Hello Uchenna,

I would like to help you, but my assistance might be too limited because I am not able to get a sufficiently clear idea about your investigation.  My first impression / recommendation is that you should keep each of your three studies separate from each other, however, or you will end up with all kinds of complications.  (I think, from your description, that you essentially have three studies.)

if we take just your first study, it seems that you want to make 80 comparisons:  i.e., comparing the three groups on each of your 80 days. If I understand the setup correctly in assuming this, you could theoretically conduct 80 one-way ANOVAs but that would be highly irregular, even improper, because you are likely to attract Type I errors in so doing - i.e., conducting so many analyses that some of them are likely to be statistically significant merely by chance.  I think you should determine which points in time are important to analyse, and then just do ANOVAs across your three groups for those points in time, perhaps making some kind of Bonferroni-type adjustment to guard against Type I errors.

One other thing to consider is whether you took baseline measures of A, B, and C. Ideally that should have occurred in case they were different to start with.  If you had done that, you might consider using ANCOVA to analyse your data - but if you use ANCOVA you should ensure that your data satisfy the necessary assumptions, and I think you should analyse your data at only a very minimal number of time points - e.g., at the start versus in the middle and at the end..  

SPSS should be able to provide the necessary wherewithal to conduct the analyses, but you should ensure that you are doing the right thing within it.

If other people would like to make suggestions that are more advanced that I have written above, or even contradict my advice, please feel free to do so.  I am trying to give Uchenna at least something to work with and think about.

Robert.

Thanks very much Robert

Hello again Uchennae,

First up, sorry for mis-spelling your name in my post above.

I have been thinking about your research since that post, and am wondering whether you might also consider using several one-way repeated-measures ANOVAs, for example one set of these analyses for Situation 1A above, another set for Situation 1B above, and so on. If the results for those were significant, you could go back and do post hoc comparisons with appropriate adjustments to avoid Type I errors.

One problem I see with your analyses is that you could end up doing a lot of "fishing" - i.e., conducting a lot of analyses and then deciding that the results that turned out to be statistically significant were the ones you had been interested in investigating at the start of your investigation.  This is not good science!  Because of the sheer amount of data that you have collected (if I understand your investigation correctly), there is a temptation to do this.  You should resist this temptation.

There might be some other more sophisticated form of time-series analyses that you could do, but I am afraid they are beyond my level of expertise (which certainly has its limits).  As I wrote in my first post above, I would not resent it if other people recommended something different from what I have done.

Much depends on what your initial research questions are.  It is really important to be clear about them before you decide on which statistical analyses are most appropriate.

Furthermore, there might be different ways of conducting analyses so that you can address the research issues from several different angles, each of them useful for providing different insights.

Robert.

Thanks again Roberts. My research questions seeks to find out if the addition of X and Y to the reactors has statistically significant effect on the mean volume of gas produced. Ordinary, from start i.e. day 0 - 25 are usually unstead, but any time above day 25 and beyond could be used to judge this effect.  Your suggestions have been so great. I will try them out.  However,  I have also attached one of my excel workbooks to enable you see the kind of data am talking about. Many thanks. Uchenna

If you look at the Experiments separately, each is basically an unreplicated complete block design, where each day is a block, and each reactor is a treatment.  This could be analyzed with traditional ANOVA; mixed effects model; or with the non-parametric test Quade.  If you are using anova, make sure you are doing the F-test appropriate for unreplicated block design.

It will probably be better to combine the three experiments. But it also matters if Day 1 in Experiment 1 is similar to Day 1 in Experiment 2.  That is, does it make sense to assume that Day 1 is e.g. always the lowest no matter on what actual date the experiment was performed?   Or does the weather on the date matter more?

Hello, Salvatore,  Thanks for your help.  Day one is usually the lowest in all cases. When each experiment is started, the production starts gradually and increases with days untill it gets to steady state say usually from say day 25 -80. The gradual increase in all cases are shown in the excel sheet I attached.  Many thanks.  Uchenna

Hello again Uchenna,

First up - again - I don't know how to spell your name, so I hope either way is OK.

