Applied Biostatistics - Science topic

It is a misconception that you would "testify the hypothesis of normal distribution". Tests use to "check for normal distribution" are actually tests to check for deviations from the normal distribution (the null hypothesis is that the data are realizations from iid normal random variables). Failing to reject this hypothesis is not a "proof" that the hypothesis is correct (it's only[!!] indicating that the sample size is not large enough to clearly see that the real-world data will not perfectly align with some idealized theoretical model!), and being able to reject it is no "proof" that the deviation is of any relevance (it only demonstrates that the sample size is large enough to clears see some deviation, but that might not at all be relevant).

The assumptions are based on understanding of the data-generating process. Often, this understanding is rather limited, and one needs to make some educated guess. Of course it is then appropriate (and recommended) to cross check if the data is roughly behaving "as expected" under these asumptions - provided the sample size is large enough to get some reliable idea about the distribution. This is not achieved by tests like Shapiro-Wilk, Anderson-Darling, Kolmogorov-Smirnov or others. It's better to check residual diagnostic plots, one of which is a normal-quantile-quantile plot (normal-QQ plot). Provided the sample size is large enough, this plot can show systematic deviations and give an impression of the kind and severity of the deviations. Together with other diagnostic plots one may make a judgement if the (set of) assumptions is reasonable / acceptable or if there ought to be a better set of more resonable assumptions.

With n < 30 (or so), you should not consider the CLT as argument that the assumption (normal distribution) is irrelevant. Even for n > 30 relying on the CLT can be probelmatic when the distribution is strongly skewed (as often in biology! E.g. for gene and protein expression, concentrations of biomolecules, titers of pathogens, etc.).

A particularily stupid example of applying a test to check for normal distribution is if the data are clearly binomial (yes/no, present/absent) data is coded as 0/1, or proportions (bound between 0 and 1, or as percentages bound between 0% and 100%). Similarily stupid is the example of testing data from counts and from strictly positive variables (concentrations, intensities). In all these cases the assumption of a normal distribution can possibly be okay, but we have very obviously more appropriate distributional assumptions (binomial, beta, Poisson, neg-binomial, gamma aor log-normal) and we have obviously more appropriate tools to deal with these kinds of distributions.

You can see with www.Stats4Edu.com

Thanks for this valuable share!!

Daniel Ruivo Marques, pardon me for the late response. Thank you for your guidance.

Bonjour! C'est ce que j'ai proposé au début (la méthode des moindres carrés - corrélations linéaires, elle convient le mieux.

Le signe négatif en mathématique en général et en statistique en particulier a toute son importance notamment au niveau des commentaires des résultats (exemple: la corrélation positive s'oppose à la corrélation linéaire négative). Les signes sont à respecter.

Hi,

Look at the distribution of scores whether they are skewed. Correct for the skew by some transformation and again look for the distribution. You may be able to do further analysis. Conversely, frequencies in the categories can be compared.

Here are a few references on the Psychometric properties and its use in some studies of the scale:

Morin CM, Belleville G, Bélanger L, Ivers H. The Insomnia Severity Index: psychometric indicators to detect insomnia cases and evaluate treatment response. Sleep. 2011 May 1;34(5):601-8. doi: 10.1093/sleep/34.5.601

Schulte, T., Hofmeister, D., Mehnert-Theuerkauf, A. et al. Assessment of sleep problems with the Insomnia Severity Index (ISI) and the sleep item of the Patient Health Questionnaire (PHQ-9) in cancer patients. Support Care Cancer 29, 7377–7384 (2021). https://doi.org/10.1007/s00520-021-06282-x

Yusufov M, Zhou ES, Recklitis CJ. Psychometric properties of the Insomnia Severity Index in cancer survivors. Psychooncology. 2019 Mar;28(3):540-546. doi: 10.1002/pon.4973

Ohayon MM. Epidemiology of insomnia: what we know and what we still need to learn. Sleep Med Rev. 2002 Apr;6(2):97-111. doi: 10.1053/smrv.2002.0186

Okajima I, Miyamoto T, Ubara A, Omichi C, Matsuda A, Sumi Y, Matsuo M, Ito K, Kadotani H. Evaluation of Severity Levels of the Athens Insomnia Scale Based on the Criterion of Insomnia Severity Index. Int J Environ Res Public Health. 2020 Nov 26;17(23):8789. doi: 10.3390/ijerph17238789

