normal_two_sample
Computes statistics for mean and variance inferences using samples from two normal populations.
Synopsis
#include <imsls.h>
float imsls_f_normal_two_sample (int n1_observations, float x1[], int n2_observations, float x2[], ..., 0)
The type double function is imsls_d_normal_two_sample.
Required Arguments
int n1_observations (Input)
Number of observations in the first sample, x1.
float x1[] (Input)
Array of length n1_observations containing the first sample.
int n2_observations (Input)
Number of observations in the second sample, x2.
float x2[] (Input)
Array of length n2_observations containing the second sample.
Return Value
Difference in means, x1_mean − x2_mean.
Synopsis with Optional Arguments
#include <imsls.h>
float imsls_f_normal_two_sample (int n1_observations, float x1[], int n2_observations, float x2[],
IMSLS_MEANS, float *x1_mean, float *x2_mean,
IMSLS_CONFIDENCE_MEAN, float confidence_mean,
IMSLS_INTERMEDIATE_RESULTS, float stats[],
IMSLS_UNION, float stats1[], float stats2[],
IMSLS_FINAL_RESULTS, float final_stats[],
IMSLS_CI_DIFF_FOR_EQUAL_VARS, float *lower_limit, float *upper_limit,
IMSLS_CI_DIFF_FOR_UNEQUAL_VARS, float *lower_limit, float *upper_limit
IMSLS_T_TEST_FOR_EQUAL_VARS, int *df, float *t, float *p_value,
IMSLS_T_TEST_FOR_UNEQUAL_VARS, float *df, float *t, float *p_value,
IMSLS_T_TEST_NULL, float mean_hypothesis_value,
IMSLS_POOLED_VARIANCE, float *pooled_variance,
IMSLS_CONFIDENCE_VARIANCE, float confidence_variance,
IMSLS_CI_COMMON_VARIANCE, float *lower_limit, float *upper_limit,
IMSLS_CHI_SQUARED_TEST, int *df, float *chi_squared, float *p_value,
IMSLS_CHI_SQUARED_TEST_NULL, float variance_hypothesis_value,
IMSLS_STD_DEVS, float *x1_std_dev, float *x2_std_dev,
IMSLS_CI_RATIO_VARIANCES, float *lower_limit, float *upper_limit,
IMSLS_F_TEST, int *df_numerator, int *df_denominator, float *F, float *p_value,
0)
Optional Arguments
IMSLS_MEANS, float *x1_mean, float *x2_mean (Output)
Means of the first and second samples.
IMSLS_CONFIDENCE_MEAN, float confidence_mean (Input)
Confidence level for two-sided interval estimate of the mean of x1 minus the mean of x2, in percent. Argument confidence_mean must be between 0.0 and 100.0 and is often 90.0, 95.0, or 99.0. For a one-sided confidence interval with confidence level c (at least 50 percent), set confidence_mean = 100.0 − 2.0 × (100.0 − c).
Default: confidence_mean = 95.0
Note: The following three optional arguments allow the analysis to be applied to subsets of the original data sets and then later combined for final results. These optional arguments may be useful when analyzing data sets too large to fit into memory, and also allow subsets of the data to be analyzed in separate threads and later combined for final results.
IMSLS_INTERMEDIATE_RESULTS, float stats[] (Input/Output)
Array of length 25 containing intermediate results. On input, stats contains intermediate statistics about a previous function invocation. When invoking the function the first time, set all stats elements to 0.0. On output, imsls_f_normal_two_sample combines the results on the current data sets and the intermediate statistics in stats.
This optional argument can be applied to separate blocks of data when physical memory cannot hold the entire data sets.
Note that when IMSLS_INTERMEDIATE_RESULTS optional argument is used, imsls_f_normal_two_sample function outputs are valid only if IMSLS_FINAL_RESULTS optional argument is specified to calculate the final statistics. See Example 3.
Default: stats = NULL.
