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### IMSL_ANOVAFACT

IMSL_ANOVAFACT

The IMSL_ANOVAFACT function analyzes a balanced factorial design with fixed effects.

This routine requires an IDL Advanced Math and Stats license. For more information, contact your sales or technical support representative.

The IMSL_ANOVAFACT function performs an analysis for an n-way classification design with balanced data. For balanced data, there must be an equal number of responses in each cell of the n-way layout. The effects are assumed to be fixed effects. The model is an extension of the two-way model to include n factors. The interactions (two-way, three-way, up to n-way) can be included in the model, or some of the higher-way interactions can be pooled into error. The keyword Order specifies the number of factors to be included in the highest-way interaction. For example, if three-way and higher-way interactions are to be pooled into error, set Order = 2.

By default, Order = N_ELEMENTS (n_levels) – 1 with the last subscript being the replicates subscript. Keyword Pure_Error indicates there are repeated responses within the n-way cell; Pool_Inter indicates otherwise.

The IMSL_ANOVAFACT function requires the responses as input into a single vector y in lexicographical order, so that the response subscript associated with the first factor varies least rapidly, followed by the subscript associated with the second factor, and so forth. Hemmerle (1967, Chapter 5) discusses the computational method.

## Examples

### Example 1

A two-way analysis of variance is performed with balanced data discussed by Snedecor and Cochran (1967, Table 12.5.1, p. 347). The responses are the weight gains (in grams) of rats that were fed diets varying in the source (A) and level (B) of protein.

The model is:

for

where

for

and

for i = 0, 1. The first responses in each cell in the two-way layout are given in the table that follows.

 Protein Level (B) Protein Source (A) Beef Cereal Pork High 73, 102, 118, 104, 81, 107, 100, 87, 117, 111 98, 74, 56, 111, 95, 88, 82, 77, 86, 92 94, 79, 96, 98, 102, 102, 108, 91, 120, 105 Low 90, 76, 90, 64, 86, 51, 72, 90, 95, 78 107, 95, 97, 80, 98, 74, 74, 67, 89, 58 49, 82, 73, 86, 81, 97, 106, 70, 61, 82
`n = [3, 2, 10]`
`  y = [73.0, 102.0, 118.0, 104.0,	81.0, \$`
`  107.0, 100.0,	87.0, 117.0, 111.0, \$`
`  90.0,	76.0,	90.0,	64.0,	86.0, \$`
`  51.0,	72.0,	90.0,	95.0,	78.0, \$`
`  98.0,	74.0,	56.0, 111.0,	95.0, \$`
`  88.0,	82.0,	77.0,	86.0,	92.0, \$`
`  107.0,	95.0,	97.0,	80.0,	98.0, \$`
`  74.0,	74.0,	67.0,	89.0,	58.0, \$`
`  94.0,	79.0,	96.0,	98.0, 102.0, \$`
`  102.0, 108.0,	91.0, 120.0, 105.0, \$`
`  49.0,	82.0,	73.0,	86.0,	81.0, \$`
`  97.0, 106.0,	70.0,	61.0,	82.0]`
`p_value = IMSL_ANOVAFACT(n, y, Anova_Table = anova_table) `
`PRINT, 'p-value = ', p_value`
` `
`p-value =	0.00229943`

### Example 2: Two-way ANOVA

In this example, the same model and data are fit as in the initial example, but keywords are used for a more complete analysis. First, a procedure to output the results is defined.

