p-Value Calculation

What p-values mean in Statsig experiments, how they are computed, and how to interpret them alongside confidence intervals and lift estimates.

In Null Hypothesis Significance Tests, the p-value is the probability of observing an effect larger than or equal to the measured metric delta, under the assumption that the null hypothesis is true. A p-value below the pre-defined Type I Error threshold ( $\alpha$ ) serves as evidence of a true effect.

The methodology for p-value calculation depends on the number of degrees of freedom (

\nu

). A two-sample z-test is appropriate for most experiments. Statsig uses Welch's t-test for smaller experiments with

\nu < 100

. In both cases, the p-value depends on the metric mean and variance computed for the test and control groups.Typically, a p-value below the threshold

\alpha

occurs only when the confidence interval doesn't cross 0. However, an exception can occur in the Statsig UI: when the p-value of the difference between test and control is statistically significant, but uncertainty in the control causes a relative delta confidence interval to cross zero (using The Delta Method) or be represented as a point estimate (using Fieller Intervals), while the absolute difference's p-value is statistically significant.

Two-sample tests

Two-sided z-test

You can compute the z-statistic (a.k.a. z-score) of a two-sample z-test in multiple equivalent formats:

\begin{split} Z &= \frac{\overline X_t - \overline X_c}{\sqrt{var(\overline X_t)+ var(\overline X_c)}} \\ &= \frac{\overline X_t - \overline X_c}{\sqrt{var(\Delta \overline{X})}} \\ &= \frac{\overline X_t - \overline X_c}{\sqrt{\sigma_{\overline{X}_t}^2 + \sigma_{\overline{X}_c}^2}} \end{split}

where:

$Z$ is the observed z-statistic (not the z-critical value $Z_{\alpha/s}$ )
$var(\Delta \overline{X})$ is the variance of the absolute delta of means
$var(\overline{X}_i)$ is the variance of sample means either control or treatment group (details here)
$\sigma_{\overline{X}_t}$ is the standard error of the mean of either control or treatment group (these are the terms you can find in Pulse under the Statistics tab of a metric)

The two-sided p-value comes from the standard normal cumulative distribution function:

p-value = 2 \cdot \frac{1}{\sqrt{2\pi}} \int \limits _{-\infty}^{-|Z|}{e^{-t^2/2}dt}

Welch's t-test

For smaller sample sizes, Welch's t-test is preferred because it produces lower false positive rates in cases of unequal sizes and variances. In Pulse, Statsig automatically applies Welch's t-test when the degrees of freedom $\nu < 100$ .

Statsig computes the t-statistic (also known as t-score) identically to the two-sample z-statistic above. Statsig computes the degrees of freedom $\nu$ using:

\nu = \frac{\left(var(\overline X_t) + var(\overline X_c)\right)^2}{\frac{var(\overline X_t)^2}{N_t - 1}+\frac{var(\overline X_c)^2}{N_c - 1}} := \frac{var(\Delta\overline{X})^2}{\frac{var(\overline X_t)^2}{N_t - 1}+\frac{var(\overline X_c)^2}{N_c - 1}}

Statsig then obtains the p-value from the t-distribution with $\nu$ degrees of freedom.

One-sided z-test

The procedure for a one-sided z-test computes the z-statistic $Z$ in the same way as the two-sided test above.

The one-sided p-value comes from the standard normal cumulative distribution function, but with the following differences:

p-value = \begin{cases} 1 - \frac{1}{\sqrt{2\pi}} \int \limits _{-\infty}^{Z}{e^{-t^2/2}dt} &\text{if right-hand test}\\ \frac{1}{\sqrt{2\pi}} \int \limits _{-\infty}^{Z}{e^{-t^2/2}dt} &\text{if left-hand test} \end{cases}

where:

$Z$ is computed as shown in the two-sided test above. This uses the signed z-statistic, not the absolute value used in the two-sided p-value.

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