The nph package includes functions to model survival distributions in terms of piecewise constant hazards and to simulate data from the specified distributions.

The package is available from CRAN and can be installed directly from R.

```
install.packages("nph")
# For dev version
# install.packages("devtools")
devtools::install_github("repo/nph")
```

Basically, there are three mechanisms for non-proportionality available in this package:

- Disease progression
- Different effect by time intervals
- Subgroups

These scenarios are illustrated in the following figures. Note that the hazard ratio is not constant across time.

The functions of the package can be grouped according to their functionality.

Functions for modelling/setting the underlying survival model:

`hazVFfun`

`subpop_pchaz`

`pop_pchaz`

Functions for generating simulated dataset given for a specified survival model:

`sample_fun`

`sample_conditional_fun`

Functions for performing statistical tests:

`logrank.test`

`logrank.maxtest`

Plotting functions:

`plot.mixpch`

`plot_diagram`

`plot_shhr`

The basic underlying model for the survival mechanism assumes that each patient can be in one of three states: Alive with no progression of disease, Alive with progression of disease, and Dead.

The first step is to create the population model with the `pop_pchaz`

function.
As the previous figure shows, there are three hazard rates that need to be defined: the hazard of disease progression \(\lambda_P(t)\), the hazard of death given no progression \(\lambda_P(t)\), and the hazard of death given progression \(\lambda_P(t)\). The arguments `lambdaProgMat`

, `lambdaMat1`

, and `lambdaMat2`

in the `pop_pchaz`

function correspond to the three hazard rates, respectively.

The hazard rates are assumed piecewise constant functions across \(k\) time intervals \([t_{j-1}, t_j)\), \(j=1, \ldots,k\) with \(0=t_0`T`

that is a vector to specify \(t_0,t_1,\ldots,t_k\). When `T`

is of length 2 (and therefore only one time interval) `lambdaProgMat`

, `lambdaMat1`

, and `lambdaMat2`

are scalars. If `T`

is of length > 2, then the `lambdaProgMat`

, `lambdaMat1`

, and `lambdaMat2`

are matrices where the number of columns is equal to the number of time intervals
\[\begin{bmatrix}\lambda^{[t_0-t_1)} & \lambda^{[t_1-t_2)} & \ldots &\lambda^{[t_{k-1}-t_k)}\end{bmatrix}.\]

For example, if the patients are followed for two years but the hazards change after the first year, then `T`

should be specified as `c(0, 365, 2*365)`

. If we assume a hazard rate for death of 0.02 and 0.04 for the first and second year respectively, the we should specify `lambdaMat1 = matrix(c(0.02, 0.04), ncol = 2)`

.

Finally, is is also possible to specify different hazard rates for subgroups. The `pop_pchaz`

has the argument `p`

which is intended to specify the subgroup prevalences. Given \(m\) subgroups with relative sizes \(p_1, p_2, \ldots, p_m\), then the `p`

argument should be specified as `c(p_1, p_2, ..., p_m)`

. The `lambdaProgMat`

, `lambdaMat1`

, and `lambdaMat2`

then should have the number of row equal to the number of defined subgroups:

\[ \begin{bmatrix} \lambda_1 \ \lambda_2 \ \ldots \ \lambda_m \end{bmatrix}. \]

For example, if patients can be divided into two subgroup with prevalences 0.2 and 0.8 with hazard rates a hazard rate for death of 0.02 and 0.03 thoughout a one year interval, then we define `T = c(0, 365)`

, `p = c(0.2, 0.8)`

and `lambdaMat1 = matrix(c(0.02, 0.03), nrow = 2)`

.

