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p-Value Calculation

In hypothesis testing, 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. In practice, a p-value that's lower than your pre-defined threshold (α\alpha) is treated as evidence for there being a true effect.

The methodology used for p-value calculation depends on the number of degrees of freedom (ν\nu). A two-sample z-test is appropriate for most experiments. Welch's t-test is used for smaller experiments with ν<100\nu < 100. In both cases, the p-value depends on the metric mean and variance computed for the test and control groups.

Two-Sample Tests​

Two-Sided z-Test​

The z-statistic (a.k.a. z-score) of a two-sample z-test can be computed in multiple equivalent formats:

Z=X‾t−X‾cvar(X‾t)+var(X‾c)=X‾t−X‾cvar(ΔX‾)=X‾t−X‾cσX‾t2+σX‾c2\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:

  • ZZ is the observed z-statistic (not the z-critical value Zα/sZ_{\alpha/s})
  • var(ΔX‾)var(\Delta \overline{X}) is the variance of the absolute delta of means
  • var(X‾i)var(\overline{X}_i) is the variance of sample means either control or treatment group (details here)
  • σX‾t\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 is obtained from the standard normal cumulative distribution function:

p−value=2⋅12π∫−∞−∣Z∣e−t2/2dtp-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 the preferred statistical test for lower false positive rates in cases of unequal sizes and variances. In Pulse, Welch's t-test is automatically applied when the degrees of freedom ν<100\nu < 100.

We compute the t-statistic (a.k.a. t-score) identically as the two-sample z-statistic above. Additionally, we compute the degrees of freedom ν\nu using:

ν=(var(X‾t)+var(X‾c))2var(X‾t)2Nt−1+var(X‾c)2Nc−1=var(ΔX‾)2var(X‾t)2Nt−1+var(X‾c)2Nc−1\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}}

The p-value is then obtained from the t-distribution with ν\nu degrees of freedom.

One-Sided Z-Test​

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

The one-sided p-value is obtained from the standard normal cumulative distribution function as well, but with slight differences:

p−value={1−12π∫−∞Ze−t2/2dtif right-hand test12π∫−∞Ze−t2/2dtif left-hand testp-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:

  • ZZ is computed above in the two-sided test. Note: this uses the signed z-statistic, not the absolute value of the z-statistic as in the two-sided p-value