SOP: Nested Analysis of Variance for Canned Pea Grade Scores

GIFSQ — Global Institute for Food Safety & Quality

Principle

In a nested (sub-grouping) sampling design — lots containing sublots, sublots containing cases, cases containing cans — the total variation in the quality scores is broken down step by step into the components due to lots, sublots within lots, cases within sublots, and cans within cases (the replicate error). Each variance ratio (F-ratio) is tested against the component below it, revealing at which level the quality genuinely varies.

Materials

• The nested sampling plan: number of lots, sublots per lot, cases drawn per sublot, cans analysed per case
• The grade score of every can analysed
• The book’s F-distribution tables

Procedure

Draw the nested sample: from each lot take the planned number of sublots, from each sublot the planned number of cases, and from each case the planned number of cans for analysis

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Record the grade score of each can and add them level by level: can scores into case totals, case totals into sublot totals, sublot totals into lot totals, lot totals into the grand total

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Compute the correction factor CF = (grand total)²/N

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Compute the uncorrected sums of squares at each level (sum of squared totals divided by the number of cans each total represents) and obtain the corrected sums of squares by successive subtraction using the expressions below

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Divide each corrected sum of squares by its degrees of freedom to obtain the mean squares

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Compute the F-ratios: lots against sublots, sublots against cases, cases against cans (replicate error), and compare each with the tabulated F-values

Calculation

CF = (Σx)² / N

Corrected SS_total = (uncorrected SS of cans) − CF

Corrected SS_lots = (uncorrected SS of lots) − CF

Corrected SS_sublots = (uncorrected SS of sublots) − CF − SS_lots

Corrected SS_cases = (uncorrected SS of cases) − CF − SS_lots − SS_sublots

Corrected SS_cans (error) = SS_total − SS_lots − SS_sublots − SS_cases

F_lots = MS_lots / MS_sublots

F_sublots = MS_sublots / MS_cases

F_cases = MS_cases / MS_cans

Where,

• Σx = grand total of the grade scores
• N = total number of cans analysed
• MS = mean square (sum of squares ÷ degrees of freedom)
• Degrees of freedom: lots = lots − 1; sublots = lots × (sublots per lot − 1); cases = sublots × (cases per sublot − 1); cans = cases × (cans per case − 1)

Worked example — uniformity of canned peas in brine

A canner supplies 3,200 cases in 4 lots; each lot contains 4 sublots. From each sublot 2 cases are drawn and from each case 2 cans are analysed (64 cans in all). The grade scores (ISI Standard 3245-1965 for canned peas; grade A overall score not less than 85, grade B not less than 75) are recorded in a nested table.

Sampling plan check: the total number of cans in the order is 4 × 4 × 200 × 24 = 76,800 cans; 29 cans are required to be sampled, and the 64 cans drawn for analysis are more than adequate.

Calculations (as printed):

• Total number of cans N = 64
• Sum of grades Σx = 5,331; mean grade = 5,331/64 = 83.30
• Uncorrected SS of cans = 446,035
• Uncorrected SS of lots = (1327² + 1271² + 1412² + 1321²)/16 = 444,697.19
• Uncorrected SS of sublots = 445,549.75
• Uncorrected SS of cases = 445,832.50
• Correction factor CF = 5331²/64 = 444,055.64
• Corrected SS_total = 446,035 − 444,055.64 = 1,979.36
• Corrected SS_lots = 444,697.19 − 444,055.64 = 641.55
• Corrected SS_sublots = 445,549.75 − 444,055.64 − 641.55 = 852.56
• Corrected SS_cases = 445,832.50 − 444,055.64 − 641.55 − 852.56 = 282.75
• Corrected SS_cans (error) = 1,979.36 − 641.55 − 852.56 − 282.75 = 202.50

Analysis of variance (recomputed from the printed sums of squares):

The F-ratio for cases tested against the replicate error is highly significant; the mean square between sublots is also highly significant, confirming variation in quality from case to case and from sublot to sublot. The difference in quality between lots approaches the 0.05 significance level.

Note: the book’s printed analysis-of-variance table is illegible in the available copy; the sums of squares and degrees of freedom above recompute exactly as printed, and the mean squares, F-ratios and conclusions are consistent with them.

Precautions

• Each variance ratio must be tested against the component immediately below it in the nesting (lots against sublots, not against the replicate error), since that is the proper error term for the level.

References

Ranganna, S., Handbook of Analysis and Quality Control for Fruit and Vegetable Products, 2nd edition.