Using individual and occasion covariates in the same DM

[simone77]

Hi all,

I have got a somehow weird result when implementing two models, {phi(COV1+weight*t) p(sex*t)} and {phi(COV2+weight*t)}.
COV1 and COV2 are two occasion specific covariates representing the population prevalence of some diseases. The inp file includes the individual covariate weight as measured at the first capture of each individual. There are 9 occasions
This is as the DM looks like (just the phi side), even though in the image decimals appear like comas it isn't the case in the real DM:


When I run both models, by changing the values of COV1 and COV2 and letting the rest be just the same (there are no interactions neither with time nor with individual covariate) i get the same identical result in terms of AIC and Deviance.
COV1 is:
0.65
0.11
0.23
0.48
0.27
0.09
0.36
0.15
COV2 is:
0.48
0.27
0.18
0.05
0.33
0.61
0.73
0.56

I am not sure if I am doing something wrong in the DM or if this has a statistical explication.
Any suggestion would be appreciated.

Simone

P.D.: Unfortunately the DM appears cut at the right side, anyway it seems to me that the important part is visualized correctly.

[abreton]

I can offer a few thoughts, I'll be curious to see what others might have to say. First, you've integrated two time-varying covariates (in your example, cov1 and weight) in a model that includes categorical time effects. Note that the categorical time effects accommodate ALL time-varying covariates (in the universe) including those that you've fitted in your example. Metaphorically speaking, by including the time-varying covariates with the categorical time-effects you've set up a competition...the covariates on one side and categorical effects on the other...each of these contestants attempting to explain the same variance.

Structurally, you've integrated redundancy into your model (over-fit the data). I suspect that if you look at the betas provided by MARK you'll discover that many of the parameters are not estimable given the data and the model structure. This explains why cov1 and cov2 models give the same AICc and deviance. Have a look at the betas, any problems?

I suggest that you replace the 1's in columns B4-B10 with 'weight'. Delete columns 3 and 11-16. You'll be left with a design matrix with 9 columns for Phi: intercept+cov1+weight:time. The latter (using RMark notation) refers to a unique weight effect for each occasion WITHOUT the main effects of year or weight. I think that's what you were aiming for in your model? I'm assuming p is time-varying without any covariates.

I went searching in the linear models chapter in the MARK book for a reference to fitting time-varying covariates with a fully-time-dependent model, I wasn't successful but I know somewhere Evan discusses this issue...in this chapter or elsewhere or in more than one location. In the meantime, open your analysis in MARK and go to help>ten commandments. See commandment #1.

Lastly, if I were in your shoes I'd be thinking about variance components and random effects. See appendix 4 in the MARK manual and Loison et al. 2002 for an example,

Loison A, Saeligther B-E, Jerstad K, Roslashstad OW (2002) Disentangling the sources of variation in the survival of the European dipper. J Appl Stat 29:289–304.

You may be able to locate a more recent application, please post a reply if you find one.

andre

[simone77]

Thank you very much for answering, it has been very useful to me.
OK, I believe I have understood your commentaries on redundancy and on using time varying covariates in a fully time dependent model.
I have followed your suggestion and I have passed from this DM:

to this:

Also, I have checked the beta estimates before and after this change and I get, for the original (wrong) model this estimates (hope it is visible enough):

and after the change, this:


It doesn't seem to me that there were huge problems in the estimates before and that after they are improved. I would believe that some problems with the estimates was related or to be very close to the boundary (see below) or to lack of data.


These days I will go deep inside the variance components and random factors material you have suggested to me, I knew about their existence but I need to really study them.

Anyway, I have a couple of other things I'd like to discuss.
1) Once I was told that it is statistically incorrect to build a model with the interaction between two variables and without their fixed effects, in other words I believed I could not build a model like {phi= intercept+cov1+weight:time} because I should have maintained the interaction variables alone, like this {phi= intercept+cov1+weight+time+weight:time}.
2) I am not sure to understand the first Commandment (probably it makes me an heretic!) because in the MARK book (11.4; 11-19) I see an example where it seems to me that it use a temporal covariate in a fully time-varying model (I suspect I am wrong!).

Thank you for any help,

Simone

[abreton]

As I was working through my response I realized that the design matrix (dm) I suggested needs a revision. Open the dm for the revised model and (right click) insert a column between B2 and B3 and then right-click again and choose re-order columns. In the first row of this new column, add the weight covariate (looks like you called this 'peso' in the image). Alternatively, you could update all cells in this column for rows phi1-8 with 'peso'. In this form, this new column (B3) is the main effect of weight (just like in your previous model) and the diagonals (B4-B10) are time-specific weight offsets. Combine the main effect and an offset, e.g., B3+B4 to get the slope of the relationship between survival in year 2 and weight.

1) Once I was told that it is statistically incorrect to build a model with the interaction between two variables and without their fixed effects, in other words I believed I could not build a model like {phi= intercept+cov1+weight:time} because I should have maintained the interaction variables alone, like this {phi= intercept+cov1+weight+time+weight:time}.

Reply: As long as the structure of the model is consistent with your hypotheses/predictions then any combination of effects could potentially be valid? No doubt there are some exceptions but I don't think your revised model structure is one of those. The structure of your revised model (after making changes I just described above) makes explicit two predictions. First, survival is a function of the time-dependent covariate 'cov1'. Note this is a time-varying covariate, so we want to consider dropping categorical time-effects from this model as you've done in the revision. Next, your revised model structure implies an effect of weight that can vary from year-to-year (another time-varying relationship). For example, the weight effect in year2 may be higher or lower than the weight effect in year3. At this point, I suggest that you label your columns in the design matrix - open the design matrix for the revised model and then choose Appearance>Label Columns. I'm going to assume you made the first set of changes that I described above (peso in just the first cell of the new column),

B1: intercept
B2: cov1
B3: weight_yr1
B3: weight_yr2
B4: weight_yr3
(etc)
B10: weight_yr8

Columns B3-B10 are slopes that add or subtract from the intercept a time-specific 'effect' of weight.

