Experimental with one treatment factor

Simple Experiment (Randomized Complete Design)

Balanced

Statistical Model is
$y_{ij} = \mu + \tau_i + \epsilon_{ij},i=1,\cdots,a,j = 1,\cdots,n$yij​=μ+τi​+ϵij​,i=1,⋯,a,j=1,⋯,n

$\mu$μ is the grand mean, $\tau_i$τi​ is the treatment effect of factor level $i$i.

Assumptions:

1. Fixed Effect: $\sum_{i=1}^a \tau_i = 0$∑i=1a​τi​=0
2. Homogenous normality: $\epsilon_{ij} \sim_{iid} N(0,\sigma^2)$ϵij​∼iid​N(0,σ2)

ANOVA table is ($N=na$N=na)

Hypothesises of ANOVA F test is
$H_0:\tau_1 = \cdots = \tau_a = 0 \\ H_a:at\ least\ one\ is\ not\ zero$H0​:τ1​=⋯=τa​=0Ha​:at least one is not zero

And the rejection rule is $F_0>F_{1-\alpha,a-1,N-a}$F0​>F1−α,a−1,N−a​.

OLS estimators for the parameters are

Unbalanced

Statistical Model is
$y_{ij} = \mu + \tau_i + \epsilon_{ij},i=1,\cdots,a,j = 1,\cdots,n_i$yij​=μ+τi​+ϵij​,i=1,⋯,a,j=1,⋯,ni​

$\mu$μ is the grand mean, $\tau_i$τi​ is the treatment effect of factor level $i$i.

Assumptions:

1. Fixed Effect: $\sum_{i=1}^a \tau_i = 0$∑i=1a​τi​=0
2. Homogenous normality: $\epsilon_{ij} \sim_{iid} N(0,\sigma^2)$ϵij​∼iid​N(0,σ2)

RCBD

$y_{ij} = \mu + \tau_i + \beta_j + \epsilon_{ij}\\ \epsilon_{ij} \sim_{iid}N(0,\sigma^2);i=1,\cdots,a;j=1,\cdots,b$yij​=μ+τi​+βj​+ϵij​ϵij​∼iid​N(0,σ2);i=1,⋯,a;j=1,⋯,b

where $\tau_i$τi​ means treatment effect and $\beta_j$βj​ means blocking effect.

Assumptions:

1. Fixed Effect: $\sum_{i=1}^a \tau_i = 0,\sum_{i=1}^b \beta_j = 0$∑i=1a​τi​=0,∑i=1b​βj​=0
2. Homogenous normality: $\epsilon_{ij} \sim_{iid} N(0,\sigma^2)$ϵij​∼iid​N(0,σ2)

ANOVA table is

Source	SS	df	MS	F
Treatment	$SSM$SSM	a-1	$MSM = \frac{SSM}{df_M}$MSM=dfM​SSM​	$F_M=MSM/MSE$FM​=MSM/MSE
Blocking	$SSB$SSB	b-1	$MSB=\frac{SSB}{df_B}$MSB=dfB​SSB​	$F_B = MSB/MSE$FB​=MSB/MSE
Residuals	$SSE$SSE	(a-1)(b-1)	$MSE = \frac{SSE}{df_E}$MSE=dfE​SSE​
Total	$SST$SST	N-1	$MST = \frac{SST}{df_T}$MST=dfT​SST​

$\sum_{i=1}^a \sum_{j=1}^b (y_{ij}+\bar{y}_{..} - \bar{y}_{i.} - \bar{y}_{.j})^2 = \sum_{i=1}^a \sum_{j=1}^b e_{ij}^2 = SSE \\ \sum_{i=1}^a \sum_{j=1}^b (\bar{y}_{.j}-\bar{y}_{..})^2 = \sum_{i=1}^a \sum_{j=1}^b \hat{\beta}_j^2 = SSB\\ \sum_{i=1}^a \sum_{j=1}^b (\bar{y}_{i.}-\bar{y}_{..})^2 = \sum_{i=1}^a \sum_{j=1}^b \hat{\tau}_i^2 = SSM \\ SST=SSM+SSB+SSE$i=1∑a​j=1∑b​(yij​+yˉ​..​−yˉ​i.​−yˉ​.j​)2=i=1∑a​j=1∑b​eij2​=SSEi=1∑a​j=1∑b​(yˉ​.j​−yˉ​..​)2=i=1∑a​j=1∑b​β^​j2​=SSBi=1∑a​j=1∑b​(yˉ​i.​−yˉ​..​)2=i=1∑a​j=1∑b​τ^i2​=SSMSST=SSM+SSB+SSE

Parameter estimation:
$y_{i.} = \sum_{j=1}^b y_{ij}, \bar{y}_{i.} = \frac{y_{i.}}{b} \\ y_{.j} = \sum_{i=1}^a y_{ij}, \bar{y}_{.j} = \frac{y_{.j}}{a} \\ y_{..} = \sum_{i=1}^a y_{i.}=\sum_{j=1}^b y_{.j}, \bar{y}_{..} = \frac{y_{..}}{N},N=ab \\ \hat{\tau}_i = \bar{y}_{i.}-\bar{y}_{..},\hat{\beta}_j=\bar{y}_{.j}-\bar{y}_{.j},e_{ij}=y_{ij} - \bar{y}_{i.} - \bar{y}_{.j} + \bar{y}_{..}$yi.​=j=1∑b​yij​,yˉ​i.​=byi.​​y.j​=i=1∑a​yij​,yˉ​.j​=ay.j​​y..​=i=1∑a​yi.​=j=1∑b​y.j​,yˉ​..​=Ny..​​,N=abτ^i​=yˉ​i.​−yˉ​..​,β^​j​=yˉ​.j​−yˉ​.j​,eij​=yij​−yˉ​i.​−yˉ​.j​+yˉ​..​

Latin Square Design

Graeco-Latin Square Design

BIBD

Experimental with multiple treatment factors

Factorial design with fixed factors

Factorial design with random factors

Factorial design with mixed factors

Nested design

Split-plot design