Gagnon C, Bélanger L, Ivers H, Morin CM. Validation of the Insomnia Severity Index in primary care. J Am Board Fam Med. 2013 Nov-Dec;26(6):701-10. doi: 10.3122/jabfm.2013.06.130064

Kraepelien M, Blom K, Forsell E, Hentati Isacsson N, Bjurner P, Morin CM, Jernelöv S, Kaldo V. A very brief self-report scale for measuring insomnia severity using two items from the Insomnia Severity Index - development and validation in a clinical population. Sleep Med. 2021 May;81:365-374. doi: 10.1016/j.sleep.2021.03.003

chi- square is a good approach

This probably does not help you, but I thought that I would have a look at the original Student (Gossett) paper of 1918 as the test was specifically designed for (very) small samples:

"if our sample be small, we have two sources of uncertainty: (1) owing to the “error of random sampling” the mean of our series

of experiments deviates more or less widely from the mean of the population,

and (2) the sample is not sufficiently large to determine what is the law of

distribution of individuals. It is usual, however, to assume a normal distribution,

because, in a very large number of cases, this gives an approximation so close

that a small sample will give no real information as to the manner in which

the population deviates from normality: since some law of distribution must

he assumed it is better to work with a curve whose area and ordinates are

tabled, and whose properties are well known. This assumption is accordingly

made in the present paper, so that its conclusions are not strictly applicable to

populations known not to be normally distributed; yet it appears probable that

the deviation from normality must be very extreme to load to serious error. " My emphasis

" Section X. Conclusions

1. A curve has been found representing the frequency distribution of stan-

dard deviations of samples drawn from a normal population.

2. A curve has been found representing the frequency distribution of the

means of the such samples, when these values are measured from the mean of

the population in terms of the standard deviation of the sample.

3. It has been shown that the curve represents the facts fairly well even

when the distribution of the population is not strictly normal." Again my emphasis.

The are several examples with a sample size below 10 in the paper.

When I used to teach this stuff (1st year geography students), I would demonstrate the Fisher Randomization and permutation test for very small numbers as the students could do this by hand and thereby see the underlying logic of the test. I would show that you could permute the data of the two variables under the null hypothesis of no difference and see how extreme a result you could get 'by chance' and then compare the observed value to this; no normality assumptions were needed in coming to some sort of judgement.

You may try this by using Chart Builder.

NEW FILE.

DATASET CLOSE ALL.

GET FILE "C:\SPSSdata\bankloan.sav".

DATASET NAME raw.

* OMS.

DATASET DECLARE logistic.

OMS

/SELECT TABLES

/IF COMMANDS=['Logistic Regression'] SUBTYPES=['Variables in the Equation']

/DESTINATION FORMAT=SAV NUMBERED=TableNumber_

OUTFILE='logistic' VIEWER=YES

/TAG = 'logistic'.

LOGISTIC REGRESSION VARIABLES default

/METHOD=ENTER age employ address income debtinc

/PRINT=CI(95)

/CRITERIA=PIN(0.05) POUT(0.10) ITERATE(20) CUT(0.5).

OMSEND TAG = ["logistic"].

DATASET ACTIVATE logistic.

COMPUTE Vfilter = Var2 NE "Constant".

FILTER by Vfilter.

VARIABLE LABELS Var2 "Variable".

* Chart Builder.

GGRAPH

/GRAPHDATASET NAME="graphdataset" VARIABLES=Var2 MAXIMUM(Upper)[name="MAXIMUM_Upper"]

MINIMUM(Lower)[name="MINIMUM_Lower"] MEAN(ExpB)[name="MEAN_ExpB"] MISSING=LISTWISE REPORTMISSING=NO

/GRAPHSPEC SOURCE=INLINE.

BEGIN GPL

SOURCE: s=userSource(id("graphdataset"))

DATA: Var2=col(source(s), name("Var2"), unit.category())

DATA: MAXIMUM_Upper=col(source(s), name("MAXIMUM_Upper"))

DATA: MINIMUM_Lower=col(source(s), name("MINIMUM_Lower"))

DATA: MEAN_ExpB=col(source(s), name("MEAN_ExpB"))

COORD: rect(dim(1,2), transpose())

GUIDE: axis(dim(1), label("Variable"))

GUIDE: axis(dim(2), label("Odds Ratio & 95% CI"))

SCALE: linear(dim(2), include(0))

ELEMENT: interval(position(region.spread.range(Var2*(MINIMUM_Lower+MAXIMUM_Upper))),

shape(shape.ibeam))

ELEMENT: point(position(Var2*MEAN_ExpB), shape(shape.circle))

END GPL.

