/** \brief Estimate by ML the effect size of the genotype, the std deviation
* of the errors and the std error of the estimated effect size in the
* multiple linear regression Y = XB + E with E~MVN(0,sigma^2I)
* \note genotype supposed to be 2nd column of X
*/
void FitSingleGeneWithSingleSnp(const gsl_matrix * X,
const gsl_vector * y,
double & pve,
double & sigmahat,
double & betahat_geno,
double & sebetahat_geno,
double & betapval_geno)
{
size_t N = X->size1, P = X->size2, rank;
double rss;
gsl_vector * Bhat = gsl_vector_alloc(P);
gsl_matrix * covBhat = gsl_matrix_alloc(P, P);
gsl_multifit_linear_workspace * work = gsl_multifit_linear_alloc(N, P);
gsl_multifit_linear_svd(X, y, GSL_DBL_EPSILON, &rank, Bhat, covBhat,
&rss, work);
pve = 1 - rss / gsl_stats_tss(y->data, y->stride, y->size);
sigmahat = sqrt(rss / (double)(N-rank));
betahat_geno = gsl_vector_get(Bhat, 1);
sebetahat_geno = sqrt(gsl_matrix_get (covBhat, 1, 1));
betapval_geno = 2 * gsl_cdf_tdist_Q(fabs(betahat_geno / sebetahat_geno),
N-rank);
gsl_vector_free(Bhat);
gsl_matrix_free(covBhat);
gsl_multifit_linear_free(work);
}
double rsComputePValueFromTValue(const double T, const int df)
{
if ( T < 0 ) {
return -gsl_cdf_tdist_P(T, df);
} else {
return gsl_cdf_tdist_Q(T, df);
}
}
/* Purpose: calculate z test for proportions, two-tailed for two-sided hypothesis H1 != H0
Parameters: p_sample = proportion of events in sample [0..1]
p_population = proportion of events in population [0..1]
n = sample size
Returns: Probability of making an error when rejecting the NULL hypothesis
*/
double gstats_test_ZP2 (double p_sample, double p_population, int n) {
double z;
double result = 0.0;
if ((p_sample < 0.0) || (p_sample > 1.0)) {
gstats_error ("z-stat: sample proportion must be given as 0.0 - 1.0.");
}
if ((p_population < 0.0) || (p_population > 1.0)) {
gstats_error ("z-stat: population proportion must be given as 0.0 - 1.0.");
}
if (n < 2) {
gstats_error ("z-stat: size of population must be a positive integer > 0.");
}
z = (p_sample - p_population) /
(sqrt ( (p_population * (1-p_population)) / n) );
/* determine direction of numeric integration */
if ( p_sample > p_population ) {
/* we use a normal distribution, if n is at least 30, */
/* t-distribution with n - 1 degress of freedom, otherwise */
if (n >= 30) {
result = gsl_cdf_ugaussian_Q (z);
} else {
result = gsl_cdf_tdist_Q (z, (double) n-1);
}
}
if ( p_sample < p_population ) {
/* we use a normal distribution, if n is at least 30, */
/* t-distribution with n - 1 degress of freedom, otherwise */
if (n >= 30) {
result = gsl_cdf_ugaussian_P (z);
} else {
result = gsl_cdf_tdist_P (z, (double) n-1);
}
}
if ( p_sample == p_population ) {
result = 0.5;
}
return (result * 2);
}
int main(){
apop_db_open("data-census.db");
gsl_vector *n = apop_query_to_vector("select in_per_capita from income "
"where state= (select state from geography where name ='North Dakota')");
gsl_vector *s = apop_query_to_vector("select in_per_capita from income "
"where state= (select state from geography where name ='South Dakota')");
double n_count = n->size,
n_mean = apop_vector_mean(n),
n_var = apop_vector_var(n),
s_count = s->size,
s_mean = apop_vector_mean(s),
s_var = apop_vector_var(s);
double stat = fabs(n_mean - s_mean)/ sqrt(n_var/ (n_count-1) + s_var/(s_count-1));
double confidence = 1 - (2 * gsl_cdf_tdist_Q(stat, n_count + s_count -2));
printf("Reject the null with %g%% confidence\n", confidence*100);
}
double
significance_of_correlation (double rho, double w)
{
double t = w - 2;
/* |rho| will mathematically always be in the range [0, 1.0]. Inaccurate
calculations sometimes cause it to be slightly greater than 1.0, so
force it into the correct range to avoid NaN from sqrt(). */
t /= 1 - MIN (1, pow2 (rho));
t = sqrt (t);
t *= rho;
if (t > 0)
return gsl_cdf_tdist_Q (t, w - 2);
else
return gsl_cdf_tdist_P (t, w - 2);
