/***************************** * Density Function Examples * *****************************/ /* normal(z) returns the cumulative standard normal distribution, e.g. the proportion of the gaussian normal distribution falling below z. */ display normal(-1) /* Z < -1 */ display normal( 1) /* Z < 1 */ display normal( 1) - normal(-1) /* -1 < Z < 1 */ display normal(1.96) /* Z < 1.96 */ display 1- normal(1.96) /* Z > 1.96 */ display normal(-1.2) /* Z < -1.2 */ display 1 - normal(-1.2) /* Z > -1.2 */ display normal(1.2) - normal(-.5) /* -.5 < Z < 1.2 */ display normal(1.2) - normal( .5) /* .5 < Z < 1.2 */ /* invnormal(p) returns the inverse cumulative standard normal distribution; normal(z) = p then invnormal(p) = z. Given a cumulative density or proportion, what is the associated z-score? */ display invnormal(0.500) /* z-score for p=0.5 */ display invnormal(0.975) /* 2-tail */ display invnormal(0.950) /* 1-tail */ display round(invnormal(0.975),.01) /* 2-tail */ /* Using 1-alpha and 1-alpha/2 */ display %4.2f invnormal(1-(.05/2)) /* 2-tail */ display %4.2f invnormal(1-(.05)) /* 1-tail */ /* ttail(n,t) returns the reverse cumulative (upper-tail) Student's t distribution with n > 0 degrees of freedom. This function returns probability T>t. */ di ttail(3000,1.96) /* 1-tail */ di ttail(3000,1.96)*2 /* 2-tail */ /* invttail(n,p) returns the inverse reverse cumulative Student's t distribution. If ttail(n,t) = p, then invttail(n,p) = t. */ display invttail(3000,.050) /* 1-tail */ display invttail(3000,.025) /* 2-tail */ display invttail(3000,(.05/2)) /* 2-tail */ local n 11 local alpha 0.05 display invttail(`n',(`alpha'/2)) /* 2-tail */ /* tden(n,t) returns the probability density function of Student's t distribution with n > 0 degrees of freedom. */ display tden(3000,1.96)/2 /* 1-tail */ display tden(3000,1.96) /* 2-tail */ /* F(n1,n2,f) returns the cumulative F distribution with n1 numerator and n2 denominator degrees of freedom, for n1, n2 > 0; returns 0 unless f > 0. For example, in a regression model with one IV and 25 cases there are 1 and n-k-1 df. The second expression is the p-value associated with F=1.74. Ftail(n1,n2,f) returns the reverse cumulative (upper-tail, survival) F distribution with n1 numerator and n2 denominator degrees of freedom. Ftail(n1,n2,f) = 1 - F(n1,n2,f). */ di F(1,23,1.74) /* Cumulative Up to F */ di 1-F(1,23,1.74) /* Cumulative Beyond F */ di F(1,23,1.74) + 1-F(1,23,1.74) /* Area bt. +- 0/Inf. */ di Ftail(1,23,1.74) /* Cumulative Beyond F */ di Ftail(1,1,161.44764) /* Cumulative Beyond F */ local df_num=1 local df_den=1 local F=161 di F(`df_num',`df_den',`F') /* Cumulative Up to F */ di 1-F(`df_num',`df_den',`F') /* Cumulative Beyond F */ di Ftail(`df_num',`df_den',`F') /* Cumulative Beyond F */ di Ftail(`df_num',`df_den',161.44764) /* Cumulative Beyond F */ di invFtail(`df_num',`df_den',.05) /* Binomial(n,k,p) returns the probability of k or more successes in n trials when the probability of a success on a single trial is p. Here n > k > 0 must be non-negative integers, and p > 0. invbinomial(n,k,P) returns the inverse binomial: for P<0.5, probability p such that the probability of observing k or more successes in n trials is P; for P>0.5, probability p such that the probability of observing k or fewer successes in n trials is 1-P. */ di binomial(10,5,.5) /* p-value for 5 successes */ di binomial(21,11,1/3) /* p-value for 11 successes i n21 trials with p=1/3 */ di invbinomial(10,5,.5) /* 1-(p-value) */ /* chi2(n,x) returns the cumulative chi-squared distribution with n degrees of freedom for n > 0; chi2(n,x) = gammap(n/2,x/2). Returns 0 unless x>0. chi2tail(n,x) returns the reverse cumulative (upper-tail, survival) chi-squared distribution with n degrees of freedom. chi2tail(n,x) = 1 - chi2(n,x). */ di chi2(3,10) /* Cumulative probability */ di chi2tail(3,10) /* p-value */ /* Graph of Normal Density Function, 2-tail; alpha=0.05 */ #delimit ; twoway (function y=normalden(x), range(-4 -1.96) color(gs12) recast(area)) (function y=normalden(x), range( 1.96 4) color(gs12) recast(area)) (function y=normalden(x), range(-4 4) lstyle(foreground) lwidth(thick)), title("Two-tail, alpha=0.05") ysca(off) xsca(noline) legend(off) xlabel(-4 "-4 sd" -3 "-3 sd" -2 "-2 sd" -1 "-1 sd" 0 "mean" 1 "1 sd" 2 "2 sd" 3 "3 sd" 4 "4 sd") ylabel(, nogrid) xtitle("") xline(0, lw(thick) lp(dash)) text(-.05 0 "Z") text(.012 1.99 "0.025", placement(east) size(small)) text(.012 -1.99 "0.025", placement(west) size(small)) graphregion(color(white)) scheme(s2mono) name(twotail, replace); #delimit cr /* Graph of Normal Density Function, 1-tail; alpha=0.05 */ #delimit ; twoway (function y=normalden(x), range(1.65 4) color(gs12) recast(area)) (function y=normalden(x), range(-4 4) lstyle(foreground) lwidth(thick)), title("One-tail, alpha=0.05") ysca(off) xsca(noline) legend(off) xlabel(-4 "-4 sd" -3 "-3 sd" -2 "-2 sd" -1 "-1 sd" 0 "mean" 1 "1 sd" 2 "2 sd" 3 "3 sd" 4 "4 sd") ylabel(, nogrid) xtitle("") xline(0, lw(thick) lp(dash)) text(.018 1.99 "0.05") text(-.05 0 "Z") graphregion(color(white)) scheme(s2mono) name(onetail, replace); #delimit cr /* Graph of Normal Density Function, 2-tail; alpha=0.05 */ #delimit ; twoway (function y=normalden(x), range( 1.65 4) color(gs8) recast(area)) (function y=normalden(x), range(-4 -1.96) color(gs12) recast(area)) (function y=normalden(x), range( 1.96 4) color(gs12) recast(area)) (function y=normalden(x), range(-4 4) lstyle(foreground) lwidth(thick)), plotregion(style(none)) title("One- and Two-tail" " alpha=0.05") ysca(off) xsca(noline) legend(off) xlabel(-4 "-4 sd" -3 "-3 sd" -2 "-2 sd" -1 "-1 sd" 0 "mean" 1 "1 sd" 2 "2 sd" 3 "3 sd" 4 "4 sd" , tlwidth(medthick)) ylabel(, nogrid) xtitle("") xline(0, lw(thick) lp(dash)) text(-.06 0 "Z") text(.008 2.08 "0.025", placement(east) size(small)) text(.008 -2.08 "0.025", placement(west) size(small)) text(.008 1.98 "0.05" , placement(west) size(small)) legend(on order(1 "one-tail" 2 "two-tail") cols(1) ring(0) position(1)) graphregion(color(white)) scheme(s2mono) name(combined, replace); #delimit cr