`.RUN`
`PRO print_results, anova_table, test_effects, means `
`anova_labels = ['df for among groups', \$`
`  'df for within groups', 'total (corrected) df', \$`
`  'ss for among groups', 'ss for within groups', \$`
`  'total (corrected) ss', 'mean square among groups', \$`
`  'mean square within groups', 'F-statistic', \$`
`  'P-value', 'R-squared (in percent)', \$`
`  'adjusted R-squared (in percent)', \$`
`  'est. std of within group error', 'overall mean of y', \$`
`  'coef. of variation (in percent)'] `
`effects_labels = ['A	', 'B	', 'A*B'] `
`means_labels = ['grand', 'A1', 'A2', \$`
`  'A3', 'B1', 'B2', 'A1*B1', 'A1*B2', \$`
`  'A2*B1', 'A2*B2', 'A3*B1', 'A3*B2'] `
`PRINT, '	* *Analysis of Variance * *' `
`FOR i = 0, 14 DO PM, anova_labels(i), \$`
`  anova_table(i), FORMAT = '(a40,f15.2)'`
`PRINT`
`; Print the analysis of variance table.`
`PRINT, '	* * Variation Due to the Model * *' `
`PRINT, 'Source	DF	SS	MS	P-value'`
`FOR i = 0, 2 DO PM, effects_labels(i), test_effects(i, *) `
`PRINT`
`PRINT, ' * * Subgroup Means * *'`
`FOR i = 0, 11 DO PM, means_labels(i), \$`
`  means(i), FORMAT = '(a5,f15.2)'`
`END`
`n = [3, 2, 10]`
`y = [73.0, 102.0, 118.0, 104.0,	81.0, \$`
`  107.0, 100.0,	87.0, 117.0, 111.0, \$`
`  90.0,	76.0,	90.0,	64.0,	86.0, \$`
`  51.0,	72.0,	90.0,	95.0,	78.0, \$`
`  98.0,	74.0,	56.0, 111.0,	95.0, \$`
`  88.0,	82.0,	77.0,	86.0,	92.0, \$`
`  107.0,	95.0,	97.0,	80.0,	98.0, \$`
`  74.0,	74.0,	67.0,	89.0,	58.0, \$`
`  94.0,	79.0,	96.0,	98.0, 102.0, \$`
`  102.0, 108.0,	91.0, 120.0, 105.0, \$`
`  49.0,	82.0,	73.0,	86.0,	81.0, \$`
`  97.0, 106.0,	70.0,	61.0,	82.0]`
`p_value = IMSL_ANOVAFACT(n, y, Anova_Table = anova_table, \$ `
`  Test_Effects = test_effects, Means = means)`
`print_results, anova_table, test_effects, means`
` `
`* * Analysis of Variance * *`
`  df for among groups                 5.00`
`  df for within groups               54.00`
`  total (corrected) df               59.00`
`  ss for among groups              4612.93`
`  ss for within groups            11586.00`
`  total (corrected) ss            16198.93`
`  mean square among groups          922.59`
`  mean square within groups         214.56`
`  F-statistic                         4.30`
`  P-value                             0.00`
`  R-squared (in	percent)             28.48`
`  adjusted R-squared (in percent)    21.85`
`  est. std of within group error     14.65`
`  overall mean of y                  87.87`
`  coef. of variation (in percent)    16.67`
`* * Variation Due to the Model * *`
`Source      DF         SS        MS        P-value`
`A        2.00000    266.533   0.621128   0.541132`
`B        1.00000   3168.27   14.7667     0.000322342`
`A*B      2.00000   1178.13    2.74552    0.0731880`
`* * Subgroup	Means * *`
`grand     87.87`
`A1        89.60`
`A2        84.90`
`A3        89.10`
`B1        95.13`
`B2        80.60`
`A1*B1    100.00`
`A1*B2     79.20`
`A2*B1     85.90`
`A2*B2     83.90`
`A3*B1     99.50`
`A3*B2     78.70`

### Example 3: Three-way ANOVA

This example performs a three-way analysis of variance using data discussed by John (1971, pp. 91–92). The responses are weights (in grams) of roots of carrots grown with varying amounts of applied nitrogen (A), potassium (B), and phosphorus (C). Each cell of the three-way layout has one response. Note that the ABC interactions sum of squares (186) is given incorrectly by John (1971, Table 5.2.)

The three-way layout is given below:

 A0 A1 A2 B0 B1 B2 B0 B1 B2 B0 B1 B2 C0 88.76 91.41 97.85 94.83 100.49 99.75 99.90 100.23 104.51 C1 87.45 98.27 95.85 84.57 97.20 112.30 92.98 107.77 110.94 C2 86.01 104.20 90.09 81.06 120.80 108.77 94.72 118.39 102.87
`.RUN`
`PRO print_results, anova_table, test_effects, means `
`  anova_labels = ['df for among groups', \$`
`    'df for within groups', 'total (corrected) df', \$`
`    'ss for among groups', 'ss for within groups', \$`
`    'total (corrected) ss', 'mean square among groups', \$`
`    'mean square within groups', 'F-statistic', \$`
`    'P-value', 'R-squared (in percent)', \$`
`    'adjusted R-squared (in percent)', \$`
`    'est. std of within group error', \$`
`    'overall mean of y', 'coef. of variation (in percent)'] `
`  effects_labels = ['A	', 'B	', 'C	', 'A*B', 'A*B', 'A*C'] `
`  PRINT, '	* *Analysis of Variance * *'`
`  FOR i = 0, 14 DO PM, anova_labels(i), \$`
`    anova_table(i), FORMAT = '(a40,f15.2)' `
`  PRINT`
`  PRINT, '	* * Variation Due to the Model * *' `
`  PRINT, 'Source	DF	SS	MS	P-value'`
`  FOR i = 0,5 DO PM, effects_labels(i), test_effects(i, *)`
`END`
`n = [3, 3, 3]`
`y = [88.76, 87.45, 86.01, 91.41, 98.27, 104.20, 97.85, \$`
`  95.85, 90.09, 94.83, 84.57, 81.06, 100.49, 97.20, \$`
`  120.80, 99.75, 112.30, 108.77, 99.90, 92.98, 94.72, \$`
`  100.23, 107.77, 118.39, 104.51, 110.94, 102.87]`
`p_value = IMSL_ANOVAFACT(n, y, Anova_Table = anova_table, \$`
`  Test_Effects = test_effects, /Pool_Inter)`
`print_results, anova_table, test_effects`
` `
`* *Analysis of Variance * *`
`  df for among groups                18.00`
`  df for within groups                8.00`
`  total (corrected) df               26.00`
`  ss for among groups              2395.73`
`  ss for within groups              185.78`
`  total (corrected) ss             2581.51`
`  mean square among groups          133.10`
`  mean square within groups          23.22`
`  F-statistic                         5.73`
`  p-value                             0.01`
`  R-squared (in percent)             92.80`
`  adjusted R-squared (in percent)    76.61`
`  est. std of within group error      4.82`
`  overall mean of y                  98.96`
`  coef. of variation (in percent)     4.87`
`* * Variation Due to the Model * *`
`  Source   DF       SS       MS       p-value`
`  A     2.00000   488.368  10.5152    0.00576699`
`  B     2.00000  1090.66   23.4832    0.000448704`
`  C     2.00000    49.1484  1.05823   0.391063`
`  A*B   4.00000   142.586   1.53502   0.280423`
`  A*B   4.00000    32.3474  0.348241  0.838336`
`  A*C   4.00000   592.624   6.37997   0.0131252`

## Syntax

Result = IMSL_ANOVAFACT(N_levels, Y [, ANOVA_TABLE=variable] [, /DOUBLE] [, MEANS=variable] [, ORDER=value] [, /PURE_ERROR] [, /POOL_INTER] [, TEST_EFFECTS=variable])

## Return Value

The p-value for the overall F-test.

## Arguments

### N_levels

One-dimensional array containing the number of levels for each of the factors and the number of replicates for each effect.

### Y

One-dimensional array of length:

n_levels (0) * n_levels (1) * ... * ((N_ELEMENTS (n_levels) – 1))

containing the responses. Argument Ymust not contain NaN for any of its elements, i.e., missing values are not allowed.

## Keywords

### ANOVA_TABLE (optional)

Named variable into which the analysis of variance table is stored. The analysis of variance statistics are as follows:

• 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)

### DOUBLE (optional)

If present and nonzero, then double precision is used.

### MEANS (optional)

Named variable into which an array of length (n_levels(0) + 1) x (n_levels(1) + 1) x... ... x (n_levels(n–1) + 1) containing the subgroup means is stored.

See the keyword TEST_EFFECTS for a definition of n. If the factors are A, B, C, and replicates, the ordering of the means is grand mean, A means, B means, C means, AB means, AC means, BC means, and ABC means.

### ORDER (optional)

Number of factors included in the highest-way interaction in the model. ORDER must be in the interval [1, N_ELEMENTS (n_levels) – 1]. For example, an ORDER of 1 indicates that a main-effect model is analyzed, and an ORDER of 2 indicates that two- way interactions are included in the model. Default: N_ELEMENTS(N_levels) – 1)

### PURE_ERROR (optional)

If present and nonzero, the default option of PURE_ERROR indicates all the main effect and the interaction effects involving the replicates, the last element in n_levels, are pooled together to create the error term. The POOL_INTER keyword indicates (ORDER + 1)- way and higher-way interactions are pooled together to create the error. The keywords PURE_ERROR and POOL_INTER cannot be used together.

### POOL_INTER (optional)

If present and nonzero, the default option of PURE_ERROR indicates all the main effect and the interaction effects involving the replicates, the last element in n_levels, are pooled together to create the error term. This keyword indicates (ORDER + 1)- way and higher-way interactions are pooled together to create the error. The keywords PURE_ERROR and POOL_INTER cannot be used together.

### TEST_EFFECTS (optional)

Named variable into which an array of size nef x 4 containing statistics relating to the sums of squares for the effects in the model is stored. Here:

where n is given by N_ELEMENTS(n_levels) if POOL_INTER is specified; otherwise, N_ELEMENTS(N_levels) – 1.

Suppose the factors are A, B, C, and error. With ORDER = 3, rows 0 through nef – 1 correspond to A, B, C, AB, AC, BC, and ABC. The columns of TEST_EFFECTS are as follows:

• 0: Degrees of freedom
• 1: Sum of squares
• 2: F-statistic
• 3: p-value

## Version History

 6.4 Introduced

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