Naturally, it is possible to combine multiple time intervals and subgroups, then the hazard matrices have the form:

Below, we consider an example where there two subgroups and two time intervals. In practice, this situation correspond to the case where there is a delayed effect of the drug. Note that for specifying the hazard matrices, we use the median time to death/progression and use the function `m2r`

(also provided in the package) to obtain the hazard rates.

```
times <- c(0, 100, 5 * 365) # Time interval boundaries, in days
t_resp <- c(0.2, 0.8) #Proportion of subgroups
B5 <- pop_pchaz(
T = times,
lambdaMat1 = m2r(matrix(c(11, 30,
11, 18), byrow = TRUE, nrow = 2)),
lambdaMat2 = m2r(matrix(c( 9, 20,
9, 11), byrow = TRUE, nrow = 2)),
lambdaProgMat = m2r(matrix(c( 5, 15,
5, 9), byrow = TRUE, nrow = 2)),
p = t_resp, discrete_approximation = TRUE
)
```

The results object is of class `mixpch`

, which has a dedicated plotting function to visualize the survival and hazard functions.

```
plot(B5, main = "Survival function")
plot(B5, fun = "haz", main = "Hazard function")
plot(B5, fun = "cumhaz", main = "Cumulative Hazard function")
```

\newpage

The sample_fun function is designed to generate a simulated dataset that would be obtained from a parallel group randomised clinical trial.

The first step is to create two objects with the (theoretical) survival functions for the treatment and control groups using `pop_pchaz`

:

```
times <- c(0, 100, 5 * 365) # Time interval boundaries, in days
# Treatment group
B5 <- pop_pchaz(T = times,
lambdaMat1 = m2r(matrix(c(11, 30,
11, 18), byrow = TRUE, nrow = 2)),
lambdaMat2 = m2r(matrix(c( 9, 20,
9, 11), byrow = TRUE, nrow = 2)),
lambdaProgMat = m2r(matrix(c( 5, 15,
5, 9), byrow = TRUE, nrow = 2)),
p = c(0.2, 0.8),#Proportion of subgroups
discrete_approximation = TRUE
)
# Control group
K5 <- pop_pchaz(T = times,
lambdaMat1 = m2r(matrix(c(11, 11), nrow = 1)),
lambdaMat2 = m2r(matrix(c( 9, 9), nrow = 1)),
lambdaProgMat = m2r(matrix(c( 5, 5), nrow = 1)),
p = 1, discrete_approximation = TRUE
)
```

Then, using the resulting objects, we use them to generate a dataset with the `sample_fun`

function:

```
# Study set up and Simulation of a data set until interim analysis at 150 events
set.seed(15657)
dat <- sample_fun(K5, B5,
r0 = 0.5, # Allocation ratio
eventEnd = 450, # maximal number of events
lambdaRecr = 300 / 365, # recruitment rate per day (Poisson assumption)
lambdaCens = 0.013 / 365, # censoring rate per day (Exponential assumption)
maxRecrCalendarTime = 3 * 365,# Maximal duration of recruitment
maxCalendar = 4 * 365.25) # Maximal study duration
head(dat)
#> group inclusion y yCalendar event adminCens cumEvents
#> 777 1 26 3 29 TRUE FALSE 1
#> 17 1 25 18 43 TRUE FALSE 2
#> 367 1 9 40 49 TRUE FALSE 3
#> 64 0 42 9 51 TRUE FALSE 4
#> 25 0 34 40 74 TRUE FALSE 5
#> 708 1 103 7 110 TRUE FALSE 6
tail(dat)
#> group inclusion y yCalendar event adminCens cumEvents
#> 889 1 755 207 962 FALSE TRUE 450
#> 893 0 205 757 962 FALSE TRUE 450
#> 896 0 814 148 962 FALSE TRUE 450
#> 900 1 510 452 962 FALSE TRUE 450
#> 907 1 225 737 962 FALSE TRUE 450
#> 908 0 717 245 962 FALSE TRUE 450
```

\newpage

The weighted log-rank test is implemented using the function `logrank.test`

, which uses the statistic:

\[ z = \sum_{t\in\mathcal{D}} w(t)(d_{t, ctr} - e_{t,ctr}) / \sqrt{\sum_{t\in\mathcal{D}} w(t)^2 var(d_{t, ctr})}. \]

where \(w(t)\) are the Fleming-Harrington \(\rho-\gamma\) family weights, such that \(w(t)=\widehat{S}(t)^{\rho}(1-\widehat{S}(t))^{\gamma}\). Under the the least favorable configuration in \(H_0\), the test statistic is asymptotically standard normally distributed and large values of \(z\) are in favor of the alternative.