IMSLS_UNION, float stats1[], float stats2[] (Input)
stats1 and stats2 are arrays of length 25 containing the intermediate results about previous computations. stats1 and stats2 are the variables provided to the IMSLS_INTERMEDIATE_RESULTS optional argument in previous function invocations.
imsls_f_normal_two_sample combines the results on the current data sets and the intermediate statistics in stats1 and stats2. stats1 and stats2 can be NULL. See Example 3.
This option would typically be used in conjunction with the IMSLS_INTERMEDIATE_RESULTS option to process a large data set using separate threads or compute nodes. For example, a data set could be split into two subsets, where each subset of data is passed into a separate thread or compute node and processed through imsls_f_normal_two_sample with the IMSLS_INTERMEDIATE_RESULTS option. The output from each thread is then saved and input to a final call of imsls_f_normal_two_sample using option IMSLS_UNON and IMSLS_FINAL_RESULTS.
Default: stats1 = NULL and stats2 = NULL.
IMSLS_FINAL_RESULTS, float final_stats[] (Output)
Array of length 25 containing the final statistics. See Example 3.
Elements of final_stats are:
|
index |
final_stats[i] |
|
0 |
Mean of the first sample. |
|
1 |
Mean of the second sample. |
|
2 |
Variance of the first sample. |
|
3 |
Variance of the second sample. |
|
4 |
Number of observations in the first sample. |
|
5 |
Number of observations in the second sample. |
|
Note: final_stats[6] through final_stats[13] depend on the assumption of equal variances. |
|
|
6 |
Pooled variance. |
|
7 |
t value, assuming equal variances. |
|
8 |
Probability of a larger t in absolute value, assuming normality, equal means, and equal variances. |
|
9 |
Degrees of freedom assuming equal variances. |
|
10 |
Lower confidence limit for the mean of the first population minus the mean of the second, assuming equal variances. |
|
11 |
Upper confidence limit for the mean of the first population minus the mean of the second, assuming equal variances. |
|
12 |
Lower confidence limit for the common variance. |
|
13 |
Upper confidence limit for the common variance. |
|
Note: final_stats[14] through final_stats[18] use approximations that do not depend on an assumption of equal variances. |
|
|
14 |
t value, assuming unequal variances. |
|
15 |
Approximate probability of a larger t in absolute value, assuming normality, equal means, and unequal variances. |
|
16 |
Degrees of freedom assuming unequal variances, for Satterthwaite's approximation. |
|
17 |
Approximate lower confidence limit for the mean of the first population minus the mean of the second, assuming equal variances. |
|
18 |
Approximate upper confidence limit for the mean of the first population minus the mean of the second, assuming equal variances. |
|
19 |
F value (greater than or equal to 1.0). |
|
20 |
Probability of a larger F in absolute value, assuming normality and equal variances. |
|
21 |
Lower confidence limit for the ratio of the variance of the first population to the second. |
|
22 |
Upper confidence limit for the ratio of the variance of the first population to the second. |
|
23 |
Number of missing values of first sample. |
|
24 |
Number of missing values of second sample. |
IMSLS_CI_DIFF_FOR_EQUAL_VARS, float *lower_limit, float *upper_limit (Output)
Argument lower_limit contains the lower confidence limit, and upper_limit contains the upper limit for the mean of the first population minus the mean of the second, assuming equal variances.
IMSLS_CI_DIFF_FOR_UNEQUAL_VARS, float *lower_limit, float *upper_limit (Output)
Argument lower_limit contains the approximate lower confidence limit, and upper_limit contains the approximate upper limit for the mean of the first population minus the mean of the second, assuming unequal variances.
IMSLS_T_TEST_FOR_EQUAL_VARS, int *df, float *t, float *p_value (Output)
A t test for μ1 − μ2 = c, where c is the null hypothesis value. (See the description of IMSLS_T_TEST_NULL.) Argument df contains the degrees of freedom, argument t contains the t value, and argument p_value contains the probability of a larger t in absolute value, assuming equal means. This test assumes equal variances.
