Chapter 4: Analysis of Variance and Designed Experiments

anova_oneway

Analyzes a one-way classification model.

Synopsis

#include <imsls.h>

float imsls_f_anova_oneway (int n_groups, int n[], float y[], ..., 0)

The type double function is imsls_d_anova_oneway

Required Arguments

int n_groups   (Input)
Number of groups.

int n[]   (Input)
Array of length n_groups containing the number of responses for each group.

float y[]   (Input)
Array of length n [0] + n [1] + … + n [n_group − 1] containing the responses for each group.

Return Value

The p-value for the F-statistic.

Synopsis with Optional Arguments

#include <imsls.h>

float imsls_f_anova_oneway (int n_groups, int n[], float y[],
IMSLS_ANOVA_TABLE, float **anova_table,
IMSLS_ANOVA_TABLE_USER, float anova_table[],
IMSLS_GROUP_MEANS, float **means,
IMSLS_GROUP_MEANS_USER, float means[],
IMSLS_GROUP_STD_DEVS, float **std_devs,
IMSLS_GROUP_STD_DEVS_USER, float std_devs[],
IMSLS_GROUP_COUNTS, int **counts,
IMSLS_GROUP_COUNTS_USER, int counts[],
IMSLS_CONFIDENCE, float confidence,
IMSLS_TUKEY, float **ci_diff_means, or
IMSLS_DUNN_SIDAK, float **ci_diff_means, or
IMSLS_BONFERRONI, float **ci_diff_means, or
IMSLS_SCHEFFE, float **ci_diff_means, or
IMSLS_ONE_AT_A_TIME, float **ci_diff_means,
IMSLS_TUKEY_USER, float ci_diff_means[], or
IMSLS_DUNN_SIDAK_USER, float ci_diff_means[], or
IMSLS_BONFERRONI_USER, float ci_diff_means[], or
IMSLS_SCHEFFE_USER, float ci_diff_means[], or
IMSLS_ONE_AT_A_TIME_USER, float ci_diff_means[],
0)

Optional Arguments

IMSLS_ANOVA_TABLE, float **anova_table   (Output)
Address of a pointer to an internally allocated array of size 15 containing the analysis of variance table. The analysis of variance statistics are as follows:

Element

Analysis of Variance Statistics

0

degrees of freedom for the model

1

degrees of freedom for error

2

total (corrected) degrees of freedom

3

sum of squares for the model

4

sum of squares for error

5

total (corrected) sum of squares

6

model mean square

7

error mean square

8

overall F-statistic

9

p-value

10

R2 (in percent)

11

adjusted R2 (in percent)

12

estimate of the standard deviation

13

overall mean of y

14

coefficient of variation (in percent)

IMSLS_ANOVA_TABLE_USER, float anova_table[]   (Output)
Storage for array anova_table is provided by the user. See IMSLS_ANOVA_TABLE.

IMSLS_GROUP_MEANS, float **means   (Output)
Address of a pointer to an internally allocated array of length n_groups containing the group means.

IMSLS_GROUP_MEANS_USER, float means[]   (Output)
Storage for array means is provided by the user. See IMSLS_GROUP_MEANS.

IMSLS_GROUP_STD_DEVS, float **std_devs   (Output)
Address of a pointer to an internally allocated array of length n_groups containing the group standard deviations.

IMSLS_GROUP_STD_DEVS_USER, float std_devs[]   (Output)
Storage for array std_devs is provided by the user. See IMSLS_STD_DEVS.

IMSLS_GROUP_COUNTS, int **counts   (Output)
Address of a pointer to an internally allocated array of length n_groups containing the number of nonmissing observations for the groups.

IMSLS_GROUP_COUNTS_USER, int counts[]   (Output)
Storage for array counts is provided by the user. See IMSLS_COUNTS.