Good luck

It sounds like you already tested your hypothesis with the linear model and used up your p value. So, you can't do more p-value tests.

As far as choosing a model, that isn't a question for the stats folks, but is related to the theory you have (stats folks can say how to implement or approximate the model).

Cohen's d isn't a test. It is a measure of effect size.

It allows you to express the difference between two groups in terms of the naturally-occurring variation in the thing you are measuring. The variation is measured by using information from both groups and pooling it as the pooled standard deviation.

The trouble with Cohen's d is that people tend to convert it to tee-shirt sizes – small, medium, large. This seems very vague when you go to the bother of doing all those calculations, somehow. And studies looking at typical values of d in different research areas suggest that it's not appropriate to have the same definitions of small, medium and large for all disciplines. I have a reference somewhere that I can post if I locate it!

How can I found out X ( microorganisms concentration- biomass concentration) and S ( substrate concentration) from blackwater characteristic data? I mean, COD or VSS does represent these?

Frank T. Edelmann yes sure, thanks for your good discussion. we create this journal based on the other multidisciplinary journal, as example Journal of King Saud University - Science (scopus indexed) publishes peer-reviewed research articles in the fields of physics, astronomy, mathematics, statistics, chemistry, biochemistry, earth sciences, life and environmental sciences.

Other example, PERIÓDICO TCHÊ QUÍMICA (scopus indexed) also publishes peer-reviewed research article in same fields with our journal.

http://environmentalcomputing.net/introduction-to-mvabund/

As I understand, the cells were randomly distributed over the 6 different concentrations, but each cell was treated for 12, 24 and 48 hours. If this is correct, then you have a Between (6 levels of Concentration) x Within (3 levels of Time (hours)) design. First run a factorial repeated measures ANOVA. If the there are sign. effects (of Concentration, Time and/or the interaction) then you can run Post Hoc tests. Before running these tests, check the distributions of the measures on the treated cells with histograms. If they obviously do not look normal, then run a non-parametric test such as the f1.LD.f1 from the parLD R package (https://www.jstatsoft.org/article/view/v050i12 )

I suppose you normalize the images of the cells like in this stdudy: https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSM1224512

In the two way table D+ D-

E+ a b

E- c d

disease odds ratio = (a/b )/(c/d) = ad/bc

exposure odds ratio = (a/c)/(b/d) = ad/bc

You can, but discriminant analysis assumes that the independent variables are normally distributed. Further, it assumes that the variances are equal across groups of the outcome (i.e., multivariate normality assumption). Fisher and Van Belle (Biostatistics: A Methodology for the Health Sciences) have a nice review. Logistic regression is a far-more flexible approach, since it doesn't make normality assumptions about the dependent or independent variables.

-p

Hi Alok

My suggestion is that, you could use one-way ANOVA or two way ANOVA followed by a Dunnettes test as post hoc test if you want to compare every mean to the control or you could use tukeys test if you want to compare every mean with every other mean using graphpad prism.

Please see

Hello,

If I well undertood your question, I think here you can use the Bayes Theorme, based on clinic litterature if you have about this specific test. we can consider the ultrasound results as a priori information about the clinic test, and the Gold-standard one as the real data (x).

I can reformulate your problem as :

If your test results come back positive when using ultrasound method, what are your chances that children actually have the disease (with G-S method) ?

CHELLAI. F

Dear Muhammad Adnan

If you are to follow up your cases for the two scoring methodologies over a certain period of time to see mortality then i guess you must have in mind certain number of patients being followed up. This will be your cohort and i interpret it will be a cohort study. It will be then prospective in nature.

Regards

Hi there,

In principle your calculation will tell you if the fold change in the mutant is higher or lower than those in the WT. You may loose important information about significant of such a change though. If you have a bit of programming knowledge this is what I would do. You have 4 experiments and 3 replicates for each. Calculate your log2(ddCT_MUT/ddCT_WT) as you did and then for 1000 times randomly shuffle the values of the expression of A among all the 12 groups. Each time calculate the log2(ddCT_MUT/ddCT_WT) and store the result. You should get a distribution of values that you can use to compare with your initial one. If your initial value is higher (or lower) than 95% of the random values then there is probably a signal there. ........ just an idea!

cluster.stat function in R coding

Yes, I think it is that simple. It would have been a great help if you had posted the graph.