}
bool TTestReductionPop2(WiggleIterator * wi) {
if (wi->done)
return true;
SetComparisonData * data = (SetComparisonData *) wi->data;
Multiset * multi = data->multi;
if (multi->done) {
wi->done = true;
return true;
}
// Go to first position where both of the sets have at least one value
while (!multi->inplay[0] || !multi->inplay[1]) {
popMultiset(multi);
if (multi->done) {
wi->done = true;
return true;
}
}
wi->chrom = multi->chrom;
wi->start = multi->start;
wi->finish = multi->finish;
// Compute measurements
double sum1, sum2, sumSq1, sumSq2;
int count1, count2;
int index;
sum1 = sum2 = 0;
sumSq1 = sumSq2 = 0;
count1 = multi->multis[0]->count;
count2 = multi->multis[1]->count;
for (index = 0; index < multi->multis[0]->count; index++) {
if (multi->multis[0]->inplay[index]) {
sum1 += multi->values[0][index];
sumSq1 += multi->values[0][index] * multi->values[0][index];
}
}
for (index = 0; index < multi->multis[1]->count; index++) {
if (multi->multis[1]->inplay[index]) {
sum2 += multi->values[1][index];
sumSq2 += multi->values[1][index] * multi->values[1][index];
}
}
// To avoid divisions by 0:
if (count1 == 0 || count2 == 0) {
popMultiset(multi);
pop(wi);
return false;
}
double mean1 = sum1 / count1;
double mean2 = sum2 / count2;
double meanSq1 = sumSq1 / count1;
double meanSq2 = sumSq2 / count2;
double var1 = meanSq1 - mean1 * mean1;
double var2 = meanSq2 - mean2 * mean2;
// To avoid divisions by 0:
if (var1 + var2 == 0) {
popMultiset(multi);
return false;
}
// T-statistic
double t = (mean1 - mean2) / sqrt(var1 / count1 + var2 / count2);
if (t < 0)
t = -t;
// Degrees of freedom
double nu = (var1 / count1 + var2 / count2) * (var1 / count1 + var2 / count2) / ((var1 * var1) / (count1 * count1 * (count1 - 1)) + (var2 * var2) / (count2 * count2 * (count2 - 1)));
// P-value
wi->value = 2 * gsl_cdf_tdist_Q(t, nu);
// Update inputs
popMultiset(multi);
return true;
}
double
gsl_cdf_tdist_Qinv (const double Q, const double nu)
{
double x, qtail;
if (Q == 0.0)
{
return GSL_POSINF;
}
else if (Q == 1.0)
{
return GSL_NEGINF;
}
if (nu == 1.0)
{
x = tan (M_PI * (0.5 - Q));
return x;
}
else if (nu == 2.0)
{
x = (1 - 2 * Q) / sqrt (2 * Q * (1 - Q));
return x;
}
qtail = (Q < 0.5) ? Q : 1 - Q;
if (sqrt (M_PI * nu / 2) * qtail > pow (0.05, nu / 2))
{
double xg = gsl_cdf_ugaussian_Qinv (Q);
x = inv_cornish_fisher (xg, nu);
}
else
{
/* Use an asymptotic expansion of the tail of integral */
double beta = gsl_sf_beta (0.5, nu / 2);
if (Q < 0.5)
{
x = sqrt (nu) * pow (beta * nu * Q, -1.0 / nu);
}
else
{
x = -sqrt (nu) * pow (beta * nu * (1 - Q), -1.0 / nu);
}
/* Correct nu -> nu/(1+nu/x^2) in the leading term to account
for higher order terms. This avoids overestimating x, which
makes the iteration unstable due to the rapidly decreasing
tails of the distribution. */
x /= sqrt (1 + nu / (x * x));
}
{
double dQ, phi;
unsigned int n = 0;
start:
dQ = Q - gsl_cdf_tdist_Q (x, nu);
phi = gsl_ran_tdist_pdf (x, nu);
if (dQ == 0.0 || n++ > 32)
goto end;
{
double lambda = - dQ / phi;
double step0 = lambda;
double step1 = ((nu + 1) * x / (x * x + nu)) * (lambda * lambda / 4.0);
double step = step0;
if (fabs (step1) < fabs (step0))
{
step += step1;
}
if (Q < 0.5 && x + step < 0)
x /= 2;
else if (Q > 0.5 && x + step > 0)
x /= 2;
else
x += step;
if (fabs (step) > 1e-10 * fabs (x))
goto start;
}
}
end:
return x;
}
int main(int argc, char **argv){
distlist distribution = Normal;
char msg[10000], c;
int pval = 0, qval = 0;
double param1 = GSL_NAN, param2 =GSL_NAN, findme = GSL_NAN;
char number[1000];
sprintf(msg, "%s [opts] number_to_lookup\n\n"
"Look up a probability or p-value for a given standard distribution.\n"
"[This is still loosely written and counts as beta. Notably, negative numbers are hard to parse.]\n"
"E.g.:\n"
"%s -dbin 100 .5 34\n"
"sets the distribution to a Binomial(100, .5), and find the odds of 34 appearing.\n"
