> ## Documentation Index
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> Use this file to discover all available pages before exploring further.

# Technical CUPED details

> Technical details of CUPED variance estimates

Here we document the technical details behind GrowthBook CUPED variance estimates. We begin with mean metrics, and then move to [ratio metrics](/statistics/cuped-technical#ratio-metric).

## Mean or Binomial Metric

We use the notation below. We describe our approach in terms of revenue, but any Mean or Binomial metric can be substituted.

1. Define $Y_{C}$ $\left(Y_{T}\right)$ as the observed post-exposure revenue for a user exposed to control (treatment).
2. Define $X_{C}$ $\left(X_{T}\right)$ as the observed pre-exposure revenue for a user exposed to control (treatment).
3. Define $Y$ ($X$) as the post-exposure (pre-exposure) revenue for all users collectively in the experiment.
4. Define $\bar{Y}_{C}$ $\left(\bar{Y}_{T}\right)$ as the sample average post-exposure revenue for users exposed to control (treatment).
5. Define $\mu_{C}$ $\left(\mu_{T}\right)$ as the population average post-exposure revenue for users exposed to control (treatment).
6. Define $N_{C}$ $\left(N_{T}\right)$ as the number of users exposed to control (treatment).

## Mean or Binomial Metric, Absolute case

For absolute inference, our target parameter is

$$
\begin{align}
\Delta_{A}&=\mu_{Y}-\mu_{YC}.
\end{align}
$$

As described in Equation 4 of ([Deng et al. 2013](https://exp-platform.com/Documents/2013-02-CUPED-ImprovingSensitivityOfControlledExperiments.pdf)), we find the optimal $\theta$ using user data across both control and treatment:
$\theta = cov(Y, X) / var(X).$
Our estimate of $\Delta_{A}$ is the difference in adjusted means

$$
\begin{align}
\hat{\Delta}_{A} &= \left(\bar{Y}_{T} - \theta\bar{X}_{T}\right)  - \left(\bar{Y}_{C} - \theta\bar{X}_{C}\right).
\end{align}
$$

Under a superpopulation framework and independence of random assignment, the adjusted means $\left(\bar{Y}_{T} - \theta\bar{X}_{T}\right)$ and $\left(\bar{Y}_{C} - \theta\bar{X}_{C}\right)$ are statistically independent.\
Therefore, the variance of the difference in adjusted means is the sum of the variances of the adjusted means.\
We denote these variances as $V_{adj, C}$ and $V_{adj, T}$, respectively, and they are defined as
Define the control (treatment) population covariance between post-exposure and pre-exposure revenue as $\sigma_{XY,C}$ ($\sigma_{XY,T}$).

$$
\begin{align}
V_{adj, C} &= \frac{\sigma^{2}_{YC} + \theta^{2}\sigma^{2}_{XC} - 2\theta\sigma_{XY,C}}{N_{C}}\\
V_{adj, T} &= \frac{\sigma^{2}_{YT} + \theta^{2}\sigma^{2}_{XT} - 2\theta\sigma_{XY,T}}{N_{T}}.
\end{align}
$$

Our estimated variance of $\hat{\Delta}_{A}$ is $\hat{\sigma}^{2}_{\Delta_{A}} = V_{adj, C} + V_{adj, T}$.

Formulae for variances of the statistics can be found in our sections on [proportion metrics](/statistics/details#proportion-metrics) and [mean metrics](/statistics/details#mean-metrics).
When estimating the covariance between two binomial variables $X$ and $Y$, we use the following formula:

$$
\begin{align*}
\sigma_{XY} &= E(XY) - E(X)E(Y)
\\ &= P(X=1, Y=1) - P(X=1)P(Y=1)
\\ &= P(X=1) * P(Y=1|X=1) - P(X=1)P(Y=1)
\\ &= P(X=1) * (P(Y=1|X=1) - P(Y=1)).
\end{align*}
$$

For mean metrics, we use the usual covariance formula:

$$
\begin{align*}
\sigma_{XY} &= \frac{1}{N-1} \sum_{i=1}^{N} (X_i - \bar{X})(Y_i - \bar{Y}).
\end{align*}
$$

## Mean or Binomial Metric, Relative case

For relative inference (i.e., estimating lift), the parameter of interest is

$$
\begin{align}
\Delta_{R}&=\frac{\mu_{T}-\mu_{C}}{\mu_{C}}.
\end{align}
$$

Our estimate of $\Delta_{R}$ is the difference in adjusted means divided by the control mean:

$$
\begin{align}
\hat{\Delta}_{R} = \frac{\left(\bar{Y}_{T} - \theta\bar{X}_{T}\right)  - \left(\bar{Y}_{C} - \theta\bar{X}_{C}\right)}{\bar{Y}_{C}}.
\end{align}
$$

To derive $\hat{\sigma}^{2}_{\Delta_{R}}$, the estimated variance of $\hat{\Delta}_{R}$, we use the delta method.

