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R(�'" Program Power by C. P. Doncaster uses a good approximation by Winer et al"
R2� " (1991 Statistical Principles in Experimental Design, 3rd Edn, pp 136-138)"
Q<�� " to calculate statistical power of any effects in a balanced ANOVA design"
NF� " with a required or given threshold ratio of treatment to error effect"
QP� " sizes (the standard deviations of variability due to treatments only and"
EZ� " to random error), or an F value obtained from a pilot study."
d:
Gn�'" Critical Type-I error for the proposed test (default=0.05)... "
Gx� " Proposed test (numerator) degrees of freedom................. "
G�� " Proposed error (denominator) degrees of freedom.............. "
G�� " Effective sample size........................................ "
G�� " Fixed (F) or random (R) effect (default=F)................... "
G�� " EITHER desired power for the proposed test................... "
G�� " OR threshold ratio of treatment to error effect sizes.... "
G�� " OR observed F value obtained from a pilot study.......... "
G�� " and number of replicates per sample in pilot study.... "
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��:@%=11:H%=63:V%=7
��:鑺H%,V%+1)ALPHA:� ALPHA<1
�� ALPHA<=0 � ALPHA=0.05
�駣H%,V%+1)ALPHA;" "
?H%=63:V%=7:@%=11:s%=0:p%=0:q%=0:THETA=0:pF=0:pn%=0:SIGMA2=1
,�:鑺H%,V%+2)p%:� p%>0:駣H%,V%+2)p%;" "
,�:鑺H%,V%+3)q%:� q%>0:駣H%,V%+3)q%;" "
,"�:鑺H%,V%+4)s%:� s%>1:駣H%,V%+4)s%;" "
D,�:鑺H%,V%+5)FR$:� FR$="F" � FR$="f" � FR$="R" � FR$="r" � FR$=""
46� FR$="" � FR$="f" � FR$="F" � FR$="F" � FR$="R"
@駣H%,V%+5)FR$;" "
J:
=T�:鑺H%,V%+6)WPOWER:� (WPOWER>ALPHA € WPOWER<1) � WPOWER=0
H^� WPOWER>0 � 騝lear(7,9):駣H%,V%+6)WPOWER;" ":� 峵xA � 騝lear(6,6)
h:
r鑺H%,V%+7)THETA$
K|� THETA$<>"" � 騝lear(8,9):THETA=�(THETA$):駣H%,V%+7)THETA;" ":� 峵xA
�騝lear(7,7):鑺H%,V%+8)pF$
"�� pF$<>"" � pF=�(pF$) � � 岲rA
(�� pF>0 � 駣H%,V%+8)pF;" " � � 岲rA
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�:
6�駣0,V%+11)�75," "):駣0,V%+11)" ...calculating...";
/�� pF=0 � THETA2=THETA^2 � THETA2=(pF-1)/pn%
�� PRINT THETA2
�� Find critical F
�騟xact
*�� WPOWER>0 � 騠ind_theta � 騠ind_power
&�� Fc>=10 � DP%=&20108 � DP%=&20208
0�@%=11:�", with F[";ALPHA;:@%=DP%:�"] = ";Fc;
� �30," ");:@%=11
� 峊DA
�
&:
0蒡find_theta
#:� FR$="R" � 騮heta_for_random:�
'D� Calculate dZ for the desired beta
BNZ=-5:dZOLD=�(1/(2*�))*�(-0.5*Z^2):I=10^-4:cZcrit=1-WPOWER:cZ=0
X�
bZ=Z+I
ldZ=�(1/(2*�))*�(-0.5*Z^2)
vcZ=cZ+I*(dZOLD+dZ)/2
€dZOLD=dZ
�� cZ>=cZcrit
W�� Iterate RATV = LOG(theta^2 / sigma^2) from the Winer et al. (1991) formula for dZ
�RATV=3:J=10:B=Fc*p%/q%
�� I%=1 � 5
(�J=J/10:�:RATV=RATV-J:騠ind_IT:� IT>0
(�J=J/10:�:RATV=RATV+J:騠ind_IT:� IT<0
��
S�@%=&20208:駣0,V%+11)" Ratio of treatment to error effect sizes = ";10^(RATV/2);
��
�:
�蒡find_IT
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4IT=(�((2*q%-1)*B)-�(2*p%*R1-R2/R1))/�(B+R2/R1)-Z
