Performs a chi-squared goodness-of-fit test.
#include <imsls.h>
float imsls_f_chi_squared_test (float user_proc_cdf(), int n_observations, int n_categories, float x[], ..., 0)
The type double function is imsls_d_chi_squared_test.
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)
Number of data elements input in
x.
int n_categories
(Input)
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.
The p-value for the goodness-of-fit chi-squared statistic.
#include <imsls.h>
float
imsls_f_chi_squared_test (float
user_proc_cdf(), int n_observations, int
n_categories, float
x[],
IMSLS_N_PARAMETERS_ESTIMATED, int
n_parameters,
IMSLS_CUTPOINTS, float
**cutpoints,
IMSLS_CUTPOINTS_USER, float
cutpoints[],
IMSLS_CUTPOINTS_EQUAL,
IMSLS_CHI_SQUARED, float
*chi_squared,
IMSLS_DEGREES_OF_FREEDOM, float
*df,
IMSLS_FREQUENCIES, float
frequencies[],
IMSLS_BOUNDS, float
lower_bound, float
upper_bound,
IMSLS_CELL_COUNTS, float
**cell_counts,
IMSLS_CELL_COUNTS_USER, float
cell_counts[],
IMSLS_CELL_EXPECTED, float
**cell_expected,
IMSLS_CELL_EXPECTED_USER, float
cell_expected[],
IMSLS_CELL_CHI_SQUARED, float
**cell_chi_squared,
IMSLS_CELL_CHI_SQUARED_USER, float
cell_chi_squared[],
IMSLS_FCN_W_DATA, float fcn(),
void
*data,
0)
IMSLS_N_PARAMETERS_ESTIMATED, int
n_parameters (Input)
Number of parameters estimated in
computing the cumulative distribution function.
IMSLS_CUTPOINTS, float
**cutpoints (Output)
Address of a pointer to an internally
allocated array of length n_categories − 1 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 IMSLS_CUTPOINTS_EQUAL
is specified, equal probability cutpoints are computed and returned in cutpoints.
IMSLS_CUTPOINTS_USER, float cutpoints []
(Input/Output)
Storage for array cutpoints is provided
by the user. See IMSLS_CUTPOINTS.
IMSLS_CUTPOINTS_EQUAL
If
IMSLS_CUTPOINTS_USER
is specified, then equal probability cutpoints can still be used if, in
addition, the IMSLS_CUTPOINTS_EQUAL
option is specified. If IMSLS_CUTPOINTS_USER
is not specified, equal probability cutpoints are used by default.
IMSLS_CHI_SQUARED, float
*chi_squared (Output)
If specified, the chi-squared test
statistic is returned in *chi_squared.
IMSLS_DEGREES_OF_FREEDOM, float *df
(Output)
If specified, the degrees of freedom for the chi-squared
goodness-of-fit test is returned in *df.
IMSLS_FREQUENCIES, float
frequencies[] (Input)
Array with n_observations
components containing the vector frequencies for the observations stored in
x.
IMSLS_BOUNDS, float lower_bound, float
upper_bound (Input)
If IMSLS_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 IMSLS_BOUNDS is used.
By convention, lower_bound is
excluded from the first interval, but upper_bound is
included in the last interval.
IMSLS_CELL_COUNTS, float
**cell_counts (Output)
Address of a pointer to an
internally allocated array of length n_categories
containing the cell counts. The cell counts are the observed frequencies in each
of the n_categories cells.
IMSLS_CELL_COUNTS_USER, float
cell_counts[] (Output)
Storage for array cell_counts is
provided by the user. See IMSLS_CELL_COUNTS.
IMSLS_CELL_EXPECTED, float
**cell_expected (Output)
Address of a pointer to an
internally allocated array of length n_categories
containing the cell expected values. The expected value of a cell is the
expected count in the cell given that the hypothesized distribution is
correct.
IMSLS_CELL_EXPECTED_USER, float
cell_expected[] (Output)
Storage for array cell_expected is
provided by the user. See IMSLS_CELL_EXPECTED.
IMSLS_CELL_CHI_SQUARED, float
**cell_chi_squared (Output)
Address of a pointer to an
internally allocated array of length n_categories
containing the cell contributions to chi-squared.
IMSLS_CELL_CHI_SQUARED_USER, float
cell_chi_squared[] (Output)
Storage for array cell_chi_squared is
provided by the user. See IMSLS_CELL_CHI_SQUARED.
IMSLS_FCN_W_DATA, float user_proc_cdf
(float
y), void
*data, (Input)
User-supplied function that returns the hypothesized,
cumulative distribution function, 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 the Introduction, Passing Data to
User-Supplied Functions at the beginning of this manual for more
details.
Function imsls_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 can be continuous, discrete, or a mixture of discrete and continuous distributions, is specified by the user-defined function user_proc_cdf. Because the user is allowed to give a range for the observations, a test that is conditional on 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 imsls_f_chi_squared_test, but intervals that are not equiprobable can be specified through the use of optional argument IMSLS_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 imsls_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 IMSLS_BOUNDS
is specified, the endpoints are user-defined by 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 user-specified endpoints.
• If the cutpoints are determined by imsls_f_chi_squared_test, then the cumulative probability at xi, F(xi), is computed by the function user_proc_cdf.
