ANOVA and Normally distributed random variables

[soil1999]

Hello,
I am carrying out two-ways ANOVA analysis on different random variables and I need your comments please to make sure that I am (being not statistician or probability guy) not doing an error

1- Should these random variables be normally distributed in order for ANOVA analysis and the following regression analysis applied on these variables to be valid

2- I have a random variable (water depth below the ground) with Mean value of 1 m and Minimum value of 0 and maximum value of 2.5 m . Since I don't have sufficient data on the variability of this variable(other than the minimum and max values) I am trying to represent this variable ( with above mean and min and max values ) by a normally distributed function. Is that assumption valid,,?
If yes how can I calculate the standard deviation of this variable

Thanks

 

[GregLocock]

2 I think that would be unwise if the results are important. First off you have no sensible way of fitting a normal distribution to your 3 values. Second do you have any evidence that a normal distribution is valid?

Diving into some weird statistical analysis on the basis of just 3 numbers is just bad statistics.

How could anybody establish a mean without knowing more than the two limits? Go and find the data that was used for the mean and plot it as a histogram.




Cheers

Greg Locock


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[soil1999]

Thx Greg

-Actually the variable is natural variable. Also the values given are the mean (the most likely value of this variable based on some annual observation) and the minimum and maximum values that are physically possible for this variable.
but my question is , theoretically can a normal distribution function of this variable be defined for given mean , min , and max value.

- Also, any comment on my first question above...

Thx

 

[soil1999]

Thx Greg

-Actually the variable is natural variable. Also the values given are the mean (the most likely value of this variable based on some annual observation) and the minimum and maximum values that are physically possible for this variable.
but my question is , theoretically can a normal distribution function of this variable be defined for given mean , min , and max value.

- Also, any comment on my first question above...

Thx

 

[GregLocock]

No, I'm not going to bother answering your complicated question when even the simple question is ridiculous. Until you know whether a normal distribution is a reasonable model then any analysis you do is just fairydust. And the only way you can do that is to look at the data.

This should be chapter 1 in any stats manual, first plot the data.





Cheers

Greg Locock


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[soil1999]

Grag, You did not seem to understand my first email (No data is there , what one can do). Well I am going to model it as normally distributed variable in the absence of data. ( this is a hypothetical case anyway). Thank you for your polite answering manner.

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[GregLocock]

OK, many statisticians would do that. When the revolution comes they will be first up against the wall. If this is actually going to be used as the basis of any design work you'd better state what you have assumed at the top in big letters.

Now, how are you going to fit a symmetric distribution like a normal curve to your assymetric limits? how many standard deviations is that?



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Greg Locock


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[IRstuff]

Seems to me that ANOVA ought to be able to handle a lopsided distribution.

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[soil1999]

"Now, how are you going to fit a symmetric distribution like a normal curve to your assymetric limits? how many standard deviations is that"

Well that is really the question of the day for me. I have reviewed quite a lot of materials on this. (As I said this is a hypothetical case and this statistics issue is just a small part related to a deterministic problem I am investigating, but I have to cover it unfortunately).

For the Mean, Max, Min values I can't find an answer to your question above.
But if for the Min and Max values given, the use of the normally distributed curve is not theoretically possible, I will be willing to adopt any other distribution for this variable as long as the ANOVA and Least Square Regression approach ( which was used for optimizing the dependent response function of that variable) are not violated.


Background on my problem:
I am just carrying out MCS on an optimized equation which has many variables one of which is a variable with no data (with specified mean , max , min). I came up with that optimized function after carrying out 2ways-Factorial and ANOVA analysis( I did sampling at mean, min , and max values) and Least Square regression analysis. If these ANOVA and Regression do not imply that the analyzed variables have to be normally distributed, I am open to use any suitable distribution function fitting my concerning variable for my MCS. But the point is that I don't want to have discrepancy. I continue to search and review the related statistics. Thx a lot for help

 

[GregLocock]

Selecting the number of standard deviations to fit this data will obviously have some effect, so try SD=range/6, and SD=range/10, and repeat the analysis. in both cases you will have some extremes where the waterline is abobve ground level but that is the sort of thing that you get with made up data.

A final check would be to run the analysis with a uniform distribution as well, and see if it makes much difference.


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Greg Locock


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[soil1999]

I don't know if it is crazy to try various standard deviations and specify the given minimum and maximum by truncating the normally distributed function till fitting the mean, min, and max I have . I check if i can do that with the software I haveRisk

By the way does using uniform distribution (or say any other function different from normally distributed one) for this variable in the MCS violates the assumptions impeded in the ANOVA and least Square optimization (the normal distribution of the residuals and their homoscedastic variance) which are used to obtain the optimized Polynomial that is function to that variable.
Maybe at the end of the day I will end up using generalized spline shape function ( which is closer representative to the random nature of the variable than the uniform function, is not it..?)

 

[GregLocock]

ANOVA assumptions

the data points must be independent from each other

the distributions must be normal (though small departures can be accommodated)

the variances of the samples are not different (though some departures are accommodated)

all individuals must be selected at random from the population

all individuals must have equal chance of being selected

sample sizes should be as equal as possible but some differences are allowed



Cheers

Greg Locock


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[soil1999]

Please see attachment showing normal plot of the subject variable (water table depth below ground surface)) with mean of 1 m, min = 0 m, max= 2.5 m and assumed COV = 0.3 (which is reasonable given some literature associated with this variable.
Any notes on this is welcome