In many studies, it is observed that the geochemical and environmental data do not follow a normal distribution. This may be due to the samples from different populations or origins.
The basic statistics (mean, standard deviations etc.) are sometimes computed considering these data which may lead to bias or wrong results. Because, the statistical methods (classical) are always based on the assumptions of normal distribution of data.
For these types of data, can we compute median (as a measure of location) and median absolute deviation (as a measure of spread) instead of mean and standard deviation?
Can we use non-parametric methods for multivariate statistics or statistical tests for such type of data?
What are the suggestions of statisticians, environmentalists and geochemists?
1) I don't know, actually, but I suppose yes, because factor analysis is based on the same principles as linear models.
2) Yes... for the purpose of testing expected values.
3) Example:
in https://www.iii.org/fact-statistic/facts-statistics-lightning you find the numbers of lightning fatalities in the US states in 2016. The data is
number of cases: 0, 1, 2, 3, 4, 9 number of states: 31, 8, 5, 1, 2, 1
Imagine the data was summarized as mean and standard deviation:
mean number of cases per state: m = 0.792
standard deviation: s = 1.597
Assuming that the number of cases would be normally distributed, one would expect that most states would have about 1 case, but in fact the vast majority of states don't have a single case. One would further expect that there shouldn't be states with more than 5 cases (m+3s = 5.6, and based on the normal probability model P(x>5.6) < 0.001, what is pretty unlikely), but we observed a state with 9 cases. The probability, under the nomal model, to get x>=9 is lower than 0.000001! We would never expect that to happen if we assume the normal probability model. Finally, under the normal probability model the probability P(x mean): the negative-binomial model that can be parametrizes with a location parameter m' and a dispersion parameter s' . If one fits the parameters of that model to the observed data, one gets m' = m = 0.792 and s' = 0.397 (that's not a standard deviation; it is the estimated value of dispersion parameter of the negative binomial model). Thus, reporting that the number of cases was negative-binomial distributed with mean 0.792 and dispersion 0.397 would be the most instructive summary of the data.
A violation of normality assumption may warrant the adoption of nonparametric methods rather than their parametric counterparts. The median may be used instead of the mean and the mean absolute deviation instead of the standard deviation.
However if the samples are sufficiently large the Central Limit Theorem which guarantees approximate normal distribution for the mean can be applied for the adoption of parametric methods.
The calculated mean and the standard deviation are not wrong for non-normal distributed data, nor do they lead to wrong results, as you wrote. The problem is that the numbers are usually interpreted as if they were calculated from normally distributed data, and this wrong interpretation may lead to wrong conlusions.
The median together with the inter-quartile range can give a good idea about the location, spread, and the skewness of the data (assuming that the distribution is unimodal). Another option is to give the mode and a skewness measure. Ideally one can simply show the data, so there is no need to give summaries at all.
You can use non-parametric tests, but you should be aware that many of them test other hypotheses than you might think they do. If the amount of data is large, you can use bootstrap methods to test exactly those hypotheses that you deem interesting, but you can also use parametric tests because of the central limit theorem (as Ette already noted). My favorite it to identify an adequate probability model. We don't need to be fixed with the normal distribution. Stats software allows to fit models for a wide variety of probability models effortlessly.
You can also try to normilize your data, if you have enough data it will be possible, then use parametric methods, which may work properly even with some non-normal data.
non-parametric could be used if you have few data.
Prof. Ette Etuk - Thanks for your valuable comment on the topic.
Thanks also to Dr.Ofelia for addressing the question.
Dr. Jochen Wilhelm,
The points I understood/not understood:
1) For large sample sizes (> 30 or 40), the violation of the normality assumption should not cause major problems; this means that parametric procedures may be applied even when the data are not normally distributed. Is it applicable for multivariate statistics (say, factor analysis) also ?
2) If the sample size is sufficiently large ( say >100), the distribution of the data can be ignored (Central Limit Theorem ).
3) Your statement " ..... is that the numbers are usually interpreted as if they were calculated from normally distributed data, and this wrong interpretation may lead to wrong conclusions". Could you please explain this with one example?
Regards
1) I don't know, actually, but I suppose yes, because factor analysis is based on the same principles as linear models.
2) Yes... for the purpose of testing expected values.
3) Example:
in https://www.iii.org/fact-statistic/facts-statistics-lightning you find the numbers of lightning fatalities in the US states in 2016. The data is
number of cases: 0, 1, 2, 3, 4, 9 number of states: 31, 8, 5, 1, 2, 1
Imagine the data was summarized as mean and standard deviation:
mean number of cases per state: m = 0.792
standard deviation: s = 1.597
Assuming that the number of cases would be normally distributed, one would expect that most states would have about 1 case, but in fact the vast majority of states don't have a single case. One would further expect that there shouldn't be states with more than 5 cases (m+3s = 5.6, and based on the normal probability model P(x>5.6) < 0.001, what is pretty unlikely), but we observed a state with 9 cases. The probability, under the nomal model, to get x>=9 is lower than 0.000001! We would never expect that to happen if we assume the normal probability model. Finally, under the normal probability model the probability P(x mean): the negative-binomial model that can be parametrizes with a location parameter m' and a dispersion parameter s' . If one fits the parameters of that model to the observed data, one gets m' = m = 0.792 and s' = 0.397 (that's not a standard deviation; it is the estimated value of dispersion parameter of the negative binomial model). Thus, reporting that the number of cases was negative-binomial distributed with mean 0.792 and dispersion 0.397 would be the most instructive summary of the data.
Dr. Jochen Wilhelm,
I have read your detailed explanation - really informative and useful. Thanks and regards
Dr. Nina Nikolic,
You have explained the matter in the perspective of an environmentalist. The researchers on environment will be benefited in the application of multivariate statistics. Regards
Hi everybody,
I have a dataset with a skewness 1.7e+03 0.0000 when using sktest in Stata and I get 0.90, when using Shapiro-Wilk W test for normal data and moments2 gives 1.04, when try Jarque-Bera normality test, it gives 296.5 Chi(2) 4.1e-65. All result are strongly statistically significant. The variable I am testing is age.
My question is how should look at this situation, which test is more reliable?
What would you suggest do further analysis, use regression in this descriptive report? If so which type of rgression, do do suggest?
Many thanks in advance,
/Homayoun
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