XLSTAT: Parametric Illness-Death model in Excel

 

Background

Illness-Death model is a 3-state survival model. It enables us to analyze the survival of individuals and to model the evolution between different states. Explanatory variables can be added to the model to determine the impact of variables on patient survival time.

This model is based on a maximum of likelihood estimation and applies to survival data: time series data with censoring and truncation.

 

Dataset for Example

A sample of 1000 individuals from the Paquid dataset is used for this Illness-Death application. Collected between 1988 to 2018, the Paquid dataset is composed of 3777 individuals from the South-West of France and aged 65 years old and older. The main objective of this collection was to help dementia research by finding some of the causes of these diseases, especially Alzheimer’s.

The dataset contains 7 variables:

 Status indicators:

• dementia:

  • 1 if the individual is demented, 

  • 0 otherwise

• death:

  • 1 if the individual is dead,

  • 0 otherwise

 Time data (age):

• entry: age of the individual at the beginning of the study,

• left censoring:

  • age before the diagnostic if the individual is demented, 

  • agre at the last visit otherwise.

• right censoring:

  • age at the time of the diagnostic if the individual is demented,

  • age at the last visit otherwise.

• end:

  • age at the time of of death,

  • otherwise, age at which the individual was seen for the last time.

 Explanatory variable:

• certif:

  • 1 if the individual did not have a certificate of primary education,

  • 0 otherwise.

The aim of this tutorial is to answer the following questions:

• Does dementia increase the risk of mortality?

• Does having primary education decrease the risk of dementia?

We need to model the transition between the healthy, dementia and death states and evaluate the effect of the certification on these transitions.

 

Setting Up

• Open XLSTAT.

• Click on the Life Sciences/ Survival analysis menu and select the Illness-Death model feature.

• The Illness-Death model dialox box will appear. 

In the General tab:

• Select the dementia and death states in the Status indicators.

• Select the 4 age variables in the Time data field.

In the Covariates tab:

• Enter the "certif" variable in the Qualitative explanatory field.

• The option Different per transition allows to select different covariates for each transition. By choosing only Covariates, the “Certif” covariate will be evaluated for all transitions.

In the Options tab:

• Set the convergence criteria, the number of iterations and the maximum computation time of the optimization algorithm.

• Click on the OK button to launch the computations.

• The results are displayed on a new Excel sheet.

 

Interpreting the Results

The first table displays a summary for the time data and status indicators:

In the transition summary output, we can observe the number of individuals per transition.

We can see that most of the patients die without getting dementia (597 individuals).

The Weibull and regression coefficients tables show the model parameters. In the regression coefficients table, the significance of the covariates is indicated using the Wald-test and its p-value. A covariate is significant and influences a transition if its p-value is lower than a threshold called alpha = 0.05 by default. The value of alpha can be modified in the Options tab.

In our case, primary education influences only the transition from the initial to the dementia state because the p-value is equal to 0.010 < 0.05. Therefore, the certification affects this disease but not mortality.

In an Illness-Death model, the coefficient values do not provide relevant information. Hazard ratios are used to interpret the relative risk of individuals with a primary education certification compared to those who don't.

The interpretation of the coefficient is described below:

• Hazard ratio = 1: No effect / Same risk,

• Hazard ratio < 1: Reduction in the hazard / An uncertified individual has less risk than a certified individual,

• Hazard ratio > 1: Increase in Hazard / An uncertified individual has more risk than a certified individual.

For the first transition, the risk of getting dementia for an uncertified individual is only 0.596 of the risk for certified individuals. In this case, having a certificate increases the risk of dementia as 0.596 < 1 and thus the risk of uncertified individuals is reduced.

For each transition, the Weibull parameters table gives the shape and scale parameters of the Weibull distribution.

 

The baseline transition intensities are computed using these previous parameters:

The transition intensities chart also provides useful information. First, since all the curves are increasing, we deduce that the risk of dementia and death increases with age. Then, the risk of death for a demented individual, is greater than the risk of death for a healthy individual. Finally, from the age of 90, the risk of dementia highly increases.

Other quantities such as transition probabilities or survival distributions  can be observed. These quantities provide additional information and ease the interpretation.

Note that the survival function from the initial to dementia state, represented in black, does not represent survival in the common sense (staying alive) but survival in good health (staying alive and healthy). It represents the probability of remaining in the initial state.

 

Conclusion

We may suggest that having a primary education certification has a significant effect on the dementia state but not mortality. Certified individuals have higher risk to get dementia than non-certified ones. Furthermore, we see that demented individuals have a lower survival probability than healthy individuals and that the risk of dementia highly increases after the age of 90.