I presume you are doing indirect standardization by age only, comparing the incidence of any second primary malignancy in those with a diagnosis of lung cancer to the incidence of any primary malignancy in the general population.

You appear to have an identified cohort of lung cancer patients.You know which of the patients developed a second primary after they developed lung cancer.The person years at risk would be calculated from the time of first cancer. The SIR for any second primary would be actual number od second primaries/ expected no of these as a first primay. The expected number would be calculated from the average age specific incidences for the general population over the period 1990-2014.

assuming the general population is quite large (eg national) , the age specific rates for all cancer combined are unlikely to have varied much between 1990 and 1999. So you can use the 1999-2014 general population age specific rates for the whole follow up period 1990-2014.

Although the question is few years old, I’ll add an answer just in case someone finds this relevant.

For comparing between treatments, you need to fit single curve for each individual light curve measured, not a single curve for a treatment, as your graph above implies. Then you can extract the parameters for all the curves and compare these between the treatments, as suggested by Denis.

There is a nice R package called “phytotools” (Silsbe & Malkin 2015) which I have used and found very convenient. It has four different light curve models: Eilers & Peeters 1988, Jassby & Platt 1976, Platt, Gallegos & Harrison 1980 and Webb et al. 1974. The manual also provides the equations for the models.

As the functions for fitting are provided in the package, fitting the light curves is straightforward. The package has basic programming examples also. One needs to do a bit of coding to loop through the data containing the curves and fitting a model. Each of the measured light curves needs to have a unique id. When looping through the ids, you need to extract the parameters (e.g. alpha, beta, so on) from the model fitted, along with the info of your treatments for that particular curve into a data frame.

After this it should be easy to perform the analyses e.g. with ANOVA or something else, depending whether your data shows heterogeneity etc...

You can use the meta package in R.

First step, perform the metaprop command in an object (e.g. metaprop_results). The metaprop function needs three arguments: the number of events per study (event), the study population (n), the dataframe in which they are stored (df).

metaprop_results <- metaprop(event = your_events, n = your_n, data = df)

Second step plot your results as forest plot

forest(metaprop_results)

The objective (i.e. what is wanted to know) is not clear. It is required to describe the objective clearly and specifically.

The data, shown in the table, is a kind of time series data. It may be possible to extract information from the data. However, the size of data is small. It will be convenient if more data are collected.

Hello Mohsen,

Generally, yes … all variables being binary. However, I would argue (as you have in your opening question) that you need to transform your (logit) coefficients into probabilities. This helps in discussing and comparing the parameters in a way that the reader can understand. It’s next to impossible to communicate different types of non-linear parameters (like logit, cubic, probit, etc) in a write-up and make sense of them.

So, I think you are on the right track with transforming the regression parameters into probabilities. Keep in mind, that you don’t really have to break them down into all possible groupings. Each transformed parameter, is the increase/decrease in probability for the DV for that independent parameter =1 (assuming your binary variables are 0 or 1), HOLDING all other variables in the equation equal.

However, because these predictors have covariance (as one would expect), the probabilities are not simply additive (e.g., the probability for x & y =1 is not the same as adding up the probabilities of when only x=1 and separately when y=1; in your example at the top).

The odds ratio is a better way of showing the magnitude of for each independent parameter.

I have added a spreadsheet for doing the logit conversions, in case it is helpful.

Wishing you well,

Hi Akila,

You can use a model which best fit to your data. For example if your data have exponential trend and we want to use linear trend then we can get misleading results. Look at your data graphs you can get a lot of information to decide which trend line is best for your data.

You can find some basic information from the link below:

All the best

NMA, sure.

In the first instance, you should be trying Poisson regression, with the population count in each area as the exposure variable. Counts are not normally distributed but tend to follow a Poisson or negative binomial distribution.

You can try the following tools called "venny". Cheer~

As the others point out, it is possible to incorporate an ordinal variable. It also seems you are interested in whether you can treat it as continuous and that is another matter. There are several considerations.

