multiple_comparisons
Performs multiple comparisons of means using one of Student-Newman-Keuls, LSD, Bonferroni, or Tukey’s procedures.
Synopsis
#include <imsls.h>
int *imsls_f_multiple_comparisons (int n_groups, float means[], int df, float std_error, …, 0)
The type double function is imsls_d_multiple_comparisons.
Required Arguments
int n_groups (Input)
Number of groups i.e., means, being compared.
float means[] (Input)
Array of length n_groups containing the means.
int df (Input)
Degrees of freedom associated with std_error.
float std_error (Input)
Effective estimated standard error of a mean. In fixed effects models, std_error equals the estimated standard error of a mean. For example, in a one-way model
where s2 is the estimate of σ2 and n is the number of responses in a sample mean. In models with random components, use
where sedif is the estimated standard error of the difference of two means.
Return Value
Pointer to the array of length n_groups - 1 indicating the size of the groups of means declared to be equal. Value equal_means [I] = J indicates the I-th smallest mean and the next J - 1 larger means are declared equal. Value equal_means [I] = 0 indicates no group of means starts with the I-th smallest mean.
Synopsis with Optional Arguments
#include <imsls.h>
int *imsls_f_multiple_comparisons (int n_groups, float means [], int df, float std_error,
IMSLS_ALPHA, float alpha,
IMSLS_SNK, or
IMSLS_LSD, or
IMSLS_TUKEY, or
IMSLS_BONFERRONI,
IMSLS_RETURN_USER, int *equal_means,
0)
Optional Arguments
IMSLS_ALPHA, float alpha (Input)
Significance level of test. Argument alpha must be in the interval [0.01, 0.10].
Default: alpha = 0.01
IMSLS_RETURN_USER, int *equal_means (Output)
If specified, equal_means is an array of length n_groups - 1 specified by the user. On return, equal_means contains the size of the groups of means declared to be equal. Value equal_means [I] = J indicates the i‑th smallest mean and the next J - 1 larger means are declared equal. Value equal_means [I] = 0 indicates no group of means starts with the i‑th smallest mean.
IMSLS_SNK
or
IMSLS_LSD
or
IMSLS_TUKEY
or
IMSLS_BONFERRONI
|
Argument |
Method |
|
IMSLS_SNK |
Student-Newman-Keuls (default) |
|
IMSLS_LSD |
Least significant difference |
|
IMSLS_TUKEY |
Tukey’s w-procedure, also called the honestly significant difference procedure. |
|
IMSLS_BONFERRONI |
Bonferroni t statistic |
Description
Function imsls_f_multiple_comparisons performs a multiple comparison analysis of means using one of Student-Newman-Keuls, LSD, Bonferroni, or Tukey’s procedures. The null hypothesis is equality of all possible ordered subsets of a set of means. The methods are discussed in many elementary statistics texts, e.g., Kirk (1982, pp. 123–125).
The output consists of an array of n_groups –1 integers that describe grouping of means that are considered not statistically significantly different.
For example, if n_groups=4 and the returned array is equal to {0, 2, 2} then we conclude that:
| 1. | The smallest mean is significantly different from the others. |
| 2. | The second and third smallest means are not significantly different from one another. |
| 3. | The second and fourth means are significantly different. |
| 4. | The third and fourth means are not significantly different from one another. |
These relationships can be depicted graphically as three groups of means:
|
Smallest |
Group |
Group |
Group |
|
1 |
x |
|
|
|
2 |
|
x |
|
|
3 |
|
x |
x |
|
4 |
|
|
x |
Examples
Example 1
A multiple-comparisons analysis is performed using data discussed by Kirk (1982, pp. 123-125). The results show that there are three groups of means with three separate sets of values: (36.7, 40.3, 43.4), (40.3, 43.4, 47.2), and (43.4, 47.2, 48.7).
In this case, the ordered means are {36.7, 40.3, 43.4, 47.2, 48.7} corresponding to treatments {1, 5, 3, 4, 2}. Since the output table is:
we can say that within each of these three groups, means are not significantly different from one another.
|
Treatment |
Mean |
Group |
Group |
Group |
|
1 |
36.7 |
x |
|
|
|
5 |
40.3 |
x |
x |
|
|
3 |
43.4 |
x |
x |
x |
|
4 |
47.2 |
|
x |
x |
|
2 |
48.7 |
|
|
x |
#include <imsls.h>
int main ()
{
int n_groups = 5;
int df = 45;
float std_error = 1.6970563;
float means[5] = {36.7, 48.7, 43.4, 47.2, 40.3};
int *equal_means;
/* Perform multiple comparisons tests */
equal_means = imsls_f_multiple_comparisons(n_groups, means, df,
std_error, 0);
/* Print results */
imsls_i_write_matrix("Size of Groups of Means", 1, n_groups-1,
equal_means, 0);
}
Output
Size of Groups of Means
1 2 3 4
3 3 3 0
Example 2
This example uses the same data as the previous example but also uses additional methods by specifying optional arguments.
Example 2 uses the same data as Example 1: Ordered treatment means correspond to treatment order {1,5,3,4,2}.
The table produced for Bonferroni is:
Thus, these are two groups of similar means.
|
Treatment |
Mean |
Group |
Group |
|
1 |
36.7 |
x |
|
|
5 |
40.3 |
x |
x |
|
3 |
43.4 |
x |
x |
|
4 |
47.2 |
|
x |
|
2 |
48.7 |
|
x |
#include <imsls.h>
int main()
{
int n_groups = 5;
int df = 45;
float std_error = 1.6970563;
float means[5] = {36.7, 48.7, 43.4, 47.2, 40.3};
int equal_means[4];
/* Student-Newman-Keuls */
imsls_f_multiple_comparisons(n_groups, means, df, std_error,
IMSLS_RETURN_USER, equal_means,
0);
imsls_i_write_matrix("SNK ", 1, n_groups-1, equal_means,
0);
/* Bonferroni */
imsls_f_multiple_comparisons(n_groups, means, df, std_error,
IMSLS_BONFERRONI,
IMSLS_RETURN_USER, equal_means,
0);
imsls_i_write_matrix("Bonferonni ", 1, n_groups-1, equal_means,
0);
/* Least Significant Difference */
imsls_f_multiple_comparisons(n_groups, means, df, std_error,
IMSLS_LSD,
IMSLS_RETURN_USER, equal_means,
0);
imsls_i_write_matrix("LSD ", 1, n_groups-1, equal_means,
0);
/* Tukey's */
imsls_f_multiple_comparisons(n_groups, means, df, std_error,
IMSLS_TUKEY,
IMSLS_RETURN_USER, equal_means,
0);
imsls_i_write_matrix("Tukey ", 1, n_groups-1, equal_means,
0);
}
Output
SNK
1 2 3 4
3 3 3 0
Bonferonni
1 2 3 4
3 4 0 0
LSD
1 2 3 4
2 2 3 0
Tukey
1 2 3 4
3 3 3 0