�
:
蒡find_power
(*� FR$="R" � 騟xact_for_random:� 峝hC
4LAMBDA=p%*s%*THETA2/SIGMA2
,>AW=(p%+2*LAMBDA)/(p%+LAMBDA):BW=p%*Fc/q%
-H� Calculate z for the normal distribuiton
3RZ=(�((2*q%-1)*BW)-�(2*(p%+LAMBDA)-AW))/�(BW+AW)
\I=10^-4:cZ=0
5f� Calculate the probability density function at z
pdZOLD=�(1/(2*�))*�(-0.5*Z^2)
.z� PRINT LAMBDA,",",A,",",B,",",Z,",",dZOLD
�:
`�� Calculate cumulative probability from the area under the left-hand tail of the dZ function
��
�Z=Z-I
�dZ=�(1/(2*�))*�(-0.5*Z^2)
�cZ=cZ+I*(dZOLD+dZ)/2
�dZOLD=dZ
�� Z<-5
�� PRINT dZ
�POWER=1-cZ
;�@%=&20208:駣0,V%+11)" Power of proposed test = ";POWER;
��
�:
蒡exact
"� Find the value of critical F
&� p%>2 € q%>3 � I=0.001 � I=0.0001
$F=9.95*10^-6:cF=0
#.a=p%/2+q%/2-1:b=p%/2-1:c=q%/2-1
8騠actorial(a):A=N
B騠actorial(b):B=N
L騠actorial(c):C=N
IVdFOLD=�(A-B-C+0.5*(p%*�(p%)+q%*�(q%)+(p%-2)*�(F)-(p%+q%)*�(q%+p%*F)))
`�
jF=F+I
6tD=p%*�(p%)+q%*�(q%)+(p%-2)*�(F)-(p%+q%)*�(q%+p%*F)
~dF=�(A-B-C+0.5*D)
�cF=cF+I*(dFOLD+dF)/2
�dFOLD=dF
�� PRINTTAB(0,V%+4)dF,,cF,,F
�� cF>=1-ALPHA
�Fc=F
��
�:
�蒡factorial(n)
�� n<-0.4 � N=�(�)/2:�
�n%=(n+0.1) � 1
(�� n-n%>0.4 � 騢alfinteger � 騣nteger
��
:
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� Integer factorial
N=0
(� I%=1 � n%
2N=N+�(I%)
<�
F�
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Z蒡halfinteger
d� Half integer factorial
nN=�(�)/2
x� I%=0 � n%
�N=N+�(2*I%+1)-�(2)
��
��
�:
�蒡clear(I1%,I2%)
%�� I%=I1% � I2%:駣H%,V%+I%)�(15):�
��
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�蒡exact_for_random
S�� Find the value of beta at critical F, using F as a fraction of the expected F
�EF=(s%*THETA2+SIGMA2)/SIGMA2
&�� p%>2 € q%>3 � I=0.001 � I=0.0001
�F=9.95*10^-6:cF=0
#a=p%/2+q%/2-1:b=p%/2-1:c=q%/2-1
騠actorial(a):A=N
騠actorial(b):B=N
"騠actorial(c):C=N
O,dFOLD=�(A-B-C+0.5*(p%*�(p%)+q%*�(q%)+(p%-2)*�(F/EF)-(p%+q%)*�(q%+p%*F/EF)))
6�
@F=F+I
=Fc
|POWER=1-cF/EF
��
�:
�蒡theta_for_random
k�� Find the value of theta at critical F that gives power = 0.8, using F as a fraction of the expected F
%�� This has to be done iteratively
G�@%=&20208:駣0,V%+11)" Ratio of treatment to error effect sizes = ";
�EF=0.1
��
�EF=EF+1
�騠ind_power_for_random
�� POWER>WPOWER
��
�EF=EF-0.1
騠ind_power_for_random
� POWERWPOWER
MD@%=&20208:駣0,V%+11)" Ratio of treatment to error effect sizes = ";THETA;
N�
X:
b蒡find_power_for_random
&l� p%>2 € q%>3 � I=0.001 � I=0.0001
vF=9.95*10^-6:cF=0
#€a=p%/2+q%/2-1:b=p%/2-1:c=q%/2-1
�騠actorial(a):A=N
�騠actorial(b):B=N
�騠actorial(c):C=N
O�dFOLD=�(A-B-C+0.5*(p%*�(p%)+q%*�(q%)+(p%-2)*�(F/EF)-(p%+q%)*�(q%+p%*F/EF)))
��
�F=F+I
<�D=p%*�(p%)+q%*�(q%)+(p%-2)*�(F/EF)-(p%+q%)*�(q%+p%*F/EF)
�dF=�(A-B-C+0.5*D)
�cF=cF+I*(dFOLD+dF)/2
�dFOLD=dF
�� F>=Fc
�POWER=1-cF/EF
THETA=�((cF/(1-POWER)-1)/s%)
d@%=&20208:駣44,V%+11);THETA; :�" Theta^2 = ";(cF/(1-POWER)-1)/s%;"; EF = ";EF;"; Power = ";POWER
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