SOP: Comparing More Than Two Lot Means by One-Way Analysis of Variance

GIFSQ — Global Institute for Food Safety & Quality

Principle

When the means of more than two groups of observations are to be compared, the t-test is not applicable and the analysis of variance (F-test) is used. Using the additive property of variance, the observed variance is broken down into the component between lots and the residual component within lots; the ratio of the two mean squares (the F-ratio) is compared with tabulated F-values to determine whether there is any significant difference among the lots.

Materials

• The replicate observations for each lot, arranged in a table
• The book’s F-distribution tables for the between-lots and within-lots degrees of freedom

Procedure

Arrange the observations in rows (lots) and columns (replicates) and compute the total of each lot and the grand total of all observations

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Square every observation and add to obtain the uncorrected sum of squares Σx²

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

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Compute the corrected total sum of squares, the corrected sum of squares between lots, and the corrected sum of squares within lots using the expressions below

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

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Compute the F-ratio as the mean square between lots divided by the mean square within lots, and compare with the tabulated F-values at the 5% and 1% levels

Calculation

Correction factor:

CF = (Σx)² / N

Corrected total sum of squares:

SS_total = Σx² − CF

Corrected sum of squares between lots:

SS_between = Σ(T_lot² / n) − CF

Corrected sum of squares within lots:

SS_within = SS_total − SS_between

Mean squares and F-ratio:

MS_between = SS_between / (number of lots − 1)

MS_within = SS_within / (N − number of lots)

F = MS_between / MS_within

Where,

• Σx = grand total of all observations
• N = total number of observations
• T_lot = total of a lot
• n = number of replicates per lot
• Degrees of freedom: total = N − 1; between lots = number of lots − 1; within lots = total df − between-lots df

Worked example — gel strength of jams from three lots

Five replicate gel-strength determinations from each of 3 lots (15 observations):

Uncorrected sum of squares: Σx² = 77,129

Correction factor: CF = 1073²/15 = 76,755.27

Corrected total SS = 77,129.00 − 76,755.27 = 473.73 (as printed)

Uncorrected SS between lots = (355² + 376² + 342²)/5 = 76,873.00

Corrected SS between lots = 76,873.00 − 76,755.27 = 117.73

SS within lots = 473.73 − 117.73 = 356.00 (as printed)

Analysis of variance:

The F-value found is less than the tabulated value of F for 2 and 12 degrees of freedom (3.74 at 5%, 6.51 at 1%, as printed); hence there is no appreciable variation in the gel strength of the three lots.

Notes: (1) the corrected total SS recomputes as 373.73 from the printed data (77,129 − 76,755.27), not 473.73 as printed; (2) the within-lots SS recomputes as 256.00, not 356.00 as printed; (3) the printed mean square within (25.43) does not follow from the printed SS within ÷ 12 (356.00/12 = 29.67); recomputed from the data, MS_within = 21.33 and F = 2.76; (4) the printed critical values (3.74, 6.51) differ slightly from standard F-table values for (2, 12) (3.89, 6.93). All figures are carried as printed; the conclusion — no appreciable variation — is unaffected, since the recomputed F (2.76) is also below the 5% critical value.

Precautions

• One-way analysis of variance is used for variations within replicates under conditions such as replicate determinations by several analysts on the same material, differences due to processing conditions, and lot-to-lot variations in shipments by the same supplier.
• The t-test and the F-test should not be used for acceptance of lots.

References

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