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For normality, actual Shapiro-Wilk has good power in fairly small samples.

The main competitor in studies that I have seen is the more general Anderson-Darling, which does fairly well, but I wouldn't say it was better. If you can clarify what alternatives interest you, possibly a better statistic would be more obvious. [edit: if you estimate parameters, the A-D test should be adjusted for that.]

[I strongly recommend against considering Jarque-Bera in small samples (which probably better known as Bowman-Shenton in statistical circles - they studied the small sample distribution). The asymptotic joint distribution of skewness and kurtosis is nothing like the small-sample distribution - in the same way a banana doesn't look much like an orange. It also has very low power against some interesting alternatives - for example it is powerless to pick up a symmetric bimodal distribution that has kurtosis close to that of a normal distribution.]

Frequently people test goodness of fit for what turn out to be not-particularly-good reasons, or they're answering a question other than the one that they actually want to answer.

For example, you almost certainly already know your data aren't really normal (not exactly), so there's no point in trying to answer a question you know the answer to - and the hypothesis test doesn't actually answer it anyway.

Given you know you don't have exact normality already, your hypothesis test of normality is really giving you an answer to a question closer to "is my sample size large enough to pick up the amount of non-normality that I have", while the real question you're interested in answering is usually closer to "what is the impact of this non-normality on these other things I'm interested in?". The hypothesis test is measuring sample size, while the question you're interested in answering is not very dependent on sample size.

There are times when testing of normality makes some sense, but those situations almost never occur with small samples.

Why are you testing normality?

For normality, actual Shapiro-Wilk has good power in fairly small samples.

The main competitor in studies that I have seen is the more general Anderson-Darling, which does fairly well, but I wouldn't say it was better. If you can clarify what alternatives interest you, possibly a better statistic would be more obvious. [edit: if you estimate parameters, the A-D test should be adjusted for that.]

[I strongly recommend against considering Jarque-Bera in small samples (which probably better known as Bowman-Shenton in statistical circles - they studied the small sample distribution). The asymptotic joint distribution of skewness and kurtosis is nothing like the small-sample distribution - in the same way a banana doesn't look much like an orange. It also has very low power against some interesting alternatives - for example it has low power to pick up a symmetric bimodal distribution that has kurtosis close to that of a normal distribution.]

Frequently people test goodness of fit for what turn out to be not-particularly-good reasons, or they're answering a question other than the one that they actually want to answer.

For example, you almost certainly already know your data aren't really normal (not exactly), so there's no point in trying to answer a question you know the answer to - and the hypothesis test doesn't actually answer it anyway.

Given you know you don't have exact normality already, your hypothesis test of normality is really giving you an answer to a question closer to "is my sample size large enough to pick up the amount of non-normality that I have", while the real question you're interested in answering is usually closer to "what is the impact of this non-normality on these other things I'm interested in?". The hypothesis test is measuring sample size, while the question you're interested in answering is not very dependent on sample size.

There are times when testing of normality makes some sense, but those situations almost never occur with small samples.

Why are you testing normality?

For normality, actual Shapiro-Wilk has good power in fairly small samples.

The main competitor in studies that I have seen is the more general Anderson-Darling, which does fairly well, but I wouldn't say it was better. If you can clarify what alternatives interest you, possibly a better statistic would be more obvious.

[I strongly recommend against considering Jarque-Bera in small samples (which probably better known as Bowman-Shenton in statistical circles - they studied the small sample distribution). The asymptotic joint distribution of skewness and kurtosis is nothing like the small-sample distribution - in the same way a banana doesn't look much like an orange. It also has very low power against some interesting alternatives - for example it is powerless to pick up a symmetric bimodal distribution that has kurtosis close to that of a normal distribution.]

Frequently people test goodness of fit for what turn out to be not-particularly-good reasons, or they're answering a question other than the one that they actually want to answer.

For example, you almost certainly already know your data aren't really normal (not exactly), so there's no point in trying to answer a question you know the answer to - and the hypothesis test doesn't actually answer it anyway.

Given you know don't have exact normality already, your hypothesis test is giving you an answer to a question closer to "is my sample size large enough to pick up the amount of non-normality that I have", while the real question is usually closer to "what is the impact of this non-normality on these other things I'm interested in?".

