# Log Metrics

### Use cases

Log metrics are useful for understanding if the distribution of a log-normal or tail-driven metric has shifted. Statsig calculates the metric as a conditional mean: a ratio metric where the numerator is the sum of unit-level log values and the denominator is 1 for units with a valid log. Statsig filters out records with a 0 denominator, because imputing 0s for logs doesn't work without a treatment such as an inverse hyperbolic sine function.

Common uses include revenue, time spent, or other metrics where a small portion of users drives most of the value but bulk improvements matter. Log metrics measure relative change per unit: an increase of 1 corresponds to a multiplication by the log's base.

## Calculation

At the unit level, count metrics run a COUNT(1) or SUM(`value`) across their metric source.

At the group level, Statsig calculates the mean as the SUM of the log of the unit-level value, divided by the count of units with a unit-level value that log is valid for (exists, and is greater than 0).

The SQL for a count metric looks like the following:

```sql expandable
-- Unit Level
SELECT
  source_data.unit_id,
  exposure_data.group_id,
  COUNT(1) as value
FROM source_data
JOIN exposure_data
ON
  -- Only include users who saw the experiment
  source_data.unit_id = exposure_data.unit_id
  -- Only include data from after the user saw the experiment
  -- In this case exposure_data is already deduped to the "first exposure"
  AND source_data.timestamp >= exposure_data.timestamp
GROUP BY unit_id, group_id;

-- Group Level
SELECT
  group_id,
  -- divide the sum of the logged values by the count of participating units
  SUM(LOG(value, <base>))/COUNT(1) as mean
FROM unit_data
WHERE value > 0
-- the filter is implicit from the CTE, but let's make it explicit
-- a sum metric might have negative values
GROUP BY group_id;
```

### Methodology notes

Log metrics can be difficult to interpret and to extrapolate to topline values. Use log metrics together with the raw or winsorized SUM and COUNT metric.

There are a few ways to handle 0s in a log metric. A transformation like [IHS](https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions) can approximate the behavior of log for large values while accepting 0s as inputs. Alternatively, you can scope the analysis to non-zero units. Statsig uses the second approach for ease of interpretation, because people broadly understand log properties.

Scoping to non-zero units introduces a potential confounding factor: participation rate. To mitigate this confounding factor, Statsig presents results the same as for [ratio](https://docs.statsig.com/statsig-warehouse-native/metrics/ratio) metrics, including statistics for the overall result and the implicit numerators and denominators.

## Options

Non-log options depend on whether the metric is a Sum or Count.

* Custom log Base
  * You can configure a custom base for the log operation. Defaults to LN.