2) I am not sure to understand the first Commandment (probably it makes me an heretic!) because in the MARK book (11.4; 11-19) I see an example where it seems to me that it use a temporal covariate in a fully time-varying model (I suspect I am wrong!).

Reply: The dm on page 11.15 does not include an intercept, in contrast to the dm on page 11.17 which DOES contain an intercept. The former structure specifies a unique intercept for each year; the latter assumes that the intercept is the same for all years. The revised model that I proposed to you incorporates an intercept...so it assumes this is the same for all years...but you could drop the intercept and build a model similar to the one shown on page 11.15 though I'd have to think about how to interpret cov1 (or cov2) if the intercept was removed.

If you haven't already, consider reviewing the chapter on Linear Models in the gentle intro to mark.

andre

[simone77]

Sorry: can't delete this!

[simone77]

First of all, thanks for taking your time to help me with this.
I have modified the DM as you suggested:
DM1

This is the DM in the MARK book I was referring to:
DM2

That, apart from making me answer why the first commandment in the help file tells that you have not to use a time varying covariate in a fully time varying model, was also my inspiration for the way I built the DM at the beginning, i.e. this (same as in the first post):
DM3

My first feeling was that "weight", as an individual covariate, is not necessarily a time dependent variable and, for this reason, it should be used together with a dummy variable representing the time variation if you want to test if "weight" is having a time varying effect on survival.
My second feeling is that the DM3 is {phi(cov1+weight*t)different intercepts} while DM1 would be {phi(cov1+weight*t)common intercept}, isn't it?
Is it possible that the first commandment (you have not to use a time varying covariate in a fully time varying model) is referring to environmental covariates (necessarily time varying covariates) and not to individual covariates?

Simone

P.D.: I have been looking at the m-array and it seems that Phi in intervals 4 and 5 are simply very very high, I have been trying to deal with that by using the "data cloning" function but I was not able to fix this problem.

[abreton]

As you suspected, DM2 does not violate the 1st commandment (sounds like a sunday sermon!) because mass is measured only one time (time-invariant). You could build this identical model using your effect of weight. But note DM3 combines the time-varying cov1 with categorical time-effects, the problem I described previously (redundancy) applies to this model. If you delete B4-B10 from this model then the redundancy has been removed (result in DM1). I should not have cast 'weight' as time-varying, that was my mistake.

Regarding, "Is it possible that the first commandment (you have not to use a time varying covariate in a fully time varying model) is referring to environmental covariates (necessarily time varying covariates) and not to individual covariates?"

It's referring to any time-varying covariate (individual or sampling...the latter is measured on each occasion such as your cov1). Many of these would be environmental. However, consider, for example, a study involving the detection of fish using electrofishing. The researcher suspects that detection is a function of fish length because larger fish interact with a larger proportion of the electrical field. Using a function, such as von Bertalanffy, and length on initial capture they predict length on all other passes and fit this to their model as a time-varying individual covariate.

DM3 does specify different intercepts for each phi parameter. The problem is that the B4-B10 columns accommodate all time-varying effects including cov1. DM1 integrates only a common intercept as you suspected.

Very very high? If every marked animal survived the interval then you could fix these to 1 using the 'fix parameter' option on the form 'Setup Numerical Estimation Run' (this is the form that pops up when you click 'run model'). I've not used the data cloning option, can't help you there.

[abreton]

Another thought, ideally the categorical time effects (B4-B10, DM3) would be specified as random. From this point, estimates of these random effects could be used to estimate a variance component (sigma^2) describing the variance among years in survival. Next, you could use your fixed effects (cov1, cov2 and mass) to explain this variance. Using this approach you could, e.g., quantify the proportion of the variance in annual survival that is explained by cov1. All of this can be accomplished using MARK. It's considered advanced but it's not too difficult. See Loison et al. and the MARK chapter I suggested previously. Keep in mind this is my preference...analysis is an art not a recipe.

[simone77]

I have really appreciated your help because it made me think about redundancy in DM and time related issues: I have learned quite a few things.
Unfortunately, I believe I can't use the random effects with my data because of unequal time intervals (as pointed out in the Appendix D, D-43, point 2), I don't know if it depends on that but I was doing some trials and I get very very high values of deviance.
In addition, I haven't any evidence that every individual survived those intervals whose survival rates are very high: I just know that a vast majority of them have been seen again subsequent to those occasions. I am not sure I can fix them in this case and neither if it would improve the estimates of the other parameters in terms of precision and/or accuracy.

Simone

[abreton]

I should have taken more time to think about my responses, sorry for initially misleading you on one or two points. With this in mind, I'm glad to know, thank u, that you've learned quite a few things!

It makes sense, intuitively, that variance components for time-varying random effects would not be valid when interval lengths were not equal. The idea is to capture the process variance (after subtracting the sampling variance) in the time-varying estimates of survival (or any other parameter). If 'time' was not equal among intervals then the variance would be confounded by these unequal time-intervals.

Have a look at the discussion on 'parameter estimates on a boundary' in the MARK book starting on page 6-23, note you may have to adjust your parameter counts. Also, I suggest that you try re-running the model with the parameter estimates(s) that are on a boundary using simulated annealing. See page 10-38 in the MARK book for more details. Sim annealing takes longer to run than the default optimization method but it may provide you with estimates and standard errors for those parameter estimates that are on a boundary. If you haven't already, download the 'entire book' from http://www.phidot.org/software/mark/docs/book/. Then you can conveniently search for key-words (such as boundary, simulated annealing, etc.).