IMSLS_CONFIDENCE, float confidence   (Input)
Confidence level for the simultaneous interval estimation.
If IMSLS_TUKEY is specified, confidence must be in the range [90.0, 99.0). Otherwise, confidence is in the range [0.0, 100.0).
Default: confidence = 95.0

IMSLS_TUKEY, float **ci_diff_means   (Output), or

IMSLS_DUNN_SIDAK, float **ci_diff_means   (Output), or

IMSLS_BONFERRONI, float **ci_diff_means   (Output), or

IMSLS_SCHEFFE, float **ci_diff_means   (Output), or

IMSLS_ONE_AT_A_TIME, float **ci_diff_means   (Output)
Function imsls_f_anova_oneway computes the confidence intervals on all pairwise differences of means using any one of six methods: Tukey, Tukey-Kramer, Dunn-Šidák, Bonferroni, Scheffé, or Fisher’s LSD (One-at-a-Time). If IMSLS_TUKEY is specified, the Tukey confidence intervals are calculated if the group sizes are equal; otherwise, the Tukey-Kramer confidence intervals are calculated.

            On return, ci_diff_means contains the address of a pointer to a

            internally allocated array containing the statistics relating to the difference of means.

Column

Description

0

group number for the i-th mean

1

group number for the j-th mean

2

difference of means (i-th mean) − (j-th mean)

3

lower confidence limit for the difference

4

upper confidence limit for the difference

IMSLS_TUKEY_USER, float ci_diff_means[]   (Output), or

IMSLS_DUNN_SIDAK_USER, float ci_diff_means[]   (Output), or

IMSLS_BONFERRONI_USER, float ci_diff_means[]   (Output), or

IMSLS_SCHEFFE_USER, float ci_diff_means[]   (Output), or

IMSLS_ONE_AT_A_TIME_USER, float ci_diff_means[]   (Output)
Storage for array ci_diff_means is provided by the user.

Description

Function imsls_f_anova_oneway performs an analysis of variance of responses from a oneway classification design. The model is

yij = μi + ɛij        i = 1, 2, …, k; j = 1, 2, …, ni

where the observed value yij constitutes the j-th response in the i-th group,
μi denotes the population mean for the i-th group, and the ɛij arguments are errors that are identically and independently distributed normal with mean 0 and variance σ2. Function imsls_f_anova_oneway requires the yij observed responses as input into a single vector y with responses in each group occupying contiguous locations. The analysis of variance table is computed along with the group sample means and standard deviations. A discussion of formulas and interpretations for the one-way analysis of variance problem appears in most elementary statistics texts, e.g., Snedecor and Cochran (1967, Chapter 10).

Function imsls_f_anova_oneway computes simultaneous confidence intervals on all

pairwise comparisons of k means μ1 μ2, …, μk in the one-way analysis of variance model. Any of several methods can be chosen. A good review of these methods is given by Stoline (1981). The methods are also discussed in many elementary statistics texts, e.g., Kirk (1982, pp. 114−127).

Let s2 be the estimated variance of a single observation. Let v be the degrees of freedom associated with s2. Let

The methods are summarized as follows:

Tukey method: The Tukey method gives the narrowest simultaneous confidence intervals for all pairwise differences of means μi − μj in balanced
(n1 = n2 = … = nk = n) one-way designs. The method is exact and uses the Studentized range distribution. The formula for the difference μi − μj is given by

where q1−a;k,v is the (1 − α) 100 percentage point of the Studentized range distribution with parameters k and v.

Tukey-Kramer method: The Tukey-Kramer method is an approximate extension of the Tukey method for the unbalanced case. (The method simplifies to the Tukey method for the balanced case.) The method always produces confidence intervals narrower than the Dunn-Šidák and Bonferroni methods. Hayter (1984) proved that the method is conservative, i.e., the method guarantees a confidence coverage of at least (1 − α) 100. Hayter’s proof gave further support to earlier recommendations for its use (Stoline 1981). (Methods that are currently better are restricted to special cases and only offer improvement in severely unbalanced cases; see, for example, Spurrier and Isham 1985.) The formula for the difference μi − μj is given by the following:

Dunn-Šidák method: The Dunn-Šidák method is a conservative method. The method gives wider intervals than the Tukey-Kramer method. (For large v and small α and k, the difference is only slight.) The method is slightly better than the Bonferroni method and is based on an improved Bonferroni (multiplicative) inequality (Miller 1980, pp. 101, 254−255). The method uses the t distribution (see function imsls_f_t_inverse_cdf, Chapter 11, “Probability Distribution Functions and Inverses. The formula for the difference μi − μj is given by

where tf ;v is the 100f percentage point of the t distribution with ν degrees of freedom.