The tally for xi is made in interval number ⌊mF(xi) + 1⌋, where m = n_categories 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 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.
This example illustrates the use of imsls_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 imsls_f_normal_cdf (Chapter 11, Probability Distribution Functions and Inverses.p<.CSCH11.DOC!NORMAL_CDF;510;) as the hypothesized distribution function. In this example, the null hypothesis is not rejected.
#include <imsls.h>
#define
SEED
123457
#define
N_CATEGORIES
10
#define
N_OBSERVATIONS
1000
main()
{
float *x, p_value;
imsls_random_seed_set(SEED);
/* Generate Normal deviates */
x = imsls_f_random_normal
(N_OBSERVATIONS,
0);
/* Perform chi squared test */
p_value =
imsls_f_chi_squared_test (imsls_f_normal_cdf,
N_OBSERVATIONS,
N_CATEGORIES, x,
0);
/* Print results */
printf ("p-value = %7.4f\n",
p_value);
}
p-value = 0.1546
In this example, optional arguments are used for the data in the initial example.
#include <imsls.h>
#define
SEED
123457
#define
N_CATEGORIES
10
#define
N_OBSERVATIONS
1000
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"};
imsls_random_seed_set(SEED);
/* Generate normal deviates
*/
x = imsls_f_random_normal (N_OBSERVATIONS,
0);
/* Perform chi squared test */
chi_squared_statistics[2] =
imsls_f_chi_squared_test
(imsls_f_normal_cdf,
N_OBSERVATIONS,
N_CATEGORIES, x,
IMSLS_CUTPOINTS, &cutpoints,
IMSLS_CELL_COUNTS, &cell_counts,
IMSLS_CELL_CHI_SQUARED, &cell_chi_squared,
IMSLS_CHI_SQUARED,
&chi_squared_statistics[0],
IMSLS_DEGREES_OF_FREEDOM,
&chi_squared_statistics[1],
0);
/* Print results */
imsls_f_write_matrix ("\nChi Squared
Statistics\n", 3, 1,
chi_squared_statistics,
IMSLS_ROW_LABELS, stat_row_labels,
0);
imsls_f_write_matrix ("Cut Points", 1, N_CATEGORIES-1,
cutpoints,
0);
imsls_f_write_matrix ("Cell Counts", 1, N_CATEGORIES,
cell_counts,
0);
imsls_f_write_matrix ("Cell Contributions to
Chi-Squared", 1,
N_CATEGORIES,
cell_chi_squared,
0);
}
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
In this example, a discrete Poisson random sample of size 1,000 with parameter θ = 5.0 is generated by function imsls_f_random_poisson (Chapter 12, Random Number Generation”;). In the call to imsls_f_chi_squared_test, function imsls_f_poisson_cdf (Chapter 11, “Probability Distribution Functions and Inverses”;) is used as function user_proc_cdf.
#include <imsls.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);
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"};
imsls_random_seed_set(SEED);
/* Generate the data */
poisson =
imsls_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] =
imsls_f_chi_squared_test(user_proc_cdf, N_NUMBERS,
N_CATEGORIES,
x,
IMSLS_CUTPOINTS_USER,
cutpoints,
IMSLS_CELL_COUNTS_USER,
&cell_statistics[0][0],
IMSLS_CELL_EXPECTED_USER, &cell_statistics[1][0],
IMSLS_CELL_CHI_SQUARED_USER,
&cell_statistics[2][0],
IMSLS_CHI_SQUARED,
&chi_squared_statistics[0],
IMSLS_DEGREES_OF_FREEDOM,
&chi_squared_statistics[1],
0);
/* Print results */
imsls_f_write_matrix("\nChi-squared
Statistics\n", 3, 1,
&chi_squared_statistics[0],
IMSLS_ROW_LABELS,
stat_row_labels,
0);
imsls_f_write_matrix("\nCell Statistics\n", 3,
N_CATEGORIES,
&cell_statistics[0][0],
IMSLS_ROW_LABELS,
cell_row_labels,
IMSLS_COL_LABELS,
cell_col_labels,
IMSLS_WRITE_FORMAT,
"%9.1f",
0);
}
float user_proc_cdf(float k)
{
float
cdf_v;
cdf_v = imsls_f_poisson_cdf ((int) k,
THETA);
return cdf_v;
}
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
Function user_proc_cdf must be supplied with calling sequence user_proc_cdf(y), which returns the value of the cumulative distribution function at any point y in the (optionally) specified range. Many of the cumulative distribution functions in Chapter 11, “Probability Distribution Functions and Inverses,” 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.
IMSLS_EXPECTED_VAL_LESS_THAN_1 An expected value is less than 1.
IMSLS_EXPECTED_VAL_LESS_THAN_5 An expected value is less than 5.
IMSLS_ALL_OBSERVATIONS_MISSING All observations contain missing values.
IMSLS_INCORRECT_CDF_1 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 1.
IMSLS_INCORRECT_CDF_2 Function user_proc_cdf is not a cumulative distribution function. The probability of the range of the distribution is not positive.
IMSLS_INCORRECT_CDF_3 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.
IMSLS_INCORRECT_CDF_4 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.
IMSLS_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.
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