Maybe by adding dummy variables or using variables with lag(1), (2)

I did not use the Kaplan-Meier test. In my opinion, you thought right. There are many statistical methods in such cases.

QLR test (Quandt likelihood ration) for unknwon "breaks" in your data might help as well (e.g. check out the Stock/Watson nonstationary chapter).

You can use exact tests to compare the numbers individuals that survive under different treatments at a given time. For spawning you could also run ANVOA with time as the covariate but you may need to transform the data to meet the assumption of linearity...

Hope this helps

If you decide to use ANOVA, you must perform Bonferrroni correction for not accepting the alternative hypothesis when it is false. To choose the appropriate statistical test for evaluating the effect of the X toxic on spawning, you must investigate the potential targets (tissues) of the studied toxic. For example, it is not the same as the toxic produces male infertility or that the toxic affects female genital system and consequently, affects the eggs produced. I think that you can use the total number of eggs, number of individuals after hatching or total spawning weight at the experiment end. If you use initial and final experimental times, you can use standard statistical test.

Thanks all for your answers.

In many cases, I don't know the test you proposed, I'll search for details.

@Israel: using data at the end is efficient (in the case I'm working on right now) but if for example spawning onset is delayed but more efficient for one condition than for the other you may obtain a "wrong" result. This is why I was looking for a more general way to assess difference. I'll have a look to ANCOVA usage.

The use of ANCOVA may be useful. Obviously, I do not know their experimental design, but in an assay using specific toxic for your animal model may have different outcomes (goals) in his research (and hence, you must define that statistical tests are most appropriate). Also, If you have a expiermental design with a lot of studied variables, you coul be apply mulitvariate analysis.

*Survival: To my knowledge, Kaplan-Mayer is ok. Cox proportional hazards model was another possible solution, allowing to control for covariates.

*Spawn: The cumulated numbers can be expressed as (empirical) cumulative distribution functions. Their (in)equality can be tested with the Kolmogorov–Smirnov test. Another way was to define some few time-intervals in which the (cumulative) number of eggs is determined. This will also give you frequency distributions that can be compared with a binomial model (logistic regression, chi-squared test) or better with a proportional-odds model (ordinal regression).

I would recommend the book of Abraham Wald, "Sequential Analysis" from 1947 (Wiley) to your problem.

An alternative to statistical tests is biology based modelling where the growth and reproduction of an individual is specified by the model. Each model parameter quantifies a physiological process (growth, somatic maintenance, reproduction ...). Dynamic Energy Budget (DEB) theory (S. A. L. M Kooijman. Dynamic Energy Budget Theory for metabolic organization. Cambridge, 2010) is such a theory.

DEB theory supposes three ranges of internal concentrations of every compound: too little, enough and too much. In the case where we are studying effects of a toxic compound, we assume all compound are in the enough range except the toxic element (which we call call Q) of interest.

As soon as Q exceeds the upper boundary of the enough range, toxic effects start. Such effects are captured by modifying (a) parameter value(s).

This upper boundary of the enough range is called the No Effect Internal concentration (NEC). in the neighbourhood of the NEC is is possible to assume a single model parameter is affected; this parameter constitutes the predominant Mode of action of compound Q. Since we are in the ecotox business (chronic long term effect of comounds) this in the interesting area. As we increase the interanlized amount of Q more and more parameters can become affected.

So first DEB model parameters need to be estimated for the blank which amounts to characterizing the eco-physiological performance of the blanc. Next one must estimate the NEC and determine the the mode of action (i.e. which parameter is affected). This amounts to quantifying deviations from the blank.

There are several very strong advantages to this approach:

- the NEC is a time independant statistic

- thus such an approach takes into account that effects vary in time

- it is possible to extrapolate individual level effect to population level effects.

If you have an idea of the size of your organisms in combination with the spawning then uncertainty as to the mode of action decreases. Dilution by growth will occur for organism which triple in size during the exposure period, larger organisms will accumulate less relative to smaller organisms etc. because effects are dependant of the amount of internalized compound. Another thing: does your compound eliminate through the eggs? Antoher thing: reproduction rate is size dependant ... so if you have data for individuals better use that than averaging over means.

Another very strong advantage of biology based modelling is to be able to test all of these things and come up with interesting hypothesis, what type of data should be acquired in the future and robust extraboliations .... it is always a very a wise strategy is consider uptake - elimination behaviour no matter what; effect do depend on this. The coupling of (simple) toxico-kinetic models to indivial biology based model which specifies food uptake and use for maintenance growth and reproduction is already well worked out within the DEB theory framework.

A great website with all of the necessary information is http://www.debtox.info/

You can find on the page, a very accessible backround, methods and concrete case studies for the analysis for reproduction & survial data using DEB theory (DEBtox methodology).

I hope this helps

Best wishes,

starrlight

Hello Starrlight - You know I know (a litlle bit about) DEB theory ;)

Thanks a lot for your detailed answer. I didn't figure out DEB could be used to evaluate these variables.

Some questions/remarks… In some cases, the toxic is accumulated (and transferred to eggs) and can be measured (the easy case) but in other they are metabolized and therefore are not transferred and cannot be measured… In this latter case, how can you feed the model with toxic information? Can given dose be used?

About individual data, I know this point and we have produced such data (growth data with some rare spawning events for an accumulated toxicant) but I've neither information nor results… since (as you know) it is tedious data to acquire I will stay on my mean/group data… Can DEB be used in this case?