Bonferroni method: The Bonferroni method is a conservative method based on the Bonferroni (additive) inequality (Miller, p. 8). The method uses the t distribution. The formula for the difference μi − μj is given by the following:

Scheffé method: The Scheffé method is an overly conservative method for simultaneous confidence intervals on pairwise difference of means. The method is applicable for simultaneous confidence intervals on all contrasts, i.e., all linear combinations

where the following is true:

This method can be recommended here only if a large number of confidence intervals on contrasts in addition to the pairwise differences of means are to be constructed. The method uses the F distribution (see function imsls_f_F_inverse_cdf, Chapter 11, “Probabilty and Distribution Functions and Inverses”.p<.CSCH11.DOC!F_INVERSE_CDF;507;). The formula for the difference μi − μj is given by

where F1−a; ( k−1),v is the (1 − α) 100 percentage point of the F distribution with
k − 1 and ν degrees of freedom.

One-at-a-Time t method (Fisher’s LSD): The One-at-a-Time t method is appropriate for constructing a single confidence interval. The confidence percentage input is appropriate for one interval at a time. The method has been used widely in conjunction with the overall test of the null hypothesis
μ1 = μ2 = … = μk by the use of the F statistic. Fisher’s LSD (least significant difference) test is a two-stage test that proceeds to make pairwise comparisons of means only if the overall F test is significant. Milliken and Johnson (1984, p. 31) recommend LSD comparisons after a significant F only if the number of comparisons is small and the comparisons were planned prior to the analysis. If many unplanned comparisons are made, they recommend Scheffé’s method. If the F test is insignificant, a few planned comparisons for differences in means can still be performed by using either Tukey, Tukey-Kramer, Dunn-Šidák,or Bonferroni methods. Because the F test is insignificant, Scheffé’s method does not yield any significant differences. The formula for the difference μi − μj is given by the following:

Examples

Example 1

This example computes a one-way analysis of variance for data discussed by Searle (1971, Table 5.1, pp. 165−179). The responses are plant weights for six plants of three different types—three normal, two off-types, and one aberrant. The responses are given by type of plant in the following table:

Normal

Off-Type

Aberrant

101

84

32

105

88

 

94

 

 


#include <imsls.h>
main()
{
    int     n_groups=3;
    int     n[] = {3, 2, 1};
    float   y[] = {101.0, 105.0, 94.0, 84.0, 88.0, 32.0};
    float   p_value;
    p_value = imsls_f_anova_oneway (n_groups, n, y, 0);
    printf ("p-value = %6.4f", p_value);
  }

Output

p-value = 0.002

Example 2

The data used in this example is the same as that used in the initial example. Here, the anova_table is printed.

#include <imsls.h>
main()
{
    int     n_groups=3;
    int     n[] = {3, 2, 1};
    float   y[] = {101.0, 105.0, 94.0, 84.0, 88.0, 32.0};
    float   p_value;
    float   *anova_table;
    char    *labels[] = {
                   "degrees of freedom for among groups",
                   "degrees of freedom for within groups",
                   "total (corrected) degrees of freedom",
                   "sum of squares for among groups",
                   "sum of squares for within groups",
                   "total (corrected) sum of squares",
                   "among mean square",
                   "within mean square", "F-statistic",
                   "p-value", "R-squared (in percent)",
                   "adjusted R-squared (in percent)",
                   "est. standard deviation of within error",
                   "overall mean of y",
                   "coefficient of variation (in percent)"};

                      /* Perform analysis */
    p_value = imsls_f_anova_oneway (n_groups, n, y,
        IMSLS_ANOVA_TABLE, &anova_table,
        0);
                      /* Print results */
    imsls_f_write_matrix("* * * Analysis of Variance * * *\n", 15, 1,
        anova_table,
        IMSLS_ROW_LABELS, labels,
        IMSLS_WRITE_FORMAT, "%9.2f",
        0);
}

Output

         * * * Analysis of Variance * * *
degrees of freedom for among groups           2.00
degrees of freedom for within groups          3.00
total (corrected) degrees of freedom          5.00
sum of squares for among groups            3480.00
sum of squares for within groups             70.00
total (corrected) sum of squares           3550.00
among mean square                          1740.00
within mean square                           23.33
F-statistic                                  74.57
p-value                                       0.00
R-squared (in percent)                       98.03
adjusted R-squared (in percent)              96.71
est. standard deviation of within error       4.83
overall mean of y                            84.00
coefficient of variation (in percent)         5.75

Example 3

Simultaneous confidence intervals are generated for the following measurements of cold-cranking power for five models of automobile batteries. Nelson (1989, pp. 232−241) provided the data and approach.

