In mixed models, as well fixed effects as in Chapter 5, random effects as in Chapter 6 also occur, i.e. mixed models are models where in the model equation at least one, but not all, effects are random variables.
In mixed models, we discuss problems of variance component estimation and of estimating and testing fixed effects.
Therefore, we need expected mean squares from the tables of Chapter 5 if the effect defining a row is fixed and the tables of Chapter 6 if the effect defining a row is random.
An interaction is random if at least one of the factors involved is random. In discussing the different classifications, we use the same procedure as in Chapters 5 and 6. Of course, in a one‐way classification a mixed model is impossible. We therefore start with the two‐way classification.
We discuss here the cross‐classification where we consider the factor A to be fixed without loss of generality. If the factor B is fixed, we rename both factors. In the nested classification, two‐mixed models occur when the super‐ordinate factor or the nested factor is random. We write random factors with their elements in bold.
We consider two cross‐classified factors A (fixed) and B (random) and their interactions AB.
The model equation of the balanced case is
with side conditions
and
or
If we use (7.4) the term vanishes, and (a,b)ij and are correlated; the covariance is
Because the covariance between the (a,b)ij and is .
In case II we define var(bj) = for all j.
Table 7.1 Expectations of the MS in Table 5.10 for a Mixed model (Levels of A fixed) for two side conditions.
Source of Variation | df | MS | E(MS) Case I | E(MS) Case II |
Between rows (A) | a − 1 | |||
Between columns (B) | b − 1 | |||
Interactions | (a − 1)(b − 1) | |||
Within classes (residual) | ab(n − 1) | σ2 | σ2 |
Searle (1971) and Searle et al. (1992) clearly recorded the relations between the two cases. He showed that in case I and in case II changed in their meaning.
In the two‐way nested classification, we have two model equations depending on which factor is random.
We use the model equation if the factor B is random, the nested factor A is fixed
The side conditions are
Let the levels of A be randomly selected from the level population and the levels of B fixed, the model equation is then
with corresponding side conditions.
Expectations of all random variables are zero, for all i; var(ek(i, j)) = σ2 for all i,j,k; all covariances between different random variables on the right‐hand side of (7.12) are zero.
The columns SS, df and MS in the corresponding ANOVA table are model independent and given in Table 5.13. The expectations of the MS for both models are in Table 7.5
Table 7.5 E(MS) for balanced nested mixed models.
A fixed | A random | |
Source of variation | B random | B fixed |
Between A levels | σ2 + n | σ2 + bn |
Between B levels within A | σ2 + n | |
Residual | σ2 | σ2 |
We are interested here in models with at least one fixed factor and for determining the minimum size of the experiment; we are interested only in hypotheses about fixed factors.
For three‐way analysis of variance, we have the following types of classification:
For the examples, we use six levels of a factor A : a = 6, α = 0.05; δ = 0.1, and δ = σ. If possible, we fix the number of fixed factors B and C by b = 5 and c = 4 respectively.
The observations yijkl in the combinations of factor levels are as follows (model I is used for all factors fixed in Chapter 5 and model II for all factors random in Chapter 6):
The missing model II is one with three random factors and will not be discussed here (this is without loss of generality because we can rename the factors so that the first one(s) is (are) fixed).
For model III the model equation is given by:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have, with the same suffixes, equal variances, and all fixed effects besides μ over j and k sum up to zero.
For model IV the model equation is given by:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have with the same suffixes equal variances and that the ck add up to zero. We consider in the sequel mainly the balanced case with ai and nijk = n.
The analysis of variance table for both models is Table 7.8.
Table 7.8 ANOVA table – three‐way ANOVA – cross‐classification, balanced case.
Source of variation | SS | df | MS |
Main effect A | a − 1 | ||
Main effect B | b − 1 | ||
Main effect C | c − 1 | ||
Interaction A × B | (a – 1)(b − 1) | ||
Interaction A × C | (a − 1)(c − 1) | ||
Interaction B × C | (b − 1)(c − 1) | ||
Interaction A × B × C | SSABC = SST − SSA − SSB − SSC − SSAB − SSAC − SSBC − SSR | (a − 1)(b − 1) (c − 1) | |
Residual | abc(n − 1) | ||
Total | N − 1 |
In Table 7.9 the expected mean squares for model III and model IV are given.
Table 7.9 Expected mean squares for the three‐way cross‐classification – balanced case.
Mixed model | Mixed model | |
A, B fixed, | A fixed, | |
Source of variation | C random (model III) | B,C random (model IV) |
Main effect A | ||
Main effect B | ||
Main effect C | ||
Interaction A × B | ||
Interaction A × C | ||
Interaction B × C | ||
Interaction A × B × C | ||
Residual | σ2 | σ2 |
To find the appropriate F‐test for testing the hypothesis
we will demonstrate the algorithm for these models step by step; in the next sections we present only the results.
Model III
Table 7.9, second column.
Main effect A.
becomes if the hypothesis is true.
For the three‐way nested classification with at least one fixed factor, we have the following seven models
For all models, the ANOVA table is the same and given as Table 7.10
Table 7.10 ANOVA table of the three‐way nested classification – unbalanced case.