Bye - Xavier

The question for me is not so much if DEB can be used, but what is the noise in the reproduction data signal? And does the background noise mask the effects of the toxicant?

Spawning data are typically quite noisy; however I proposed a method to test such spawning data by using the DEB model to test if (i) cumulated investement in reproduction is affected or (ii) alternatively only the [spawning] behaviour or (iii) perhaps both?

Cumulative investment in reprodution is I think much less variable between individuals than the spawning behaviour. Of course, we get the cumulative investment by looking at the slope of cumulative number of eggs against time BUT your organism may or may not want to spawn even if he/she is healthy :-) making the whole thing very confusing to analyse.

The advantage of working with a fully parameterzied DEB model for the blank of your test species is that with limited inforamtion such as inital or final mass in combination with length you can decompose what is the relative contribution of reproduction material total biomass (with associated uncertainties); thus even if the organism does not spawn the reproduction material should still built up inside the body and contribute to the biomass. By looking at what the model predicts the indivual should have invested to what it actually does we can test for (i);

concerning (ii) the spawning behaviour. It would seem that parameters relative to "reproduction buffer rules", i.e. when to spawn, how many eggs in a spawn and when is ripening of immature oocytes initiated is quite variable among individuals. So I would recommend looking at distributions of values for the parameters specifying "reproduction buffer handling" among control organisms and then comparing against the distributions for the exposed ones. Again such a "buffer handling rules" module must be built (there are published examples) and then parameterized against the data. One can alway try and find a mean value for each parameter which best explains the data for each condtion. Further reasearch is need to really understand the inter-individual variability in the blank.

About means VS individuals. The problem is in intial differences in the 'condition' of the fish. If you can standardize initial reserve density, initial size as well as ensure that there is (hardly) and ripe reproduciton buffer material then working with means should be quite nice and effective in terms of effort and costs. Otherwise, my results suggest that it may not be possible to get a clear signal to answer question (i), (ii) and (iii) althought the analysis can of course still be done.

I would imagine such standardization of initial conditions can be achieved during the acclimation proces: speparate out different size groups (say 2 different groups of a same size) feed each group two different levels of nice nutritous food; and make pairs who spawn. For the experiment shoose (females) from each of the two groups who spawned several times just before the experiment. Then you will have two groups with empty buffer, same amount of initial structure and two different reserve densities .... Of course this has not (to lmy knowledge) been carried out but it is what I think needs to be done based on the analysis of reproduction data using the DEB model.

I also have the question about male zebrafish :-) So far, nothing suggest that males allocate to reproduction differently than females so the question is where does the mass allocated to reproduction in the males go? A first possiblity, which is simple, is to assume that dissipation linked to sperm production is much higher than that linked to egg production (kappa_R parameter) . So perhaps it is to best take the effective males from the standarzied initial conditions process described above for the eventual exeriment ....

A third aspect about the means VS individuals concerns the length of the expsoure and reproducion experiment. Initial conditions of the females (namely length, and initial amount of reproduction material) imact the performace of the individuals less when the experiment is LONG and alsmost completely condition the reproductive performance when the experiment is short. ...

Since we need short and effort/ cost effective experiements this brings us back to investing effort in standardizing initial condtions....

Finally, to improve the clarity of the signal, it is better to work with smaller fish than with bigger ones. If we start to approach ultimate size then the minimum ingestion level for which somatic maintenance is paid is higher relative to smaller individuals. If working with smaller mature individuals then there is a food level (and reserve density) for which growth will hardly occur while some reproduction still does and another higher food level for which growth does occur as well as reproduction. The effect of the chemical on individuals from both of such groups would together (theoretically) give a very interesting and less variable signal.

Smaller fish also spawn less and have (in general) lower inter-spawn internals.

Food for thought :-)

"Cumulated number of eggs often give nice curves that seems different. How should we test this?"

If possible, I would linearize the curves (logit if sigmoide), pick the time when 50% eggs are produced and test those times with ANOVA, to see if your xenobiotic advance or delay spawning. I would also try the same thing with mortality.

Maybe the following articles might be helpful for your analysis:

Toxicokinetic-toxicodynamic modeling of quantal and graded sublethal endpoints: A brief discussion of concepts

Ashauer R, Agatz A, Albert C, Ducrot V, Galic N, Hendriks J, Jager T, Kretschmann A, O'Connor I, Rubach MN, Nyman A-M, Schmitt W, Stadnicka J, van den Brink PJ, Preuss TG

2011 | Environmental Toxicology and Chemistry 30(11):2519-2524

Jager T, Albert C, Preuss TG, Ashauer R (2011): General Unified Threshold Model of Survival - a Toxicokinetic-Toxicodynamic Framework for Ecotoxicology. Environ Sci Technol 45, 2529-2540

Kaplan-Meyer is good for survival analysis. You can use log-rank test if you want to compare survival by category and cox regression to adjust for more covariates.

For spawning, could you determine a time point when you have the maximum/minimum value? In that case, you can use the same methodologies to assess time to event.

Time series models are applicable when outcome at successive times point is dependent or correlated with the outcome at previous time point. If in your experiment, number of individuals surviving at time t1 are dependent on number of individuals at time t0 and you want to predict for survival at any future time tn, you can use ARIMA model