Chapter 15 ANOVA for Balanced Split-Plot Experiments
Model: yijk=μij+bk+wik+eijk, Genotype i=1,2,3, Fertilizer j=1,2,3,4, Block k=1,2,3,4.
- Because the experiment is balanced, the GLS estimator is equal to the OLS estimator for any estimable Cβ: CˆβΣ=Cˆβ.
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there are no terms in our model corresponding to Block×Fert.
E(MSBlocks×Fert)=E(MSBlocks×Geno×Fert)=σ2e.
E(MSGeno )=sbw−1w∑i=1E(ˉyi..−ˉy…)2=sbw−1w∑i=1E(ˉμi−ˉμ..+ˉwi−ˉw..+ˉei.−ˉe…)2=sb{∑wi=1(ˉμi−ˉμ..)2w−1+E[∑wi=1(ˉwi−ˉw..)2w−1]+E[∑wi=1(ˉei..−ˉe…..)2w−1]}=sb∑wi=1(ˉμi−ˉμ..)2w−1+sbσ2wb+sbσ2esb=sb∑wi=1(ˉμi−ˉμ.)2w−1+sσ2w+σ2e
E(MSBlock × Geno )=s(w−1)(b−1)w∑i=1b∑k=1E(ˉyi⋅k−ˉyi..−ˉy⋅⋅k+ˉy…)2=s(w−1)(b−1)w∑i=1b∑k=1E(wik−ˉwi.−ˉw⋅k+ˉw..+ˉei⋅k−ˉei..−ˉe..k+ˉe…)2=s(w−1)(b−1)E[w∑i=1b∑k=1(wik−ˉwi.)2−2w∑i=1b∑k=1(wik−ˉwi)(ˉw⋅k−ˉw..)+w∑i=1b∑k=1(ˉw⋅k−ˉw..)2+e2 sum ]=s(w−1)(b−1)E[w∑i=1b∑k=1(wik−ˉwi.)2−wb∑k=1(ˉw⋅k−ˉw..)2+e2sum]=s(w−1)(b−1)[w(b−1)σ2w−w(b−1)σ2w/w+E(e2 sum )]=sσ2w+σ2e
where
E(e2sum)=E[w∑i=1b∑k=1(ˉei⋅k−ˉei.−ˉe..k+ˉe…)2]=(w−1)(b−1)sσ2e
Inference for Whole-Plot
Test for whole-plot factor main effects:
H0:ˉμ1.=…=ˉμw.⇔H0:sbw−1w∑i=1(ˉμi.−ˉμ..)2=0
We compare MSGenoMSBlock×Geno to a central F-test with w−1 and (w−1)(b−1) degrees of freedom.
Comparison for whole-plot marginal means:
H0:ˉμ1.=ˉμ2.
Note that Var(ˉy1..−ˉy2..)=Var(ˉμ1.−ˉμ2.+ˉw1.−ˉw2.+ˉe1..−ˉe2⋅.)=2σ2wb+2σ2esb=2sb(sσ2w+σ2e)=2sbE(MSBlock× Geno )
so that ^Var(ˉy1..−ˉy2..)=2sbMSBlock×Geno. We can use t=ˉy1..−ˉy2..−(ˉμ1.−ˉμ2.)√2sbMSBlock × Geno ∼t(w−1)(b−1)
to test H0.
Multiple test for whole-plot marginal means: H_{0}: \boldsymbol{C}\left[\begin{array}{c}\bar{\mu}_{1.} \\\vdots \\\bar{\mu}_{w.}\end{array}\right]=\mathbf{0},
we can use an F statistics with q and (w-1)(b-1) degrees of freedom: F=\frac{\left(\boldsymbol{C}\left[\begin{array}{c}\bar{y}_{1..} \\\vdots \\\bar{y}_{w..}\end{array}\right]\right)^{\prime}\left[\frac{M S_{\text {Block } \times \text{Gen}}}{s b} \boldsymbol{C} \boldsymbol{C}^{\prime}\right]^{-1}\left(\boldsymbol{C}\left[\begin{array}{c}\bar{y}_{1..} \\\vdots \\\bar{y}_{w..}\end{array}\right]\right)}{q} where C is a matrix whose rows are contrast vectors so that C1= 0.
Inference for Split-Plot
\begin{aligned}E\left(M S_{F e r t}\right) &=\frac{w b}{s-1} \sum_{j=1}^{s} E\left(\bar{y}_{.j .}-\bar{y}_{...}\right)^{2} \\&=\frac{w b}{s-1} \sum_{j=1}^{s} E\left(\bar{\mu}_{. j}-\bar{\mu}_{..}+\bar{e}_{. j.}-\bar{e}_{...}\right)^{2} \\&=\frac{w b}{s-1} \sum_{j=1}^{s}\left(\bar{\mu}_{. j}-\bar{\mu}_{..}\right)^{2}+\sigma_{e}^{2} \\ E(MS_{error}) & = \sigma_e^2 \end{aligned}
Test for split-plot main effects:
H_{0}: \bar{\mu}_{.1}=\cdots=\bar{\mu}_{. s} \Longleftrightarrow H_{0}: \frac{w b}{s-1} \sum_{j=1}^{s}\left(\bar{\mu}_{. j}-\bar{\mu}_{..}\right)^{2}=0
We compare \frac{MS_{Fert}}{MS_{error}} to a central F distribution with s-1 and w(s-1)(b-1) degrees of freedom.