1. Define the control (treatment) population post-exposure variance as $\sigma^{2}_{YC}$ ($\sigma^{2}_{YT}$).
2. Define the control (treatment) population pre-exposure variance as $\sigma^{2}_{XC}$ ($\sigma^{2}_{XT}$).
3. Define the covariance of the sample control means $ \boldsymbol{\Lambda}_{C} = \text{Cov}\left[\bar{Y}_{C}, \bar{X}_{C}\right] =\begin{pmatrix}
   \sigma^{2}_{Y,C} & \sigma*{XY,C}\\
   \sigma*{XY,C} & \sigma^{2}_{X,C}
   \end{pmatrix}/ N_{C}$.
4. Define the covariance of the sample treatment means $ \boldsymbol{\Lambda}_{T} = \text{Cov}\left[\bar{Y}_{T}, \bar{X}_{T}\right] =\begin{pmatrix}
   \sigma^{2}_{Y,T} & \sigma*{XY,T}\\
   \sigma*{XY,T} & \sigma^{2}_{X,T}
   \end{pmatrix}/ N_{T}$.
5. Define the vector of population means $\boldsymbol{\beta}_{0} = \left[\mu_{YT}, \mu_{XT}, \mu_{YC}, \mu_{XC} \right].$
6. Define their sample counterparts as $\hat{\boldsymbol{\beta}} = \left[\bar{Y}_{T}, \bar{X}_{T}, \bar{Y}_{C}, \bar{X}_{C} \right].$
7. Define $\boldsymbol{\Lambda} = \text{Cov}\left(\hat{\boldsymbol{\beta}}\right) = \begin{pmatrix}
   \boldsymbol{\Lambda}_{T} & \textbf{0}\\
   \textbf{0} & \boldsymbol{\Lambda}_{C}
   \end{pmatrix},$
   where $\textbf{0}$ is a $2 \times 2$ matrix of zeros.

By the multivariate central limit theorem:

$$
\begin{align}
\hat{\boldsymbol{\beta}}
\stackrel{}{\sim}\mathcal{MVN}\left(\boldsymbol{\beta}_{0},\boldsymbol{\Lambda}\right).
\end{align}
$$

For vector $\boldsymbol{\beta}$, define its $k^{\text{th}}$ element as $\beta[k]$.\
Define the function
$g(\boldsymbol{\beta}; \theta) = \frac{\left(\beta[1] - \theta\beta[2]\right)  - \left(\beta[3] - \theta\beta[4]\right)}{\beta[3]}.$

Define the vector of partial derivatives as $\boldsymbol{\nabla}_{r} = \frac{\partial g(\boldsymbol{\beta})}{\partial\boldsymbol{\beta}}$, where the individual elements are

$$
\begin{align*}
\boldsymbol{\nabla}[1] &= \frac{1}{\boldsymbol{\beta}[3]}
\\\boldsymbol{\nabla}[2] &= \frac{-\theta}{\boldsymbol{\beta}[3]}
\\\boldsymbol{\nabla}[3] &= \frac{-\boldsymbol{\beta}[3]
-
\left(\boldsymbol{\beta}[1] - \theta\boldsymbol{\beta}[2] - \boldsymbol{\beta}[3] + \theta\boldsymbol{\beta}[4]\right)
}{\boldsymbol{\beta}[3]^{2}}
= \frac{
-\boldsymbol{\beta}[1] + \theta\boldsymbol{\beta}[2]  - \theta\boldsymbol{\beta}[4]
}{\boldsymbol{\beta}[3]^{2}}
\\\boldsymbol{\nabla}[4] &= \frac{\theta}{\boldsymbol{\beta}[3]}.
\end{align*}
$$

By the delta method,
$\hat{\Delta}_{r} = g(\hat{\boldsymbol{\beta}}) \stackrel{}{\sim}\mathcal{N}\left(\Delta_{r} = g\left(\boldsymbol{\beta}\right), \boldsymbol{\nabla}_{r}^{\top}\Lambda\boldsymbol{\nabla}_{r} \right)$.

