This article provides notes on different approaches to computing
social jetlag (\(SJL\)) for the Munich
ChronoType Questionnaire (MCTQ). It also explains how the
`method`

argument in the `sjl_rel()`

,
`sjl()`

, `sjl_sc_rel()`

, and `sjl_sc()`

functions works.

It’s helpful to have the standard MCTQ questionnaire and the guidelines for standard MCTQ variable computation open while reading this article. This will enhance your understanding of the data objects discussed. You can download the MCTQ full standard version here and the guidelines for standard MCTQ variables here.

## The two intervals problem

According to Roenneberg et al. (2012), the relative social jetlag (\(SJL_{ rel}\)), i.e., the discrepancy between social and biological time, must be computed as the difference between \(MSF\) (local time of mid-sleep on work-free days) and \(MSW\) (local time of mid-sleep on workdays).

\[SJL_{rel} = MSF - MSW\]

This simple equation seems trivial until you consider that it deals with two time values detached from a timeline. In other words, \(MSW\) and \(MSF\) represent two moments in different contexts (workdays and work-free days).

If you dive into the MCTQ articles, you can see that this computation have two objectives:

Represent the distance between \(MSW\) and \(MSF\) (i.e., the discrepancy).

Establish what value comes before or after the other, representing that with a \(+/-\) signal. That is, when \(MSW\) comes before \(MSF\), \(SJL_{rel}\) must be positive, and when \(MSW\) comes after \(MSF\), \(SJL_{rel}\) must be negative.

To represent the distance between \(MSW\) and \(MSF\) (i.e., the discrepancy).

To establish what value comes before or after the other, representing that with a \(+/-\) signal. That is, when \(MSW\) comes before \(MSF\), \(SJL_{rel}\) must be positive, and when \(MSW\) comes after \(MSF\), \(SJL_{rel}\) must be negative.

You can find the rationale about the \(SJL_{rel}\) signal in Roenneberg et al. (2019) (see item “3.2 Social Jetlag Computation”).

Most people have some trouble understanding this. To illustrate what we mean, let’s visualize a timeline overlapping an \(MSW\) and \(MSF\) value:

```
day 1 day 2
MSF MSW MSF MSW
05:00 21:00 05:00 21:00
-----|------------------|---------|------------------|----->
16h 8h 16h
longer int. shorter int. longer int.
```

Note that, while doing the representation above, we’re dealing with the assumption that \(MSW\) and \(MSF\) can be represented in a two-day timeline since people don’t usually sleep more than 24 hours (basic assumption).

As you can see, by overlapping two time values in a two-day timeline, we need to make a choice of what interval to use. For most people \(MSF\) and \(MSW\) are close to each other, so, usually, we are looking for the shorter interval between the two. But, in some extreme cases, usually when dealing with shift workers, \(MSW\) and \(MSF\) distance can surpass 12 hours, making the longer interval the correct answer.

To obtain the \(SJL_{rel}\) signal we must check the start value of the interval. If the interval between \(MSW\) and \(MSF\) starts with \(MSW\), that means that \(MSW\) comes before \(MSF\), hence, the signal must be positive. Else, if the interval between \(MSW\) and \(MSF\) starts with \(MSF\), that means that \(MSW\) comes after \(MSF\), hence, the signal must be negative.

- Example 1: when \(MSF - MSW\) makes
a
**positive**\(SJL_{rel}\)

```
day 1 day 2
MSW MSF
21:00 05:00
------------------------|---------|------------------------>
```

- Example 2: when \(MSF - MSW\) makes
a
**negative**\(SJL_{rel}\)

```
day 1 day 2
MSF MSW
21:00 05:00
------------------------|---------|------------------------>
```

We call this the **two intervals problem**. It
represents an unsolvable mathematical scenario, if you deprive it of the
respondent context. That can generate minor errors when computing \(SJL\), especially if you’re dealing with
large datasets.

## Methods for computing \(SJL\)

the `sjl_rel()`

, `sjl()`

,
`sjl_sc_rel()`

, and `sjl_sc()`

functions provides
an argument called `method`

that allows you to choose three
different methods to deal with the two intervals problem. Here’s how
they work.

