# SMR Analysis

    library(Colossus)
    library(data.table)
    library(survival)
    library(dplyr)

## External Rate Comparisons

At times, it is important to determine if the event rate in a cohort is statistically different from the event rate in a reference population. This has traditionally been done by calculating a Standardized Mortality Ratio (SMR). The analysis assumes the researcher has a table of event rates for different combinations of ages or categories in a reference population, which are then applied to the collected data to estimate the number of events if the collected data was subject to the same background event rate as the reference population. The goal is generally to determine if the observed events are statistically different than the expected number of events, and to investigate how much of an effect different covariates have.

### SMR Calculation

One method to approximate this is to fit a model to the SMR. The simplest model is to estimate the true rate (\(\lambda_i\)) as a multiple of the expected rate (\(\lambda^{*}_{i}\)). This provides an SMR for the full cohort, assuming one has an estimated rate for every row of data.

\[ \begin{aligned} \lambda_i = \beta * \lambda^{*}_{i} \end{aligned} \]

For this analysis, we will use the veteran lung cancer trial data from the survival package. We will treat events to be status equal to 2, scale the survival time to be per 100 days, and assume every row has an expected event rate of 0.5, roughly double the average rate in the data. In practice, this could be a constant population average or a row-specific rate taken from mapping a table of external events onto the data.

    data(cancer, package = "survival")
    cancer %>% setDT()
    df <- copy(cancer)

    cancer$status <- as.integer(cancer$status == 2)
    cancer$time <- cancer$time / 100

    cancer$erate <- 0.5

The analysis proceeds identically to a standard Poisson regression. We are interested in a linear relationship between the expected event rate and true event rate, and set the initial parameter estimate to 1.

    a_n <- c(1)
    control <- list(
      "ncores" = 1, "maxiter" = 20, "halfmax" = 5, "epsilon" = 1e-9,
      "deriv_epsilon" = 1e-9, "verbose" = 2
    )
    e <- PoisRun(Poisson(time, status) ~ linear(erate), cancer, a_n = a_n, control = control)
    print(e)
    #> |-------------------------------------------------------------------|
    #> Final Results
    #>    Covariate Subterm Term Number Central Estimate Standard Error 2-tail p-value
    #>       <char>  <char>       <int>            <num>          <num>          <num>
    #> 1:     erate     lin           0        0.4741856      0.0369153   9.147657e-38
    #>
    #> Poisson Model Used
    #> -2*Log-Likelihood: 576.22,  Deviation: 246.22,  AIC: 248.22,  BIC: 581.65
    #> Iterations run: 13
    #> maximum step size: 1.74e-10, maximum first derivative: 1.24e-06
    #> Analysis did not converge, check convergence criteria or run further
    #> Run finished in 0.01 seconds
    #> |-------------------------------------------------------------------|

In this case, we found the SMR to be 0.474, analysis of the confidence interval (either Wald or Likelihood-based) could be performed to check if the results are statistically significant. Suppose we wanted to take the analysis a step further and investigate the effect of difference covariates. The SMR equation can be adjusted to either an additive or multiplicative effect, similar to any other Poisson regression model. Let us assume we are interested in the effects of biological sex. We can add another element to our model and rerun the regression.

    a_n <- c(1, 1)

    e <- PoisRun(Poisson(time, status) ~ linear(erate, 0) + linear(sex, 1), cancer, a_n = a_n, control = control)
    print(e)
    #> |-------------------------------------------------------------------|
    #> Final Results
    #>    Covariate Subterm Term Number Central Estimate Standard Error 2-tail p-value
    #>       <char>  <char>       <int>            <num>          <num>          <num>
    #> 1:     erate     lin           0        0.3696187      0.1111424   0.0008821847
    #> 2:       sex     lin           1        0.1904852      0.2571859   0.4589044455
    #>
    #> Poisson Model Used
    #> -2*Log-Likelihood: 582.87,  Deviation: 252.87,  AIC: 256.87,  BIC: 593.72
    #> Iterations run: 7
    #> maximum step size: 9.31e-10, maximum first derivative: 1.42e+01
    #> Analysis did not converge, check convergence criteria or run further
    #> Run finished in 0.01 seconds
    #> |-------------------------------------------------------------------|

In this case, we found the SMR for sex=0 to be 0.37 and the SMR for sex=1 to be 0.56. Once again we could further investigate the confidence intervals to determine if these estimates are statistically significant. This demonstrates how one can use Poisson regressions and external event rates to estimate differences in event rates between external populations and studied populations.