Decompose $\boldsymbol{\nabla}_{r}$ into $\boldsymbol{\nabla}_{r} = \left[
\boldsymbol{\nabla}_{r}[1:2], \boldsymbol{\nabla}_{r}[3:4]
\right].$

Then the final variance

$$
\begin{align*}
\boldsymbol{\nabla}_{r}^{\top}\Lambda\boldsymbol{\nabla}_{r} &=
\boldsymbol{\nabla}_{r}[1:2]^{\top}
\boldsymbol{\Lambda}_{T}
\boldsymbol{\nabla}_{r}[1:2]
+
\boldsymbol{\nabla}_{r}[3:4]^{\top}
\boldsymbol{\Lambda}_{C}
\boldsymbol{\nabla}_{r}[3:4]
\nonumber\\&=
\left[\frac{1}{\boldsymbol{\beta}[3]}, \frac{-\theta}{\boldsymbol{\beta}[3]}\right]
^{\top}
\begin{pmatrix}
\sigma^{2}_{Y,T}  & \sigma_{XY,T} \\
\sigma_{XY,T} & \sigma^{2}_{X,T}
\end{pmatrix}/ N_{T}
\left[\frac{1}{\boldsymbol{\beta}[3]}, \frac{-\theta}{\boldsymbol{\beta}[3]}\right]
\nonumber\\&+ \left[\frac{
-\boldsymbol{\beta}[1] + \theta\boldsymbol{\beta}[2]  - \theta\boldsymbol{\beta}[4]
}{\boldsymbol{\beta}[3]^{2}},  \frac{\theta}{\boldsymbol{\beta}[3]}\right]
\begin{pmatrix}
\sigma^{2}_{Y,C}  & \sigma_{XY,C} \\
\sigma_{XY,C} & \sigma^{2}_{X,C}
\end{pmatrix}/ N_{C}
\left[\frac{
-\boldsymbol{\beta}[1] + \theta\boldsymbol{\beta}[2]  - \theta\boldsymbol{\beta}[4]
}{\boldsymbol{\beta}[3]^{2}},  \frac{\theta}{\boldsymbol{\beta}[3]}\right]
\nonumber\\&=
\frac{1}{N_{T}\boldsymbol{\beta}[3]^{2}}\left[1, -\theta\right]
^{\top}
\begin{pmatrix}
\sigma^{2}_{Y,T}  & \sigma_{XY,T} \\
\sigma_{XY,T} & \sigma^{2}_{X,T}
\end{pmatrix}
\left[1, -\theta\right]
\nonumber\\&+
\frac{1}{N_{C}\boldsymbol{\beta}[3]^{2}}
\left[\frac{
-\boldsymbol{\beta}[1] + \theta\boldsymbol{\beta}[2]  - \theta\boldsymbol{\beta}[4]
}{\boldsymbol{\beta}[3]},  \theta\right]
\begin{pmatrix}
\sigma^{2}_{Y,C}  & \sigma_{XY,C} \\
\sigma_{XY,C} & \sigma^{2}_{X,C}
\end{pmatrix}
\left[\frac{
-\boldsymbol{\beta}[1] + \theta\boldsymbol{\beta}[2]  - \theta\boldsymbol{\beta}[4]
}{\boldsymbol{\beta}[3]},  \theta\right]
\nonumber\\&=\frac{
\sigma^{2}_{Y,T}
+\theta^{2}\sigma^{2}_{X,T}
-2\sigma_{XY,T}
}{N_{T}\boldsymbol{\beta}[3]^{2}}
\nonumber\\&+
\frac{1}{N_{C}\boldsymbol{\beta}[3]^{2}}
\left[\frac{
-\boldsymbol{\beta}[1] + \theta\boldsymbol{\beta}[2]  - \theta\boldsymbol{\beta}[4]
}{\boldsymbol{\beta}[3]},  \theta\right]
\begin{pmatrix}
\frac{\sigma^{2}_{Y,C}  \left(
-\boldsymbol{\beta}[1] + \theta\boldsymbol{\beta}[2]  - \theta\boldsymbol{\beta}[4]
\right)}{
\boldsymbol{\beta}[3]
}
+ \theta\sigma_{XY,C} \\
\frac{\sigma_{XY,C}
\left(
-\boldsymbol{\beta}[1] + \theta\boldsymbol{\beta}[2]  - \theta\boldsymbol{\beta}[4]
\right)}{\boldsymbol{\beta}[3]}
+\theta\sigma^{2}_{X,C}
\end{pmatrix}
\nonumber\\&=\frac{
\sigma^{2}_{Y,T}
+\theta^{2}\sigma^{2}_{X,T}
-2\sigma_{XY,T}
}{N_{T}\boldsymbol{\beta}[3]^{2}}
\nonumber\\&+
\frac{1}{N_{C}\boldsymbol{\beta}[3]^{2}}
\left(
\frac{\sigma^{2}_{Y,C}  \left(
-\boldsymbol{\beta}[1] + \theta\boldsymbol{\beta}[2]  - \theta\boldsymbol{\beta}[4]\right)^{2}}{\boldsymbol{\beta}[3]^{2}}
+2\frac{\theta\sigma_{XY,C}
\left(-\boldsymbol{\beta}[1] + \theta\boldsymbol{\beta}[2]  - \theta\boldsymbol{\beta}[4]\right)}{\boldsymbol{\beta}[3]}
+\theta^{2}\sigma^{2}_{X,C}
\right)
\nonumber\\&=\frac{
\sigma^{2}_{Y,T}
+\theta^{2}\sigma^{2}_{X,T}
-2\sigma_{XY,T}
}{N_{T}\bar{Y}_{C}^{2}}
\nonumber\\&+
\frac{1}{N_{C}\bar{Y}_{C}^{2}}
\left(
\frac{\sigma^{2}_{Y,C}  \left(
-\bar{Y}_{T} + \theta\bar{X}_{T}  - \theta\bar{X}_{C}
\right)^{2}}{\bar{Y}_{C}^{2}}
+2\frac{\theta\sigma_{XY,C}
\left(-\bar{Y}_{T} + \theta\bar{X}_{T}  - \theta\bar{X}_{C}\right)}{\bar{Y}_{C}}
+\theta^{2}\sigma^{2}_{X,C}
\right),
\end{align*}
$$