Source of variation | SS | df | MS |
Between A | a − 1 | ||
Between B in A | B. − a | ||
Between C in B and A | C. . − B. | ||
Residual | N − C. . |
The model equation is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have with the same suffixes equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares gives Table 7.11.
Table 7.11 Expected mean squares for the balanced case of model III.
Source of variation | Model III E(MS) |
Between A levels | |
Between B levels within A levels | |
Between C levels within B levels | |
Residual | σ2 |
From this table, we can derive the test statistics by using steps 1–7 from Section 7.3.1. We find for testing
and this is under HB0 bj(i) = 0 ∀ j, i F[a(b − 1); abc(n − 1)]‐distributed with a (b − 1) and abc(n − 1) degrees of freedom.
To test HC0 : ck(i, j) = 0, ∀ i, j, k against HC0 : ck(i, j) ≠ 0 for at least one combination of i,j,k we find, using steps 1–7 from Section 7.3.1, that
can be used, which, under HC0 : ck(i, j) = 0, ∀ i, j, k, is F[a(b − 1); abc(n − 1)]‐distributed with ab[c − 1] and abc(n − 1) degrees of freedom.
The model equation is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have with the same suffixes equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are given in Table 7.12.
Table 7.12 Expected mean squares for balanced case of model IV.
Source of variation | Model IV E(MS) |
Between A levels | |
Between B levels within A levels | |
Between C levels within B levels | |
Residual | σ2 |
From this table, we can derive the test statistic for testing
H0A : ai = 0 ∀ i against HAA: at least one ai ≠ 0.
by using steps 1–7 from Section 7.3.1. From this we find
is, under H0A, F((a − 1); ab(c − 1))‐distributed with a − 1 and ab(c − 1) degrees of freedom.
The model equation is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have with the same suffixes equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are given in Table 7.13
Table 7.13 Expected mean squares for model V.
Source of variation | Model V E(MS) |
Between A levels | |
Between B levels within A levels | |
Between C levels within B levels | |
Residual | σ2 |
From this table, we can derive the test statistic for testing H0 : ai = 0 ∀ i; against HA: at least one ai ≠ 0 in the usual way.
From this we find
is, under H0, F((a − 1); ab(c − 1))‐distributed with a − 1 and ab(c − 1) degrees of freedom.
The model equation is given by:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have with the same suffixes equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are give in Table 7.14.
Table 7.14 Expected mean squares for model VI.
Source of variation | Model VI E(MS) |
Between A levels | |
Between B levels within A levels | |
Between C levels within B levels | |
Residual | σ2 |
From this table, we can derive the test statistic for testing
We find
is, under H0, F((a − 1); ab(c − 1))‐distributed with a − 1 and ab(c − 1) degrees of freedom.
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have with the same suffixes equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are given in Table 7.15.
From this table we can derive the test statistic for testing
We find is, under H0, F(a(b − 1); ab (c − 1))‐distributed with a(b − 1) and abc − 1) degrees of freedom.
The model equation is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have with the same suffixes equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are given in Table 7.16.
Table 7.16 Expected mean squares for model VIII.
Source of variation | Model VIII E(MS) |
Between A levels | |
Between B levels within A levels | |
Between C levels within B levels | |
Residual | σ2 |
From this table, we can derive the test statistic for testing
We find is, under H0, F(ab(c − 1); abc(n − 1))‐distributed with ab(c − 1) and abc(n − 1) degrees of freedom.
We have four mixed models in this classification. The ANOVA table is independent of the models and given in Table 7.17 for the balanced case.
Table 7.17 ANOVA table for the balanced three‐way analysis of variance – mixed classification – (A × B) ≻ C.
Source of variation | df | MS | |
Between A levels | a − 1 | ||
Between B levels | b − 1 | ||
Between C levels within A × B levels | ab(c − 1) | ||
Interaction A × B | SSAB= | (a − 1) (b − 1) | |
Residual | N − abc |
We now consider the four models of this classification.
The model equation is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have with the same suffixes equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are given in Table 7.18.
Table 7.18 Expected mean squares for model III.
Source of variation | Model III E(MS) |
Between A levels | |
Between B levels | |
Between C levels within A × B | |
Interaction A × B | |
Residual | σ2 |
From this table, we can derive the test statistic for testing
We find
is, under H0, F(a − 1; (a − 1)(b − 1))‐distributed with a − 1 and (a − 1)(b − 1) degrees of freedom.
The model equation is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have, with the same suffixes, equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares for the balanced case are given in Table 7.19.
Table 7.19 Expected mean squares for model IV.
Source of variation | Model IV E(MS) |
Between A levels | |
Between B levels | |
Between C levels within A × B | |
Interaction A × B | |
Residual | σ2 |
From this table, we can derive the test statistic for testing
We find
is, under H0, F[ab(c − 1); abc(n − 1)]‐distributed with ab(c − 1) and abc(n − 1) degrees of freedom.
The model equation is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have, with the same suffixes, equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares for the balanced case are given in Table 7.20.
Table 7.20 Expected mean squares for the balanced model V.