IMSLS_T_TEST_FOR_UNEQUAL_VARS, float *df, float *t, float *p_value (Output)
A t test for μ1 − μ2 = c, where c is the null hypothesis value. (See the description of IMSLS_T_TEST_NULL.) Argument df contains the degrees of freedom for Satterthwaite’s approximation, argument t contains the t value, and argument p_value contains the approximate probability of a larger t in absolute value, assuming equal means. This test does not assume equal variances.
IMSLS_T_TEST_NULL, float mean_hypothesis_value (Input)
Null hypothesis value for the t test.
Default: mean_hypothesis_value = 0.0
IMSLS_POOLED_VARIANCE, float *pooled_variance (Output)
Pooled variance for the two samples.
IMSLS_CONFIDENCE_VARIANCE, float confidence_variance (Input)
Confidence level for inference on variances. Under the assumption of equal variances, the pooled variance is used to obtain a two-sided confidence_variance percent confidence interval for the common variance if IMSLS_CI_COMMON_VARIANCE is specified. Without making the assumption of equal variances, the ratio of the variances is of interest. A two-sided confidence_variance percent confidence interval for the ratio of the variance of the first sample to that of the second sample is computed and is returned if IMSLS_CI_RATIO_VARIANCES is specified. The confidence intervals are symmetric in probability.
Default: confidence_variance = 95.0
IMSLS_CI_COMMON_VARIANCE, float *lower_limit, float *upper_limit (Output)
Argument lower_limit contains the lower confidence limit, and upper_limit contains the upper limit for the common, or pooled, variance.
IMSLS_CHI_SQUARED_TEST, int *df, float *chi_squared, float *p_value (Output)
The chi-squared test for
is the common, or pooled, variance, and
is the null hypothesis value. (See description of IMSLS_CHI_SQUARED_TEST_NULL.) Argument df contains the degrees of freedom, argument chi_squared contains the chi-squared value, and argument p_value contains the probability of a larger chi-squared in absolute value, assuming equal means.
IMSLS_CHI_SQUARED_TEST_NULL, float variance_hypothesis_value (Input)
Null hypothesis value for the chi-squared test.
Default: variance_hypothesis_value = 1.0
IMSLS_STD_DEVS, float *x1_std_dev, float *x2_std_dev (Output)
Standard deviations of the first and second samples.
IMSLS_CI_RATIO_VARIANCES, float *lower_limit, float *upper_limit (Output)
Argument lower_limit contains the approximate lower confidence limit, and upper_limit contains the approximate upper limit for the ratio of the variance of the first population to the second.
IMSLS_F_TEST, int *df_numerator, int *df_denominator, float *F, float *p_value (Output)
The F test for equality of variances. Argument df_numerator and df_denominator contain the numerator and denominator degrees of freedom, argument F contains the F test value, and argument p_value contains the probability of a larger F in absolute value, assuming equal variances.
Description
Function imsls_f_normal_two_sample computes statistics for making inferences about the means and variances of two normal populations, using independent samples in x1 and x2. For inferences concerning parameters of a single normal population, see function imsls_f_normal_one_sample.
Let μ1 and 
be the mean and variance of the first population, and let μ2 and 
be the corresponding quantities of the second population. The function contains test confidence intervals for difference in means, equality of variances, and the pooled variance.
The means and variances for the two samples are as follows:
and
Inferences about the Means
The test that the difference in means equals a certain value, for example, μ0, depends on whether or not the variances of the two populations can be considered equal. If the variances are equal and mean_hypothesis_value equals 0, the test is the two-sample t test, which is equivalent to an analysis-of-variance test. The pooled variance for the difference-in-means test is as follows:
The t statistic is as follows:
Also, the confidence interval for the difference in means can be obtained by specifying IMSLS_CI_DIFF_FOR_EQUAL_VARS.