where in the last step we move away from $\boldsymbol{\beta}$ notation and use sample mean notation.

For estimating uncertainty in production, we use

$$
\begin{align}
\hat{\sigma}^{2}_{\Delta_{R}}&=\frac{
\sigma^{2}_{Y,T}
+\theta^{2}\sigma^{2}_{X,T}
-2\sigma_{XY,T}
}{N_{T}\bar{Y}_{C}^{2}}
\nonumber\\&+
\frac{1}{N_{C}\bar{Y}_{C}^{2}}
\left(
\frac{\sigma^{2}_{Y,C}  \left(
-\bar{Y}_{T}
\right)^{2}}{\bar{Y}_{C}^{2}}
+2\frac{\theta\sigma_{XY,C}
\left(-\bar{Y}_{T}\right)}{\bar{Y}_{C}}
+\theta^{2}\sigma^{2}_{X,C}
\right),
\end{align}
$$

which leverages the fact that the pre-exposure revenue population means are equal due to randomization.

## Ratio Metric

Throughout define the $k^{\text{th}}$ element of vector $\textbf{x}$ as $\textbf{x}[k]$.\
Below we define parameters.

1. Under control, define the numerator (denominator) post-exposure population mean as $\mu_{MYC}$ ($\mu_{DYC}$).
2. Under treatment, define the numerator (denominator) post-exposure population mean as $\mu_{MYT}$ ($\mu_{DYT}$).
3. Under control, define the numerator (denominator) pre-exposure population mean as $\mu_{MXC}$ ($\mu_{DXC}$).
4. Under treatment, define the numerator (denominator) pre-exposure population mean as $\mu_{MXT}$ ($\mu_{DXT}$).

Due to randomization, $\mu_{MXC}$ equals $\mu_{DXC}$ and $\mu_{MXT}$ $\mu_{DXT}$, but for bookkeeping purposes, it is easier to have separate parameters.

## Ratio Metric, Absolute case

For ratio metrics the absolute parameter of interest is

$$
\begin{align}
\Delta_{a}&=
\frac{\mu_{MYT}}{\mu_{DYT}} - \frac{\mu_{MYC}}{\mu_{DYC}}.
\end{align}
$$

Below we define statistics.

1. Under control, define the numerator (denominator) post-exposure sample mean as $\bar{M}_{YC}$ ($\bar{D}_{YC}$).
2. Under treatment, define the numerator (denominator) post-exposure sample mean as $\bar{M}_{YT}$ ($\bar{D}_{YT}$).
3. Under control, define the numerator (denominator) pre-exposure sample mean as $\bar{M}_{XC}$ ($\bar{D}_{XC}$).
4. Under treatment, define the numerator (denominator) pre-exposure sample mean as $\bar{M}_{XT}$ ($\bar{D}_{XT}$).

