole of society or base on the centeral limit theorem.

Consequently, by using SmartPLS I took an acceptable CFA, especially AVE.

However, any suggestion will be appreciated.

Best regards,

@Jochen : I respectfully but nonetheless strongly disagree with what you say. I suspect you confuse distribution of a random variable and distribution of the means of a random variable. For one thing, if I draw 350 observation from, say, a binomial distribution of probability value 0.01 (why a binomial and why p=0.01 ? well why not,  I just give a counter-exemple to your claim), the distribution I'll get is by no means symmetrical, and by no means normal. You can check this in R:

 a

@Behzad: "Serban, Maybe by using data winsorizing, the data give me a different story, but what would you prefer to do instead of this?"

My point in telling you that using winsorizing will change the story your data tell is that you will never know the "true" story your data would have told you were they not winsorized.

On a more constructive note, I concur with Shane McGee McMahon that "It is still unclear (at least to me) what your data look like, what question(s) you're asking, and what methods you're applying", so I don't see how I can help any further without knowing some more on the goals you have when looking at your data.

One suggestion for the future however: Do not use Lickert scales ever again. Lickert scales are a pleague, since they yield the kind of response distribution you are dealing with. And it is annoying to have to deal with that, as you see.

Use instead visuo-analogic scales (i.e., a line of defined length and with a left and a right tick) along with indications on what the left tick means (e.g., "I don't agree at all") and what the right tick means (e.g., "I fully agree"). Do NOT substitute these indications (i.e., "I don't agree at all" and "I fully agree") by numbers. Once the participants have answered, you will have to measure where the particpant put a line through the line of defined length and convert this into a number, but there are automated ways to do this. If the items you present the participants with do not entail too extreme reactions, the distributions of responses will be normal.

Cheers,

Serban

Serban, thank you for your comments. It is not the values being normally distributed, it's the sampling distribution of the mean. In your example:

m = replicate(10000, mean(rbinom(350,350,0.01)))

hist(m)

qqnorm(m)

That's pretty well symmetric.

Dear Serban,

thank you for your great interpretation.

I was wondering if you could tell me more about this:

((there are automated ways to do this. If the items you present the participants with do not entail too extreme reactions, the distributions of responses will be normal.))

What does it mean?

@Jochen: It is the sample distribution of the mean indeed that is normally distributed (thank you for your nice R code and sorry for my newbie one, that's the best I could think of...). But I do not see what this brings to the topic at hand. I mean Behzad has an only sample of about 350 values for each question, and as far as I understand the different questions do not measure the same construct. So the thing is I don't see the link with the sample distribution of the mean, because i) the mean is not a good index for a non-normal distribution and ii) there is only one measure (granted, on 350 participants) so ... really I don't see what you think of and I'm left with the impression I am missing something. I would really appreciate you tell more on what you have in mind.

@Behzad: There are automated ways for you to get the answer for each question from the sheet of paper the participant responded on without having to measure with a ruler the distance from the left tick and use the rule of three to compute the answer value.

The interesting thing about visuo-analogic scales is that they yield responses that are quite well normally distributed EXCEPT when the question draw people to answer in a very extreme way. Just think of it this way: If the question *does* provoke extreme responses (e.g., everybody responding extremely close or even on the right-hand tick mark), the distribution of responses on that question will be all but normal, with a very high probability density near the tick mark, a probability density of 0 above the tick mark (that's because participants cannot respond above the right-hand tick mark), and a probability density that is nearly of 0 (but not quite 0) as you go left from the right-hand tick mark (called an asymmetric i-shape distribution).

Let me know if my explanation is clear to you. HTH

Cheers,

SCM

Dear Serban C. Musca and Shane McGee McMahon,

Thank your for your comments. Here there are more info about my research:

Firstly, I have three questionnaires that one of them had used before in Iranian pupulation and it gives an acceptable results in validity and reliability.

All of questionnaires were in 5 or 7 Likert point.

