Kipp Johnson

Implements and extends the fragility index calculation as described in Walsh M, Srinathan SK, McAuley DF, et al. *The statistical significance of randomized controlled trial results is frequently fragile: a case for a Fragility Index*. Journal of clinical epidemiology. 67(6):622-8. 2014.

As originally defined, the fragility index is the number of patients with a different outcome it would require to change a result from significant to non-significant. Consider for example the following example situation: In a clinical trial, there are two groups of patients. In group 1, 15/40 patients have an adverse event. In group 2, 5/40 patients have the adverse event. We can test this for statistical significance in the following way:

```
> mat1 <- matrix(c(15,6,25,34), nrow=2)
> mat2
[,1] [,2]
[1,] 15 25
[2,] 6 34
> fisher.test(mat2)$p.value
[1] 0.04060921
```

However, what if a single additional patient in the second group had an event?

```
> mat2 <- matrix(c(15,7,25,33), nrow=2)
> mat2
[,1] [,2]
[1,] 15 25
[2,] 7 33
> fisher.test(mat2)$p.value
[1] 0.07836101
```

This result is no longer statistically significant at the alpha=0.05 level! This is a "fragile" statistical result, despite the moderately low initial p value. Because it took only a single additional patient to make the result non-significant, we can say this clinical trial has a fragility index of 1. If it had taken two patients, this test would would have a fragility index of 2, and so on.

This package contains functions to automatically calculate fragility indices in several situations, as explained below.

We recommend installing from Github to ensure you have the latest version of the R package. The most up-to-date version can be installed from this github repository with the following commands:

```
install.packages("devtools") # If you do not have the devtools package
library(devtools)
install_github('kippjohnson/fragilityindex')
library(fragilityindex)
```

CRAN accepts only periodic submissions, and thus R packages cannot be frequently updated. Older versions of the package can also be installed from CRAN as follows:

```
install.packages("fragilityindex")
library(fragilityindex)
```

```
fragility.index(15, 5, 40, 40)
```

For a dichotomous outcome, fragility index is the additional number of patients with an event it would take to make a significant result at a given P-value non-significant. A smaller index means the observed result is more fragile to small variations.

```
revfragility.index(6,5,50,50, verbose=TRUE, print.mat=FALSE)
```

This package also contains a function to compute the "reverse fragility index," or the number of patients it would require who if they did not experience an event would take a conclusion from non-significant to significant. This may be applied to analyze the sensitivity of non-inferiority trials to small differences in event counts. A smaller index means the observed result is more fragile to small variations.

We present a new method to calculate logistic regression coefficient fragility, or how many events it would take to change a signficiant logistic regression coefficient to non-significant at the given confidence level. To do this, we replace responses (which should be binary events, i.e. 0 or 1) with the opposite event until the coefficient is nonsignificant. If the initial regression coefficient (beta) is positive, we change a 1 event to a 0. If the regression coefficient is negative, we change a 0 event to a 1.

We then count the number of times this replacement must be and then obtain a fragility index. To account for variability in the response replacement, we then repeat this process a number of times and take the mean of all of the computed fragility indices to obtain a single fragility index.

Examining fragility of a single covariate:

```
mydata <- read.csv("http://www.ats.ucla.edu/stat/data/binary.csv")
mydata$rank <- factor(mydata$rank)
logisticfragility(admit ~ gre + gpa + rank, data = mydata, covariate="gre", niter=100)
```

Or looking at all covariates in one step:

```
logisticfragility(admit ~ gre + gpa + rank, data = mydata, covariate="all", niter=100, progress.bar=TRUE)
```

Example output:

```
[1] "Doing (Intercept)..."
|++++++++++++++++++++++++++++++++++++++++++++++++++| 100% elapsed = 15s
[1] "Doing gre..."
|++++++++++++++++++++++++++++++++++++++++++++++++++| 100% elapsed = 03s
[1] "Doing gpa..."
|++++++++++++++++++++++++++++++++++++++++++++++++++| 100% elapsed = 08s
[1] "Doing rank2..."
|++++++++++++++++++++++++++++++++++++++++++++++++++| 100% elapsed = 05s
[1] "Doing rank3..."
|++++++++++++++++++++++++++++++++++++++++++++++++++| 100% elapsed = 28s
[1] "Doing rank4..."
|++++++++++++++++++++++++++++++++++++++++++++++++++| 100% elapsed = 28s
coefficient fragility.index
1 (Intercept) 65.05
2 gre 11.25
3 gpa 29.65
4 rank2 19.02
5 rank3 120.62
6 rank4 123.35
```

We also present a new method to calculate fragility index for survival analysis in a similar way to the calculation performed for the 2x2 and logistic regression fragility index calculations. We use the survdiff() function from the survival package in R to compute survival tests from the G-rho family of tests.

For this calculation, we randomly swap a 0/1 (survive/die) outcome in the survival dataset and continue then repeat this process, counting the number of swaps until the survival test is no longer significant. The number of swaps is the fragility index for this single iteration. We repeat this process a number of times and take the average of each resulting fragility index to obtain a more stable result.

Example:

```
library(survival)
head(lung) # example survival data
lung$status <- lung$status - 1 # we require 0/1 outcomes; this variable originally is coded as 1/2
survivalfragility(Surv(time, status) ~ pat.karno + strata(inst), data=lung, niter=100, progress.bar = TRUE)
```

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