Define:

1. $\boldsymbol{\beta}_{0} = \left[\mu_{MYT}, \mu_{DYT}, \mu_{MXT}, \mu_{DXT}, \mu_{MYC}, \mu_{DYC}, \mu_{MXC}, \mu_{DXC}\right]$.
2. $\hat{\boldsymbol{\beta}} = \left[\bar{M}_{YT}, \bar{D}_{YT}, \bar{M}_{XT}, \bar{D}_{XT},
   \bar{M}_{YC}, \bar{D}_{YC}, \bar{M}_{XC}, \bar{D}_{XC}
   \right]$.
3. $\boldsymbol{\Lambda}_{T} = \text{Cov}\left[\bar{M}_{YT}, \bar{D}_{YT}, \bar{M}_{XT}, \bar{D}\_{XT},\right]
   N*{T}^{-1}
   \begin{pmatrix}
   \text{Var}\left(M_{YT}\right) & & &\\
   \text{Cov}\left(M_{YT}, D_{YT}\right) & \text{Var}\left(D_{YT}\right) & & \\
   \text{Cov}\left(M_{YT}, M_{XT}\right) & \text{Cov}\left(D_{YT}, M_{XT}\right) & \text{Var}\left(M_{XT}\right) & \\
   \text{Cov}\left(M_{YT}, D_{XT}\right) & \text{Cov}\left(D_{YT}, D_{XT}\right) & \text{Cov}\left(M_{XT}, D_{XT}\right) & \text{Var}\left(D_{XT}\right) \\
   \end{pmatrix}$
4. $\boldsymbol{\Lambda}_{C} = \text{Cov}\left[\bar{M}_{YC}, \bar{D}_{YC}, \bar{M}_{XC}, \bar{D}\_{XC},\right]
   N*{C}^{-1}
   \begin{pmatrix}
   \text{Var}\left(M_{YC}\right) & & &\\
   \text{Cov}\left(M_{YC}, D_{YC}\right) & \text{Var}\left(D_{YC}\right) & & \\
   \text{Cov}\left(M_{YC}, M_{XC}\right) & \text{Cov}\left(D_{YC}, M_{XC}\right) & \text{Var}\left(M_{XC}\right) & \\
   \text{Cov}\left(M_{YC}, D_{XC}\right) & \text{Cov}\left(D_{YC}, D_{XC}\right) & \text{Cov}\left(M_{XC}, D_{XC}\right) & \text{Var}\left(D_{XC}\right) \\
   \end{pmatrix}$
5. $\boldsymbol{\Lambda} = \text{Cov}\left(\hat{\boldsymbol{\beta}}\right)
   \begin{pmatrix}
   \boldsymbol{\Lambda}_{T} & \underset{4\times4}{\textbf{0}}\\
   \underset{4\times4}{\textbf{0}} & \boldsymbol{\Lambda}_{C} \\
   \end{pmatrix}$

Define the function

$$
\begin{align}
g\left(\boldsymbol{\beta}; \theta\right)_{a}&= \left(\frac{\boldsymbol{\beta}[1]}{\boldsymbol{\beta}[2]} - \theta\frac{\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]} \right)
-\left(\frac{\boldsymbol{\beta}[5]}{\boldsymbol{\beta}[6]} - \theta\frac{\boldsymbol{\beta}[7]}{\boldsymbol{\beta}[8]} \right).
\end{align}
$$

The CUPED estimator is

$$
\begin{align}
\hat{\Delta}_{a}&=g_{a}(\hat{\boldsymbol{\beta}}; \theta)
\\&=
\left(\frac{\hat{\boldsymbol{\beta}}[1]}{\hat{\boldsymbol{\beta}}[2]} - \theta\frac{\hat{\boldsymbol{\beta}}[3]}{\hat{\boldsymbol{\beta}}[4]} \right)
-\left(\frac{\hat{\boldsymbol{\beta}}[5]}{\hat{\boldsymbol{\beta}}[6]} - \theta\frac{\hat{\boldsymbol{\beta}}[7]}{\hat{\boldsymbol{\beta}}[8]} \right).
\end{align}
$$

Define the vector of partial derivatives as $\boldsymbol{\nabla}_{a}\left(\boldsymbol{\beta}; \theta\right) = \frac{\partial g_{a}(\boldsymbol{\beta})}{\partial\boldsymbol{\beta}}$.

$$
\begin{align*}
\boldsymbol{\nabla}[1] &= \frac{1}{\boldsymbol{\beta}[2]}
\\\boldsymbol{\nabla}[2] &= \frac{-\boldsymbol{\beta}[1]}{\boldsymbol{\beta}[2]^2}
\\\boldsymbol{\nabla}[3] &= \frac{-\theta}{\boldsymbol{\beta}[4]}
\\\boldsymbol{\nabla}[4] &= \frac{\theta\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]^2}
\\\boldsymbol{\nabla}[5] &= \frac{-1}{\boldsymbol{\beta}[6]}
\\\boldsymbol{\nabla}[6] &= \frac{\boldsymbol{\beta}[5]}{\boldsymbol{\beta}[6]^2}
\\\boldsymbol{\nabla}[7] &= \frac{\theta}{\boldsymbol{\beta}[8]}
\\\boldsymbol{\nabla}[8] &= \frac{-\theta\boldsymbol{\beta}[7]}{\boldsymbol{\beta}[8]^2}.
\end{align*}
$$