Secondly, I have translate two questionnaires from English to Persian for the first time, that the first questionnaire includes 2 factors and 10 items and the seconds includes 1 factor and 21 items!

Thirdly, I have gather more than 350 data, and the descriptive statistical have showed that some items (questions) have not a compatible Skewness and Kurtosis, as well as all of items have not normally distributions, by Shapiro Wilk and KrusKal Wallis tests. There are so many exterme values in the data in some of questions, so that, when I wanted to remove outliers, I must to remove more than 50 data, so it remains 300 data, that I would not perefer to do.

In the next place, I have used the data  Winsorizing that Serban noted its give me a different story, that I do agree pretty with this,

After that, the next (I think) problem is that the resluts of CFA analysis showed that this data had not a compatible range in CFI, RMSEA, RMR and TLI and P values, due to non-normally distributed.

but I do not know, what is the best way for analysing with this data!?

When, I have used SmartPLS for measuring CFA, I could take an accpetable range,  but when I ought to remove one of the items in the first questionnaire and it remains 9 items, (the AVE was more than .45, with 9 items), that from the theory view that I used, I think it is not a good choise.

Finaly, I want to measure the relationships between these questionnairs that the first of them is independent variable and the others are dependent variables.

So, any suggestion in how I can take an acceptable range for Validity (by CFA) in this data, by which method or in which software or anythings that you would perefer to do, will be appreciated.

regards,

Removing outliers for the sake of generating 'acceptable' kurtosis or skewness can be misleading. Better to study the data properties and distribution features, applying validation methods robust to outliers and figure out the true picture of what your data tries to convey. Fitting to distribution models is not always possible, in particular to data with chaos phenomenon.

Dear Sherban, to me it looks as if you are really missing an important point. But before coming to this, I want to stress the point that all we two are discussing is actually not applicable for the data of Behzad (he is dealing with ordinal data, and statistics like sums, differences, means and thus skewness and kurtosis do not have any sensible meaning on such data).

Now let us assume that we were talking about continuous/metric data where sensible mean values can be calculated. An experiment was carried out and it provided 350 values or some response variable. These values tell us what we should reasonably expect about the values the variable can take. The fact that we cannot predict the observed values with certainty tells us that we are ignorant of some information that might enable us to correctly predict each of these values. The (sample) mean is an estimate of the expected value. This value describes a centre for which we can say that we equally likely expect any of the values being higher or lower that this centre. It is an estimate, based on these known 350 values. It is often important to give some indication of how good or precise this estimate is. This is, for instance, of particular interest if we wish to investigate if/how strong the expectation about the response depends on another variable (e.g. a treatment etc.). The precision can be expressed as a range for the estimate that is not too unexpected in such experiments. In “such experiments” means experiments providing another 350 values from which another mean is calculated. The sampling theory and the central limit theorem tell us what to expect, based on this set of 350 observed values from this single experiment. Think of this question: how much different may the mean of 350 values be if this experiment would be replicated? This is answered by the sampling distribution of the mean. The shape of this distribution depends on the distribution of the variable (what might be unknown or arbitrary), but it approaches the normal distribution with increasing sample size (this is what the central limit theorem ensures).

You must distinguish the questions “in what range can we expect the data to be?” and “in what range can we expect to get means from such experiments?”. The former question requires knowledge about the distribution of the data, the latter can be answered without knowing the data distribution and relying on the central limit theorem when the sample is large (and 350 is quite large). [there might only by some strange, rare, and rather pathological cases where this may not be the case]

I hope I could make the point clear.

Ordinal variables with many categories, such as 7-point Likert-type scales of agreement (such as the one used here), are usually safely treated as “continuous.” In practice, most researches treat ordinal variables with 5 or more categories as continuous, and there is evidence to suggest this is not likely to result in much practical impact on results (e.g., Babakus, Ferguson, & Joreskog, 1987; Dolan, 1994; Johnson & Creech, 1983; Hutchinson & Olmos, 1998). If ordinal variables with many categories are nonnormal, then data analytic techniques for nonnormal continuous variables should be used. 

