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Survival studies mainly deal with distribution of time to event. Often in such studies researchers are interested in comparing several treatment or prognostic groups. At the time of analysis, there is an unmeasured chance of making type I error, or finding a falsely significant difference between any two groups. The chance of making type I error is increased, if multiple groups are compared simultaneously. In this paper, survival analysis with Bonferroni correction is explained in easy way to cope up with this issue. The DLHS-3 data are taken to explain this methodology in the context of neonatal survival. Kaplan-meier plot with three survival comparison test is used to elaborate the application of Bonferroni correction.

Several biological, epidemiological and clinical studies have “time to an event” as their endpoint. Survival analysis approaches are used to find any conclusion from these studies. Survival Analysis is a statistical procedure for data analysis in which the outcome of interest is time until an event occurs [

In these tests, the probability of making a type I error or α, an “acceptable” risk of type I errors, conventionally set at 0.05. Problems arise, when researchers perform several hypothesis tests instead of one. This is because each test again has a probability of producing a type I error, and performing a large number of hypothesis tests factually guarantees the presence of type I errors among the findings. Often such analyses are done without any adjustment for multiple comparisons, resulting in an excess of type I errors. A more appropriate criterion to control when making several comparisons is the family wise error (FWE) rate, which is the chance of making at least one type I error among all treatment comparisons being made.

The key goal of multiple testing methods is to control, or at least to quantify, the overflow of type I errors that arise when many hypothesis tests are performed simultaneously. There are different techniques of doing this as proposed by different researcher. In recent time more than twenty techniques are available. Several post-hoc procedures for pairwise comparison like Boneferroni [

The above mentioned correction methods are being used frequently in Analysis of Variance (ANOVA). In another sense comparison of mean is done in more than two categories of a variable by using above correction methods. But the use of post-hoc correction methods in survival analysis is hardly seen. This is the main motivation behind this endeavour to explore the post hoc comparison in survival analysis where Kaplan-Meier plot and log rank test are used to compare the survival status in different group.

In this paper, survival analysis with multiple testing has been performed on neo-natal survival status. In child mortality estimates the neonatal mortality plays a vital role because majority of deaths occurring in this age group is contributed by neonatal mortality. Neonatal survival is a very sensitive indicator of population growth and socio-economic development. For these reasons, the issue of neonatal deaths is a serious national health concern. The neonatal mortality is defined as probability of death of a newborn within 30 days from the date of birth.

Kaplan Meier, log rank test and post hoc adjustment are described, to complete the flow of survival analysis with post hoc comparison.

The Kaplan-Meier estimate [^{th} is denoted

during the interval from

estimated probability of survival through that interval is then

have i(i = 1,2,3 ・・・) no. of group to be compared by survival probability then the generalized probability of survival through that interval for each group is

based on hypergeometric distribution of the number of events at distinct event times. The generalized test statistic for comparison of survival pattern among groups is as follows

And

^{th} distinct observed time. ^{th} distinct observed time.

In this study above mentioned three tests as well as KM plot are obtained. For pairwise or multiple comparison bonferroni correction is used. The boneferroni correction procedure is as follows:

Let _{i}, that is, of making at least one Type I error. The Bonferroni correction rejects

the null hypothesis for each

Bonferroni correction assumes null hypothesis true for all test in consideration. Hence it lacks power. When the number of comparisons becomes large, the test may become too conservative and no longer allows you to find anything significant [

The Data selected to describe the survival analysis in post hoc setup is taken from District Level Household and Facility Survey (DLHS-3) [^{st} January 2004 to the date of survey for the state Uttar Pradesh [^{th} day.

The two independent variable Birth order and Age of mother are taken in study to find out their effect on neo-natal survival. Both variables are divided into three categories. Birth order has first category as birth order 1, second category defined those female who have birth order between 2 - 4 and third category covers the other than above mentioned two categories. Mother age (in years) have three category first “≤19”, second “20 - 34” and third “≥35”.

Descriptive analysis of selected variable is given in

Variables | Death (1054, 2.8%) | Survival (36,626, 97.2%) | ||
---|---|---|---|---|

No. | Percentage | No | Percentage | |

Birth order | ||||

1 (1) | 322 | 4.1 | 7603 | 95.9 |

2^{nd}-4^{th} (2) | 465 | 2.4 | 18,914 | 97.6 |

Else (3) | 267 | 2.6 | 10,109 | 97.4 |

Mother age | ||||

≤19 (1) | 161 | 3.7 | 4201 | 96.3 |

20 - 34 (2) | 785 | 2.6 | 29,196 | 97.4 |

≥35 (3) | 108 | 3.2 | 3229 | 96.8 |

Kaplan-meier curve is portrayed (

the width among curve became wider.

To find out that whether these differences occurred by chance or the difference is really significant, According to our methodology all three test were performed with their posthoc comparison for each pair of group in every variable. The posthoc adjust p value are calculated by bonferroni correction. Both variables have shows the overall significant difference among group. To find out which pairs of groups are significant different all the three tests are done without correction and with correction by bonferroni (p value adjustment).

Adjustment of p values in multiple hypothesis testing is the concern of various statisticians [

Variables | p-value among group for the difference in survival | ||
---|---|---|---|

Log-Rank Test | Breslow Test | Tarone-Ware test | |

Birth Order | <0.001 | <0.001 | <0.001 |

Mother Age | <0.001 | <0.001 | <0.001 |

Pair-wise comparison (pairs who found significant in post hoc test) | |||

Birth Order | (1, 2)* (1, 3)* | (1, 2)* (1, 3)* | (1, 2)* (1, 3)* |

Mother Age | (1, 2)* (2, 3) | (1, 2)* (2, 3) | (1, 2)* (2, 3) |

*After adjusting by bonferroni correction.

words KM curve, all three test shows there is a difference in survival among categories of birth order and if we go for posthoc or multiple comparison KM curve shows a clear difference in category 1, 2 as well as 1, 3 and these finding are also supported by selected survival test with non adjusted and adjusted p values. The variable age of mother shows the significant difference in neo-natal survival among categories of mother age and this finding supported by KM curve for variable mother age, but in case of multiple/post-hoc comparison category 1, 2 shows clear difference in survival pattern and by test p values in adjusted and for not adjusted case are also significant. When we test the pair 2, 3 it shows the survival pattern differ by all three test for non adjusted p value even the KM curve also shows the difference but slightly close pattern in both group in starting of survival curve. Now p value adjusted by Bonferroni correction for comparing pair 2, 3 and it was found insignificant difference between group 2 and 3 for neo-natal survival. So this pair gives an example of correction of p value in multiple testing and it also shows the importance of p-value adjustment in multiple testing for draw a right conclusion.

Tripathi, A. and Pandey, A. (2017) Post-Hoc Comparison in Survival Analysis: An Easy Approach. Jour- nal of Biosciences and Medicines, 5, 112- 119. https://doi.org/10.4236/jbm.2017.53012