# Models and model comparisons

# Implemented model types

The risks package implements all major regression models that have been proposed for relative risks and risk differences. By default (`approach = "auto"`), `riskratio()` and `riskdiff()` estimate the most efficient valid model that converges; in more numerically challenging cases, they default to marginal standardization, which does not return parameters for covariates.

The following models are implemented in the risks package:

#1 | `approach =` | RR | RD | Model | Reference
--- | --- | --- | --- | --- | ---
1 | `glm` | `riskratio` | `riskdiff` | Binomial model with a log or identity link | Wacholder S. Binomial regression in GLIM: Estimating risk ratios and risk differences. Am J Epidemiol 1986;123:174-184.
2 | `glm_startp` | `riskratio` | `riskdiff` | Binomial model with a log or identity link, convergence-assisted by starting values from Poisson model | Spiegelman D, Hertzmark E. Easy SAS calculations for risk or prevalence ratios and differences. Am J Epidemiol 2005;162:199-200.
3 | `margstd_delta` | `riskratio` | `riskdiff` | Marginally standardized estimates using binomial models with a logit link (logistic model) with standard errors calculated via the delta method. | This package.
4 | `margstd_boot` | `riskratio` | `riskdiff` | Marginally standardized estimates using binomial models with a logit link (logistic model) with bias-corrected accelerated (BCa) confidence intervals from parametric bootstrapping (see Marginal standardization). | This package. For marginal standardization with nonparametric bootstrapping, see: Localio AR, Margolis DJ, Berlin JA. Relative risks and confidence intervals were easily computed indirectly from multivariable logistic regression. J Clin Epidemiol 2007;60(9):874-82.
– | `glm_cem` | `riskratio` | — | Binomial model with log-link fitted via combinatorial expectation maximization instead of Fisher scoring | Donoghoe MW, Marschner IC. logbin: An R Package for Relative Risk Regression Using the Log-Binomial Model. J Stat Softw 2018;86(9).
– | `glm_cem` | — | `riskdiff` | Additive binomial model (identity link) fitted via combinatorial expectation maximization instead of Fisher scoring | Donoghoe MW, Marschner IC. Stable computational methods for additive binomial models with application to adjusted risk differences. Comput Stat Data Anal 2014;80:184-96.
– | `robpoisson` | `riskratio` | `riskdiff` | Log-linear (Poisson) model with robust/sandwich/empirical standard errors | Zou G. A modified Poisson regression approach to prospective studies with binary data. Am J Epidemiol 2004;159(7):702-6
– | `duplicate` | `riskratio` | – | Case-duplication approach, fitting a logistic model with cluster-robust standard errors | Schouten EG, Dekker JM, Kok FJ, Le Cessie S, Van Houwelingen HC, Pool J, Vandenbroucke JP. Risk ratio and rate ratio estimation in case-cohort designs: hypertension and cardiovascular mortality. Stat Med 1993;12:1733–45.
– | `glm_startd` | `riskratio` | — | Binomial model with a log link, convergence-assisted by starting values from case-duplication logistic model | This package.
– | `logistic` | `riskratio`, for comparison only | — | Binomial model with logit link (i.e., the logistic model), returning odds ratios | Included for comparison purposes only.

1 Indicates the priority with which the legacy modelling strategy (`approach = "legacy"`) attempts model fitting (`glm_startp`: only for RR).

Which model was fitted is always indicated in the first line of the output of `summary()` and in the `model` column of `tidy()`. In methods sections of manuscripts, the approach can be described in detail as follows: “Risk ratios (or risk differences) were obtained via (method listed in the first line of model `summary.risks(...)`) using the `risks` R package (reference to this package and/or the article listed in the column ‘reference’).”

For example: “Risk ratios were obtained from binomial models with a log link, convergence-assisted by Poisson models (ref. Spiegelman and Hertzmark, AJE 2005), using the `risks` R package (https://stopsack.github.io/risks/).”

# Model choice

By default, automatic model fitting (`approach = "auto"`) reports results from marginal standardization using a logistic model with delta method standard errors (equivalent to `approach = "margstd_delta"`). An exception is made if interaction terms between exposure and confounders are included. This case, confidence intervals are calculated using bootstrapping (equivalent to requesting `approach = "margstd_boot"`). Alternatively, any of the options listed under `approach =` in the table can be requested directly. However, unlike with `approach = "auto"` (the default) or `approach = "legacy"`, the selected model may not converge.

We load the same example data as in the Get Started vignette.

    library(risks)  # provides riskratio(), riskdiff(), postestimation functions
    library(dplyr)  # For data handling
    library(broom)  # For tidy() model summaries
    data(breastcancer)

We then select a binomial model with starting values from the Poisson model:

    riskratio(formula = death ~ stage + receptor,
              data = breastcancer,
              approach = "glm_startp")
    #>
    #> Risk ratio model
    #> Call:  stats::glm(formula = death ~ stage + receptor, family = stats::binomial(link = "log"),
    #>     data = breastcancer, start = "(from Poisson model)")
    #>
    #> Coefficients:
    #>    (Intercept)   stageStage II  stageStage III     receptorLow
    #>        -2.3521          0.9314          1.7695          0.4436
    #>
    #> Degrees of Freedom: 191 Total (i.e. Null);  188 Residual
    #> Null Deviance:       228.1
    #> Residual Deviance: 185.9     AIC: 193.9

However, the binomial model without starting values (`approach = "glm"`) does not converge, as expected.

