You can take the mid-range (average of min and max) per cell. Given the distribution of the (unmeasured) underlying variable is roughly normal, the distribution of the midranges is also roughly normal, and the average of several of such mid-ranges (for sereval cells) will have a distribution that is an even better approximation of the normal distribution. Thus, you can use a standard linear model for the mid-ranges.

http://en.wikipedia.org/wiki/Mid-range

Bland, R. P., Gilbert, R. D., Kapadia, C. H., and Owen, D. B., “On the Distributions of the Range and Mean Range for Samples from a Normal Distribution,”Biometrika, 53, (1966), 245–248. CrossRef

Consistency of the Midrange: Distributions Whose Tails are Both Light and Smooth

Tim Robertson and F. T. Wright. The American Statistician Vol. 28, No. 4 (Nov., 1974), pp. 135-137

Very interesting case . I am not aware of any systematic way to do the comparison, I would build confidence intervals for the minimums and for the maximums and then reason in that terms

One way is finding the underlying joint-distribution of max and min for observed data, then based on accepted theoretical distribution, find the best hypothesis. 

Or you can define new variable, like rang=Max-Min, and then use the most appropriate statistical tests.

Hi Mohammad, about option #2 I don't think is appropriate, because you can get same ranges with totally different values of min and max (e.g., minA=10, minB=50, maxA=20,maxB=60, range is 10 in both cases). Does it make sense ?

You can take the mid-range (average of min and max) per cell. Given the distribution of the (unmeasured) underlying variable is roughly normal, the distribution of the midranges is also roughly normal, and the average of several of such mid-ranges (for sereval cells) will have a distribution that is an even better approximation of the normal distribution. Thus, you can use a standard linear model for the mid-ranges.

http://en.wikipedia.org/wiki/Mid-range

Bland, R. P., Gilbert, R. D., Kapadia, C. H., and Owen, D. B., “On the Distributions of the Range and Mean Range for Samples from a Normal Distribution,”Biometrika, 53, (1966), 245–248. CrossRef

Consistency of the Midrange: Distributions Whose Tails are Both Light and Smooth

Tim Robertson and F. T. Wright. The American Statistician Vol. 28, No. 4 (Nov., 1974), pp. 135-137

@Antonio: Thanks.

@Jochen: Thank you for sharing it.

Thank you all for your helpful comments. Someone also suggested I looked into ‘interval-valued data’. When googling it, the words ‘fuzzy data’ came up quite a few times. I am currently trying to find out more about statistical analysis methods for such fuzzy data.

Use the manova, not just for comparison between the minimun and the maximum (SIC), but for comparing the two POPULATIONS from which the two groups have been obtained.

In addition, since there are only two groups yuo can do the Hotellint T2 test. Of course, the sample size has to be greater than 30 (at least) in each groups for avoiding the Gaussian assumption by referring to the central limit theorem.

Best

BMCesana

@manuel would be fine in your opinion to check first whether the correlation between min and max is moderate and if so proceed with MANOVA ?

I would try Jochen's answer first. You could also use a nonparametric test on only the minimum and then only the maximum values. I would also test to see if the treatment influences the range.

In a perfect world, the first three tests will give the same answer and the latter will be nonsignificant.The results from the other tests would help support or refute the assumptions made in the initial analysis

I agree that you don't have two independent measurements, but the problem is more that you don't know what the distribution is between them. If the distribution is skewed, then it is possible to get a different answer using the midpoint, versus minimum, versus maximum. If the treatment changes the variance structure of your data, then you could get a significant difference in the range.

Adding to Timothy: It may be possible to recognize an assymtery (skewdness) by comparing the variance of the minimum values with the variance of the maximum values (within the same group). If, for instance, the variance of the maxima is much larger than the variance of the minima, this would indicate a right-skewed distribution (and in this case using the midrange technique based on the logs would likely be most appropriate).

If you have a sample size of 3, you will have to calculate the variances that Jochen suggests, and then guess if they are different. If you have a sample size of 20, then you could make a test using a randomization technique. For both treatments break your data into two groups of 10, by randomly assigning individuals. For one group calculate the difference between the variance in the minimum and variance in maximum. Do this over and over to get a frequency distribution (one distribution for the treatment and one distribution for the control group). Is the observed value more extreme than 97.5% of the random values? Is the answer the same for the control and the treatment?

I would not try this with 12 or fewer replications. At 14 replications I would use a permutation test because there are only 3432 possible combinations. At 18 or more replicates you are fine with a randomization test. You should work with 10,000 to 20,000 randomizations to get a decent frequency distribution.

I am astonished as a so "simple problem" can receive so many (absolutely not shearable) suggestions!. I repeat that the comparison between two groups with more than one variable has to be made by means of the Hotelling's T2 (squared) and by Manova / Mancova in the case of more than two groups. All the doubts / uncertainties raised to my proposal are meaningless: you can compare two (or more) population by considering jointly the Systolic and the Diastolic blood pressure and, of course one is always more that the other!. Of course, the assumptions have to be fulfilled, but with more than 30 observations the Gaussian distribution of the variables is not very important owing to the Central Limit Theorem.