Thank you for making your Excel data available.  That has helped me (and perhaps others) get a better picture of things.  It would appear that, for each condition, you have only one reading on each day.  Is that correct?  Please let us know. If so, my earlier advice is awry. I had been assuming that you had a number of readings for each condition on each day, but that now appears to be incorrect. In order to conduct an ANOVA you would need a number of readings on each day, not just one, within each group.

Could you, therefore confirm what the nature of you data was please? Was there really only one reading per day for each condition? It would also help if you could indicate exactly what your research questions were.  That might help us to help you more.

At the moment, I feel that there are problems if you have only one observation per day for each condition, but there might be some way of overcoming that problem.  I need more information first, however.

Robert

Hello Robert,

The experimental procedure involved measuring the volume of gas produced in each reactor on daily basis for 80 days. Thus, for each reactor, I have 80 readings corresponding to 80 days measurement. Same applies to all other reactors.  None was done in duplicate for any of the experiments.  The only unifying factors were the supplements X and Y added to the different experimental runs.  Note: it is also desirable to compare the effects of the additives on each reactor after ever 20 days. The pure explanation goes like this, assuming that we have three men from UK(a,b,c),  USA (e,f,g) and Canada(h,i,j) representing different ethnic groups in this study and we are required to find out the effects of administering a new drug X on b,f,I and another drug Y on c,g,j over a period of 80 days by measuring the volume (m3) of urine produced by each man on daily basis!.  My experiments seek find out (1). the effect of these new drugs on these men countrywise compared to the control based on the mean volume of urine produced over 80 days and also (2) compare the effects of the drugs X and Y across the 3 countries (representing 3 races) using the mean of the volumes of urines produced by each man for a period of 80 days. Assume that people from each of the countries are distinct and genetically different.  Thanks.  Uchenna

The unreplicated nature of the experiment shouldn't cause problems.  But it may be challenging to build the correct model.

I've attached a file with R code for the first 10 days, as a possible example.

For this example I followed a split-plot design approach, treating Grass (which is Experiment) as the plot, and Day as the subplot.  This doesn't really capture the repeated measures nature of the measurements.  It is a mixed effects model.

I'd be happy to hear any other suggestions for an appropriate model.

NOTES ON ATTACHED R FILE:

If you have access to R, it's worth running my file to see sample results and sample plots

The code will install several packages.  If you have fast and unlimited internet, it will take only a minute.

Thank you very much Salvatore.  I sincerely appreciate your help.  I try it on R, although I am not am expert with R.  I will let you know how far am able to go with the codes. Many thanks once again.  Uchenna

Hello Uchenna,

Thanks for the extra information.  I think I'm working on a different planet from Salvatore in terms of how we are conceiving of your research, and he certainly seems to be more sophisticated than I am in terms of statistics.  I won't feel at all hurt if you decide to pursue what he is suggesting!

However, in case some things that I suggest are helpful, I'll continue.  As I had begun to suspect, you are really dealing with only a sample of 1 for each of your conditions.  (Your parallel example of men and urine helped. Thanks.) That creates problems in terms of quantitative stats because they rely on samples of more than 1.  ANOVA, for example, is partly dependent on the variability WITHIN each group, but you can't have variability within a single person / thing.

I could be getting things wrong (and I'm quite happy for you or others to indicate that if so), but one thing that is occurring to me to get around this problem of your N - 1 situation is to reconceive the "design" of your investigation (I think it's probably too late to go out and get a lot of extra observations).  I notice from your Excel file that there is some variation in the results from day to day.  This is inevitable in real world situations.  I wonder, therefore, if you could/shoulc choose, say, three broad points in time and regard those points as having several observations within each of them. That would bring your N up. For example, you could take your first 10 days of intervention as comprising 10 samples (it might be best to omit Day 1).  Then you could take Days 25 to 34 as comprising another 10 samples, and another 10 days at the end (Days 71 to 80) as a third set of samples. Then you could perhaps do 3 one-way ANOVAs (one for each of the 3 periods) with 10 in each group, or a single 2-way ANOVA in which you used the period as your second independent variable.  It seems that you want to regard each "country" / feedstock separately, so you could repeat each of the above separately.  To bundle them all into a single analysis might create difficulties for interpretation.

Does that make sense?  If it does, you would need to add some codes to your data. 

