HERITABILITY AND RUST RESISTANCE IX WHITE PINE 407 



APPENDIX 



DERIVATION OF BINOMIAL MODEL AND TRANSFORMATION' 



BINOMIAL MODEL 



Each seedling within a plot is a sample from a population. The 

 populations differ from each other in their probability of a seedling 

 being healthy because of genetic segregation (seedlings are full sibs to 

 one another) and because of environmental effects (seedlings are planted 

 in different spots within the plot) . The basic datum is the proportion 

 of healthy seedlings divided by the total number of seedlings per plot. 

 Because each seedling has a different probability of being healthy the 

 standard binomial model does not apply. 



If we can use the number of healthy seedlings per plot as an obser- 

 vation (n/j7>) , then the variance of this observation is 



a 2 = 



n 



«=1 



where p^ is the probability of the hth seedling being healthy and q}i is 

 1 - p^, h = 1,2,... m within a plot. 



mm mm 



h \Pjflh = *Ph (1 "Pa 3 = * Ph ~ J-fh > 



h=l h=l n=l n=l 



mm m m 



h=l h=l h=l h=l 



= m p - m p 2 - m a 2 , ; 



where p is the mean of the p's in the different populations; a 2 7 is the 

 variance due to the populations within a plot being different; and m is 

 the total number of seedlings per plot. 



m 



lp h q h =mpq -mo* d . 



P.. , is the proportion of healthy seedlings in a plot [ith tester, 

 jth candidate, and feth replication) and is calculated by 



n . ., 



ijk 



The variance of p for a plot is the variance of the number of healthy 

 seedlings per plot divided by the square of the total number of individuals 

 (m 2 ) 



1 m 



m 2 ^ % 



