{"id":370,"date":"2017-05-06T03:56:46","date_gmt":"2017-05-06T03:56:46","guid":{"rendered":"http:\/\/sites.warnercnr.colostate.edu\/gwhite\/?page_id=370"},"modified":"2017-05-06T05:14:16","modified_gmt":"2017-05-06T05:14:16","slug":"design-matrix-advanced-applications","status":"publish","type":"page","link":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/design-matrix-advanced-applications\/","title":{"rendered":"Design Matrix — Advanced Applications"},"content":{"rendered":"

Contents<\/a> – Index<\/a><\/span><\/p>\n

\n

Advanced applications of the Design Matrix<\/a> involve constraining parameters across groups.\u00a0 Suppose that a set of recapture data have 2 groups, say males and females.\u00a0 Then, the 4 parameter matrices for a time model for each parameter for each group might look like the following.<\/span><\/p>\n

\u00a0 Apparent Survival Group 1<\/span>
\n\u00a0 1\u00a0 2\u00a0 3\u00a0 4\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0 2\u00a0 3\u00a0 4\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3\u00a0 4\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4\u00a0<\/span><\/p>\n

\u00a0 Apparent Survival Group 2<\/span><\/p>\n

\u00a0 5\u00a0 6\u00a0 7\u00a0 8\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0 6\u00a0 7\u00a0 8\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 7\u00a0 8\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 8\u00a0<\/span><\/p>\n

\u00a0 Recapture Probabilities Group 1<\/span>
\n\u00a0 9\u00a0 10\u00a0 11\u00a0 12\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0 10\u00a0 11\u00a0 12\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 11\u00a0 12\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 12\u00a0<\/span><\/p>\n

\u00a0 Recapture Probabilities Group 2<\/span><\/p>\n

\u00a0 13\u00a0 14\u00a0 15\u00a0 16\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0 14\u00a0 15\u00a0 16\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 15\u00a0 16\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 16\u00a0<\/span><\/p>\n

An identity design matrix, or Full Design Matrix, results in the model {Phi(g*t<\/i>) p<\/i>(g*t<\/i>)}.\u00a0 The identity matrix for this example would look like the following.<\/span><\/p>\n

1\u00a0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\u00a0<\/span>
\n0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0\u00a0<\/span>
\n0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0\u00a0<\/span>
\n0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0\u00a0<\/span>
\n0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0\u00a0<\/span>
\n0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0\u00a0<\/span>
\n0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0\u00a0<\/span>
\n0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0\u00a0<\/span>
\n0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0\u00a0<\/span>
\n0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0\u00a0<\/span>
\n0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\u00a0<\/span>
\n0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0\u00a0<\/span>
\n0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0\u00a0<\/span>
\n0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0\u00a0<\/span>
\n0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0\u00a0<\/span>
\n0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1\u00a0<\/span><\/p>\n

Each row corresponds to a parameter, and in this case, each column corresponds to a parameter.<\/span><\/p>\n

Suppose that the model {Phi(t<\/i>) p<\/i>(t<\/i>)} is desired, but with no change in the parameter matrices, shown above.\u00a0 Thus, 16 parameters need to be constrained to 8 parameters.\u00a0 The following design matrix will make these constraints.\u00a0 There are still 16 rows in the matrix, because there is still 16 parameters in the model.\u00a0 However, the rows of the matrix are identical for the first parameter and the fifth\u00a0 parameter, because these 2 parameters are now the apparent survival rate for the survival across the first interval.\u00a0 Likewise, row 2 and\u00a0 row 6 are identical, to model the separate apparent survival rates for interval 2 for each group as the same parameter.\u00a0 This process is continued for all 8 parameters with the following matrix.<\/span><\/p>\n

1\u00a0 0 0 0 0 0 0 0\u00a0<\/span>
\n0 1 0 0 0 0 0 0\u00a0<\/span>
\n0 0 1 0 0 0 0 0\u00a0<\/span>
\n0 0 0 1 0 0 0 0\u00a0<\/span>
\n1 0 0 0 0 0 0 0\u00a0<\/span>
\n0 1 0 0 0 0 0 0\u00a0<\/span>
\n0 0 1 0 0 0 0 0\u00a0<\/span>
\n0 0 0 1 0 0 0 0\u00a0<\/span>
\n0 0 0 0 1 0 0 0\u00a0<\/span>
\n0 0 0 0 0 1 0 0\u00a0<\/span>
\n0 0 0 0 0 0 1 0\u00a0<\/span>
\n0 0 0 0 0 0 0 1\u00a0<\/span>
\n0 0 0 0 1 0 0 0\u00a0<\/span>
\n0 0 0 0 0 1 0 0\u00a0<\/span>
\n0 0 0 0 0 0 1 0\u00a0<\/span>
\n0 0 0 0 0 0 0 1\u00a0<\/span><\/p>\n

This model could also be developed by manipulating the parameter matrices.\u00a0 To demonstrate this, the following 4 parameter matrices would result in the same estimates at the above design matrix would generate with the original 4 parameter matrices with 16 parameters.\u00a0 Note that the parameter matrix is identical for each group, hence, the parameter estimates are identical for each group, and thus model {Phi(t<\/i>) p<\/i>(t<\/i>)}results.<\/span><\/p>\n

\u00a0 Apparent Survival Group 1<\/span>
\n\u00a0 1\u00a0 2\u00a0 3\u00a0 4\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0 2\u00a0 3\u00a0 4\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3\u00a0 4\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4\u00a0<\/span><\/p>\n

\u00a0 Apparent Survival Group 2<\/span><\/p>\n

\u00a0 1\u00a0 2\u00a0 3\u00a0 4\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0 2\u00a0 3\u00a0 4\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3\u00a0 4\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4\u00a0<\/span><\/p>\n

\u00a0 Recapture Probabilities Group 1<\/span>
\n\u00a0 5\u00a0 6\u00a0 7\u00a0 8\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0 6\u00a0 7\u00a0 8\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 7\u00a0 8\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 8\u00a0<\/span><\/p>\n

\u00a0 Recapture Probabilities Group 2<\/span><\/p>\n

\u00a0 5\u00a0 6\u00a0 7\u00a0 8\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0 6\u00a0 7\u00a0 8\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 7\u00a0 8\u00a0<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 8\u00a0<\/span><\/p>\n

See Design Matrix Advanced 2<\/a> for examples of building additive effects models.<\/span><\/p>\n

See Design Matrix Scaling Covariates<\/a> for why “reasonable” values should be used in the design matrix.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Contents – Index Advanced applications of the Design Matrix involve constraining parameters across groups.\u00a0 Suppose that a set of recapture data have 2 groups, say males and females.\u00a0 Then, the 4 parameter matrices for a time model for each parameter …<\/p>\n

Design Matrix — Advanced Applications<\/span> Read More »<\/a><\/p>\n","protected":false},"author":117,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-370","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/370","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/users\/117"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/comments?post=370"}],"version-history":[{"count":4,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/370\/revisions"}],"predecessor-version":[{"id":431,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/370\/revisions\/431"}],"wp:attachment":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/media?parent=370"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}