1) Continuous scales assume equivalent differences between intervals on the sale. This might not be valid if all you have is the rank order from the ordinal scale. For example, can you reasonably assume that the distance between "agree" and "strongly agree" is the same for all respondents to your questionnaire?

2) I would also consider the purpose of the study. How high are the stakes in your results?

3) How has the scale been treated in your discipline? It is very common in social science research to treat ordinal scales as continuous. What about the scale you are using? How have others treated it?

In the strictest sense you have an ordinal scale. On the other hand, what are typically truly ordinal scales are often treated as continuous in social science research. Though it's not often explicitly discussed, I think one should be explicit about the treatment of the scale as continuous if it is novel in your field. Knowing nothing about your research, to be precise I would treat the variable as ordinal and, if you are curious about that particular scale, conduct research on how reasonable it is to treat it as continuous if such research doesn't already exist for your scale and population. Stevens (1946) is a starting point on this issue.

Best wishes

Experimental unit: This is the (field plot/animal/gear/whatever) to which "treatments" are applied. Treatments can be directly applied (like a dose of insecticide to an insect) or they can be observational (sex, weather, disease). If you randomize, you typically randomize the experimental units. Note that there are some additional terms: subsample, techincal replicate, pseudoreplicate. These three are terms used when multiple samples are taken from a single experimental unit.

Replicate: A replicate is one experimental unit in one treatment. The number of replicates is the number of experimental units in a treatment.

Total sample size: My guess is that this is a count of the number of experimental units in all treatments. This is not very informative, and leads to trouble if the design is unballanced. Afterall, I could say that my total sample size was 100 in four treatments. Sounds good, unless I reveal that one treatment had 60 replicates, one treatment had 30 replicates and the other two had five each.

Treatment size: I am not sure, but I would guess that this is the number of experimental units in each treatment.

Recognize that a huge variety of people use statistics. They may not all use all of the same terminology in exactly the same way.

The problem with examples is that they can be aranged in a multitude of different ways. Methods sections often don't provide sufficient detail to replicate an experiment. In your example there is a treatment (herbicide) and a control (water). I want to evaluate herbicide resistance in 6 genotypes, 20 plants per genotype.

1) I place all of the plants in one box. I mix up a tank of herbicide, and spray half the box. Technically, this is one replicate, no matter how I process the plants.

2) I place 12 plants in a box. Half get sprayed. This is two plants from every genotype. I still made up a single tank, so this is one replicate.

3) I place 12 plants in a box, just like in 2. However, I mix up a new solution each time. If I am really good, I might use a different sprayer/nozzles. I will end with 10 replicates.

People often argue that I really don't care about the sprayer effect, nor the effect of the person using it. So I will go with plan 2 (above), and treat this as 10 replicates. This is great so long as you assume that the sprayer that you are using is typical, and no errors were made in mixing up the herbicide. With those untestable assumptions holding true, this is fine.

In this case, it may be useful to calculate, not only the median, but a set of percentile points; for example: 0.1%, 0.5%, 1%, 5%, 10%, 20%, 50%, 80%, 90%, 95%, 99%, 99.5%, 99.9% points.

I still do not know if you just want to set some benchmarks or if you have additional research goals.

With this amount of data, I do not think it would be useful to apply inferential procedures to either means or medians.

Yes, this is correct, as you want to see systematic differences in miRNA expression between groups.

You save money and you pay for this by a slight loss of statistical power and the inability to identify individual profiles (so it is impossible to identify outliers, cluster individuals, do a subgroup analysis etc).

I suggest not to pool more than 5 individuals, so that you have 10 pools/group.

Pooling all the samples (of each group, getting 1 pool/group) is usually not a good idea because then you would loose any information about the variability so that you can not judge the statistical significance of your results (you will have *no* statistical power). However, if you need to screen a large number of miRNAs it might be ok to run a first round on such "total pools" to pick the miRNAs that showed the largest change in a second round with pooling fewer individuals (5, say) to check the significance. If this is a preliminary experiment to select candidates for further experiments, significance may not be important and you can directly proceed with the experiments testing the biological relevance of your selected candidates.

Thanx Jochen. Have a nice weekend.

Sorry but none of these answers meet the question.

Why hsa-miR-4454+hsa-miR-7975 in NanoString gives rise to high counts. However, if you perform RTqPCR or miRNASeq this microRNA is not detectable.

Sorry, Ali but I do not have an answer yet.