The `sjl()`

function will be used in the examples, but the
same logic apply to the other `sjl`

functions.

###
`method = "difference"`

By using `method = "difference"`

, `sjl()`

will
do the exact computation proposed by the MCTQ authors, i.e., \(SJL\) will be computed as the linear
difference between \(MSF\) and \(MSW\).

Let’s see some examples using this method.

- Example 3: using the
`"difference"`

method

\(MSW = \text{04:00}\)

\(MSF = \text{06:00}\)

\(\text{Real difference: + 02:00}\)

\(MSF - MSW = \text{06:00} - \text{04:00} =
\text{+ 02:00}\) (**right**)

- Example 4: using the
`"difference"`

method

\(MSW = \text{23:00}\)

\(MSF = \text{03:00}\)

\(\text{Real difference: + 04:00}\)

\(MSF - MSW = \text{03:00} - \text{23:00} =
\text{- 20:00}\) (**wrong**)

As you can see with the second example, the `"difference"`

method uses a linear time frame approach, creating problems regarding
the circularity of time.

###
`method = "shorter"`

(default method)

By using `method = "shorter"`

, `sjl()`

uses the
shorter interval between \(MSW\) and
\(MSF\).

This is the most reliable method we found to compute \(SJL\), considering the context of the MCTQ
data. However, it comes with a limitation: when \(MSW\) and \(MSF\) values distance themselves by more
than 12 hours, `sjl()`

can return a wrong output. From our
experience with MCTQ data, a \(SJL\)
greater than 12 hours is highly improbable.

Let’s see some examples using this method.

- Example 5: using the
`"shorter"`

method

\(MSW = \text{04:00}\)

\(MSF = \text{06:00}\)

\(\text{Real difference: + 02:00}\)

```
day 1 day 2
MSF MSW MSF MSW
06:00 04:00 06:00 04:00
-----|------------------|---------|------------------|----->
22h 2h 22h
longer int. shorter int. longer int.
```

By using the shorter interval, \(MSW\) comes before \(MSF\), so \(SJL_{rel}\) must be equal to \(\text{+ 02:00}\)
(**right**).

- Example 6: using the
`"shorter"`

method

\(MSW = \text{23:00}\)

\(MSF = \text{03:00}\)

\(\text{Real difference: + 04:00}\)

```
day 1 day 2
MSF MSW MSF MSW
03:00 23:00 03:00 23:00
-----|------------------|---------|------------------|----->
20h 4h 20h
longer int. shorter int. longer int.
```

By using the shorter interval, \(MSW\) comes before \(MSF\), so \(SJL_{rel}\) must be equal to \(\text{+ 04:00}\)
(**right**).

- Example 7: when the
`"shorter"`

method fails

\(MSW = \text{12:00}\)

\(MSF = \text{23:00}\)

\(\text{Real difference: - 13:00}\)

```
day 1 day 2
MSW MSF MSW
12:00 23:00 12:00
-----|-----------------------|------------------------|----->
11h 13h
shorter int. longer int.
```

By using the shorter interval, \(MSW\) comes before \(MSF\), so \(SJL_{rel}\) must be equal to \(\text{+ 11:00}\)
(**wrong**).

You can see example 7 in the `shift_mctq`

dataset provided
by the `mctq`

package (ID 39, on and after night shifts).
That’s the only MCTQ\(^{Shift}\) case
in `shift_mctq`

where we think that the
`"shorter"`

method would fail.

###
`method = "longer"`

By using `method = "longer"`

, `sjl()`

uses the
longer interval between \(MSW\) and
\(MSF\). It’s just the opposite of the
`"shorter"`

method showed above.

## So, what method should I use?

We recommend that you always use the `"shorter"`

method
when computing \(SJL_{rel}\) or \(SJL\) (the default `sjl()`

method).

In our tests, the `"shorter"`

method demonstrated to be
almost fail-safe. You just need to worry about the \(SJL\) computation if you are dealing with
shift workers.

When dealing with a large MCTQ\(^{Shift}\) dataset, it will be very difficult to identify \(SJL\) errors, unless you look case by case and check the results with your respondents. This is usually not a viable option. We recommend that you mention which method you use to compute \(SJL\) and add it as a possible limitation of your results.