# Cumulative incidence

Cumulative incidence allows estimating the risk of disease at a certain period of time. Available in Excel using the XLSTAT statistical software.

## What is cumulative incidence

**Cumulative incidence** investigates disease frequency at a certain period of time. It is often called competing risks case.

Technically, for a given period, the cumulative incidence is the probability that an observation still included in the analysis at the beginning of this period will be affected by an event during the period. It is especially appropriate in the case of competing risks, that is to say, when several types of events may occur.

This technique is used for the analysis of survival data, whether individuals (cancer research, for example) or products (resistance time of a production tool, for example): some individuals die (in this case we will have 2 causes of death: from the disease or an other cause), the products break (in this case we can model different breaking points), but others leave the study because they heal, you lose track of them (moving for example) or because the study was discontinued. The first type of data is usually called** failure data**, or **event data**, while the second is called **censored data**.

Time intervals should not necessarily be regular. XLSTAT allows the treatment of censored data in competing risks and to compare different groups within the population.

You can compare groups of individuals with XLSTAT using the Gray test. Groups are compared based on cumulative incidence for each type of event.

## Censoring of data for cumulative incidence

There are several types of censoring of survival data:

- Left censoring: when an event is reported at time t=t(i), we know that the event occurred at t * t(i).
- Right censoring: when an event is reported at time t=t(i), we know that the event occurred at t * t(i), if it ever occurred.
- Interval censoring: when an event is reported at time t=t(i), we know that the event occurred during [t(i-1); t(i)].
- Exact censoring: when an event is reported at time t=t(i), we know that the event occurred exactly at t=t(i).

The cumulative incidence method requires that the observations are independent. Second, the censoring must be independent: if you consider two random individuals in the study at time t-1, if one of the individuals is censored at time t, and if the other survives, then both must have equal chances to survive at time t. There are four different types of independent censoring:

- Simple type I: all individuals are censored at the same time or equivalently individuals are followed during a fixed time interval.
- Progressive type I: all individuals are censored at the same date (for example, when the study terminates).
- Type II: the study is continued until n events have been recorded.
- Random: the time when a censoring occurs is independent of the survival time.

When working with competing risks, the different types of events can happen only once, after the event has occurred, the observation is withdrawn from the analysis. We can calculate the risk of occurrence of an event in the presence of competitive events. XLSTAT allows you to compare the types of events but also to take account of groups of observations (depending on the treatment administered, for example).

## Results for the cumulative incidence in XLSTAT

### Cumulative incidence table

This table displays the various results obtained from the analysis, including:

- Interval start lime: lower bound of the time interval.
- At risk: number of individuals that were at risk.
- Events i: number of events of type i recorded.
- All types of events: number of events of all types recorded.
- Censored: number of censored data recorded.
- Cumulative incidence: Cumulative incidence obtained for event I at the considered time.
- Cumulative incidence standard error.
- Cumulative incidence confidence interval.

### Cumulative Survival function

The Cumulative Survival function table displays the various results obtained from the analysis, including:

- Interval start lime: lower bound of the time interval.
- At risk: number of individuals that were at risk.
- Events i: number of events of type i recorded.
- All types of events: number of events of all types recorded.
- Censored: number of censored data recorded.
- Cumulative survival function: Cumulative survival function obtained for event i at the considered time.
- Cumulative survival function standard error.
- Cumulative survival function confidence interval.

### Charts for cumulative incidence

XLSTAT offers two charts for this method:

- Cumulative incidence
- and cumulative survival function.

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