If the population variances are not equal, the ordinary t statistic does not have a t distribution and several approximate tests for the equality of means have been proposed. (See, for example, Anderson and Bancroft 1952, and Kendall and Stuart 1979.) One of the earliest tests devised for this situation is the Fisher-Behrens test, based on Fisher’s concept of fiducial probability. A procedure used if IMSLS_T_TEST_FOR_UNEQUAL_VARS and/or IMSLS_CI_DIFF_FOR_UNEQUAL_VARS are specified is the Satterthwaite’s procedure, as suggested by H.F. Smith and modified by F.E. Satterthwaite (Anderson and Bancroft 1952, p. 83).
The test statistic is
where
Under the null hypothesis of μ1 − μ2 = c, this quantity has an approximate t distribution with degrees of freedom df (in IMSLS_T_TEST_FOR_UNEQUAL_VARS), given by the following equation:
Inferences about Variances
The F statistic for testing the equality of variances is given by 
, where 
is the larger of 
and 
. If the variances are equal, this quantity has an F distribution with n1 − 1 and n2 − 1 degrees of freedom.
It is generally not recommended that the results of the F test be used to decide whether to use the regular t test or the modified tʹ on a single set of data. The modified tʹ (Satterthwaite’s procedure) is the more conservative approach to use if there is doubt about the equality of the variances.
Examples
Example 1
This example, taken from Conover and Iman (1983, p. 294), involves scores on arithmetic tests of two grade-school classes. The question is whether a group taught by an experimental method has a higher mean score. Only the difference in means is output. The data are shown below.
|
Scores for Standard Group |
Scores for Experimental Group |
|
72 |
111 |
|
75 |
118 |
|
77 |
128 |
|
80 |
138 |
|
104 |
140 |
|
110 |
150 |
|
125 |
163 |
|
|
164 |
|
|
169 |
#include <imsls.h>
#include <stdio.h>
int main()
{
#define N1_OBSERVATIONS 7
#define N2_OBSERVATIONS 9
float diff_means;
float x1[N1_OBSERVATIONS] = {
72.0, 75.0, 77.0, 80.0, 104.0, 110.0, 125.0
};
float x2[N2_OBSERVATIONS] = {
111.0, 118.0, 128.0, 138.0, 140.0, 150.0, 163.0, 164.0, 169.0
};
/* Perform analysis */
diff_means = imsls_f_normal_two_sample(
N1_OBSERVATIONS, x1,
N2_OBSERVATIONS, x2,
0);
/* Print results */
printf("\nx1_mean - x2_mean = %5.2f\n", diff_means);
}
Output
x1_mean - x2_mean = -50.48
Example 2
The same data is used for this example as for the initial example. Here, the results of the t test are output. The variances of the two populations are assumed to be equal. It is seen from the output that there is strong reason to believe that the two means are different (t value of −4.804). Since the lower 97.5-percent confidence limit does not include 0, the null hypothesis is that μ1 ≤ μ 2 would be rejected at the 0.05 significance level. (The closeness of the values of the sample variances provides some qualitative substantiation of the assumption of equal variances.)