How do you intend to analyse your data?  Using Excel, or perhaps SPSS or some other stats package? I ask because I'm an SPSS person for a number of reasons (including the flexibility and extent of its options and the clarity of its output), and in their current state the data in your Excel spreadsheet would need to be altered a little bit to make it suitable for analyses in SPSS. That would be easy, and Excel data can be transferred across to SPSS easily.

Robt.

Hello Robert, 

I use SPSS a lot.   Your suggestions and that of Salvatore have been very helpful because Salvatore also broke the data down in groups of tens.  I have run the codes he generated with R but like you said, interpreting the results is a bit difficult for me because I am not yet 'an expert' in using R though an doing some trainings on it already.  I will be attaching the results from the script I ran so Salvatore could see if I did it properly.   Obviously, the main issue I have is to be able to statistically show the effects of my supplements in each experimental run (same grass with different additives, X and Y), and secondly accross the different experiments (different grasses but same additives X and Y).  Grouping each data in say tens or 20's like you both suggested is perfect since I also need to know compare the variability in the mean volumes along different sections in the time series to be able to establish where the experiment reached steady-state conditions.  I will try to that with SPSS and then attach the outcome too.   I  sincerely appreciate your invaluable help to me.  Many thanks. Uchenna

Hello Uchenna,

I'm delighted that Salvatore and I had made similar recommendations even though we were coming from different angles - and I'm also pleased that you are familiar with SPSS.  It will be interesting to see if R and SPSS produce the same outcomes as each other.

Enjoy your work!

Robt.

Uchenna and Robert,

A few points:

1) I am _not_ suggesting that the data be analyzed in chunks of 10 days.  The analysis I shared had 10 days only because I was too lazy to copy over all the data, and gives an idea of one way to analyze the data.  You _can_ look at the data in groups of days, but I don’t see any good reason to do so.

2) The model I used is similar to a standard analysis of variance.  I used a split plot design * , which is a traditional design used in agriculture.  The other complication is that I used a mixed effects model.  This was just because (for me) it’s easier to use a mixed effects model than the ordinary fixed effects model for the split plot design.

2b) You can look up literature on the split plot design for further information on this analysis. You should be able to do this in SPSS.

2c) If you are not comfortable with this design, you might just start with an ordinary general linear model anova, with all factors crossed.  This will be less appropriate for the design of the experiment, but should give some insight in to what's going on with the data.

3) Robert, of course you need replication for an analysis of variance, but you don’t need replication within every cell.  Uchenna’s design can be seen as an extension of an unreplicated complete block design.  This design is analyzable by analysis of variance.  Try to analyze the attached data.  Include the effect of Treatment and Block; just don’t include the Block x Treatment interaction.

4) When determining the time for steady state, I wouldn’t look at the data as groups of days either.  Why would you?  You might look at each Treatment x Grass separately, or you might look at all Treatments and Grasses together.  In either case, you might look at linear plateau model or quadratic plateau model (http://rcompanion.org/handbook/I_11.html#_Toc459810570).

* I don't think this analysis is perfect for the way the data were collected.  One thing is that in the analysis I am treating Day 1 in Experiment 1 as the same day as Day 1 in Experiment 2.  That is, I am treating the data as if there were nine reactors all running at the same time.  I'm assuming that this is a reasonable assumption.  But let's think about a different scenario.  Imagine that the experiments were run outside, and that the temperature affects the amount of gas produced.  And let's say the average temperature was 15C in Experiment 1 and was 20C in Experiment 2.  Then, obviously my assumption would be invalid.  The problem with accounting for this is that Grass 1 was used in Experiment 1 and Grass 2 was used in Experiment 2.  So, you would never know if the difference in results for Experiment 1 and Experiment 2 were because of the difference in temperature or the difference in grasses.  So, I'm assuming that the experiments were run in controlled conditions where there are no relevant differences between Experiment 1 and Experiment 2 except for the grasses used.  [Keep this paragraph for reference when questioned by reviewers.]

Hello Uchenna and Salvatore,

Thanks for getting back and clarifying what you were and were not recommending, Salvatore.  What you are recommending, and the way to go about it in Uchenna's case, is beyond my knowledge of statistics, so I think I should bow out!

Every good wish with your work, Uchenna - but if I might be able to help any more in any way, please don't hesitate to draw me back into the discussion.

Robt.