chi_squared_test

Performs a chi-squared goodness-of-fit test.

Synopsis

#include <imsl.h>

float imsl_f_chi_squared_test (float user_proc_cdf(), int n_observations, int n_categories, float x[], …, 0)

The type double function is imsl_d_chi_squared_test.

Required Arguments

float user_proc_cdf (float y) (Input)
User-supplied function that returns the hypothesized, cumulative distribution function at the point y.

int n_observations (Input)
The number of data elements input in x.

int n_categories (Input)
The number of cells into which the observations are to be tallied.

float x[] (Input)
Array with n_observations components containing the vector of data elements for this test.

Return Value

The p-value for the goodness-of-fit chi-squared statistic.

Synopsis with Optional Arguments

#include <imsl.h>

float imsl_f_chi_squared_test (float user_proc_cdf(), int n_observations, int n_categories, float x[],

IMSL_N_PARAMETERS_ESTIMATED, int n_parameters,

IMSL_CUTPOINTS, float **p_cutpoints,

IMSL_CUTPOINTS_USER, float cutpoints[],

IMSL_CUTPOINTS_EQUAL,

IMSL_CHI_SQUARED, float *chi_squared,

IMSL_DEGREES_OF_FREEDOM, float *df,

IMSL_FREQUENCIES, float frequencies[],

IMSL_BOUNDS, float lower_bound, float upper_bound,

IMSL_CELL_COUNTS, float **p_cell_counts,

IMSL_CELL_COUNTS_USER, float cell_counts[],

IMSL_CELL_EXPECTED, float **p_cell_expected,

IMSL_CELL_EXPECTED_USER, float cell_expected[],

IMSL_CELL_CHI_SQUARED, float **p_cell_chi_squared,

IMSL_CELL_CHI_SQUARED_USER, float cell_chi_squared[],

IMSL_FCN_W_DATA, float user_proc_cdf(), void *data,

0)

Optional Arguments

IMSL_N_PARAMETERS_ESTIMATED, int n_parameters (Input)
The number of parameters estimated in computing the cumulative distribution function.

IMSL_CUTPOINTS, float **p_cutpoints (Output)
The address of a pointer to the cutpoints array. On return, the pointer is initialized (through a memory allocation request to malloc), and the array is stored there. Typically, float *p_cutpoints is declared; &p_cutpoints is used as an argument to this function; and imsl_free(p_cutpoints) is used to free this array.

IMSL_CUTPOINTS_USER, float cutpoints[] (Input or Output)
Array with n_categories − 1 components containing the vector of cutpoints defining the cell intervals. The intervals defined by the cutpoints are such that the lower endpoint is not included, and the upper endpoint is included in any interval. If IMSL_CUTPOINTS_EQUAL is specified, equal probability cutpoints are computed and returned in cutpoints.

IMSL_CUTPOINTS_EQUAL
If IMSL_CUTPOINTS_USER is specified, then equal probability cutpoints can still be used if, in addition, the IMSL_CUTPOINTS_EQUAL option is specified. If IMSL_CUTPOINTS_USER is not specified, equal probability cutpoints are used by default.

IMSL_CHI_SQUARED, float *chi_squared (Output)
If specified, the chi-squared test statistic is returned in *chi_squared.

IMSL_DEGREES_OF_FREEDOM, float *df (Output)
If specified, the degrees of freedom for the chi-squared goodness-of-fit test is returned in *df.

IMSL_FREQUENCIES, float frequencies[] (Input)
Array with n_observations components containing the vector frequencies for the observations stored in x.

IMSL_BOUNDS, float lower_bound, float upper_bound (Input)
If IMSL_BOUNDS is specified, then lower_bound is the lower bound of the range of the distribution, and upper_bound is the upper bound of this range. If lower_bound = upper_bound, a range on the whole real line is used (the default). If the lower and upper endpoints are different, points outside the range of these bounds are ignored. Distributions conditional on a range can be specified when IMSL_BOUNDS is used. By convention, lower_bound is excluded from the first interval, but upper_bound is included in the last interval.