#include <imsls.h>
#include <stdio.h>
int main()
{
#define N1_OBSERVATIONS 7
#define N2_OBSERVATIONS 9
int df;
float diff_means, lower_limit, upper_limit, t, p_value, sp2;
float x1[N1_OBSERVATIONS] = {
72.0, 75.0, 77.0, 80.0, 104.0, 110.0, 125.0
};
float x2[N2_OBSERVATIONS] = {
111.0, 118.0, 128.0, 138.0, 140.0, 150.0, 163.0, 164.0, 169.0
};
/* Perform analysis */
diff_means = imsls_f_normal_two_sample(N1_OBSERVATIONS, x1,
N2_OBSERVATIONS, x2,
IMSLS_POOLED_VARIANCE, &sp2,
IMSLS_CI_DIFF_FOR_EQUAL_VARS, &lower_limit, &upper_limit,
IMSLS_T_TEST_FOR_EQUAL_VARS, &df, &t, &p_value,
0);
/* Print results */
printf("\nx1_mean - x2_mean = %5.2f\n", diff_means);
printf("Pooled variance = %5.2f\n", sp2);
printf("95%% CI for x1_mean - x2_mean is (%5.2f,%5.2f)\n",
lower_limit, upper_limit);
printf("df = %3d\n", df);
printf("t = %5.2f\n", t);
printf("p-value = %8.5f\n", p_value);
}
Output
x1_mean - x2_mean = -50.48
Pooled variance = 434.63
95% CI for x1_mean - x2_mean is (-73.01,-27.94)
df = 14
t = -4.80
p-value = 0.00028
Example 3
The same data is used for this example as for the initial example. This example illustrates the use of the IMSLS_INTERMEDIATE_RESULTS, IMSLS_UNION, and IMSLS_FINAL_RESULTS optional arguments with "x1" and "x2" divided into three sub-groups. Since there are more "x2" values than "x1" values, n1_observations is set to zero on later calls to the function.
This example demonstrates how the analysis can be applied to subsets of the original data sets and then later combined for final results. These techniques may be useful when analyzing data sets too large to fit into memory, and also allow subsets of the data to be analyzed in separate threads (though this example does not show the use of separate threads) and later combined for final results.
#include <imsls.h>
#include <stdio.h>
#define N1_OBSERVATIONS 7
#define N2_OBSERVATIONS 9
int main()
{
int n1, n2, i;
float diff_means, pooled_variance;
float x1[N1_OBSERVATIONS] = {
72.0, 75.0, 77.0, 80.0, 104.0, 110.0, 125.0
};
float x2[N2_OBSERVATIONS] = {
111.0, 118.0, 128.0, 138.0, 140.0, 150.0, 163.0,
164.0, 169.0
};
float stats1[25], stats2[25], final_stats[25];
/* Initialize variables. */
for (i = 0; i < 25; i++) {
stats1[i] = 0.0;
stats2[i] = 0.0;
final_stats[i] = 0.0;
}
/*
** Bring in first group of observations on x1 and x2.
** Save intermediate results into variable, stat1.
**
** Total second group:
** n1_observations = 3, n2_observations = 3
*/
/* First call using: n1_observations = 2, n2_observations = 2 */
n1 = 2;
n2 = 2;
imsls_f_normal_two_sample(n1, x1, n2, x2,
IMSLS_INTERMEDIATE_RESULTS, stats1,
0);
/* Second call using: n1_observations = 1, n2_observations = 1 */
n1 = 1;
n2 = 1;
imsls_f_normal_two_sample(n1, &x1[2], n2, &x2[2],
IMSLS_INTERMEDIATE_RESULTS, stats1,
0);
/*
** Bring in second group of observations on x1 and x2.
** Save intermediate results into variable, stat2.
**
** Total second group:
** n1_observations = 4, n2_observations = 4.
*/
n1 = 4;
n2 = 4;
imsls_f_normal_two_sample(n1, &x1[3], n2, &x2[3],
IMSLS_INTERMEDIATE_RESULTS, stats2,
0);
/*
** Bring in third group of observations on x1 and x2
** and combine the results in variables, stats1 and stats2,
** from the first and second groups.
**
** Total third group:
** n1_observations = 0, n2_observations = 2.
*/
n1 = 0;
n2 = 2;
diff_means = imsls_f_normal_two_sample(n1, x1, n2, &x2[7],
IMSLS_UNION, stats1, stats2,
IMSLS_FINAL_RESULTS, final_stats,
IMSLS_POOLED_VARIANCE, &pooled_variance,
0);
/* Print results */
printf("x1_mean - x2_mean = %5.2f\n", diff_means);
printf("pooled variance = %5.2f\n", pooled_variance);
}
Output
x1_mean - x2_mean = -50.48
pooled variance = 434.63