Hi Mrigaya,

I would certainly not split the analysis in multiple data frames. DESeq will estimate some negative binomial parameters based on your data, and the more data you provide, the more reliable the estimates. Also, you want to work with the same estimates for all comparisons.

There are ways of putting the 3 comparisons into one model but the specific details depend on which contrasts you are interested in. I wouldn't know the exact syntax, but I'm sure that the community can help you if you specify precisely what you want to test. 

By the way: there are many good software packages for the detection of differential expression. For more complex designs, I personally really like EBseq, which has a very straightforward syntax to test multiple hypothesis. Just saying.

Good luck!

Rik

Dear Martin,

if you have the information about the real size of the total target population (in your case Venezuelan mestizo population .....) you can calculate the  research sample size here:

But if you have don't have info about the size of this specific population than your research sample would be purposive and than the number of the respondents is not strictly defined (as much respondents as you can reach is ok)

In both the cases if the differences of the genes is important for the research than you should have (nearly) equal number of the respondents from each category (min 30 respondents from each sub-sample)

Hello Paz Aranega Bou.

When the unit of sampling can not be tightly controlled, standardization (the samples by total) may be neccesary (Clarke and Gorley 2006), this would turn abundance data to values of relative abundance (percentage).

Clarke K., Gorley R. 2006. Primer v6: User Manual/Tutorial. PRIMER-E, Plymouth, UK, 193 pp.

A priori power analysis is intended to avoid studies that can't address their primary question or studies that waste precious resources by being larger than they need to be. They also force you to define your primary question and think about clinically meaningful effect sizes. In that sense, I regard them as mandatory. As a result, I tend to reject grants or papers sent to me for review that don't include a sample size calculation. That said,  most statisticians eschew a posteriori power calculations, since once the study is done, you either saw the effect or didn't. For the latter, it's a matter of taste and philosophy.

The easiest way would be through a linear regression (you can use R, SAS, even MS Excel). All what you have to do is to code your alleles as 0-1-2 {AA, Aa, aa}, then regress the phenotype over that SNP. Simple like that: The beta coefficient is you effect of an allele substitution.

For dominance, the procedure is the same, but you code you alleles as 0-1-0 {AA, Aa, aa}. You may want to consider fit both additive and dominance effect at the same time. In which case, the regression coefficient of dominance indicate what kind of dominance this locus display (under dominance, over dominance, complete dominance, etc.).

Please be more specific about your question. Beyond that, there is no probably a "best" method, but just more appropriate.

Yes.

If you have (for instance) several measurements from per dish, several dishes per cell extraction, several extractions per animal, then you may use a multilevel (hierarchical) model that can give you the additional information at what level of (technical) replication the most variance is introduced (that may hbe halpful to plan further experiments). For your final scientific question, however, it will (should) give you the same amount of information that is obtained from the averages per animal.

Ok, then it is not "repeated measures". The best option still is a logistic regression, just that you do not need to consider "animal" (neither as random nor as fixed factor).

depends on data which transformation to apply. available in standard texts

Your best bet is to covert everything to cumulative density functions. Easy if you know the distribution (Poisson, weibull, or whatever and its parameters). The Kaplan-Meier plot is essentially a non-parametric CDF. See. e.g., this link

Stephen,

   What an interesting idea, and especially cool if there is sufficient data to support using the method. The article can be found at:

One great thing about RG is that you sometimes find articles from other disciplines that you would never ever find otherwise.

I concur with Paul Chiou on this.

Yes you can use BEST with a single response variable.  For a single response variable you use Euclidean distance to make the 'target' matrix, then search for the subset of (transformed) explanatory variables that gives the best match (with whatever the appropriate resemblance measure for those variables is). Other methods are available (as alluded to in previous answers) but each has its own set of assumptions.

Yes, I think ANOVA is the correct method with treatment as the fixed effect.  Random effect would be reps nested within treatment.  Variation among samples is the residual term.  

Alternatively, you could average the samples first and then the model would only include the fixed treatment effect.

In either case remember to verify the assumptions of normality and homogeneity of residuals before making any conclusions about the treatment effect.

Just wondering, how large was each EU?  And the distance between EUs was 25 cm? That doesn't seen very far apart.  Why 10 samples per unit?  Were you expecting a lot of variation within each unit?  You many want to consider using more reps and fewer samples in the future.  With only 3 reps the inference space is rather limited.  Were the samples taken all at the same time or over time?  If over time, then you have a repeated measures ANOVA which is a bit more complex to analyze.

