Your question seems to be, Did the treatment increase the success rate? 

What increase is necessary to say, That the cost of the treatment is worth the increase?

Knowing a necessary yield should help designing an analysis method.

It is not clear if number of eggs for a pair leads to a lower success rate. Several questions arise. Does the treatment group have more or less total eggs? Is the difference in total important. What is common for a clutch size? Why does clutch size vary?

If clutch size is usually highly variable, it could be important to sort by clutch size. Does the success rate by clutch size show a difference by treatment?

If the differences between treatment and control are small, you may not have enough subjects by clutch size to make a determination. Seeking a specialized statistical test will not help. Clutch size appears to be a major confounder. You can easily be subject to Simpson's paradox in this case.

Thanks for the insights, Joseph!

I wanted to make a few clarifications. The question at hand is whether the treatment affected hatching success. HS could have increased or decreased due to dosing. Pairs breed in cycles, each cycle producing a clutch, all cycles together producing a total number of hatched offspring, which is divided by total eggs laid (total clutch size). Large total clutch sizes could be found in both over-performing pairs (pairs with high HS) and under-performing pairs (pairs with low HS). 

The problem I was pointing out was that it seems that a binomial GzLM (which is commonly used in the analysis of hatching successes) under-represents pairs with smaller clutch sizes, when the metric of interest is simply a proportion or percentage (for example, if a pair bred for longer or shorter, i.e. had a total clutch size of 20 or 40, is not of interest). Given the HS of each pair, I would like to compare the two treatments, but don't know how to make this comparison.

You may have two questions. Does treatment affect clutch size and does treatment affect HS? This is were confounding and Simpson's paradox become a problem. Does treatment or controls have more small clutch sizes? Small clutch size might indicate better brooding conditions. See examples of Simpson's paradox to determine if you need to  consider how you represent clutch size. 

Be careful using common analysis methods. Often required or expected methods are not the best for your conditions. Reviewers will object if different methods are used so be prepared to defend why you used something different.

Thanks for the sound advice, Joseph,

I see your point about Simpson's Paradox. Though, don't you think that by having a HS value for each pair, and comparing mean HS among treatments, I will be avoiding the paradox alltogether? 

Plus, even if I knew that clutch size has no effect on the probability of hatching (as my analysis indicates), using a binomial GzLM would still give added weight to the pairs that produced more eggs (because this kind of model sums up all trials and successes in each treatment, right?). When comparing the HS of two treatments that include many pairs (experimental units) one wouldn't want that to be the case (this kind of approach wouldn't make sense, right?) - you want each pair to count equally, i.e. because you're taking a sample of all pairs that could have been exposed to these treatments.

Thanks Stephen - That definitely makes sense.

Do you have an idea for a good approach to compare HS means, which are proportions or percentages that are not normally distributed?

Please let me know if the following references/sites are helpful to you:

1.  Analysis of Proportions Data - CiteSeerX

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=44A855F806AB4BEF1FB11F781043ACE5?doi=10.1.1.25.3465&rep=rep1&type=pdf

Actual data often do not exhibit this mean variance equality. ... tion techniques for non-binomial dispersion in the analysis of proportions (or frequency, or.

2.  Advanced Statistics:Statistical Methods for Analyzing Cluster and ...

http://onlinelibrary.wiley.com/doi/10.1197/aemj.9.4.330/pdf

for analyzing clustered data and calculating sample ... for cluster data and principles for planning, analyz- ..... (means, proportions, odds ratios) for important .

3.  ANALYSIS OF PROPORTIONS (ANOP)

http://wps.prenhall.com/wps/media/objects/9431/9657451/Ch_12/levine-smume6_topic_

ANOP.pdfwill learn how to evaluate several groups of categorical data. ... larger than the overall average of all group proportions if it lies above the upper

Thanks again, Stephen, and thanks for the great article!

I think the issue of experimental units here is quite important. As Jaeger thoroughly reviews, mixed logit models have all the advantages over ANOVA. But, isn't this only the case when the experimental unit is the binomial variable (e.g. egg)? My research looks at the performance of pairs, my experimental units. Each pair has a HS value (percentage or proportion) that can be calculated for it (by dividing hatched by laid eggs per pair). 

