# Thread: Normality Assumption for Independent t-test

1. ## Normality Assumption for Independent t-test

Hi All,

This may be a fairly basic question but I am not the best at stats and so would very much appreciate your input.

I wanted to get your advice on whether you think I am able to conduct an independent t-test if one group has n = 21 and the other has n = 159.

Kolmogorov-Smirnov and Shaprio-Wilk tests indicate a violation of normality (p <.05), however the z-scores suggest that there are no univariate outliers present.

Thanks!

2. ## Re: Normality Assumption for Independent t-test

Hi, your data can be non-normally distributed even if you have no outliers. Since parametric assumptions are thus violated, you should not use the classical T-test. I would propose to use the Wilcoxon rank-sum test (wich does not assume normality of the data) which has the additional advantage that it can work with unequal group sizes.

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bozatron (01-12-2017)

4. ## Re: Normality Assumption for Independent t-test

Or, you may want to use a permutation t-test. If you are familiar with R, you may find a function, some more information, and the rationale, from this page of my website: http://cainarchaeology.weebly.com/r-...on-t-test.html

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bozatron (01-12-2017)

6. ## Re: Normality Assumption for Independent t-test

Formal tests of normality, such as the KS test for normality (or SW), are often very sensitive to immaterial departures from normality. I would recommend using a normal probability plot to determine how much your data deviate from a normal distribution. It may not be necessary to immediately jump to a nonparametric test. If you are really concerned, you can run the Wilcoxon rank sum as well and see if you're getting a very different picture between the two tests. If not, you can probably feel a little more comfortable about your conclusions in the parametric method. If you're doing research, you would want to mention that you also ran the Wilcoxon to check if the conclusion was vastly different (from an ethical point, and to bolster your findings a bit).

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bozatron (01-12-2017)

8. ## Re: Normality Assumption for Independent t-test

Thanks so much for all your responses, I'm very grateful!

I have run some normal probability plots and attached them to this post. I can see that there is some deviation from normality but was hoping to get your advice on whether you think it looks like too much of a deviation to run a t-test.

Also, I have read that when n is relatively large (30+ in each group), that the t-test is considered to be robust to violations of normality. Is this correct, and if so, does that mean that I could just reference this and run one even if there is non-normality evident?

Thanks!

9. ## Re: Normality Assumption for Independent t-test

I opened the word doc and realized no one know what you're analyzing.

Is MMSE a mini mental status exam score?

In terms of robustness, "large enough" usually depends on the underlying distribution (less normal means larger sample size is needed to be "large enough" for the CLT to apply).

10. ## Re: Normality Assumption for Independent t-test

Normality assumptions concern the distributions in the respective groups, not the distribution of the total sample.

Since (normal) distributions of the underlying populations are concerend (not the distribution of the sample data!), the graphical inspection of sample data does not solve any problem (AFAICS).

A total n of 180 ensures robustness of your t-test, even if the distributions are non-normal. With unequal sample sizes, though, inhomogenous variances could be an issue. You therefore should use the Welch test (a corrected version of the t-test).

With kind regards

K.

11. ## Re: Normality Assumption for Independent t-test

Originally Posted by Karabiner
Normality assumptions concern the distributions in the respective groups, not the distribution of the total sample.
The poster included NPPs for each group which would help in determining if the normality assumption (of the populations) is reasonable based on the data.

Originally Posted by Karabiner
Since (normal) distributions of the underlying populations are concerend (not the distribution of the sample data!), the graphical inspection of sample data does not solve any problem (AFAICS).
I think it's fairly common practice to inspect the data to see if it's reasonable that the data were drawn from a normal distribution (unless they're known not to be). If the data don't appear to be from a normal distribution, then you'd want to consider other options (again, this is assuming you don't know the distribution they're from). Would you agree this is reasonable?

Originally Posted by Karabiner
A total n of 180 ensures robustness of your t-test, even if the distributions are non-normal. With unequal sample sizes, though, inhomogenous variances could be an issue. You therefore should use the Welch test (a corrected version of the t-test).

With kind regards

K.
I agree with you on this point, too. I think a Welch's test is another alternative if a large disparity in the variances is a concern. If the poster is concerned about robustness of the inference, then he or she should take another look from a different angle (such as the Wilcoxon rank sum, as mentioned before). It can't hurt to hit a question from a few different angles to see how your conclusions hold up, so long as those methods are reasonable (and as long as you report all tests performed and why you performed them).

12. ## Re: Normality Assumption for Independent t-test

I think it's fairly common practice to inspect the data to see if it's reasonable that the data were drawn from a normal distribution (unless they're known not to be).
I don't now whether this is fairly common. I haven't seen it for some while in my domain of research, but maybe it is used elesewhere. As far as I can see, the problem whether t-test or a GLM is appropriate, is not solved.

If the poster is concerned about robustness of the inference, then he or she should take another look from a different angle (such as the Wilcoxon rank sum, as mentioned before).
I suppose s/he need not be concerned, given the sample size.