It sounds like you have only one response and 6 treatments. In this case a one way ANOVA could be used providing the data accomplish the assumptions of this test, but more details about the experimental design is necessary to give you a better and more accurate answer.

Dear Tara Stirland,

These are some of the links useful for you.

I hope this will help you.

All the best !

I discriminate 6 different type ordinary datasets and six microarray datasets using 10 discriminant functions. Download my papers after 2012.

You can estimate the standard deviation of the ratio using propagation of errors. See the link below and.see the "Multiplication or Division" row in Table 1.

Hi Umar,

If there is homogeneity, you can just add them up. since there is a confounding variable, traditional method will definitely lose some information if you just add all the k contingency tables together to make a 2 by 2 table.

i recommend Mantel-Haenszel method.

 μk = E(n11k) = n1+kn+1k/n++k

νk^2=Var(n11k) = n1+k*n2+k*n+1k*n+2k/[n++k^2(n++k −1)]

the statistic is [sum_k_(n11k)-sum_k_(μk)]^2/sum_k(νk^2)

hope this will help you.

If you want SAS code you may refer to http://www.stat.ncsu.edu/people/dickey/courses/st711/notes/Notes_pdf/Alpha%20Lattice%20designs.pdf or if you want R code you may refer to https://cran.r-project.org/web/packages/agricolae/vignettes/tutorial.pdf

What are you interested to compare? The species number and the species composition among  the zones? If it's the case a good start would be read:

Marti J. Anderson 2001. A new method for non‐parametric multivariate analysis of variance and

Chao et al 2014. Rarefaction and extrapolation with Hill numbers: a framework for sampling and estimation in species diversity studies

Hi Hardik

Form your description, a linear fixed effects model would be suitable for your first question, and a linear mixed effects model for your second. 

Another way to compare two slopes from two different groups is to use the bootstrap. Bootstrap each group, generating the two slopes. Do this 1000 or 10,000 times saving the results. You will have a distribution of the slopes and how often they overlap. Probabilities can be calculated from the distribution. Wish you well.

Perhaps this paper will help?

for comparison of two groups independent sample t-test and willcoxon signed test or mann whitney nonparametric test for continuous and categorical data  is more appropriate..

The "enrichment" of essential gene in a set can be tested with Fisher's exact test (or the chi-squared test as approximative test). This is alos called "overrepresentation analysis".

If you wanna go Bayesian you can use an hierarchical Bayesian model considering mixed effects

These kind of models behave pretty fine with missing data. For instance, if you haven't other variables besides time, you can consider a transformation of the response as Jochen Wilhelm said, e.g. logit. In this case you can consider that y represents the transfomed data and $y \sim N( m(t), \sigma )$, with $m(t) = \beta_0 + \beta_1 \times t + b_0 + b_1\times t$ being \bf{b}=(b_0,b_1)\sim N(0,\Sigma) a vector of random effectas and the relation is linear. But if you suspect that the relation is not linear you can add t^2 and t^3 and the respective random effects. From my point of view this is the simple strategy. For a complex one you can think about splines.

Thanks a lot Paul, very helpful!

-David

Stands limits are based on mean of these two ratings (generally two measurements, assessments) and standard deviation (SD) of this mean defined as

(mean - 2 SD, mean + 2 SD).

More precisely 1.96 SD and not 2 SD.  

But you could additionally define something as "logical agreement limits" which are not defined statistically but which are coming from definition of the task. 

Hi Bjorn,

There are many ways of answering this question. And of course you can do this yourself. The internet has made our lives much easier nowadays. If you have ArcGIS or any other GIS software it can be done by just typing some keywords on the web. Moreover, there are thousands of online youtube videos that can assist you further. I have the following suggestion for you to give you a head start:

1. After you start ArcMap there is a possibility to add a Basemap Layer. See if you can find Antarctica base Maps.

2. Once you have the base map, you can import the coordinates of the sampling points from an excel sheet and convert it into a point shape file.

3. Draw a buffer around each point. This is an automatic process.

The rest I think you can figure out yourself. Check the links that I have attached. 

However, the questions that I have for you are these: Are you interested in any specific dates or have you collected any satellite imagery of your study area in Antarctica? The area which you are studying, are they covered by ice all the year round or do they change with seasons?