can I do my statistics work based on the central limit theorem? I need to perform a t-test, ANOVA and multiple regression. my outcome variable is highly not normally distributed (Highly positively skewed) and my sample size N=115. I'd like to keep the non-parametric tests as a last option for me.
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$\begingroup$ Please search for the extensive discussions about this on the site. Why do you want nonparametric to not be the first option? Why are results from $n \rightarrow \infty$ of interest to you? $\endgroup$– Frank HarrellCommented Mar 7, 2014 at 22:27
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1$\begingroup$ Indeed nonparametric and semiparametric statistics are of interest to most statistician. Though theoretical statements, like normality and equal variance, for t-test and ANOVA promise an upper hand over nonparametric test. In practice parametric statistic faces a lot of problems. $\endgroup$– Chamberlain MbahCommented Mar 7, 2014 at 23:05
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$\begingroup$ I see, I already used a lot of non-parametric tests in my work. but I need to perform multiple regression and that is why the assumption of normality made a problem for me. so can I consider the central limit theorem to assume normality? $\endgroup$– Mahmoud IsmaelCommented Mar 7, 2014 at 23:18
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1$\begingroup$ Why don't you perform the regression and see just how much the residuals depart from normality? Then you can provide us much more specific and focused information to help you choose appropriate procedures. $\endgroup$– whuber ♦Commented Mar 7, 2014 at 23:45
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$\begingroup$ PLEASE INDICATE WHETHER YOU HAVE WORKED OUT THE FREQUENCY POLYGON OR YOU HAVE CONTINUOUS DATA? Moreover specify that you have outcome values in terms negative or positive or both. $\endgroup$– user10619Commented Mar 9, 2014 at 9:59
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(1) The CLT is a result in the limit as $n\to\infty$. There's no particular n that's certain to be large enough. e.g. see here which gives a method which works for constructing cases which require larger sample sizes than any $n$ you can nominate.
(2) the central limit theorem on its own is not enough. The statistics you mention rely on a ratio for which the CLT would only help with the numerator, and so you need something to hold for the denominator. The distributions also rely on independence of the numerator and denominator.
If the assumptions are reasonable, you might want to consider GLMs, perhaps (since with suitable software, regression via GLMs is almost as convenient as ordinary regression), though there are other alternatives.
There are various other nonparametric, parametric and robust alternatives.
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$\begingroup$ Dear Nick, It was intended to be comment. thanks for converting it and placing it in the right place. $\endgroup$– user10619Commented Mar 9, 2014 at 10:44
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$\begingroup$ Thanks for your thanks, but it was @mbq who did that. $\endgroup$– Nick CoxCommented Mar 9, 2014 at 10:49