MLogit Link

The MLogit link one of the many link functions available in MARK, and is used primarily to constrain a set of parameters to sum to <=1. The MLogit link is used primarily with the multi-state, open robust design, and POPAN data types. Consider the following set of parameters in a PIM for the probability of entry (pent) in a POPAN model:

61 62 63 64 65 66 67 68 69 70

The parameter-specific link would be selected in the Run Window, and the MLogit(1) link would be applied to parameters 61 through 70 to force these 10 estimates to sum to <=1. But suppose that you wanted to force all of the 10 entry probabilities to be the same. You would be tempted to specify a PIM such as

61 61 61 61 61 61 61 61 61 61

but that would be wrong. Changing the PIM and selecting the MLogit link for parameter 61 would result in parameter 61 alone summing to <=1 (i.e., just like a logit link), but would not force the sum of the 10 values of parameter 61 to sum to <=1.

To implement the proposed model, the PIM should not be changed from the top example (61 to 70), and the design matrix should be used to force the same estimate for parameters 61 to 70:

Parameter Design Matrix

61 1

62 1

63 1

64 1

65 1

66 1

67 1

68 1

69 1

70 1

Then the MLogit(1) link should be specified for the 10 parameters 61 to 70. The result is that now all 10 parameters have the same value, and 10 times this value is <=1.

Another example would be that you wanted parameters 61 and 62 to be the same value, 63 and 64 the same, 65 and 66 the same, 67 and 68 the same, and 69 and 70 the same. Again you would use the PIM

61 62 63 64 65 66 67 68 69 70

but use the design matrix to implement the constraints. The following design matrix is one example that would produce such a set of constraints.

Parameter Design Matrix

61 1 1 0 0 0

62 1 1 0 0 0

63 1 0 1 0 0

64 1 0 1 0 0

65 1 0 0 1 0

66 1 0 0 1 0

67 1 0 0 0 1

68 1 0 0 0 1

69 1 0 0 0 0

70 1 0 0 0 0

The key point with these examples is that the PIM cannot be used to constrain parameters if you want the entire set of parameters to sum to <=1. Rather, the design matrix has to be used to make the constrains, with each of the entries in the PIM given the same MLogit(x) link.