There are times when testing of normality makes some sense, but those situations almost never occur with small samples.

Why are you testing normality?

For normality, actual Shapiro-Wilk has good power in fairly small samples.

The main competitor in studies that I have seen is the more general Anderson-Darling, which does fairly well, but I wouldn't say it was better. If you can clarify what alternatives interest you, possibly a better statistic would be more obvious. [edit: if you estimate parameters, the A-D test should be adjusted for that.]

[I strongly recommend against considering Jarque-Bera in small samples (which probably better known as Bowman-Shenton in statistical circles - they studied the small sample distribution). The asymptotic joint distribution of skewness and kurtosis is nothing like the small-sample distribution - in the same way a banana doesn't look much like an orange. It also has very low power against some interesting alternatives - for example it is powerless to pick up a symmetric bimodal distribution that has kurtosis close to that of a normal distribution.]

Frequently people test goodness of fit for what turn out to be not-particularly-good reasons, or they're answering a question other than the one that they actually want to answer.

For example, you almost certainly already know your data aren't really normal (not exactly), so there's no point in trying to answer a question you know the answer to - and the hypothesis test doesn't actually answer it anyway.

Given you know you don't have exact normality already, your hypothesis test of normality is really giving you an answer to a question closer to "is my sample size large enough to pick up the amount of non-normality that I have", while the real question you're interested in answering is usually closer to "what is the impact of this non-normality on these other things I'm interested in?". The hypothesis test is measuring sample size, while the question you're interested in answering is not very dependent on sample size.

There are times when testing of normality makes some sense, but those situations almost never occur with small samples.

Why are you testing normality?

For normality, actual Shapiro-Wilk has good power in fairly small samples.

The main competitor in studies that I have seen is the more general Anderson-Darling, which does fairly well, but I wouldn't say it was better. If you can clarify what alternatives interest you, possibly a better statistic would be more obvious.

[I strongly recommend against considering Jarque-Bera in small samples (which probably better known as Bowman-Shenton in statistical circles - they studied the small sample distribution). The asymptotic joint distribution of skewness and kurtosis is nothing like the small-sample distribution. It also has very low power against some interesting alternatives - for example it is powerless to pick up a symmetric bimodal distribution that has kurtosis close to that of a normal distribution.]

Frequently people test goodness of fit for what turn out to be not-particularly-good reasons, or they're answering a question other than the one that they actually want to answer.

For example, you almost certainly already know your data aren't really normal (not exactly), so there's no point in trying to answer a question you know the answer to - and the hypothesis test doesn't actually answer it anyway.

Your hypothesis test is giving you an answer to a question closer to "is my sample size large enough to pick up the amount of non-normality that I have", while the real question is usually closer to "what is the impact of this non-normality on these other things I'm interested in?".

There are times when testing of normality makes some sense, but those situations almost never occur with small samples.

Why are you testing normality?

For normality, actual Shapiro-Wilk has good power in fairly small samples.

The main competitor in studies that I have seen is the more general Anderson-Darling, which does fairly well, but I wouldn't say it was better. If you can clarify what alternatives interest you, possibly a better statistic would be more obvious.

[I strongly recommend against considering Jarque-Bera in small samples (which probably better known as Bowman-Shenton in statistical circles - they studied the small sample distribution). The asymptotic joint distribution of skewness and kurtosis is nothing like the small-sample distribution - in the same way a banana doesn't look much like an orange. It also has very low power against some interesting alternatives - for example it is powerless to pick up a symmetric bimodal distribution that has kurtosis close to that of a normal distribution.]

Frequently people test goodness of fit for what turn out to be not-particularly-good reasons, or they're answering a question other than the one that they actually want to answer.

For example, you almost certainly already know your data aren't really normal (not exactly), so there's no point in trying to answer a question you know the answer to - and the hypothesis test doesn't actually answer it anyway.

Given you know don't have exact normality already, your hypothesis test is giving you an answer to a question closer to "is my sample size large enough to pick up the amount of non-normality that I have", while the real question is usually closer to "what is the impact of this non-normality on these other things I'm interested in?".

There are times when testing of normality makes some sense, but those situations almost never occur with small samples.

Why are you testing normality?