Note that $\boldsymbol{\nabla}_{a}\left(\hat{\boldsymbol{\beta}}; \theta\right)$

$$
\begin{align*}
 \boldsymbol{\nabla}[1] &= \frac{1}{\bar{D}_{YT}}
\\\boldsymbol{\nabla}[2] &= \frac{-\bar{M}_{YT}}{\bar{D}_{YT}^{2}}
\\\boldsymbol{\nabla}[3] &= \frac{-\theta}{\bar{D}_{XT}}
\\\boldsymbol{\nabla}[4] &= \frac{\theta\bar{M}_{XT}}{\bar{D}_{XT}^2}
\\\boldsymbol{\nabla}[5] &= \frac{-1}{\bar{D}_{YC}}
\\\boldsymbol{\nabla}[6] &= \frac{\bar{M}_{YC}}{\bar{D}_{YC}^2}
\\\boldsymbol{\nabla}[7] &= \frac{\theta}{\bar{D}_{XC}}
\\\boldsymbol{\nabla}[8] &= \frac{-\theta\bar{M}_{XC}}{\bar{D}_{XC}^2}.
 \end{align*}
$$

By the central limit theorem,
$\hat{\Delta}_{a} = g(\hat{\boldsymbol{\beta}}) \stackrel{}{\sim}\mathcal{N}\left(\Delta_{r} = g\left(\boldsymbol{\beta}\right), \boldsymbol{\nabla}_{a}^{\top}\Lambda\boldsymbol{\nabla}_{a} \right)$.

Decompose $\boldsymbol{\nabla}_{a}$ into its first four elements ($\boldsymbol{\nabla}_{a, T}$) and its last four elements ($\boldsymbol{\nabla}_{a, C}$).
Our variance of interest is

$$
\begin{align}
\hat{\sigma}^{2}_{\Delta_{a}}&=
\boldsymbol{\nabla}_{a}\left(\hat{\boldsymbol{\beta}}; \theta\right)^{\top}\boldsymbol{\Lambda}\boldsymbol{\nabla}_{a}\left(\hat{\boldsymbol{\beta}}; \theta\right)
\nonumber\\
&= \boldsymbol{\nabla}_{a, T}^{\top}\boldsymbol{\Lambda}_{T}\boldsymbol{\nabla}_{a, T}
+ \boldsymbol{\nabla}_{a, C}^{\top}\boldsymbol{\Lambda}_{C}\boldsymbol{\nabla}_{a, C}.
\end{align}
$$

All of these moments are available via CupedRatioRegressionAdjustedStatistics.

## Optimal regression coefficient for ratio metrics

The optimal $\theta$ minimizes Equation (14).
We can write

$$
\begin{align*}
\boldsymbol{\nabla}_{a, T}^{\top}\boldsymbol{\Lambda}_{T}\boldsymbol{\nabla}_{a, T}&=\boldsymbol{\nabla}_{a, T}[1:2]^{\top}\boldsymbol{\Lambda}_{T}[1:2, 1:2]\boldsymbol{\nabla}_{a, T}[1:2]
\nonumber\\&+2\boldsymbol{\nabla}_{a, T}[1:2]^{\top}\boldsymbol{\Lambda}_{T}[3:4, 1:2]\boldsymbol{\nabla}_{a, T}[3:4]+\boldsymbol{\nabla}_{a, T}[3:4]^{\top}\boldsymbol{\Lambda}_{T}[3:4, 3:4]\boldsymbol{\nabla}_{a, T}[3:4]
\nonumber\\
&=c_{T} + 2\theta
\left[\frac{1}{\boldsymbol{\beta}[2]},  \frac{-\boldsymbol{\beta}[1]}{\boldsymbol{\beta}[2]^2}  \right]^{\top}
\boldsymbol{\Lambda}_{T}[3:4, 1:2]\left[\frac{-1}{\boldsymbol{\beta}[4]}, \frac{\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]^2} \right]
+
\theta^{2}\left[\frac{-1}{\boldsymbol{\beta}[4]}, \frac{\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]^2}\right]^{\top}\boldsymbol{\Lambda}_{T}[3:4, 3:4]\left[\frac{-1}{\boldsymbol{\beta}[4]}, \frac{\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]^2}\right]
\end{align*}
$$

where $c_{T}$ is free of $\theta$.

