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

# Technical CUPED details

> How GrowthBook estimates CUPED-adjusted means and variances for experiment metrics, including the covariate model.

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

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