Unbalanced Two Way ANOVA Calculator

Unbalanced two way ANOVA calculator

Models

• Fixed effect model (A-Fixed, B-Fixed), no repeats - both factors are fixed.
• Mixed effect model (A-Random, B-Fixed), no repeats - factor A is random, factor B is fixed, each subject is measured only once.
• Mixed effect model (A-Fixed, B-Random), no repeats - factor A is fixed, factor B is random, each subject is measured only once.
• Random effect model (A-Random, B-Random), no repeats

Design

Balanced design

The balanced design has the same number of observations in each cell - each combination of factor.
Since there are no overlaps between any combination of the factors and the interaction : SS_T = SS_A + SS_B + SS_AB + SS_E.

SS_T

Unbalanced design

When the model is unbalanced, it leads to correlations. When the distribution of data between the cells matches the population distribution, the correlation will exist only between each factor and the interactions. However, when the distribution of data between the cells does not match the population distribution, correlations will occur both between each factor and the interactions, and among the factors themselves.
When correlation exists, there is overlap among the sum of squares (SS). If we calculate the SS as we do for the balanced model, the result will be incorrect, leading to a SS larger than the actual SS. Hence, you don't know how to allocate the shared SS between the two factors and between the factors and the interaction. There are several methods for dealing with the shared sum of squares.

Unbalanced design - Type I (Sequential Sum of Squares):

Type I - sequential, the first some of squares (SS) you calculate get the shared some of squares. In this case the order is matter!. Following the sum of squares formulas:
Factor A: SS_A.
Factor B: SS_B|A = SSR(y = β₀ + β₁A + β₂B) - SS_A.
Interaction AB: SS_AB|A,B = SSR(y = β₀ + β₁A + β₂B + β₃AB) - SSR(y = β₀ + β₁A + β₂B).

Unbalanced design - Type II (Partial Sum of Squares):

Type II - conservative, it assumes there is no interaction between the factors, it ignores the shared SS between the factors. Following the sum of squares formulas:
Factor A: SS_A|B = SSR(y = β₀ + β₁A + β₂B) - SS_B.
Factor B: SS_B|A = SSR(y = β₀ + β₁A + β₂B) - SS_A.
In this case we assume no interaction, but we also test it using the following:
Interaction AB: SS_AB|A,B = SSR(y = β₀ + β₁A + β₂B + β₃AB) - SSR(y = β₀ + β₁A + β₂B).

Unbalanced design - Type III (Marginal Sum of Squares)

Type III - assumes there is interaction between the factors, it ignores all the shared SS between the factors and between the factors and the interactions. Following the sum of squares formulas:
Factor A: SS_{A|B, AB} = SSR(y = β₀ + β₁A + β₂B + β₃AB) - SSR(y = β₀ + β₁B + β₂AB).
Factor B: SS_{B|A, AB} = SSR(y = β₀ + β₁A + β₂B + β₃AB) - SSR(y = β₀ + β₁A + β₂AB).
In this case we assume no interaction, but we also test it using the following:
Interaction AB: SS_AB|A,B = SSR(y = β₀ + β₁A + β₂B + β₃AB) - SSR(y = β₀ + β₁A + β₂B).

If the interaction does not exist in the population, the Type II method is a more powerful test than the Type III method"

Glossary

SS_T is the sum of squared differences between the dependent variable and the grand mean.
SS_Model = SS_A + SS_B + SS_AB + SS_E.
For balanced model and Type I, SS_Model = SS_T

Targets

• Checks if the difference between Factor A averages of two or more categories is significant
• Checks if the difference between Factor B averages of two or more categories is significant
• Checks if there is an interaction between Factor A and Factor B

Right-tailed F test, for ANOVA test you can use only the right tail. Why?

Two-way ANOVA

HypothesesThe Hypotheses of the different types are not the same

Factor A: H₀: μ₁ = .. = μ_a

There is no difference in the means of variable A categories.

Factor B: H₀: μ₁ = .. = μ_b

There is no difference in the means of variable B categories.

H₀: Interaction(A_iB_j) = 0 (∀ i = 1 to a, j = 1 to b)
There is no interaction between variable A and variable B, i.e., for all the cells, the effect of variable A on the cells' means is not depend on the effect of variable B, and vice versa.

Two-way ANOVA tests formulas

Fixed Model		Mixed Model		Random Model		Mixed Repeated
FA=	MSA	FA=	MSA	FA=	MSA	FA=	MSA
	MSE		MSAB		MSAB		MSSWA
FB=	MSB	FB=	MSB	FB=	MSB	FB=	MSB
	MSE		MSE		MSAB		MSBSWA
FAB=	MSAB	FAB=	MSAB	FAB=	MSAB	FAB=	MSAB
	MSE		MSE		MSE		MSBSWA

F distribution

Assumptions

• The dependent variable is continuous (ratio or interval)
• Two categorical independent variables
• Independent observations (no repeated measure)
• The residuals distribution is normal
• Homogeneity of variances, a similar variance for each cell

Required Sample Data

Sample data from all compared groups

Parameters

a - the number of categories in variable A, number of rows.
b - the number of categories in variable B, number of columns.
n_i - sample side of category i of variable A (row i).
n_j - sample side of category j of variable B (column j).
n_i,j - sample side of cell i,j (row i, column j). In the balance n_i,j=n/(a*b)
n - overall sample side, includes all the groups (Σn_i,j, i=1 to a, j=1 to b).
Ȳ_i - average of all the observations of category i of variable A (row i).
Ȳ_j - average of all the observations of category j of variable B (column j).
Ȳ - overall average (ΣY_i,j,k / n, i=1 to a, j=1 to b, k=1 to n_i,j).