IMSL_CELL_COUNTS, float **p_cell_counts (Output)
The address of a pointer to an array containing the cell counts. The cell counts are the observed frequencies in each of the n_categories cells. On return, the pointer is initialized (through a memory allocation request to malloc), and the array is stored there. Typically, float *p_cell_counts is declared; &p_cell_counts is used as an argument to this function; and imsl_free(p_cell_counts) is used to free this array.

IMSL_CELL_COUNTS_USER, float cell_counts[] (Output)
If specified, the n_categories cell counts are returned in the array cell_counts provided by the user.

IMSL_CELL_EXPECTED, float **p_cell_expected (Output)
The address of a pointer to the cell expected values. The expected value of a cell is the expected count in the cell given that the hypothesized distribution is correct. On return, the pointer is initialized (through a memory allocation request to malloc), and the array is stored there. Typically, float *p_cell_expected is declared; &p_cell_expected is used as an argument to this function; and imsl_free(p_cell_expected) is used to free this array.

IMSL_CELL_EXPECTED_USER, float cell_expected[] (Output)
If specified, the n_categories cell expected values are returned in the array cell_expected provided by the user.

IMSL_CELL_CHI_SQUARED, float **p_cell_chi_squared (Output)
The address of a pointer to an array of length n_categories containing the cell contributions to chi-squared. On return, the pointer is initialized (through a memory allocation request to malloc), and the array is stored there. Typically, float *p_cell_chi_squared is declared; &p_cell_chi_squared is used as an argument to this function; and imsl_free(p_cell_chi_squared) is used to free this array.

IMSL_CELL_CHI_SQUARED_USER, float cell_chi_squared[] (Output)
If specified, the cell contributions to chi-squared are returned in the array cell_chi_squared provided by the user.

IMSL_FCN_W_DATA, float user_proc_cdf (float y, void *data), void *data, (Input)
User supplied function that returns the hypothesized, cumulative distribution function at the point y, which also accepts a pointer to data that is supplied by the user. data is a pointer to the data to be passed to the user-supplied function. See Passing Data to User-Supplied Functions in the introduction to this manual for more details.

Description

The function imsl_f_chi_squared_test performs a chi-squared goodness-of-fit test that a random sample of observations is distributed according to a specified theoretical cumulative distribution. The theoretical distribution, which may be continuous, discrete, or a mixture of discrete and continuous distributions, is specified via the user-defined function user_proc_cdf. Because the user is allowed to give a range for the observations, a test conditional upon the specified range is performed.

Argument n_categories gives the number of intervals into which the observations are to be divided. By default, equiprobable intervals are computed by imsl_f_chi_squared_test, but intervals that are not equiprobable can be specified (through the use of optional argument IMSL_CUTPOINTS).

Regardless of the method used to obtain the cutpoints, the intervals are such that the lower endpoint is not included in the interval, while the upper endpoint is always included. If the cumulative distribution function has discrete elements, then user-provided cutpoints should always be used since imsl_f_chi_squared_test cannot determine the discrete elements in discrete distributions.

By default, the lower and upper endpoints of the first and last intervals are − ∞ and + ∞, respectively. If IMSL_BOUNDS is specified, the endpoints are defined by the user via the two arguments lower_bound and upper_bound.

A tally of counts is maintained for the observations in x as follows. If the cutpoints are specified by the user, the tally is made in the interval to which xi belongs using the endpoints specified by the user. If the cutpoints are determined by imsl_f_chi_squared_test, then the cumulative probability at xi, F(xi), is computed via the function user_proc_cdf. The tally for xi is made in interval number

 

and is the function that takes the greatest integer that is no larger than the argument of the function. Thus, if the computer time required to calculate the cumulative distribution function is large, user-specified cutpoints may be preferred to reduce the total computing time.

If the expected count in any cell is less than 1, then a rule of thumb is that the chi-squared approximation may be suspect. A warning message to this effect is issued in this case, as well as when an expected value is less than 5.

On some platforms, imsl_f_chi_squared_test can evaluate the user-supplied function user_proc_cdf in parallel. This is done only if the function imsl_omp_options is called to flag user-defined functions as thread-safe. A function is thread-safe if there are no dependencies between calls. Such dependencies are usually the result of writing to global or static variables

Programming Notes

The user must supply a function user_proc_cdf with calling sequence user_proc_cdf(y), that returns the value of the cumulative distribution function at any point y in the (optionally) specified range. Many of the cumulative distribution functions in Special Functions can be used for user_proc_cdf, either directly, if the calling sequence is correct, or indirectly, if, for example, the sample means and standard deviations are to be used in computing the theoretical cumulative distribution function.

