function [stat p]=circ_ttest_p_first(alpha_1,alpha_2)
%stat=circ_ttest_p_first(alpha_1,alpha_2)
%parametric paired ttest for first order analysis (adaptation from
%hotelling test
%see Zar,1999,, page 645
%written by Andre M. Cravo 07/Sep/2011

% % data is copied from Zarr, page 647, example 27.19 for means of comparing results
% alpha_1=[105 120 135 95 155 170 160 155 120 115]'%morning
% alpha_2=[205 210 235 245 260 255 240 245 210 200]'%afternoon
% alpha_1=deg2rad(alpha_1)
% alpha_2=deg2rad(alpha_2)

 %equations 27.35-36, page 646)
X=cos(alpha_2)-cos(alpha_1);
Y=sin(alpha_2)-sin(alpha_1);
   
%equations 27.24 to 27.26 (page 638)
k=numel(alpha_1);
sum_xsquare=[sum((X.^2))]-[((sum(X))^2)/k];
sum_ysquare=[sum((Y.^2))]-[((sum(Y))^2)/k];
sum_xy=[sum(X.*Y)]-[(sum(X)*sum(Y))/k];

%equations 27.27 (page 639)
k_factor=(k*(k-2))/2;
mean_X=mean(X);
mean_Y=mean(Y);
mean_X_sq=mean_X^2;
mean_Y_sq=mean_Y^2;

F_num=[(mean_X_sq*sum_ysquare)-(2*mean_X*mean_Y*sum_xy)+(mean_Y_sq*sum_xsquare)];
F_den=[(sum_xsquare*sum_ysquare)-(sum_xy)^2];

stat=k_factor*(F_num/F_den);

%critical value one-tailed F test wit df=2, k-2, App22 (Zarr)
p = 1-fcdf(stat,2,k-2);


end