Similarly we can write

$$
\begin{align*}
\boldsymbol{\nabla}_{a, C}^{\top}\boldsymbol{\Lambda}_{C}\boldsymbol{\nabla}_{a, C}&=
c_{C} + 2\theta
\left[\frac{-1}{\boldsymbol{\beta}[2]},  \frac{\boldsymbol{\beta}[1]}{\boldsymbol{\beta}[2]^2}  \right]^{\top}
\boldsymbol{\Lambda}_{C}[3:4, 1:2]\left[\frac{1}{\boldsymbol{\beta}[4]}, \frac{-\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]^2} \right]
+
\theta^{2}\left[\frac{-1}{\boldsymbol{\beta}[8]}, \frac{\boldsymbol{\beta}[7]}{\boldsymbol{\beta}[8]^2}\right]^{\top}\boldsymbol{\Lambda}_{C}[3:4, 3:4]\left[\frac{-1}{\boldsymbol{\beta}[8]}, \frac{\boldsymbol{\beta}[7]}{\boldsymbol{\beta}[8]^2}\right]
\end{align*}
$$

where $c_{C}$ is a constant free of $\theta$.

Differentiating the sum of these two equations with respect to $\theta$ and setting equal to zero shows that the minimum of this quadratic form occurs at

$$
\begin{align}
\theta_{\text{opt}} &=
-\frac{
\left[\frac{-1}{\boldsymbol{\beta}[2]},  \frac{\boldsymbol{\beta}[1]}{\boldsymbol{\beta}[2]^2}  \right]^{\top}
\boldsymbol{\Lambda}_{C}[3:4, 1:2]\left[\frac{1}{\boldsymbol{\beta}[4]}, \frac{-\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]^2} \right] +
\left[\frac{1}{\boldsymbol{\beta}[2]},  \frac{-\boldsymbol{\beta}[1]}{\boldsymbol{\beta}[2]^2}  \right]^{\top}
\boldsymbol{\Lambda}_{T}[3:4, 1:2]\left[\frac{-1}{\boldsymbol{\beta}[4]}, \frac{\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]^2} \right]
}{\left[\frac{-1}{\boldsymbol{\beta}[8]}, \frac{\boldsymbol{\beta}[7]}{\boldsymbol{\beta}[8]^2}\right]^{\top}\boldsymbol{\Lambda}_{C}[3:4, 3:4]\left[\frac{-1}{\boldsymbol{\beta}[8]}, \frac{\boldsymbol{\beta}[7]}{\boldsymbol{\beta}[8]^2}\right] +
\left[\frac{-1}{\boldsymbol{\beta}[4]}, \frac{\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]^2}\right]^{\top}\boldsymbol{\Lambda}_{T}[3:4, 3:4]\left[\frac{-1}{\boldsymbol{\beta}[4]}, \frac{\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]^2}\right]
}.
\end{align}
$$

The numerator represents the sum of the covariances, the denominator the sum of the variances.
This is the same $\theta$ as is presented in Appendix B of ([Deng et al. 2013](https://exp-platform.com/Documents/2013-02-CUPED-ImprovingSensitivityOfControlledExperiments.pdf)).

## Ratio Metric, Relative case

For relative inference much of the approach in the previous section works, we simply need to define the appropriate $g$ function and its partial derivatives.

For ratio metrics the relative parameter of interest is

$$
\begin{align}
\Delta_{r}&=\frac{\frac{\mu_{MTY}}{\mu_{DTY}} - \frac{\mu_{MCY}}{\mu_{DCY}}}{\frac{\mu_{MCY}}{\mu_{DCY}}}
\\&= \frac{\frac{\mu_{MTY}}{\mu_{DTY}}}{\frac{\mu_{MCY}}{\mu_{DCY}}} - 1.
\end{align}
$$

Define the function

$$
\begin{align}
g\left(\boldsymbol{\beta}; \theta\right)_{r}&=
\frac{
\left(\frac{\boldsymbol{\beta}[1]}{\boldsymbol{\beta}[2]} - \theta\frac{\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]} \right)
-\left(\frac{\boldsymbol{\beta}[5]}{\boldsymbol{\beta}[6]} - \theta\frac{\boldsymbol{\beta}[7]}{\boldsymbol{\beta}[8]} \right)
}{\frac{\boldsymbol{\beta}[5]}{\boldsymbol{\beta}[6]}}
\end{align}
$$

We can consistently estimate Equation (18) with the CUPED estimator
$g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r}.$

Define the numerator in Equation (18) as $g\left(\boldsymbol{\beta}; \theta\right)_{r, num}$ and the denominator as $g\left(\boldsymbol{\beta}; \theta\right)_{r, den}$.