Comparison for split-plot marginal means:
H_0: \bar \mu_{.1} = \bar \mu_{.2}
We can use t=\frac{\bar{y}_{.1.}-\bar{y}_{. 2 .}-\left(\bar{\mu}_{.1}-\bar{\mu}_{. 2}\right)}{\sqrt{\frac{2}{w b} M S_{E r r o r}}} \sim t_{w(s-1)(b-1)}
because \operatorname{Var}\left(\bar{y}_{\cdot 1 \cdot}-\bar{y}_{\cdot 2 \cdot}\right) =\operatorname{Var}\left(\bar{\mu}_{\cdot 1}-\bar{\mu}_{\cdot 2}+\bar{e}_{\cdot 1 \cdot}-\bar{e}_{\cdot 2 \cdot}\right) =\frac{2}{w b} \sigma_{e}^{2}=\frac{2}{w b} E\left(M S_{\text {Error }}\right).
Multiple test for split-plot marginal means: H_{0}: \boldsymbol{C}\left[\begin{array}{c}\bar{\mu}_{.1} \\\vdots \\\bar{\mu}_{.w}\end{array}\right]=\mathbf{0},
we can use an F statistics with q and w(s-1)(b-1) degrees of freedom: F=\frac{\left(\boldsymbol{C}\left[\begin{array}{c}\bar{y}_{.1.} \\\vdots \\\bar{y}_{.w.}\end{array}\right]\right)^{\prime}\left[\frac{M S_{Error}}{w b} \boldsymbol{C} \boldsymbol{C}^{\prime}\right]^{-1}\left(\boldsymbol{C}\left[\begin{array}{c}\bar{y}_{.1.} \\\vdots \\\bar{y}_{.1.}\end{array}\right]\right)}{q}
where C is a matrix whose rows are contrast vectors so that C1= 0.
Inference for Interactions
\begin{aligned}&E\left(M S_{\text {Geno } \times \text { Fert }}\right)=\frac{b}{(w-1)(s-1)} \sum_{i=1}^{w} \sum_{j=1}^{s} E\left(\bar{y}_{i j.}-\bar{y}_{i ..}-\bar{y}_{.j.}+\bar{y}_{...}\right)^{2} \\&=\frac{b}{(w-1)(s-1)} \sum_{i=1}^{w} \sum_{j=1}^{s} E\left(\mu_{i j}-\bar{\mu}_{i.}-\bar{\mu}_{. j}+\bar{\mu}_{..}+\bar{e}_{i j .}-\bar{e}_{i . .}-\bar{e}_{.j .}+\bar{e}_{...}\right)^{2} \\&=\quad \cdots \\&=\frac{b}{(w-1)(s-1)} \sum_{i=1}^{w} \sum_{j=1}^{s}\left(\mu_{i j}-\bar{\mu}_{i.}-\bar{\mu}_{. j}+\bar{\mu}_{..}\right)^{2}+\sigma_{e}^{2}\end{aligned}
Note that \mu_{i j}-\bar{\mu}_{i.}-\bar{\mu}_{. j}+\bar{\mu}_{..} = 0. \forall i, j is equivalent to \mu_{i j}-\bar{\mu}_{i^*j}-\bar{\mu}_{i j^*}+\bar{\mu}_{i^*j^*}, \forall i\neq i^*, j\neq j^*. Therefore, \frac{b}{(w-1)(s-1)} \sum_{i=1}^{w} \sum_{j=1}^{s}\left(\mu_{i j}-\bar{\mu}_{i.}-\bar{\mu}_{. j}+\bar{\mu}_{..}\right)^{2} = 0 can be used as null hypothesis for no interactions between genotypes and fertilizers.
Test for Whole * Split interaction Effects
H_0: \frac{b}{(w-1)(s-1)} \sum_{i=1}^{w} \sum_{j=1}^{s}\left(\mu_{i j}-\bar{\mu}_{i.}-\bar{\mu}_{. j}+\bar{\mu}_{..}\right)^{2} = 0
We can compare \frac{MS_{Geno\times Fert}}{MS_{Error}} to a central F distribution with (w-1)(s-1) and w(s-1)(b-1) degrees of freedom.
Inference for simple effects:
H_0: \mu_{11} = \mu_{12}
Note that \widehat{Var}(\bar y_{11.} -\bar y_{12.}) = \frac{2}{b}MS_{error}, we can use t=\frac{\bar{y}_{11 \cdot}-\bar{y}_{12 \cdot}-\left(\mu_{11}-\mu_{12}\right)}{\sqrt{\frac{2}{b} M S_{\text {Error }}}} \sim t_{w(s-1)(b-1)} to test H_0.
However, Var(\bar y_{11.} - \bar y_{21.}) = \frac{2}{b}(\sigma_w^2 + \sigma_e^2) which is not a constant times any expected mean square from our ANOVA table. E(\frac{1}{s}MS_{Block\times Geno}+\frac{s-1}{s}MS_{error}) = \sigma_w^2 + \frac{\sigma_e^2}{s} + \frac{(s-1)\sigma_e^2}{s} = \sigma_w^2 + \sigma_e^2. So that \widehat{Var}(\bar y_{11.} - \bar y_{21.})= \frac{2}{sb}MS_{Block\times Geno}+\frac{2s-2}{s}MS_{error}. We can use
\frac{\bar{y}_{11 \cdot}-\bar{y}_{21 \cdot}-\left(\mu_{11}-\mu_{12}\right)}{\sqrt{\frac{2}{sb}MS_{Block\times Geno}+\frac{2s-2}{s}MS_{error}}} \sim t_{d} \text{ with }d \text{ degrees of freedom}
Inference for Cell Means \mu_{ij}: Note Var(\bar y_{ij.}) = \frac{\sigma_b^2}{b} + \frac{\sigma_w^2}{b} + \frac{\sigma_e^2}{b} so that we can construct unbiased estimator with approximate degrees of freedom from Cochran-Satterthwaite. ****