Repeated measures ANOVA

s - represent the order of subject in category i (subject 1 in category 1 is different than subject 1 in category 2)
sub - number of subjects per cell, cell is one combination of variable A and variable B. For the balance design: N=a*b*sub.
Ȳ_i,s - subject's average, ΣY_i,j,s for subject i,s ,the average of all the observations of subject s of category j of variable B (column j).
Ȳ - overall average (ΣY_i,j,s / n

Results calculations

Sum of squares

The sum of squares accumulates the squared differences related to the effect we try to estimate.
SS_A - the squared differences related to the effect of variable A. You compare the average of every category to the total average. The same value as the sum of squares between groups in one way ANOVA.
SS_B - the same as SS_A, for variable B.
SS_AB - the squared differences related to the effect of the combination of variable A and variable B in each cell, Since we try to understand the influence of the interaction AB, the interaction of the specific value of variable A and the specific value of variable B, we take the average of each cell, remove the influence of variable A and variable B, and compare to the total average.
A effect = Ȳ_i - Ȳ
B effect = Ȳ_j - Ȳ
AB effect = Cell average - A effect - B effect - Total average.
= Ȳ_i,j - (Ȳ_i - Ȳ) - (Ȳ_j - Ȳ) - Ȳ.
= Ȳ_i,j - Ȳ_i - Ȳ_j + Ȳ.
Take the square of each difference
Ȳ_i,j - Ȳ_i - Ȳ_j + Ȳ)².
Count the square differences of each value in the cell, hence multiply by the sample size of each cell (n_i,j).
SS_AB=Σ_i^aΣ_j^bn_i,j(Ȳ_i,j - Ȳ_i - Ȳ_j + Ȳ)²

Fixed and Random Effects

The fixed and random effects are related to the independent variables ().

Fixed Effect

The effect is constant across individuals.

• The categories of the variable contains the entire categories' list
• The effect of this variable is interesting. The difference between the categories is important
• There is no know pattern on the difference between the categories

Random Effect

The effect vary across individuals, the individuals may be people, products.

• The categories' list is only a sample from the entire categories' list
• The effect of this variable is not interesting by itself. The difference between the categories is not important.
• There is no know pattern on the difference between the categories

Example: collecting data from several schools.
A sample from the entire groups' population.
There is no pattern about the difference between the schools, and if there will be a pattern, it will be another factor, like school's size.
Each school is not important by itself.

When you change the interaction field or the model, the following ANOVA table and diagram will be adjusted!

ANOVA table - With interaction - Type II

Source	Degrees of Freedom (DF)	Sum of Squares (SS)	Mean Square (MS)	F statistic	p-value
Between subjects Between the subjects when ignoring factor A	DFBS = a*sub - 1	SSBS = SWA2
Factor A (rows) Between the categories of factor A	DFA = a - 1	SSA = SS(A|B) = SS(A, B) - SS(B)	MSA = SSA / DFA	FA = MSA / MSE	P(x > FA)
Subject within A	DFSWA = a*(sub - 1)	SSSWA = SSBS - SSA	SSSWA / DFSWA
Within subjects	DFWS = N - a*sub	SSWS = ΣiaΣssub(Ȳi,j,s-Ȳi,s )2
Factor B (Columns) Between the categories of factor B	DFB = b - 1	SSB = SS(B|A) = SS(A, B) - SS(A)	MSB = SSB / DFB	FB = MSB / MSE	P(x > FB)
Interaction AB Between the cells after reducing factor A and factor B effects	DFAB = (a - 1)(b - 1)	SSAB = SS(AB|A, B) = SS(A, B, AB) - SS(A, B)	MSAB = SSAB / DFAB	FAB = MSAB / MSE	P(x > FAB)
B*Subject within A	DFBSWA = a*(sub - 1)	SSBSWA = SSWS - SSB - SSAB
Error Within the cells	DFE = n - a*b	SSE=ΣiaΣjbΣkni,j(Yi,j,k - Ȳi,j)2	MSE = SSE / DFE
Error Within the cells	DFE = n - a - b + 1	SSE=ΣiaΣjbΣkni,j(Yi,j,k - Ȳi - Ȳj + Ȳ)2	MSE = SSE / DFE
Total All the deviations from the average	DFT = n - 1	SST=ΣiaΣjbΣkni,j(Yi,j,k - Ȳ)2 SST=Sample Variance*(n-1)	MSE = S2 = SST / (n - 1)

Sum of squares diagram - with interaction

This is the In the following diagram you may see the differences per each observation Y_i,j,k that used to calculate the sum of squares.
A effect: Ȳ_i - Ȳ.
B effect: Ȳ_j - Ȳ.

Interaction effect (AB): Y_i,j - Ȳ_i - Ȳ_j + Ȳ.
Between Subjects effect: Y_i,j,s - Ȳ_i,s.
Within Subjects effect: Ȳ_i,s - Ȳ.
SS_SWA = SS_BS - SS_A.
SS_BSWA = SS_WS - SS_B -SS_AB.
Total effect: Between Subjects + Within Subjects.

Interaction effect (AB): Y_i,j - Ȳ_i - Ȳ_j + Ȳ.
Error: Y_i,j,k - Ȳ_i,j.

Error: Y_i,j,k - Ȳ_i - Ȳ_j + Ȳ.

Total effect: Y_i,j,k - Ȳ.