For the standard normal theory fixed effects model with equal variances for the one-way classification, the distribution of the F test statistic MS(Trt)/MSEMS(Trt)/MSE under H1H1 is noncentral F with numerator degrees of freedom a?1a?1, denominator degrees of freedom N?aN?a and noncentrality parameter ??2?ai=1ni?2i=??2?ai=1ni(?i??)2??2?i=1ani?i2=??2?i=1ani(?i??)2, where ?=?ini?i/N?=?ini?i/N.

The definition of noncentral F is as follows. Let U1U1 and U2U2 be independent random variables such that U1U1 has a noncentral chi-squared distribution with ?1?1 degrees of freedom and noncentrality parameter ?2?2 and U2U2 has a central chi-squared distribution with ?2?2 degrees of freedom. Then [U1/?1]/[U2/?2]has a noncentral F distribution with numerator degrees of freedom ?1?1, denominator degrees of freedom ?2?2 and noncentrality parameter ?2. A notation for the distribution is F??1,?2(?2)F?1,?2?(?2).

The definition of the noncentral ?2?2 is as follows. A noncentral ?2 distribution with ?? degrees of freedom and noncentrality parameter ?2?0 is defined as the distribution of WW, where

W=X21+X22+?+X2?

with Xj?N(?j,1)Xj?N(?j,1) being independent random variables and ?2=??j=1?2j?2=?j=1??j2. A notation for the distribution is ??2?(?2)???2(?2).

Note: for the noncentral F cumulative distribution function, use the R function pf() with specified parameter ncp.

or the standard normal theory fixed effects model with equal variances for the one way classification , the distribution of the Ftest statistic MS ( Tre ) MSF , under IT , is noncentral F with numerator degrees of freedom a - 1 , denominator degrees of freedom N and noncentrality parameter , ? It Ina ? La Mi( Hi - 14 ) 2 where u - LurkIN ntral chi - squared distribution with v degrees of freedom The definition of noncentral F is as follows Let U , and U be independent random variables such that U , has a noncentral chi - squared distribution with in degrees of freedom and noncentrality parameter and noncentrality parameter 1 2 and I , has a central chi -squared distribution with U 2 degrees of freedom . Then2 / 12 has a noncentral F distribution U 2 and noncentrality parameter ? A notation for the distribution is Fun ( 1 2 ) The definition of the noncentral X 2 is as follows A noncentral x 2 distribution with degrees of freedom and noncentrality parameter ? > O is defined as the distribution of W . where W - X 3 + x 2 + + x with ~ ( 8 , 1 ) being independent random variables and 2 - 8? A notation for the distribution is X 2 ( 1 2 ) Note : for the noncentral F cumulative distribution function use the R function pro with specified parameter nce art a ) Suppose a researcher is planning an experiment with a - 5 treatments and wants a power of at of at least 0 90 if the vector of a ; happens to be ( - 1 0.5 0 0 .5 1 ) and of Also suppose that the researcher wants a common sample size for each treatment , le . The - 2 for i - 1 5 . What should the sample size nn be if the " be if the probability of a type I error is to be 0.05 ? part b ) For a sensitivity analysis , determine nit of equals 15 0 . 8 and the vector of of ; is as in part ( a ) part ( ) Assume anything not mention tioned is fixed , which of the following holds ? You can do some numerical checks as a guide More than one item could be correct A . The required sample size n to achieve a specified power A decreases as of increases B . The required sample size in to ach plesize in to achieve a specified power & decreases as increases C . The required sample size in to achieve a eve a specified power s decreases as the popu lation means become further apart D . The required sample size le size n to achieve a specified power & increases as the popul lation means become further apart E. The required sample size in to achieve a specified power increases as of increases F . The required sample size in to achieve a specified power increases as increases G . None of the above