Cesana, could you please explain why the assumption of normality is justified for the minimum and the maximum values. Yes biologists often just assume and hope for the best -- or hope that no one will be able to prove otherwise. Why do this when there are alternatives? To be fair, my assumption is that biological data are never normally distributed and a failure to detect a departure from normality only indicates insufficient sample size. That is different from the philosophical argument about whether a specific departure from normality has a meaningful effect on the conclusions, and issues associated with the robustness of specific tests to departures from the assumptions of the model.

The other problem is that I assumed that the minimum and the maximum are dependent because they are being taken from the same sample and are measures of the same variable. Your example involved two means of the same process (heart function). A better example would be to build your heart model using the minimum systolic pressure and maximum systolic pressure. Take systolic pressure measurements for 5 minutes (or some other time interval). After finding the minimum and maximum discard all the other numbers so that you cannot calculate a mean.

Given that we do not know the sample size, I wonder if we can fulfill any of the assumptions of the model that you propose.

However, this discussion does suggest another option. Estimate the correlation between the minimum and maximum values. If the correlation is 1.0 then you can discard either the minimum or the maximum and proceed with a simple univariate model. The poorer the correlation the more information is retained by using both variables in some way.

First, I would like to thank everyone for their suggestions. Second, I noticed that the question about the sample size came up quite a few times. In my experiments, I usually get only a limited number of data points (around 10 cells for each condition). 

First point: Gaussian distribution. The parametric statistical methods devoted to the comparisons of means (continuous variables) have as assumption the Gaussian distribution of the analyzed variables; this is a short expression for the distribution of the means since, if a variable is Gausssianly distributed it is immediate that the distribution of all the possible sample means is Gaussian with the same mean and the same variance divided by the sample size. However, the necessary condition is actually on the distribution of the means that are in fact compared. So, it is obvious that the minimum or the maximum and also the range etc are not Gaussian distributed, but according to the Central Limit Theorem their mean tend to be Gaussianly distributed and it it possible to rely on this if the sample size is at  least more than 30. This fact has been ruled out by the communication that the sample size is only about 10 and, consequently all the parametric statistical methods, since the variables are not gaussianly distributed have to be discharged.

Second point: correlation between minimum and maximum etc. For me it is a not shearable suggestion since, in principle, significance test have to be not conditioned on the result of some other significance test. This applies to the "testing for the Gaussianity and the homoscedasticity" before an unpaired Student's t test, for example. Doing "conditionated tests" does not allow to keep the Type I error and the power (too) of the statistical test at the required levels. The above approach is typical of self made statisticians because also elementary statistical books, of course, report these assumptions at the basis of the statistical significance test, but they cannot go deeper in explaining the concept of "conditional significance tests". It is much better to rely on the well known robustness of the parametric statistical procedures, particularly if the sample sizes are equal and the variance are similar. In addition, there is "a prior" hypothesis under any research that has to be approached by the pertinent statistical test. According to your proposal, if there is a correlation (statistical significance with 10 units!?) I analyze the minimum ? the maximum ?.  According to which criterion?. The statistical methods have to be applied according to the statistical theory under which they have been obtained, not according to some empirical receipt. In addition, it is possible, in theory, having two well correlated variables (say on 30 subjects and with a Pearson's / Spearman's correlation coefficient of more than 0.8 - 1.0 is practically impossible apart from the fact that the correaltion is calculated on the same duplicated data) and obtaining a statistically significant result with the first variable and not with the second and viceversa.

Third point. After having actually known the context of this problem, do two Wicoxon test each at a 0.025 significance level (according to the "raw" Bonferroni's correction) on the minimum and on the maximum and this is all. In addition, when the researcher (Cècile Bats) do a research has to calculate the sample size for having a required power (0.80 at least) of succeeding in demonstrating the hypothesis of her interest. With 10 units it is possible to demonstrate an effect size of about 1.35-1.40 with a Student's t test for unpaired data at a significance level of 0.05 (two tails) at a power of 0.80 and the effect size will be a little more for the non-parametric Wilcoxon test. So, it is reliable the above effect size? in the answer is yes go two tests after having adjusted the significance level otherwise do an appropriate sample size calculation and repeat the experiment.

Cesana, you put forward many good points. I have three questions:

1) How can you claim that n

Dear Wilhelm,

your first question: it is a very well known result from the statstical theory reported in every statistical books (even at an elementary level) and it is not based on a particular application but the fact that the variable is not Gaussian distributed.

Your second question on a conposite statistic like the mid-range; the statistical analysis has to be done on the variables that describe the phenomenon in the most pertinent way. Perhaps the mid-range does not have a "biological interpretation" since the researcher is interested in considering both: minimum and maximum. Otherwise, if it has a sense, a composite statsitics can always be used. It seems to me the same problem about a repeated measure design. You can use a "summary statistic" such as the mean, etc but you miss the opportunity of estimating and testing the pattern that can be different between two drugs even if the means are very similar.