Wilcoxon does not compare means, therefore it deals with different question than the t-test.
This could be a problem if comparison of means is important für the OP.

With kind regards

Karabiner

13. ## Re: Normality Assumption for Independent t-test

Originally Posted by Karabiner
I don't now whether this is fairly common. I haven't seen it for some while in my domain of research, but maybe it is used elesewhere. As far as I can see, the problem whether t-test or a GLM is appropriate, is not solved.
I see your point regarding t-test or GLM. I guess it would be helpful to have a little more information about the project at hand in terms of the variables and what the actual research question is, because that could give guidance on the approach.

Originally Posted by Karabiner
I suppose s/he need not be concerned, given the sample size.

Wilcoxon does not compare means, therefore it deals with different question than the t-test.
This could be a problem if comparison of means is important für the OP.

With kind regards

Karabiner
Undoubtedly, Wilcoxon doesn't compare means as a parametric t-test does, but you can still get a general picture for central location with the t-test and Wilcoxon-- this is what I meant regarding "another look from a different angle". My apologies for the lack of clarity in my response. This does resurface the idea that knowing the OP's research question and variables would help in determining if one method is actually more appropriate for the question.

14. ## Re: Normality Assumption for Independent t-test

Why don't you put to work permutation t test (as I suggested in my earlier reply) in order to get, among other things, an idea of whether or not 'regular' t test would have worked?

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bozatron (01-18-2017)

16. ## Re: Normality Assumption for Independent t-test

I will add a little fuel to you all's conversation. I believe the MMSE is a test score based on weighted components and is on a scale between 0-30, so bounded. In addition, depending on their samples, I believe certain score are typically more likely, since certain questions can be troubling to test takers. In addition, I believe most uninhibited test-takers cluster around 30, while others could have a different distribution, since a mid-value total score can have room to vary without the impact of a bound.

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bozatron (01-18-2017)

18. ## Re: Normality Assumption for Independent t-test

Originally Posted by hlsmith
I will add a little fuel to you all's conversation. I believe the MMSE is a test score based on weighted components and is on a scale between 0-30, so bounded. In addition, depending on their samples, I believe certain score are typically more likely, since certain questions can be troubling to test takers. In addition, I believe most uninhibited test-takers cluster around 30, while others could have a different distribution, since a mid-value total score can have room to vary without the impact of a bound.
You're absolutely correct, which is why I asked OP to clarify it for us just in case it's some different acronym that looks familiar to me. I think each element of the test receives points within a section, and then the overall section scores are tallied to get the total.

On another note, I'd like to hear thoughts on this issue. I'm not sure that the MMSE score would be an interval measurement. For example, the difference between 28 to a 30 on the exam is 2 points, but these 2 points might not represent the same thing when moving from a 23 to a 25 (I see this as a similar topic as likert scales). The latter move of 2 points moves you from below the 24-point cut off to above it, indicating normal cognitive function. If this isn't an interval measurement, or if it's unclear, might it be more appropriate to target the median (or even mode), assuming the OP does care about centrality? What are the thoughts on this?

19. ## Re: Normality Assumption for Independent t-test

Agreed, there is at least 3 issues:

-not continuous
-bounded
-and perhaps not linear in interpretation as you mentioned.

So, I am guessing most people just stay naïve and use ttest. I wouldn't be surprised if they is a paper out there on how to analyze these instrument data. Some say a 7-item or larger Likert scale can be treated using parametric test, given assumption are not broken. I am guessing you would be fine with a large enough sample. The normal,..., etc. interpretation scales were probably a classification collapse of the instrument. Probably papers on that methodology. Side note, I think the MMSE gets used A LOT. We could probably find it analyzed using all types of approaches.

This probably falls into the contextual subjective category.

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bozatron (01-18-2017)

21. ## Re: Normality Assumption for Independent t-test

Originally Posted by hlsmith
Agreed, there is at least 3 issues:

-not continuous
-bounded
-and perhaps not linear in interpretation as you mentioned.

So, I am guessing most people just stay naïve and use ttest. I wouldn't be surprised if they is a paper out there on how to analyze these instrument data.
I know for a fact I've seen paired MMSE scores analyzed with a Wilcoxon signed rank test (but seeing it doesn't necessitate appropriateness in their case or this one). I'm going to take a look and see what I can find in terms of a guidance paper.

Originally Posted by hlsmith
Some say a 7-item or larger Likert scale can be treated using parametric test, given assumption are not broken. I am guessing you would be fine with a large enough sample.
I've heard this too, at least in a test utilizing ranks.

Originally Posted by hlsmith
Side note, I think the MMSE gets used A LOT. We could probably find it analyzed using all types of approaches.
This probably falls into the contextual subjective category.
Indeed, they are used quite frequently. Another is called the MoCA. Let's see what we can dig up after the OP gets back with some clarification on the questions we've posed.

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bozatron (01-18-2017)