Thanks in advance for your reply.

- Thread starter blubblub
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Thanks in advance for your reply.

Thanks in advance for your reply.

I understand parametric tests are preferred over non-parametric tests (such as the ones that use ranking), but perhaps to make my question a bit more clear, let's simply use a one-sided z-test as an example. I understand the sample size has to be reasonable large (often said n > 30) in order for the central limit theory to be applied. But many books/sites then also say in order to perform a one-sample z-test one assumption is that the population is normally distributed (for example here), and therefor one might want to test for this (for example with a Shapiro-Wilk Test). What I don't understand is where this assumption is coming from?

I was always impressed by the central limit theory to apply even if the population is not normally distributed, and then surprised that to use it I have to test if it is.

For instance, take your case of the one-sample z-test and run it a small simulation with a small sample size:

For a population coming form standard normal distribution where the null hypothesis is true you can see something like:

Code:

```
library(BSDA)
pval<-double(10000)
for (i in 1:10000){
a<- rnorm(20, mean=0, sd=1)
pval[i]<-z.test(a, mu=0, sigma.x=1)$p.val
}
sum(pval<.05)/10000
[1] 0.0502
```

Try the same scenario but we're switching our samples from a normal distribution to a chi-square distribution with 1 degree of freedom (so very skewed):

Code:

```
pval<-double(10000)
for (i in 1:10000){
a<- rchisq(20, df=1)
pval[i]<-z.test(a, mu=1, sigma.x=sqrt(2))$p.val
}
sum(pval<.05)/10000
[1] 0.0444
```

Code:

```
pval<-double(10000)
for (i in 1:10000){
a<- rnorm(100, mean=0, sd=1)
pval[i]<-z.test(a, mu=0, sigma.x=1)$p.val
}
sum(pval<.05)/10000
[1] 0.0498
```

Code:

```
pval<-double(10000)
for (i in 1:10000){
a<- rchisq(100, df=1)
pval[i]<-z.test(a, mu=1, sigma.x=sqrt(2))$p.val
}
sum(pval<.05)/10000
[1] 0.0501
```

But many books/sites then also say in order to perform a one-sample z-test one assumption is that the population is normally distributed (for example here)

Small thing is:

....

For a population coming form standard normal distribution where the null hypothesis is true you can see something like:

...

So the nominal Type 1 error rate of 5% is just off by .0002, which is very small. So we're good here.

Try the same scenario but we're switching our samples from a normal distribution to a chi-square distribution with 1 degree of freedom (so very skewed):

[1] 0.0444

[/code]

Uhm... what do we see here? When the Type 1 error rate should be 5% it is now 4.44% I mean, it's not horrible but it is still *not* 5%. ...

......

For a population coming form standard normal distribution where the null hypothesis is true you can see something like:

...

So the nominal Type 1 error rate of 5% is just off by .0002, which is very small. So we're good here.

Try the same scenario but we're switching our samples from a normal distribution to a chi-square distribution with 1 degree of freedom (so very skewed):

[1] 0.0444

[/code]

Uhm... what do we see here? When the Type 1 error rate should be 5% it is now 4.44% I mean, it's not horrible but it is still *not* 5%. ...

......

So if I understand correct, in essence if a sample size would be large enough there indeed would be no need to test for normality, but since 'large enough' is a vague limit, it's better to simply test for it. I also came across this site but will have to read that more careful.

Yeah, and a lot of textbooks aimed for methodology courses (particularly in the social sciences, which is the area I come from) are notorious for relying on procedures that perhaps made sense back in the 1970s or just prefer a cookbook approach to statistical analysis without engaging in any critical thinking. It shouldn't come as a surprise then that, after years of questionable statistical practice, psychology is finding itself in the midst of its own crisis of replicablity.

Thanks spunky for the elaboration. I think I'm getting there

Small thing is:

With the normal distribution you mention 'only .0002' but for the chi-square the .0444, I guess you meant .0056 still more than the .0002 but not as much. Anyway I get what you're saying and thanks for that simulation.

So if I understand correct, in essence if a sample size would be large enough there indeed would be no need to test for normality, but since 'large enough' is a vague limit, it's better to simply test for it. I also came across this site but will have to read that more careful.[/URL]

Small thing is:

With the normal distribution you mention 'only .0002' but for the chi-square the .0444, I guess you meant .0056 still more than the .0002 but not as much. Anyway I get what you're saying and thanks for that simulation.

So if I understand correct, in essence if a sample size would be large enough there indeed would be no need to test for normality, but since 'large enough' is a vague limit, it's better to simply test for it. I also came across this site but will have to read that more careful.[/URL]

Wow, touched a nerve? Thanks for that article link, will definitely go through it.

Wow, touched a nerve?

One of the most common problems I see with normality testing is that people don't know when it's appropriate and they often misinterpret the results. For example, they incorrectly conclude the data come from a normal distribution because the test was not significant.