# analysis of covariance and constrained parameters

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## analysis of covariance and constrained parameters

 Consider an analysis of covariance involving age and cohort. The goal is to assess whether the influence of cohort depends upon the age. The simplest case involves data as follows value Age Cohort x1       1       3 x2       1       4 x3       1       5 x4       2       3 x5       2       4 x6       2       5 etc. Age is a factor. The numeric response variable is value and Cohort is a numeric predictor. So, (pseudo-code) commands to estimate the age=specific relationship between value and Cohort could be glm(value ~ Age/Cohort -  1, family =......, data = .....) glm(value ~ Age/(Cohort + I(Cohort^2)) - 1, family =......, data = .....). The latter commands would provide estimates of the age-specific intercept, linear, and quadratic coefficients, as in value_Age1 <- intercept_Age1 + linear_Age1*Cohort + quad_Age1*Cohort^2 value_Age2 <- intercept_Age2 + linear_Age2*Cohort + quad_Age2*Cohort^2 This is standard. One would choose among the above models via analysis of variance or AIC. Now assume that I have external knowledge that tells me that there is NO influence of Cohort on value for Age1 and that there could be up to a quadratic influence for Age2. Accordingly, I would like to fit a model which estimates these relationships: value_Age1 <- intercept_Age1 (+ 0*Cohort + 0*Cohort^2)                              (which is, of course, value_Age1 <- intercept_Age1) value_Age2 <- intercept_Age2 + linear_Age2*Cohort + quad_Age2*Cohort^2 What is the glm syntax to fit this model? It is a model in which we have constraints that (two) coefficients for one level of the factor must have a particular value (0) and there is no such constraint for the second level of the factor. Please note that I understand that glm(value ~ Age/(Cohort + I(Cohort^2)) - 1, family =......, data = .....). generates point estimates of the linear and quadratic coefficients for Age1 (as above) and one could inspect them to determine whether they are statistically equivalent to 0. However, I want to incorporate the knowledge that these coefficients MUST BE 0 into my hypothesis testing. Knowing that these coefficients are 0 could influence the results of anova and AIC comparisons since it reduces the number of degrees of freedom associated with model. Many thanks for suggestions in advance! -- Steven Orzack Fresh Pond Research Institute 173 Harvey Street Cambridge, MA 02140 617 864-4307 www.freshpond.org ______________________________________________ [hidden email] mailing list -- To UNSUBSCRIBE and more, see https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.