Source of variation | Model V E(MS) |
Between A levels | |
Between B levels | |
Between C levels within A × B | |
Interaction A × B | |
Residual | σ2 |
From this table, we can derive the test statistic for testing
We find
is, under H0, F(a − 1; ab (c − 1))‐distributed with a − 1 and ab(c − 1) degrees of freedom.
The model equation is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have, with the same suffixes, equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are given Table 7.21.
Table 7.21 Expected mean squares for model VI.
Source of variation | Model VI E(MS) |
Between A levels | |
Between B levels | |
Between C levels within A × B | |
Interaction A × B | |
Residual | σ2 |
From this table, we can derive the test statistic for testing
We find
is, under H0, F(b − 1; (a − 1) (b − 1))‐distributed with a − 1 and (a − 1) (b − 1) degrees of freedom.
We have six mixed models in this classification. The ANOVA table is independent of the models and given in Table 7.22 for the balanced case.
Table 7.22 ANOVA table for the three‐way balanced analysis of variance – mixed classification (A ≻ B) × C.
Source of variation | SS | df | MS |
Between A levels | a − 1 | ||
Between B levels within A levels | a(b − 1) | ||
Between C levels | c − 1 | ||
Interaction A × C | (a − 1)(c − 1) | ||
Interaction B × C within A | a(b − 1)(c − 1) | ||
Residual | N − abc |
The model equation for the balanced case is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have, with the same suffixes, equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares shows Table 7.23.
Table 7.23 Expected mean squares for balanced model III.
Source of variation | E(MS) |
Between A levels | |
Between B levels within A levels | |
Between C levels | |
Interaction A × C | |
Interaction B × C within A | |
Residual | σ2 |
From Table 7.23, we can derive the test statistic for testing: is, under H0, F[a(b − 1); abc(n − 1)]‐distributed with a(b − 1) and abc(n − 1) degrees of freedom.
The model equation for the balanced case is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have, with the same suffixes, equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are given in Table 7.24.
Table 7.24 Expected mean squares for balanced model IV.
Source of variation | E(MS) for model IV |
Between A levels | |
Between B levels within A levels | |
Between C levels | |
Interaction A × C | |
Interaction B × C within A | |
Residual | σ2 |
The model equation for the balanced case is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have, with the same suffixes, equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are given in Table 7.25.
Table 7.25 Expected mean squares for the balanced model V.
Source of variation | E(MS) for model V |
Between A levels | |
Between B levels within A levels | |
Between C levels | |
Interaction A × C | |
Interaction B × C within A | |
Residual | σ2 |
From Table 7.25, we can derive the test statistic for testing
is, under H0, F((a − 1); (a − 1)(c − 1))‐distributed with a − 1 and (a − 1)(c − 1) degrees of freedom.
The model equation for the balanced case is given by:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have, with the same suffixes, equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are given in Table 7.26.
Table 7.26 Expected mean squares for model VI.
Source of variation | E(MS) for model VI |
Between A levels | |
Between B levels within A levels | |
Between C levels | |
Interaction A × C | |
Interaction B × C within A | |
Residual | σ2 |
From Table 7.26 we cannot derive an exact test statistic for testing H0 : ai = 0 ∀ i against HA : at least one ai ≠ 0, and this means that for this hypothesis no exact F‐test exists.
We can only, analogously to Section 7.3.1, derive a test statistic for an approximate F‐test and obtain
and have, under H0 : ai = 0 for all i, approximately a – 1 and
degrees of freedom for this approximate F‐statistics.
In addition, variance component estimation is not easy – at least with the ANOVA method and we drop this topic here. How to obtain minimal sample sizes for this model is shown in Spangl et al. (2020).
The model equation for the balanced case is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have, with the same suffixes, equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are given in Table 7.27.
Table 7.27 Expected mean squares for the balanced model VII.
Source of variation | E(MS) for model VII |
Between A levels | |
Between B levels within A levels | |
Between C levels | |
Interaction A × C | |
Interaction B × C within A | |
Residual | σ2 |
From Table 7.27, we can derive the test statistic for testing
H0 : bj(i) = 0 ∀ j, i against HA: at least one bj(i) ≠ 0
is, under H0, F[a(b − 1); a(b − 1)(c − 1)]‐distributed with a(b − 1) and a(b − 1)(c − 1) degrees of freedom.
The model equation is:
The model becomes complete under the conditions that the random variables on the right‐hand side of the equation are uncorrelated and have, with the same suffixes, equal variances, and all fixed effects besides μ over j and k sum up to zero.
The expected mean squares are given in Table 7.28.
Table 7.28 Expected mean squares for model VIII.
Source of variation | E(MS) for model VIII |
Between A levels | |
Between B levels within A levels | |
Between C levels | |
Interaction A × C | |
Interaction B × C within A | |
Residual | σ2 |
From Table 7.28, we can derive the test statistic for testing
H0 : ck = 0 ∀ k against HA: at least one ck ≠ 0.
is, under H0, F(c − 1; (a − 1)(c − 1))‐distributed with c − 1 and (a − 1)(c − 1) degrees of freedom.
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