Methods and formulas for two-way designs with normal data in Analysis of Means

For two-factor models, analysis of means is a procedure to determine if the interaction effects are significant or the main effects are different from the grand mean. For two-way analysis of means, the data must be balanced.

Formula

The average of the observations for a factor at a given level. Minitab plots the mean for each factor level on the graph.

Mean of Factor A at the ith level:
Mean of Factor B at the jth level:

Notation

Formula

The average of all observations in the sample. Minitab uses the grand mean as the center line on the graph of the main effects.

Notation

The decision limits indicate whether factor level means are different from the grand mean. Points that lie outside the upper decision limit (UDL) or lower decision limit (LDL) are statistically different from the grand mean.

The calculation of the upper and lower decision limits varies based on the number of levels in the factor and the number of observations at each level. The formulas below show the upper and lower decision limits for factor A. To calculate the decision limits for factor B, replace terms specific to factor A with equivalent terms for factor B.

Two-level factor

The upper and lower decision limits for factor A are:

• UDL_A = _...+ (.5) * h_α* Sqrt[(MSE) * (2 / n₁)]
• LDL_A = _...- (.5) * h_α* Sqrt[(MSE) * (2 / n₁)]

where ha = absolute value(t(a / 2; abn - ab), MSE = mean square error (from an ANOVA with terms A, B, and AB) and n₁ = number of observations at each level of the factor A.

Factor with more than two levels

• UDL_A = _...+ h_α* Sqrt[MSE * (a - 1) / (a * n₁)]
• LDL_A = _...- h_α* Sqrt[MSE * (a - 1) / (a * n₁)]

where MSE = mean square error (from an ANOVA with terms A, B, and AB), a = number of factor levels in factor A, and n₁= number of observations at each level of the factor. The critical value h_α depends on alpha, the number of means being plotted, and the degrees of freedom for MSE. You can find values for h_α in Table B.1 in Appendix B in Nelson¹.

For values of alpha outside the range of 0.001 and 0.1, the decision limits are:

• UDL_A = _...+ h_α* Sqrt[MSE * (a - 1) / (a * n₁)]
• LDL_A = _...- h_α* Sqrt[MSE * (a - 1) / (a * n₁)]

where MSE = mean square error (from an ANOVA with terms A, B, and AB), n₁= number of observations at each level of the factor. and h_α = absolute value (t(α2, df); where a2 = (1- (1- a )** (1 / a)) / 2 and df = n_T - ab where n_T = total number of observations in the sample,.

The decision limits indicate whether the interaction is significant. Points that lie outside the upper decision limit (UDL) or lower decision limit (LDL) indicate that the interaction is statistically significant.

Listed below are the general formulas for the upper and lower decision limits for the interaction of factors A and B. The terms are defined differently based on the number of levels and observations in each factor.

• UDL_{A B} = h_α* Sqrt[MSE * (q / (a * b * n)]
• LDL_{A B} = –h_α* Sqrt[MSE * (q / (a * b * n)]

where ha = absolute value (t(α2, dfe)), a = number of levels in factor A, b = number of levels in factor B, n = number of observations for each interaction between factors, q = degrees of freedom for interaction effects, (a - 1)(b - 1) and dfe = degrees of freedom for error, abn - ab.

Factors A and B both have two levels

• α2 = a / 2

Factor A has two levels and Factor B has more than two levels

• α2 = (1- (1- a )** (1 / b)) / 2

where a = number of levels in factor A and b = number of levels in factor B.

Factor A has more than two levels and Factor B two levels

• α2 = (1- (1- a )** (1 / a)) / 2

where a = number of levels in factor A and b = number of levels in factor B.

Factors A and B have more than two levels

• α2 = (1- (1- a )** (1 /a * b)) / 2

where a = number of levels in factor A and b = number of levels in factor B.