SOP: Fitting a Linear Regression Equation with Linearity Test

GIFSQ — Global Institute for Food Safety & Quality

Principle

Regression deals with situations where the variation of one (dependent) variable is related to the variation of a second (independent) variable — for example, the increase in sugar content with maturity. When the paired observations plotted on a scatter diagram show a linear trend, a straight line y = a + bx is fitted to them; the regression coefficient b gives the rate of increase or decrease in y per unit change in x, and an analysis of variance tests whether the regression is significantly linear.

Materials

• The n pairs of observations (x = independent variable, y = dependent variable)

Procedure

Record the n pairs of observations (x, y) and plot them on a scatter diagram to confirm a linear trend

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Compute the sums Σx, Σy, Σx², Σy² and Σxy

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Compute the corrected sum of squares for x, the corrected sum of squares for y, and the corrected sum of products using the expressions below

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Compute the regression coefficient b and the constant a, and write the regression equation y = a + bx

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Use the equation to compute the expected value of y for chosen values of x

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Test the linearity of the regression by analysis of variance: divide the corrected sum of squares for regression by the mean square error to obtain the F-ratio, and compare with the tabulated F-values

Calculation

Corrected sum of squares for x:

Σ(x − x̄)² = Σx² − (Σx)²/n

Corrected sum of squares for y:

Σ(y − ȳ)² = Σy² − (Σy)²/n

Corrected sum of products:

Σ(x − x̄)(y − ȳ) = Σxy − (Σx)(Σy)/n

Regression coefficient:

b = (corrected sum of products) / (corrected sum of squares for x)

Constant:

a = ȳ − b·x̄

Regression equation:

y = a + bx

Linearity test:

SS_regression = (corrected sum of products)² / (corrected sum of squares for x)

SS_error = (corrected sum of squares for y) − SS_regression

MS_error = SS_error / (n − 2)

F = SS_regression / MS_error

Where,

• x = independent variable
• y = dependent variable
• a = value of y when x is zero
• b = regression coefficient (slope)
• x̄, ȳ = means of x and y
• n = number of pairs of observations
• The F-ratio has (1, n − 2) degrees of freedom

Worked example — total solids vs weight-by-length ratio of French beans

Thirty-four pairs of observations of weight-by-length (w/l) ratio (x) against total solids % (y) of French beans of different maturities show a strong positive association.

n = 34; Σx = 13.42; Σy = 277.68; x̄ = 0.3947; ȳ = 8.167

Corrected SS for x = 5.6634 − 13.42²/34 = 5.6634 − 5.2970 = 0.366

Corrected SS for y = 2285.6250 − 277.68²/34 = 2285.6250 − 2267.8289 = 17.7961

Corrected sum of products = 110.6561 − (13.42 × 277.68)/34 = 110.6561 − 109.6019 = 1.0542

b = 1.0542/0.366 = 2.88

a = 8.167 − 2.88 × 0.3947 = 7.0316 (as printed)

Regression equation: y = 7.0316 + 2.88x

Note: the rounded chain “8.167 − 2.88 × 0.3947” as displayed gives 7.0303, not the printed 7.0316; the book’s expected-value table below was computed with a ≈ 7.0303 (0.29 → 7.866, 0.36 → 8.067, 0.44 → 8.297, 0.50 → 8.470, 0.60 → 8.758) — all carried as printed.

Expected values of y (as printed):

Linearity test (analysis of variance):

SS_regression = (1.0542)²/0.366 = 3.0364; SS_error = 17.7961 − 3.0364 = 14.7597; MS_error = 14.7597/32 = 0.4612; F = 3.0364/0.4612 = 6.58.

The F-ratio for (1, 32) degrees of freedom is significant (0.01 < P < 0.05); the regression equation is significantly linear — a large and significant portion of the variance of y has been explained by regression on x.

Precautions

• The regression coefficient may be positive or negative depending on the orientation of the line; b indicates the rate of increase or decrease in y per unit change in x.

References

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