In theory, the sampling distribution can often be figured out mathematically based on probability theory. In practice, this can be quite difficult or impossible for some statistics. The Central Limit Theorem is one way of providing an approximate sampling distribution for some statistics, but it is not universally applicable. The Central Limit Theorem is one of the major concepts of Statistics and is seductively useful. However, it is not a panacea. There are some statistics for which the sampling distribution is not approximately normal, no matter how large the sample size. And there are some populations that are so non-normal that astronomically large sample sizes are required before the sampling distribution is approximately normal.

The precise statement of the central limit thereom(s) is a little bit difficult to define, principally because it has been refined several times since the first version due to de Moivre. It was subsequently refined by Laplace, Chebyshev, Markov, Lyapunov, and most recently Lindeberg. The most general version, due to Lindeberg, deals with the sum of n independent random variables with finite variance for some finite n. Previous variations contained extraneous conditions, but also dealt with the sum of random variables. The CLT can be further generalized to include a ranom number of random variables, or dependent random variables, although the latter in particular is non-trivial.

The central limit theoreom is indeed not universally applicable. It describes the behavior of the mean of a sample, and generally should not be invoked to describe the behaviour of other statistics. And rigorously speaking, the CLT only applies for arbitrarily large sample sizes, so clearly some care should be taken if one only has a finite set of data. In practice, the last restriction is quite severe and could be simplified to stating that the CLT(s) should be applied with caution. The reason is obvious: the CLT(s) provide no bounds for the errors introduced by a finite sample size. For this, one may appeal to the Berry-Essen approximation thereom which provides an upper limit on the error introduced by finite sample size but requires some additional assumptions.

@Jochen: I do agree with all you say. Entirely. You say it better than the way I would have put it (unless I took a couple of hours to write it!), but I knew the gist of it. It is just that there was way much too implicit in what you said in your previous post and I was in the state of mind thinking that Behzad' data were not normally distributed, so that the mean would not have been a sensible index (not to speak of the measurement theory whereby the obtained "values" are just ordinal, not numeric, an argument you make and with which I fully agree).

Now, again, as you point out, the fact is Lickert scales yield ordinal data. Period.

However, you'd be amazed (driven crazy, rather) to see what people in many disciplines (e.g., social psychology -- and alas not only) think about data Lickert scales yield, and the way they use those data at the time of testing hypotheses through statistical procedures. The litterature in these disciplines is full of t-tests and ANOVAS on those ordinal data (and more often than not without any check for normality conditional to the levels/values of the IVs, or for homogeneity of variances)...

But thank you for having spelled out what you were thinking about so that I could understantd your reasoning.

And +1000 for reminding people that Lickert scales yield ordinal data. ;)

Cheers,

Serban

@Yohannes: "If ordinal variables with many categories are nonnormal, then data analytic techniques for nonnormal continuous variables should be used."

I beg to disagree. If anything, those data could be processed as nonnormal DISCRETE variables. My 2 cents.

Cheers,

SCM

The answer to whether "ordinal variables with many categories" can be treated as continuous depends on the definition of "many."  The most common definition in practice seems to be than "many" > 5. And the approximation becomes arbitrarily good as the number of categories approaches infinity.

I would contend that in any case where approximating ~5 categories as a continuous distribution produces a neglible impact on the analysis, it must be the case that the sum of errors from other sources is rather large. It would probably be more constructive to interpret such a result as a need to collect better data rather than suggesting that such approximations are generally valid.

If the survey were conducted using a photometric scale where they may conceivably be something ~2000 categories, I probably would not object to treating the values as if they were from a continuous distribution. But 5 categories definitely seems a little low.

@Shane: The most common definition in practice seems to be than "many"

@Jochen,

Is kurtosis even a problem? The data are still evenly balanced, so mean=median=mode. As long as only kurtosis and standard deviation change, will this have much effect on OLS models? I would still expect that 68% of the data would be within one standard deviation of the mean.

I can sort of see that one might want equality of kurtosis values as one wants homoscedasticity. So what am I missing?