Crucially, if you use a mixed binomial setup you are, by definition (I think), starting by adding up, per treatment, all the hatched eggs, adding up all the laid egg, and dividing, i.e. you begin with what seems to me to be an inappropriate procedure (if the exp. units are wrongly defined) that creates a bias toward pairs that laid more eggs, where what you're actually interested in is pair performance. Specifically, your estimated means are, from the beginning, disproportionately influences by pairs with more eggs. Yes, you have a random effect of pair in the model, but this doesn't change the "origin" of the mean. 

I'll give you an example of the numbers I'm looking at:

Mean, calculated by averaging HS mean of each pair for treatment B: 0.53

Mean, calculated by SAS in a binomial GzLM for trt B: 0.48

Mean, calculated by SAS in a binomial GzLMM for trt B: 0.50

The way I see it, if the experimental units of interest in my research are the pairs, then 0.53 is the most accurate representation of treatment B's performance. I'm searching for established methods to calculate CI's, significance, etc. under these assumptions. These may not exists... I still need to check out Dennis's links though (see above).

I'm here to learn, so feel free to scrutinize anything above, of course!

Thanks Stephen, sounds good. Following your advice, I tried the mixed setup and got a mean of 0.50 (see my previous post "Mean, calculated by SAS in a binomial GzLMM for trt B: 0.50"). It's a little disconcerting that it is not the same as the mean achieved by averaging HS mean of each pair in treatment B, which is 0.53. Do you have any idea why that is?

I argue that the binomial linear model setup, due to the way it works (i.e. adding up, per treatment, all the hatched eggs, adding up all the laid egg, and dividing) produces biased estimates in this case (biased towards pairs with more eggs since they contribute more eggs to the total). This is all because the experimental units of interest in my study are the pairs, but binomial GzLMM are meant for experimental units with a binomial response (eggs in this case).

Thanks!

 Ohad

Eggs is binomial and hatched eggs is binomial. Hatching success is a rate. The rates are not binomial. You want to know if there was an effect, but you do not know if the pairs were affected differently. Did the agent affect clutch size? Did the agent affect hatching success? You might observe that total number of eggs appears to be larger in the control and total successes are appear larger in the control. You might see the inverse or you might see total and success do not have the same direction. Do the observations tell you anything? That is, are the differences large enough to conclude an effect?

You could look at the data by plotting pair success against pair clutch size. Is there a clutch-size effect? Does it differ between control and test? You have noted a possible clutch-size problem. You must resolve this problem before proceeding.

Look at your data? Plot it several ways. Can you say there is a difference between test and control? How can you model that difference? Using the GzLM is forcing your data to model. You should be looking for a model for your data.

Thanks so much, guys!

I wanted to first bring up a few important facts I mentioned earlier:

- Clutch size does not differ among the treatments (i.e. average clutch size of the groups is very similar, and the model shows no effect of clutch size on HS).

- My main concern in this post is that I am (and occasionally others are) forcing the data into an inherently inappropriate model, and so I'm seeking alternatives.

- Clutch size can vary for many many reasons. These may include the time the pair was in the experiment, the genetics of the female, the health of the pair, the time it takes the pair to raise young - these are generally unrelated to treatment. Importantly, this is not clutch size for a single breeding cycle, but a total clutch size and total hatches, over several cycles, these are cumulative numbers, e.g. 12 hatches from 19 eggs, the product of 4 breeding cycles.

Stephen - The means I'm reporting from the binomial models are least square means already converted from the log odds estimates into the "real world".

Joseph - There is a large difference between dosed and control, but the problem is that the means are, as I'm arguing, unreliable, for example, looking at the dosed treatment:

Mean, calculated by averaging HS mean of each pair for dosed treatment: 0.53

Mean, calculated by SAS in a binomial GzLMM (random effect of pair) for dosed treatment: 0.50

Why are the means so different? Quoting myself here:

I think the what is happening is that if you use a mixed binomial setup you are, by definition (I think), starting by adding up, per treatment, all the hatched eggs, adding up all the laid egg, and dividing, i.e. you begin with what seems to me to be an inappropriate procedure that creates a bias toward pairs that laid more eggs (number of eggs varies wildly and is not related to treatment), where what you're actually interested in is pair performance. Specifically, your estimated means are, from the beginning, disproportionately influences by pairs with more eggs. Yes, you have a random effect of pair in the model, but this doesn't change the "origin" of the mean. 

What do you think?

The way I see it, if the experimental units of interest in my research are the pairs, then 0.53 is the most accurate representation of treatment B's performance. I'm searching for established methods to calculate CI's, significance, etc. under these assumptions. These may not exists... I still need to check out Dennis's links though (see above).

How would I analyze mean proportions data?.