Your third question: it is not a comparison of effect size, but of the percent of statistically significant results. If for a Student's t test we have an expected power of 0.80 with a particular sample size, the percentage of statistically significant results at the Wilcoxon is about 95% of the 0.80 (Student's t test) in the case of a completely ideal situation: Distribution Gaussian and homoscedasticity. It is very easy to obtain this result empirically from a simulation study. In any case, the performance of the nonparametric test is usually and appropriately compared against their parametric counterparts in every books on nonparametric statistics. Finally, I do not understand the meaning of your last sentence about a bootstrap test on the mean, but it seems to me not pertinent; in this case we are interested in a comparison between two populations and it is possible to rely on standard statistical procedure and it is better particularly when there are quite extreme situation with ten subjects and "particular variables".

It amazes me that a sample size of 29 is not ok, while a sample size of 30 is appropriate for invoking CLT given non-Gaussian distributed variables. I have looked in a few statistics books and couldn't find this result. So it seems to be missing from a few of the statistics books written in English. Sadly, I cannot check all the books in English, nor any of the books in other languages.

I have some trouble thinking of a criterion that would justify such a precise decision value.

Thank you for your response.

ad 1) Counter example: X ~ Uniform. The mean of 5 to 10 values already has a distribution that approximates the normal distribution good enough for any practical purpose. On the other hand, for considerably skewed data even sample sizes of several hundred may not give a sufficiently good approximation.

ad 2) Good point. As a biologist I understood that it is only the problem of the measurement (the actual extremes are not interesting but the only things that could be recorded). Here we agree.

ad 3) The problem is what exactly is to be "compared". The percent of statistically significant results can easily be tuned, when I am free to select a test. It is relevant what hypothesis the test adresses, and I think it should be the responsibility of the researcher to wisely select the hypothesis  - then the test is more or less given. If the hypothesis here is "(equal) expected values", then the Wilcoxon test is a bad choice, because this test will at most incidently test this hypothesis, whereas the t-test just precisely tests this hypothesis. If the hypothesis is "(equal) variances ", then both tests are wrong, also if the hypothesis is "(same) modality" etc. Given some constraints, the Wilcoxon test may correctly address the hypothesis "(equal) symmetry", and (as far as I know) this test always adresses the hypothesis "(equal) expectation of observing a higher(lower) values in either group" (stochastic difference). But this is not equal to the hypothesis of (equal) expected values. So what should be tested? Expected values, stochastic equality, symmetry, variability, some other feature? This must be decided in the first place. Then - if there are several tests that adress the correct hypothesis - the test with the highest power should be used. But testing some different hypothesis, just because the percent statistically significant results will be larger does not seem to be good science in my eyes.

To Ebert,

To Ebert,

You can find the above approximation of the Central Limit Theorem on an elementary statistical book: Pagano, Gauvreau, Principles of Biostatistics, second edition, Duxbury, 2000, on page 198. "if the underòying population is itself normal, samples of size 1 are large enough. Even if the population is bimodal or noticeably skewed, a sample of size 30 is often sufficient.  Of course, you are not a statistician and the affermation "quite deterministic" 30 is good and 29 is bad is a nonsense in satistics.

To Wilhelm,

I cannot agree, and I had not propose the use of a statistical test independently from the underlying hypothesis since it is the most powerful !!. I repeat that, of course, if you know that the assumptions underlying a statistical test are absolutely satisfied, you can use the Student's t test for unpaired data without any problem. The fact, in this case and always in practice, is that you do not know if the assumptions are satisfied; so you shift from a parametric test to a non-parametric one, particularly if the sample size is low. What you know is that, in the ideal situation, this shift give loosing a bit of power, but in the actual situation you can even obtain more valid results (Type I error or Type II error near to their optimal values: 0.05 under the null hypothesis and 0.80, say, under the alternative one on which the sample size has been calculated). In addition, it is well known that if the distribution is uniform you can have a satisfactory approximation even with a lower number of sample size than 30; however, the problem is that you do not know which is the distribution from which your data have been sampled. About the two different hypotheses; the problem is, of course, to compare two populations on a position parameter (the mean); if it is more sensible to use a non-parametric test the hypothesis of the equality of two distributions (on the location parameters, Wilcoxon) corresponds, in practice, to the hypothesis of equal means and equal variances (from the assumption of homoscedasticity). This fact is also for the Kruskal-Wallis , the Friedman two-way analysis of variance by ranks, etc.

BM Cesana

To Cesana,

You are the one who has used the cut-off value of 30 on multiple postings, not me. So if 29 is also ok, how about 28, 27, 26.....?

The sample size of 1 does not even warrant further comment.

However, thank you for the citation.

@Cesana,

thank you for commenting. I wonder how you can show that the assumption of equal distributions (except location shift) are satisfied. In biology, very typically,the distributions will clearly not have the same shape. Typically, when the data really seems to be skewed, samples with higher mean also have higher variance. The Wilcoxon test actually does NOT test the location there. The test is rather about some kind of "stochastic directedness". My problem is that if the researcher is interested in the expected values (or expected differences), this test gives answers to something else. I further wonder how a power can be specified when it is not clear how exactly the distributions (and, even worse, the differences between the samples) do look like. I do not see how the entire shapes of the distributions can be estimated reliably (especially with sparse data or how the assumption of "equal distibutions" can be justified or checked.