통계방법론 W7

ese2o·2024년 6월 15일

Analysis of Variance and Design of Experiments (2)

Randomized Block Design (RBD)

CRD와 달리, 실험에서 고려되지 않은 외부요인에 의한 변동을 제어

RBD includes a second variable, referred to as a blocking variable, that can be used to control for confounding (or concomitant) variable(교란 변수)

교란변수; Confounding (concomitant) variables: variables that are not being controlled by the analyst in the experiment, but can have an effect on the outcome of the treatment being studied

Blocking variable: a variable that the analyst wants to control, but is not the treatment variable of interest

RBD: formula

\begin{aligned} & \mathrm{SSC}=n \sum_{j=1}^C\left(\bar{x}_j-\bar{x}\right)^2 \\ & \mathrm{SSR}=C \sum_{i=1}^n\left(\bar{x}_i-\bar{x}\right)^2 \\ & \mathrm{SSE}^{\prime}=\sum_{i=1}^n \sum_{j=1}^C\left(x_{i j}-\bar{x}_j-\bar{x}_i+\bar{x}\right)^2 \\ & \mathrm{SST}=\sum_{i=1}^n \sum_{j=1}^C\left(x_{i j}-\bar{x}\right)^2 \\ & \end{aligned}

$i$ : block group
$j$ : treatment level
$n$ : number of observations in each treatment level C : number of treatment levels
$N = nC$ : total number of observations
$x_ij$ : individual observation
$\bar x_i$ : block mean
$\bar x_j$ : treatment mean
$\bar x$ : total mean

\begin{aligned} \mathrm{MSC} & =\frac{S S C}{C-1} \\ \mathrm{MSR} & =\frac{S S R}{n-1} \\ \mathrm{MSE}^{\prime} & =\frac{S S E^{\prime}}{N-n-C+1} \end{aligned}

\begin{aligned} \mathrm{F}_{\text {treatments }} & =\frac{\mathrm{MSC}}{\mathrm{MSE}^{\prime}} \\ F_{\text {blocks }} & =\frac{\mathrm{MSR}}{\mathrm{MSE}^{\prime}} \end{aligned}

Step 1.

가설을 2개 세운다.

Hypothesis for the treatment effect:

$H_0$ : the mean value is identical for different treatments
$H_a$ : At least one of the treatment means is different from the others

Hypothesis for the blocking effect:

$H_0$ : the mean value is identical for different blocks
$H_a$ : At least one of the blocking means is different from the others

Step 2.

compute observed values(SSE, SSR, SST, MSE, ... formula)

Step 3.

conclude the process

1) For treatment

observed value: $F_t=\frac{\mathrm{MSC}}{\mathrm{MSE}^{\prime}}$

critical value: $F_{\alpha, C-1,(C-1)(b-1)}=F_{0.01,2,8}$

Because the observed value of $F_t$ is greater than the critical $F$ value, the null hypothesis is rejected.

At least one of the population means of the treatment level is not the same as the others

2) For Blocking

observed value: $F_b=\frac{\mathrm{MSB}}{\mathrm{MSE}^{\prime}}$

critical value: $F_{\alpha, b-1,(C-1)(b-1)}=F_{0.01,4,8}$

Because the observed value of $F_b$ is greater than the critical F value, the blocking effect is significant.

추가적으로, ANOVA table을 다음과 같이 작성할 수 있다.

blocking 없는 F value는 $F=\frac{\mathrm{MSC}}{\mathrm{MSE}}$ 로 구할 수 있다.

Two-way ANOVA

: test the effect of two independent variables (or factors) on one dependent variable

예를 들어, 하나 이상의 요인(factor: 자동차 등급과 제조사)에 의해 종속변수가 결정되는 경우
Factorial: all possible combinations of the levels of the factors are investigated
interaction effect를 고려해야 한다.
조건: Each sample must be i.i.d. from a normally distributed population / Each population must have the same variance

Two-way ANOVA: formula

\begin{aligned} & \mathrm{SSR}=n C \sum_{i=1}^R\left(\bar{x}_i-\bar{x}\right)^2 \\ & \mathrm{SSC}=n R \sum_{j=1}^C\left(\bar{x}_j-\bar{x}\right)^2 \\ & \mathrm{SSI}=n \sum_{i=1}^R \sum_{j=1}^C\left(\bar{x}_{i j}-\bar{x}_i-\bar{x}_j+\bar{x}\right)^2 \\ & \mathrm{SSE}=\sum_{i=1}^R \sum_{j=1}^C \sum_{k=1}^n\left(x_{i j k}-\bar{x}_{i j}\right)^2 \\ & \mathrm{SST}=\sum_{i=1}^R \sum_{j=1}^C \sum_{k=1}^n\left(x_{i j k}-\bar{x}\right)^2 \end{aligned}

$i$ : row treatment level
$j$ : column treatment level
$k$ : cell member
$n$ : number of observations per cell
$C$ : number of column treatments
$R$ : number of row treatments
$x_{ijk}$ : individual observation
$\bar x_{ij}$ : cell mean
$\bar x_i$ : row mean
$\bar x_j$ : column mean
$\bar x$ : total mean

\begin{aligned} & \mathrm{MSR}=\frac{S S R}{R-1} \\ & \mathrm{MSC}=\frac{S S C}{C-1} \\ & \mathrm{MSI}=\frac{S S I}{(R-1)(C-1)} \\ & \mathrm{MSE}=\frac{S S E}{R C(n-1)} \end{aligned}

\begin{aligned} & \mathrm{F}_{\mathrm{R}}=\frac{\mathrm{MSR}}{\mathrm{MSE}} \\ & \mathrm{F}_{\mathrm{C}}=\frac{\mathrm{MSC}}{\mathrm{MSE}} \\ & \mathrm{F}_{\mathrm{I}}=\frac{\mathrm{MSI}}{\mathrm{MSE}} \end{aligned}