Examples

 

Example 1

This example illustrates the use of imsl_f_chi_squared_test on a randomly generated sample from the normal distribution. One-thousand randomly generated observations are tallied into 10 equiprobable intervals. The null hypothesis that the sample is from a normal distribution is specified by use of the imsl_f_normal_cdf (see Special Functions) as the hypothesized distribution function. In this example, the null hypothesis is not rejected.

 

#include <imsl.h>

#include <stdio.h>

 

#define SEED 123457

#define N_CATEGORIES 10

#define N_OBSERVATIONS 1000

 

int main()

{

float *x, p_value;

 

imsl_omp_options(

IMSL_SET_FUNCTIONS_THREAD_SAFE, 1,

0);

imsl_random_seed_set(SEED);

 

/* Generate Normal deviates */

x = imsl_f_random_normal (

N_OBSERVATIONS,

0);

 

/* Perform chi squared test */

p_value = imsl_f_chi_squared_test (imsl_f_normal_cdf,

N_OBSERVATIONS,

N_CATEGORIES, x,

0);

 

/* Print results */

printf ("p value %7.4f\n", p_value);

}

Output

 

p value 0.1546

 

Example 2

In this example, some optional arguments are used for the data in the initial example.

 

#include <imsl.h>

 

#define SEED 123457

#define N_CATEGORIES 10

#define N_OBSERVATIONS 1000

 

int main()

{

float *cell_counts, *cutpoints, *cell_chi_squared;

float chi_squared_statistics[3], *x;

char *stat_row_labels[] = {"chi-squared", "degrees of freedom",

"p-value"};

 

imsl_omp_options(IMSL_SET_FUNCTIONS_THREAD_SAFE, 1, 0);

 

imsl_random_seed_set(SEED);

/* Generate Normal deviates */

x = imsl_f_random_normal (N_OBSERVATIONS, 0);

/* Perform chi squared test */

chi_squared_statistics[2] =

imsl_f_chi_squared_test (imsl_f_normal_cdf,

N_OBSERVATIONS, N_CATEGORIES, x,

IMSL_CUTPOINTS, &cutpoints,

IMSL_CELL_COUNTS, &cell_counts,

IMSL_CELL_CHI_SQUARED, &cell_chi_squared,

IMSL_CHI_SQUARED, &chi_squared_statistics[0],

IMSL_DEGREES_OF_FREEDOM, &chi_squared_statistics[1],

0);

/* Print results */

imsl_f_write_matrix ("\nChi Squared Statistics\n", 3, 1,

chi_squared_statistics,

IMSL_ROW_LABELS, stat_row_labels,

0);

imsl_f_write_matrix ("Cut Points", 1, N_CATEGORIES-1, cutpoints, 0);

imsl_f_write_matrix ("Cell Counts", 1, N_CATEGORIES, cell_counts,

0);

imsl_f_write_matrix ("Cell Contributions to Chi-Squared", 1,

N_CATEGORIES, cell_chi_squared,

0);

}

Output

 

Chi Squared Statistics

 

chi-squared 13.18

degrees of freedom 9.00

p-value 0.15

 

Cut Points

1 2 3 4 5 6

-1.282 -0.842 -0.524 -0.253 -0.000 0.253

 

7 8 9

0.524 0.842 1.282

 

 

Cell Counts

1 2 3 4 5 6

106 109 89 92 83 87

 

7 8 9 10

110 104 121 99

 

Cell Contributions to Chi-Squared

1 2 3 4 5 6

0.36 0.81 1.21 0.64 2.89 1.69

 

7 8 9 10

1.00 0.16 4.41 0.01

Example 3

In this example, a discrete Poisson random sample of size 1000 with parameter θ = 5.0 is generated via function imsl_f_random_poisson. In the call to imsl_f_chi_squared_test, function imsl_f_poisson_cdf is used as function user_proc_cdf.