"%s -p 2 \n"
"find the area of the Normal(0,1) between -infty and 2. \n"
"\n"
"-pval Find the p-value: integral from -infinity to your value\n"
"-qval Find the q-value: integral from your value to infinity\n"
"\n"
"After giving an optional -p or -q, specify the distribution. \n"
"Default is Normal(0, 1). Other options:\n"
"\t\t-binom Binomial(n, p)\n"
"\t\t-beta Beta(a, b)\n"
"\t\t-f F distribution(df1, df2)\n"
"\t\t-norm Normal(mu, sigma)\n"
"\t\t-negative bin Negative binomial(n, p)\n"
"\t\t-poisson Poisson(L)\n"
"\t\t-t t distribution(df)\n"
"I just need enough letters to distinctly identify a distribution.\n"
, argv[0], argv[0], argv[0]);
opterr=0;
if(argc==1){
printf("%s", msg);
return 0;
}
while ((c = getopt (argc, argv, "B:b:F:f:N:n:pqT:t:")) != -1){
switch (c){
case 'B':
case 'b':
if (optarg[0]=='i')
distribution = Binomial;
else if (optarg[0]=='e')
distribution = Beta;
else {
printf("I can't parse the option -b%s\n", optarg);
exit(0);
}
param1 = atof(argv[optind]);
param2 = atof(argv[optind+1]);
findme = atof(argv[optind+2]);
break;
case 'F':
case 'f':
distribution = F;
param1 = atof(argv[optind]);
findme = atof(argv[optind+1]);
break;
case 'H':
case 'h':
printf("%s", msg);
return 0;
case 'n':
case 'N':
if (optarg[0]=='o'){ //normal
param1 = atof(argv[optind]);
param2 = atof(argv[optind+1]);
findme = atof(argv[optind+2]);
} else if (optarg[0]=='e'){
distribution = Negbinom;
param1 = atof(argv[optind]);
param2 = atof(argv[optind+1]);
findme = atof(argv[optind+2]);
} else {
printf("I can't parse the option -n%s\n", optarg);
exit(0);
}
break;
case 'p':
if (!optarg || optarg[0] == 'v')
pval++;
else if (optarg[0] == 'o'){
distribution = Poisson;
param1 = atof(argv[optind]);
findme = atof(argv[optind+1]);
} else {
printf("I can't parse the option -p%s\n", optarg);
exit(0);
}
break;
case 'q':
qval++;
break;
case 'T':
case 't':
distribution = T;
param1 = atof(argv[optind]);
findme = atof(argv[optind+1]);
break;
case '?'://probably a negative number
if (optarg)
snprintf(number, 1000, "%c%s", optopt, optarg);
else snprintf(number, 1000, "%c", optopt);
if (gsl_isnan(param1)) param1 = -atof(number);
else if (gsl_isnan(param2)) param2 = -atof(number);
else if (gsl_isnan(findme)) findme = -atof(number);
}
}
if (gsl_isnan(findme)) findme = atof(argv[optind]);
//defaults, as promised
if (gsl_isnan(param1)) param1 = 0;
if (gsl_isnan(param2)) param2 = 1;
if (!pval && !qval){
double val =
distribution == Beta ? gsl_ran_beta_pdf(findme, param1, param2)
: distribution == Binomial ? gsl_ran_binomial_pdf(findme, param2, param1)
: distribution == F ? gsl_ran_fdist_pdf(findme, param1, param2)
: distribution == Negbinom ? gsl_ran_negative_binomial_pdf(findme, param2, param1)
: distribution == Normal ? gsl_ran_gaussian_pdf(findme, param2)+param1
: distribution == Poisson ? gsl_ran_poisson_pdf(findme, param1)
: distribution == T ? gsl_ran_tdist_pdf(findme, param1) : GSL_NAN;
printf("%g\n", val);
return 0;
}
if (distribution == Binomial){
printf("Sorry, the GSL doesn't have a Binomial CDF.\n");
return 0; }
if (distribution == Negbinom){
printf("Sorry, the GSL doesn't have a Negative Binomial CDF.\n");
return 0; }
if (distribution == Poisson){
printf("Sorry, the GSL doesn't have a Poisson CDF.\n");
return 0; }
if (pval){
double val =
distribution == Beta ? gsl_cdf_beta_P(findme, param1, param2)
: distribution == F ? gsl_cdf_fdist_P(findme, param1, param2)
: distribution == Normal ? gsl_cdf_gaussian_P(findme-param1, param2)
: distribution == T ? gsl_cdf_tdist_P(findme, param1) : GSL_NAN;
printf("%g\n", val);
return 0;
}
if (qval){
double val =
distribution == Beta ? gsl_cdf_beta_Q(findme, param1, param2)
: distribution == F ? gsl_cdf_fdist_Q(findme, param1, param2)
: distribution == Normal ? gsl_cdf_gaussian_Q(findme-param1, param2)
: distribution == T ? gsl_cdf_tdist_Q(findme, param1) : GSL_NAN;
printf("%g\n", val);
}
}