Have you plotted histograms of the mean proportions? Does it look like a normal, binomial, Poisson distribution? If Poisson, then run a Poisson regression. (Yes the pairs are the observational units.)

I would suggest you try a Poisson regression (complete code in attachment):

data eggs_hatched;

input pair treatment eggs hatched;

ln = log(eggs);

rate = 100*(hatched/eggs);

datalines;/* fictitious data*/

1 0 32144 13

2 0 28022 24

3 0 23411 44

4 0 18409 43

5 1 10329 39

6 1 12669 45

7 1 14053 75

8 1 15243 108

;

proc genmod data=eggs_hatched;

class treatment;

model hatched = treatment / dist=poisson  offset=ln;

format treatment treat.;

estimate  'Beta treatment' treatment 1 -1/ exp;

run;

I forgot the histogram (also in attachment):

proc sgplot data=eggs_hatched;

histogram rate;

density rate;

density rate / type=kernel;

run;

You might want to test the normality, and if normally distributed run an ANOVA:

proc univariate data=eggs_hatched normal plot;

var rate;

run;

proc glm data=eggs_hatched;

class treatment;

format treatment treat.;

model rate = treatment / ss3;

means treatment;

run;

Dear Ohad, I browsed through your PhD thesis and realized you probably know much more about SAS than I do.Sorry for my 'intrusion'.  Anyway, I completed the SAS program with quite a few possible ways to analyze the data and they all yield quasi the same results. I again (!!) attached the (completed) SAS program.

Hi Jos - Thanks so much for the thorough response and the SAS file! It's great! I'm by no means proficient in SAS, just know how to use what I've used before. Oh, and it's my Master's thesis, by the way :)

As far as I can tell the distribution of the proportions data doesn't correspond with a normal, Poisson or NegBin distributions (proportions are not count data anyway) (see attached histogram). But, your idea of using the count of hatched and offsetting by natural log of laid in a Poisson or NegBin GzLM setup is an interesting option. I just ran this model. Deviance/DF was high but I used a scale=Pearson adjustment. 

The LS mean in this case for the dosed treatment was very close to the one calculated by simply averaging HS for that treatment, but the mean for the control group was high. It was 0.787, where the mean calculated by simply averaging the HS was 0.740. I'm assuming this is the consequence of a poor fit of the data to the Poisson/NegBin distribution (despite using the Pearson adjustment).

Stephen - SAS produced the CI's with the command CL under LSMEANS for example. But the means are the problem, they do not correspond well with the means produced by simply averaging HS of each pair for each treatment, which are the means of interest since pairs are the experimental unit. The origin for this may be the way the binomial GzLMM calculated the raw means - by adding up all hatched, adding up all laid, and dividing, therefore putting more weight on pairs that laid more. Issue of fit may contribute to the wayward means as well.

Many thanks! Keep the ideas coming (if any are left)

You state there is a large difference betwen treament and controls. What is it? How did you determine it?

You talk of means. The discussion seems about the treatment mean. What is the control mean? You have a mean of 0.53 as apposed to 0.50. This seems small, yet, you ask why are they so different?

The mean has no meaning without a distribution and a model of the distribution. Even then, the mean is one of context. Why care about the mean? The mean is a description of something, but what in this case? Is the mean important? Is the distribution important?

You have pairs that have differences in total eggs and HS. Pairs did not all have the same time for production? What is the time/production relationship?

What is the time/production relationship for controls?

What is the treatment dose? Was dose the same for all pairs? Was dose acute or chronic? If chronic what was total dose? What was the dose for pairs enetering the study at different times?

How did dose affect HS? By altering parent behavior? By altering embryos?

In their book on 'Applied Medical Statistics Using SAS', Der & Everitt illustrate how to use summary measures for binary responses. Despite the critics on using arcsin transformation of proportions, they use such ones. In the further completed SAS program in attachment, I use such transformation and then run t-tests and Wilcoxon tests on the rates and their transformations. T-tests are preferred (over ANOVA) because they yield nice distributions plots and they correct for heterogeneity of variances. Wilcoxon is 'just for curiosity'.

It might be interesting to test if, over the 20 pairs and per treatment (control vs. dosed), the total number of hatched vs. not hatched eggs is significantly different. This can be done with a chisquare test for contingency tables.  The SAS program in attachment runs this chisquare test.

Use a count model with an offset for number of eggs.

If no other variables than your treatment (control vs. doses) are involved, then you don't need a regression model.