Define the vector of partial derivatives as $\boldsymbol{\nabla}_{r}\left(\boldsymbol{\beta}; \theta\right) = \frac{\partial g_{r}(\boldsymbol{\beta})}{\partial\boldsymbol{\beta}}$.

$$
\begin{align*}
\boldsymbol{\nabla}[1] &= \frac{\frac{1}{\boldsymbol{\beta}[2]} }{g\left(\boldsymbol{\beta}; \theta\right)_{r, den}}
\\\boldsymbol{\nabla}[2] &= \frac{\frac{-\boldsymbol{\beta}[1]}{\boldsymbol{\beta}[2]^2}}{g\left(\boldsymbol{\beta}; \theta\right)_{r, den}}
\\\boldsymbol{\nabla}[3] &=
  \frac{
    \frac{-\theta}{\boldsymbol{\beta}[4]}
  }{g\left(\boldsymbol{\beta}; \theta\right)_{r, den}}
\\\boldsymbol{\nabla}[4] &= \frac{\frac{\theta\boldsymbol{\beta}[3]}{\boldsymbol{\beta}[4]^2}}{g\left(\boldsymbol{\beta}; \theta\right)_{r, den}}
\\\boldsymbol{\nabla}[5] &=
  \frac{
    \frac{-g\left(\boldsymbol{\beta}; \theta\right)_{r, den}}{\boldsymbol{\beta}[6]} - \frac{g\left(\boldsymbol{\beta}; \theta\right)_{r, num}}{\boldsymbol{\beta}[6]}
  }{g\left(\boldsymbol{\beta}; \theta\right)_{r, den}^{2}}
\\\boldsymbol{\nabla}[6] &=
  \frac{
    \frac{\boldsymbol{\beta}[5]g\left(\boldsymbol{\beta}; \theta\right)_{r, den}}{\boldsymbol{\beta}[6]^2} +
      \frac{\boldsymbol{\beta}[5]g\left(\boldsymbol{\beta}; \theta\right)_{r, num}}{\boldsymbol{\beta}[6]^2}
  }{g\left(\boldsymbol{\beta}; \theta\right)_{r, den}^{2}}
\\\boldsymbol{\nabla}[7] &=
  \frac{\frac{\theta}{\boldsymbol{\beta}[8]}
  }{g\left(\boldsymbol{\beta}; \theta\right)_{r, den}}
\\\boldsymbol{\nabla}[8] &=
  \frac{
    \frac{-\theta\boldsymbol{\beta}[7]}{\boldsymbol{\beta}[8]^2}
  }{g\left(\boldsymbol{\beta}; \theta\right)_{r, den}}.
\end{align*}
$$

Note that

$$
\begin{align*}
g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, num} &= \left(\frac{\bar{M}_{YT}}{\bar{D}_{YT}} - \theta\frac{\bar{M}_{XT}}{\bar{D}_{XT}} \right)
-\left(\frac{\bar{M}_{YC}}{\bar{D}_{YC}} - \theta\frac{\bar{M}_{XC}}{\bar{D}_{XC}} \right)
\\g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, den} &= \frac{\bar{M}_{YC}}{\bar{D}_{YC}}
\end{align*}
$$

Note that $\boldsymbol{\nabla}_{r}\left(\hat{\boldsymbol{\beta}}; \theta\right)$ is equal to

$$
\begin{align*}
\boldsymbol{\nabla}[1] &= \frac{\frac{1}{\bar{D}_{YT}}}{g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, den}}
\\\boldsymbol{\nabla}[2] &= \frac{\frac{-\bar{M}_{YT}}{\bar{D}_{YT}^{2}}}{g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, den}}
\\\boldsymbol{\nabla}[3] &=
 \frac{
   \frac{-\theta}{\bar{D}_{XT}}
 }{g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, den}}
\\\boldsymbol{\nabla}[4] &= \frac{\frac{\theta\bar{M}_{XT}}{\bar{D}_{XT}^2}}{g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, den}}
\\\boldsymbol{\nabla}[5] &=
 \frac{
   \frac{-g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, den}}{\bar{D}_{YC}} - \frac{g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, num}}{\bar{D}_{YC}}
 }{g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, den}^{2}}
\\\boldsymbol{\nabla}[6] &=
 \frac{
   \frac{\bar{M}_{YC}g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, den}}{\bar{D}_{YC}^2} +
     \frac{\bar{M}_{YC}g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, num}}{\bar{D}_{YC}^2}
 }{g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, den}^{2}}
\\\boldsymbol{\nabla}[7] &=
 \frac{\frac{\theta}{\bar{D}_{XC}}
 }{g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, den}}
\\\boldsymbol{\nabla}[8] &=
 \frac{
   \frac{-\theta\bar{M}_{XC}}{\bar{D}_{XC}^2}.
 }{g\left(\hat{\boldsymbol{\beta}}; \theta\right)_{r, den}}.
  \end{align*}
$$