Individual covariates are entered into Program MARK via the Encounter Histories File. These values follow the number of animals encountered for the encounter history, as described in the Encounter Histories Format. Individual covariates are used to incorporate information specific to the individual into the parameter estimate. Individual covariates are named when the number of individual covariates is specified during the Initial Values screen.

Individual covariates are not allowed to have missing values, because the parameter value for the animal with the missing individual covariate value would also be missing, and henced the animal could not contribute to the likelihood. Several options for handling individual covariates are available.

Interactions of individual covariates can be created with the product function.

By specifying the name of an individual covariate in a cell of the Design Matrix, you tell MARK to use the value of this covariate in the design matrix when the capture history for this individual is used in the likelihood. As an example, suppose that 2 individual covariates are included in the : age (0=subadult, 1=adult), and weight at time of initial capture. The variable names given to these variables are, naturally, AGE and WEIGHT. Their are 8 survival rates in a known fate analysis. The following design matrix would use weight as a covariate, with an intercept term.

1 weight

1 weight

1 weight

1 weight

1 weight

1 weight

1 weight

1 weight

Each of the 8 time-specific survival rates would be modeled with the same model. Thus, time is not included in the relationship. Suppose you believe that the relationship between survival and weight changes with each time interval. The following design matrix would allow 8 different weight models.

1 weight 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 weight 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 weight 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 weight 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 weight 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 weight 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 weight 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 weight

The following model would have a common intercept for each survival rate, but different slopes for the weight variable.

1 weight 0 0 0 0 0 0 0

1 0 weight 0 0 0 0 0 0

1 0 0 weight 0 0 0 0 0

1 0 0 0 weight 0 0 0 0

1 0 0 0 0 weight 0 0 0

1 0 0 0 0 0 weight 0 0

1 0 0 0 0 0 0 weight 0

1 0 0 0 0 0 0 0 weight

The Copy Value is useful for creating these design matrices, as well as Saving the design matrix to a file, editing in a spreadsheet, and copying the new design matrix to the clipboard, and then Pasting the clipboard back into the design matrix.

Note that models with individual covariates have a different set of real parameter estimates for each animal. To compute the real parameter estimates, 3 options are available from the Run Window. More details are provided in the topic discussing the real parameter estimates from models with individual covariates.

**Warning** If you use the Standardize Individual Covariates option of the Run Window to perform the default standardization method, and then build a model with a 2 different slopes but a common intercepts, e.g.,

1 weight 0

1 0 weight

you will get different estimates of the beta parameters for the standardized individual covariates than you would get with unstandardized individual covariates unless you use the identity link (which will seldom converge with individual covariates). This problem is because the non-linear link functions available cause the centering of the standardized covariate to affect the intercept differently than if unstandardized covariates were used. For this situation, using the unstandardized covariates is probably the best solution, assuming that you can get your model to converge.

**Time-Varying Individual Covariates**

You can include time-varying individual covariates in MARK files, but must have a value for every animal on every occasion, even if the animal is not captured. Typically, you can impute these values, but be sure to recognize what this imputation might do to your estimates. You implement time-varying individual covariates just like any other individual covariate, expect that you have to have a different name for each covariate corresponding to each time period. For example, suppose you have a known fate model with 5 occasions, and you have estimated the parasite load for each animal at the beginning of each of the 5 occasions. The 5 values for each animal are contained in the variables var1, var2, var3, var4, and var5. A design matrix that would estimate the effect of the parasite load assuming that the effect is constant across time would be:

1 var1

1 var2

1 var3

1 var4

1 var5

The second beta estimate is the slope parameter associated with the time-varying individual covariates. Note that you **do not** want to standardize these individual covariates, because standardizing them will cause them to no longer relate to one another on the same scale (making a common slope parameter nonsensical). Each would have a different scale after standardizing. If you need to standardize the covariates, you must do so before the values are included in a MARK encounter histories input file, and you must use a common mean and standard deviation across the entire set of variables and observations.

The following design matrix would build a model where you assume the effect of parasite load is different for each interval, but with the same survival rate for animals with no parasites (i.e., the same intercept).

.

1 var1 0 0 0 0

1 0 var2 0 0 0

1 0 0 var3 0 0

1 0 0 0 var4 0

1 0 0 0 0 var5

The following model would allow different survival rates for each interval (i.e., time-specific survival), but assumes the same impact of parasites on survival on the logit scale (assuming that a logit link function is used).

1 1 0 0 0 var1

1 0 1 0 0 var2

1 0 0 1 0 var3

1 0 0 0 1 var4

1 0 0 0 0 var5

Finally, a model like the following would allow a completely different survival rate and parasite effect for each occasion.

1 1 0 0 0 var1 0 0 0 0

1 0 1 0 0 0 var2 0 0 0

1 0 0 1 0 0 0 var3 0 0

1 0 0 0 1 0 0 0 var4 0

1 0 0 0 0 0 0 0 0 var5

Individual covariates can be used in the add and product functions of the design matrix.

The value of individual covariates specified at the time the model is run will determine the value of the real and derived parameter values. Thus, the same values should be used for all models where the real or derived parameters are to be model averaged.