Variance Components
Variance Components
This dialog box will compute an estimate of the underlying process variance, sigma^2, for a set of parameter estimates. Either the real parameters (S(i)), or the beta parameters may be used, and in some cases, even derived parameters can be used. Note that for beta parameters, the MCMC procedure provides an alternative to the method described here.
Select either the Variance Components Real Parameters or the Variance Components Beta Parameters from the Output | Specific Model Output menu choice of the Results Browser Window. The basic concept is that the S(1), S(2), …, S(n), are considered a random sample from some distribution, hence have a mean and variance. If we could directly observe (i.e., measure without error) these survival rates, we would still make use of the S(i) in our thinking and in our models on population dynamics. We would also compute a mean and the usual simple estimate of sigma^2. However, we have instead a type of measurement error variation and covariance in our values S-hat(i) . When that measurement (sampling) variation is included into the inference methods, we end up with rather more complicated estimators of these two “population” parameters. Moreover, if the estimated sampling variances and covariances, in the real parameters variance-covariance matrix, are too big we cannot with any reliability partition the total observed variance,
sum from i =1 to n (S-hat(i) – Mean(S-hat))^2 / (n – 1)
into sigma^2-hat and the average of the diagonal elements of the real parameters variance-covariance matrix.
Variance Components Parameter Indices
Once we have estimated the process variance (sigma^2-hat), we can consider an unconditional estimator of the unobserved parameters S (S-tilde) and a corresponding variance-covariance matrix.
To specify the parameters, S-hat(i), to be used in estimating sigma^2-hat, you select them from the list of parameters, or else specify their indices in the edit box, e.g.,
6 7 8 9 10
in the edit box. You can also use the “to” operator to specify a range of values, such as
6 to 10
to specify the range of values 6 through 10. The “to” operator can also be combined with the “by” operator to select steps within the range. For example
1 to 10 by 3
would select the values 1, 4, 7, and 10.
Variance Components Design Matrix
The default model for the S-hat(i) values is the mean. To select a linear trend model, click that model in the upper right corner of the dialog window. You can also specify your own design matrix, instead of accepting 1 of the 2 built-in models.
When you have specified the parameters and model, click the OK button. You can also abort the calculation by clicking the Cancel button.
The initial output from the variance component estimation will be displayed from the numerical procedure that computes the estimates. The default is to display a numerical summary of the parameter estimates (both the original estimates and the “shrunk” estimates (S-tilde)), plus a graph of the original estimates and their confidence intervals, the “shrunk” estimates (S-tilde) and their confidence intervals, and the estimates from the model used for shrinking the original estimates (i.e., either the mean, linear trend, or user-specified design matrix). However options on the Variance Components Dialog Window allow you to select what output will be displayed. Output from the variance component estimation routine will be displayed in a NotePad window, including the estimate of sigma^2 and its 95% confidence interval, sigma and its 95% confidence interval, beta (the parameters for the linear model) and its variance-covariance matrix, and S-tilde and its variance-covariance matrix and standard errors.
Variance Components Output
Several modes of output are available from the variance components estimator. Output in the NotePad Window provides you with estimates of the beta parameters of the model you specified, the S-tilde vector and associated measures of precsion, sigma^2 and associated confidence intervals, and sigma and associated confidence intervals. Additional check boxes along the right side of the dialog box allow you to have additional output.
Besides the NotePad Window, you can also view a graph of the S-hat, S-tilde, and mean values, along with confidence intervals. This plot is useful for visualizing the amount of shrinkage that has taken place.
Variance Components Random Effects Model
The random effects model represents an intermediate model (in terms of the number of parameters) between the S(t) and S(.) model. To obtain the AICc or QAICc for the random effects model, select the Random Effects Model option in the list of Output Options. The numerical estimation code of MARK will then be run, with real parameter values fixed to their S-tilde values. The correct number of parameters estimated for this model is the number of parameters estimated by MARK for this model with the parameters fixed to S-tilde, plus the trace of the G matrix from the variance components analysis. Because the trace of G is not an integer, the number of parameters estimated for this model will not be an integer in the Results Browser Window.
The output for this random effects model is saved in the Model Notes field for later viewing.