For example, the following code performs the weighted log-rank test using the simulated dataset and \(\rho = 1\) and \(\gamma = 0\).

```
logrank.test(time = dat$y,
event = dat$event,
group = dat$group,
# alternative = "greater",
rho = 1,
gamma = 0)
#> Call:
#> logrank.test(time = dat$y, event = dat$event, group = dat$group,
#> rho = 1, gamma = 0)
#>
#> N Observed Expected (O-E)^2/E (O-E)^2/V
#> 1 411 267 211 15.2 58.3
#> 2 385 183 239 13.3 58.3
#>
#> Chisq= 22 on 1 degrees of freedom, p= 3e-06
#> rho = 1 gamma = 0
# survival::survdiff(formula = survival::Surv(time = dat$y, event = dat$event) ~ dat$group)
```

For a set of \(k\) different weight functions \(w_1(t), \ldots, w_k(t)\), the maximum log-rank test statistic is \(z_{max} = \max_{i=1,\ldots,k}z_i\). Under the least favorable configuration in \(H_0\), approximately \((Z_1, \ldots, Z_k) \sim N_k(0, \Sigma)\). The \(p\)-value of the maximum test, \(P_{H_0}(Z_{max} > z_{max})\), is calculated based on this multivariate normal approximation via numeric integration.

The following code performs the maximum log-rank test using four combinations of \(\rho\) and \(\gamma\) for the weights.

```
lrmt = logrank.maxtest(
time = dat$y,
event = dat$event,
group = dat$group,
rho = c(0, 0, 1, 1),
gamma = c(0, 1, 0, 1)
)
lrmt
#> Call:
#> logrank.maxtest(time = dat$y, event = dat$event, group = dat$group,
#> rho = c(0, 0, 1, 1), gamma = c(0, 1, 0, 1))
#>
#> Two sided p-value = 4.91e-08 (Bonferroni corrected: 1.96e-07)
#>
#> Individual weighted log-rank tests:
#> Test z p
#> 1 1 5.37 7.99e-08
#> 2 2 5.24 1.58e-07
#> 3 3 4.69 2.76e-06
#> 4 4 5.45 4.91e-08
```

The individual tests can also be accesed using the `testListe`

elements in the resulting object.

```
lrmt$logrank.test[[1]]
#> Call:
#> logrank.test(time = time, event = event, group = group, alternative = alternative,
#> rho = rho[i], gamma = gamma[i])
#>
#> N Observed Expected (O-E)^2/E (O-E)^2/V
#> 1 411 267 211 15.2 28.8
#> 2 385 183 239 13.3 28.8
#>
#> Chisq= 28.8 on 1 degrees of freedom, p= 8e-08
#> rho = 0 gamma = 0
lrmt$logrank.test[[2]]
#> Call:
#> logrank.test(time = time, event = event, group = group, alternative = alternative,
#> rho = rho[i], gamma = gamma[i])
#>
#> N Observed Expected (O-E)^2/E (O-E)^2/V
#> 1 411 267 211 15.2 185
#> 2 385 183 239 13.3 185
#>
#> Chisq= 27.5 on 1 degrees of freedom, p= 2e-07
#> rho = 0 gamma = 1
lrmt$logrank.test[[3]]
#> Call:
#> logrank.test(time = time, event = event, group = group, alternative = alternative,
#> rho = rho[i], gamma = gamma[i])
#>
#> N Observed Expected (O-E)^2/E (O-E)^2/V
#> 1 411 267 211 15.2 58.3
#> 2 385 183 239 13.3 58.3
#>
#> Chisq= 22 on 1 degrees of freedom, p= 3e-06
#> rho = 1 gamma = 0
lrmt$logrank.test[[4]]
#> Call:
#> logrank.test(time = time, event = event, group = group, alternative = alternative,
#> rho = rho[i], gamma = gamma[i])
#>
#> N Observed Expected (O-E)^2/E (O-E)^2/V
#> 1 411 267 211 15.2 806
#> 2 385 183 239 13.3 806
#>
#> Chisq= 29.8 on 1 degrees of freedom, p= 5e-08
#> rho = 1 gamma = 1
```