Model 1

Model 2

Model 3

Model 4

Model 5

41

42

27

48

28

43

43

26

45

32

42

46

28

51

37

46

38

27

46

25

The Tukey method is chosen for the analysis of pairwise comparisons, with a confidence level of 99 percent. The means and their confidence limits are output.

#include <imsls.h>

void main()
{

   int    n_groups = 5;
   int    n[] = {4, 4, 4, 4, 4};
   int    permute[] = {2, 3, 4, 0, 1};
   float  y[] = {41.0, 43.0, 42.0, 46.0, 42.0,
                43.0, 46.0, 38.0, 27.0, 26.0,
                28.0, 27.0, 48.0, 45.0, 51.0,
                46.0, 28.0, 32.0, 37.0, 25.0};
   float  *anova_table, *ci_diff_means, tmp_diff_means[50];
   float  confidence = 99.0;
   char   *labels[] = {
                    "degrees of freedom for among groups",
                    "degrees of freedom for within groups",
                    "total (corrected) degrees of freedom",
                    "sum of squares for among groups",
                    "sum of squares for within groups",
                    "total (corrected) sum of squares",
                    "among mean square",
                    "within mean square", "F-statistic",
                    "p-value", "R-squared (in percent)",
                    "adjusted R-squared (in percent)",
                    "est. standard deviation of within error",
                    "overall mean of y",
                    "coefficient of variation (in percent)"};
   char   *mean_row_labels[] = {
                    "first and second",
                    "first and third",
                    "first and fourth",
                    "first and fifth",
                    "second and third",
                    "second and fourth",
                    "second and fifth",
                    "third and fourth",
                    "third and fifth",
                    "fourth and fifth"};
   char   *mean_col_labels[] = {
                    "Means",
                    "Difference of means",
                    "Lower limit",
                    "Upper limit"};
                        /* Perform analysis */
  
 imsls_f_anova_oneway(n_groups, n, y,
        IMSLS_ANOVA_TABLE, &anova_table,
        IMSLS_CONFIDENCE, confidence,
        IMSLS_TUKEY, &ci_diff_means,
        0);
                        /* Print anova_table */
   imsls_f_write_matrix("* * * Analysis of Variance * * *\n", 15,
        1, anova_table,
        IMSLS_ROW_LABELS, labels,
        IMSLS_WRITE_FORMAT, "%9.2f",
        0);
                      /* Permute ci_diff_means for printing */
   imsls_f_permute_matrix(10, 5, ci_diff_means, permute,
        IMSLS_PERMUTE_COLUMNS,
        IMSLS_RETURN_USER, tmp_diff_means,
        0);
                      /* Print ci_diff_means */
   imsls_f_write_matrix("* * * Differences in Means * * *\n", 10,
        3, tmp_diff_means,
        IMSLS_A_COL_DIM, 5,
        IMSLS_ROW_LABELS, mean_row_labels,
        IMSLS_COL_LABELS, mean_col_labels,
        IMSLS_WRITE_FORMAT, "%9.2f",
        0);
}

Output

         * * * Analysis of Variance * * *

degrees of freedom for among groups           4.00
degrees of freedom for within groups         15.00
total (corrected) degrees of freedom         19.00
sum of squares for among groups            1242.20
sum of squares for within groups            150.75
total (corrected) sum of squares           1392.95
among mean square                           310.55
within mean square                           10.05
F-statistic                                  30.90
p-value                                       0.00
R-squared (in percent)                       89.18
adjusted R-squared (in percent)              86.29
est. standard deviation of within error       3.17
overall mean of y                            38.05
coefficient of variation (in percent)         8.33
 
           * * * Differences in Means * * *

Means              Difference  Lower limit  Upper limit
                     of means                         
first and second         0.75        -8.05         9.55
first and third         16.00         7.20        24.80
first and fourth        -4.50       -13.30         4.30
first and fifth         12.50         3.70        21.30
second and third        15.25         6.45        24.05
second and fourth       -5.25       -14.05         3.55
second and fifth        11.75         2.95        20.55
third and fourth       -20.50       -29.30       -11.70
third and fifth         -3.50       -12.30         5.30
fourth and fifth        17.00         8.20        25.80

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