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The measure that combines the entire survival curve into a single effect is the Hazard Ratio (HR), which is the ratio of the hazard (instantaneous mortality) between the two groups.
In statistics, we want to use the hazard ratio as a summary measure of the survival curve between the two groups, but the value of the ratio is not necessarily constant over time. In order for the ratio to be constant over time, the two groups must have similar mortality patterns. In other words, when the value of the mortality ratio is constant over time, the hazard ratio can be used as a summary measure of the survival curves of the two groups.
Figure 1 and Figure 2 show different patterns of survival curves, but the mortality ratios appear to be the same between the two groups, both in the initial and later stages of the study. In light of this data, the Cox proportional hazards model may be used, because of the similar pattern of mortality transition between the two groups.
[Figure 1]
[Figure 2]
Figure 3 shows that the mortality rates between the two groups are the same in the initial part of the study but are clearly different in the later part. Thus, the Cox proportional hazards model cannot be used when the mortality trends are not similar for the two groups (in practice, it is often difficult to judge from the shape of the graph alone). If the hazard ratio is calculated using the Cox proportional hazards model, and is clearly large, it implies that the data is not appropriate for the model/data cannot be used with the model.
[Figure 3]
In Figure 4, the trends in mortality rates are crossed between the two groups. In such cases, the patterns between the two groups are quite different, therefore, the Cox proportional hazards model cannot be used.
[Figure 4]
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