Nuisance Removal¶

Blocking vs Randomization¶

Block what you can, randomize what you cannot

	Blocking	Covariate analysis	Randomization
	Group experiments into blocks, each having a fixed value of the variable	Put measured uncontrolled inputs into the model Ignore them when modeling is complete
	Increase $\text{cov}(T_1, T_2)$ to decrease $\sigma^2(T_1-T_2)$	Model impact of variable, and then subtract it out	Prevent unknown effect from biasing results
	Error that is same for $T_1$ and $T_2$ will cancel out		Turn systematic errors into random errors which average out to zero
Effective for	Measured uncontrolled inputs		Unmeasured uncontrolled inputs
Application	Understanding treatment effects		Allows for time-series analysis to detect drift
Example	Randomly assign each person to - one new shoe - one old shoe (randomly assigned to left/right foot)		Randomly assign - half of participants w/ new shoes - half of participants w/ old shoes

\[ \begin{aligned} \sigma^2(\text{ATE}) &= \sigma^2(T_1-T_2) \\ &= \sigma^2_{T_1} + \sigma^2_{T_2} - 2 \text{ cov}(T_1, T_2) \end{aligned} \]

For measured uncontrolled inputs, use blocking or covariate analysis to remove effect of nuisance factors, and reduce known variability

Different measurement tools, prices batches
Spatial/temporal variations

Random Treatment Assignment¶

Used for removing sources of variation due to nuisance factors

Blocking/Randomized Complete Block Design (RCBD)¶

Complete experiment is performed for each block
Each block sees each treatment exactly onec
With each block, testing order is randomized
Examples of blocks
Raw material batches
People (operators)
Process/measurement tools
Time

$\alpha_1$	$\alpha_2$	$\alpha_3$	$\alpha_4$
$T_2$	$T_1$	$T_1$	$T_3$
$T_1$	$T_3$	$T_2$	$T_2$
$T_3$	$T_2$	$T_3$	$T_1$

\[ T \in \{ T_1, T_2, T_3 \} \\ s \in \{ \alpha_1, \alpha_2, \alpha_3, \alpha_4 \} \]

Note: The term ‘blocking’ originated from agriculture, where a block is typically a set of homogeneous (contiguous) plots of land with similar fertility, moisture, and weather, which are typical nuisance factors in agricultural studies $$ y_{ij} = f(T_i) + \text{Block}j + u $$

Balanced Incomplete Block Design¶

Latin Square Design (LSD)¶

	$\alpha_1$	$\alpha_2$	$\alpha_3$
$\beta_1$	$T_1$	$T_2$	$T_3$
$\beta_2$	$T_2$	$T_3$	$T_1$
$\beta_3$	$T_3$	$T_1$	$T_2$

\[ \begin{aligned} T &\in \{ T_1, T_2, T_3 \} \\ s_1 &\in \{ \alpha_1, \alpha_2, \alpha_3 \} \\ s_2 &\in \{ \beta_1, \beta_2, \beta_3 \} \end{aligned} \]

A Latin square of order $n$ is an $n \times n$ array of cells in which $n$ treatments are placed, one per cell, such that each treatment only occurs

once in each row
once in each column

Requirements

Number of levels of each blocking var must equal number of levels of primary var
No interactions between input vars, especially b/w nuisance vars

Outcome

Not complete block
Model is orthogonal, if no interactions

\[ y_{ijk} = f(T_i) + R_j + C_k + u_{ijk} \]

Example: Test effect of amount of gasoline additive on emissions

Last Updated: 2024-05-14 ; Contributors: AhmedThahir

\(\alpha_1\)	\(\alpha_2\)	\(\alpha_3\)	\(\alpha_4\)
\(T_2\)	\(T_1\)	\(T_1\)	\(T_3\)
\(T_1\)	\(T_3\)	\(T_2\)	\(T_2\)
\(T_3\)	\(T_2\)	\(T_3\)	\(T_1\)

	\(\alpha_1\)	\(\alpha_2\)	\(\alpha_3\)
\(\beta_1\)	\(T_1\)	\(T_2\)	\(T_3\)
\(\beta_2\)	\(T_2\)	\(T_3\)	\(T_1\)
\(\beta_3\)	\(T_3\)	\(T_1\)	\(T_2\)

	Blocking	Covariate analysis	Randomization
	Group experiments into blocks, each having a fixed value of the variable	Put measured uncontrolled inputs into the model Ignore them when modeling is complete
	Increase \(\text{cov}(T_1, T_2)\) to decrease \(\sigma^2(T_1-T_2)\)	Model impact of variable, and then subtract it out	Prevent unknown effect from biasing results
	Error that is same for \(T_1\) and \(T_2\) will cancel out		Turn systematic errors into random errors which average out to zero
Effective for	Measured uncontrolled inputs		Unmeasured uncontrolled inputs
Application	Understanding treatment effects		Allows for time-series analysis to detect drift
Example	Randomly assign each person to - one new shoe - one old shoe (randomly assigned to left/right foot)		Randomly assign - half of participants w/ new shoes - half of participants w/ old shoes