 

#include <imsl.h>

 

#define SEED 123457

#define N_CATEGORIES 10

#define N_PARAMETERS_ESTIMATED 0

#define N_NUMBERS 1000

#define THETA 5.0

 

float user_proc_cdf(float);

 

int main()

{ int i, *poisson;

 

float cell_statistics[3][N_CATEGORIES];

float chi_squared_statistics[3], x[N_NUMBERS];

float cutpoints[] = {1.5, 2.5, 3.5, 4.5, 5.5, 6.5,

7.5, 8.5, 9.5};

char *cell_row_labels[] = {"count", "expected count",

"cell chi-squared"};

char *cell_col_labels[] = {"Poisson value", "0", "1", "2",

"3", "4", "5", "6", "7", "8", "9"};

char *stat_row_labels[] = {"chi-squared", "degrees of freedom",

"p-value"};

 

imsl_omp_options(IMSL_SET_FUNCTIONS_THREAD_SAFE, 1, 0);

 

imsl_random_seed_set(SEED);

/* Generate the data */

poisson = imsl_random_poisson(N_NUMBERS, THETA, 0);

/* Copy data to a floating point vector*/

for (i = 0; i < N_NUMBERS; i++)

x[i] = poisson[i];

 

chi_squared_statistics[2] =

imsl_f_chi_squared_test(user_proc_cdf, N_NUMBERS, N_CATEGORIES, x,

IMSL_CUTPOINTS_USER, cutpoints,

IMSL_CELL_COUNTS_USER, &cell_statistics[0][0],

IMSL_CELL_EXPECTED_USER, &cell_statistics[1][0],

IMSL_CELL_CHI_SQUARED_USER, &cell_statistics[2][0],

IMSL_CHI_SQUARED, &chi_squared_statistics[0],

IMSL_DEGREES_OF_FREEDOM, &chi_squared_statistics[1],

0);

/* Print results */

imsl_f_write_matrix("\nChi-squared statistics\n", 3, 1,

&chi_squared_statistics[0],

IMSL_ROW_LABELS, stat_row_labels,

0);

imsl_f_write_matrix("\nCell Statistics\n", 3, N_CATEGORIES,

&cell_statistics[0][0],

IMSL_ROW_LABELS, cell_row_labels,

IMSL_COL_LABELS, cell_col_labels,

0);

}

 

 

float user_proc_cdf(float k)

{

float cdf_v;

 

cdf_v = imsl_f_poisson_cdf ((int) k, THETA);

return cdf_v;

}

Output

 

Chi-squared statistics

 

chi-squared 10.48

degrees of freedom 9.00

p-value 0.31

 

 

Cell Statistics

 

Poisson value 0 1 2 3 4

count 41.0 94.0 138.0 158.0 150.0

expected count 40.4 84.2 140.4 175.5 175.5

cell chi-squared 0.0 1.1 0.0 1.7 3.7

 

Poisson value 5 6 7 8 9

count 159.0 116.0 75.0 37.0 32.0

expected count 146.2 104.4 65.3 36.3 31.8

cell chi-squared 1.1 1.3 1.4 0.0 0.0

Warning Errors

IMSL_EXPECTED_VAL_LESS_THAN_1	An expected value is less than 1.
IMSL_EXPECTED_VAL_LESS_THAN_5	An expected value is less than 5.

Fatal Errors

IMSL_ALL_OBSERVATIONS_MISSING	All observations contain missing values.
IMSL_INCORRECT_CDF_1	The function user_proc_cdf is not a cumulative distribution function. The value at the lower bound must be nonnegative, and the value at the upper bound must not be greater than one.
IMSL_INCORRECT_CDF_2	The function user_proc_cdf is not a cumulative distribution function. The probability of the range of the distribution is not positive.
IMSL_INCORRECT_CDF_3	The function user_proc_cdf is not a cumulative distribution function. Its evaluation at an element in x is inconsistent with either the evaluation at the lower or upper bound.
IMSL_INCORRECT_CDF_4	The function user_proc_cdf is not a cumulative distribution function. Its evaluation at a cutpoint is inconsistent with either the evaluation at the lower or upper bound.
IMSL_INCORRECT_CDF_5	An error has occurred when inverting the cumulative distribution function. This function must be continuous and defined over the whole real line.
IMSL_STOP_USER_FCN	Request from user supplied function to stop algorithm. User flag = "#".