# Model comparisons

With `approach = "all"`, all model types listed in the tables are fitted. The fitted object, e.g., `fit`, is one of the converged models. A summary of the convergence status of all models is displayed at the beginning of `summary(fit)`:

    fit_all <- riskratio(formula = death ~ stage + receptor,
                         data = breastcancer,
                         approach = "all")
    summary(fit_all)
    #>
    #> All fitted models:
    #>            Model Converged Max.prob.
    #> 1     robpoisson      TRUE 0.9052473
    #> 2            glm     FALSE        NA
    #> 3     glm_startp      TRUE 0.8702313
    #> 4         logbin      TRUE 0.8702292
    #> 5   logbin_start      TRUE 0.8702294
    #> 6   margstd_boot      TRUE 0.8158560
    #> 7  margstd_delta      TRUE 0.8158560
    #> 8       logistic      TRUE 0.8158560
    #> 9      duplicate      TRUE 0.4785245
    #> 10    glm_startd      TRUE 0.8702318
    #>
    #> Risk ratio model, fitted as Poisson model with robust covariance (robpoisson).
    #> Call:
    #> stats::glm(formula = death ~ stage + receptor, family = poisson(link = "log"),
    #>     data = breastcancer, start = "(no starting values)")
    #>
    #> Coefficients:
    #>                Estimate Std. Error z value Pr(>|z|)
    #> (Intercept)     -2.3658     0.3602  -6.569 5.08e-11 ***
    #> stageStage II    0.9246     0.3932   2.351   0.0187 *
    #> stageStage III   1.7772     0.3855   4.610 4.03e-06 ***
    #> receptorLow      0.4891     0.2129   2.297   0.0216 *
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> (Dispersion parameter for poisson family taken to be 1)
    #>
    #>     Null deviance: 137.00  on 191  degrees of freedom
    #> Residual deviance: 110.29  on 188  degrees of freedom
    #> AIC: 226.29
    #>
    #> Number of Fisher Scoring iterations: 6
    #>
    #> Confidence intervals for coefficients: (robust)
    #>                      2.5 %     97.5 %
    #> (Intercept)    -3.07172916 -1.6599084
    #> stageStage II   0.15386831  1.6952382
    #> stageStage III  1.02161187  2.5328206
    #> receptorLow     0.07183369  0.9062773

Individual models can be accessed as `fit$all_models[[1]]` through `fit$all_models[[6]]` (or `[[7]]` if fitting a risk ratio model). `tidy()` shows coefficients and confidence intervals from all models that converged:

    tidy(fit_all) %>%
      select(-statistic, -p.value) %>%
      print(n = 50)
    #> # A tibble: 34 × 6
    #>    term            estimate std.error conf.low conf.high model
    #>    <chr>          <dbl[1d]>     <dbl>    <dbl>     <dbl> <chr>
    #>  1 (Intercept)       -2.37      0.360  -3.07      -1.66  robpoisson
    #>  2 stageStage II      0.925     0.393   0.154      1.70  robpoisson
    #>  3 stageStage III     1.78      0.386   1.02       2.53  robpoisson
    #>  4 receptorLow        0.489     0.213   0.0718     0.906 robpoisson
    #>  5 (Intercept)       -2.35      0.360  -3.06      -1.65  glm_startp
    #>  6 stageStage II      0.931     0.393   0.161      1.70  glm_startp
    #>  7 stageStage III     1.77      0.385   1.01       2.52  glm_startp
    #>  8 receptorLow        0.444     0.197   0.0578     0.829 glm_startp
    #>  9 (Intercept)       -2.35      0.360  -3.06      -1.65  logbin
    #> 10 stageStage II      0.931     0.393   0.161      1.70  logbin
    #> 11 stageStage III     1.77      0.385   1.01       2.52  logbin
    #> 12 receptorLow        0.444     0.197   0.0578     0.829 logbin
    #> 13 (Intercept)       -2.35      0.360  -3.06      -1.65  logbin_start
    #> 14 stageStage II      0.931     0.393   0.161      1.70  logbin_start
    #> 15 stageStage III     1.77      0.385   1.01       2.52  logbin_start
    #> 16 receptorLow        0.444     0.197   0.0578     0.829 logbin_start
    #> 17 stageStage I       0         0       0          0     margstd_boot
    #> 18 stageStage II      0.899     0.439   0.228      1.88  margstd_boot
    #> 19 stageStage III     1.81      0.431   1.07       2.67  margstd_boot
    #> 20 stageStage I       0         0       0          0     margstd_delta
    #> 21 stageStage II      0.899     0.387   0.140      1.66  margstd_delta
    #> 22 stageStage III     1.81      0.378   1.07       2.55  margstd_delta
    #> 23 (Intercept)       -2.37      0.420  -3.19      -1.54  logistic
    #> 24 stageStage II      1.13      0.466   0.222      2.05  logistic
    #> 25 stageStage III     2.94      0.586   1.79       4.08  logistic
    #> 26 receptorLow        0.920     0.395   0.145      1.69  logistic
    #> 27 (Intercept)       -2.37      0.362  -3.08      -1.66  duplicate
    #> 28 stageStage II      0.923     0.395   0.149      1.70  duplicate
    #> 29 stageStage III     1.78      0.385   1.03       2.54  duplicate
    #> 30 receptorLow        0.504     0.224   0.0647     0.944 duplicate
    #> 31 (Intercept)       -2.35      0.360  -3.06      -1.65  glm_startd
    #> 32 stageStage II      0.931     0.393   0.161      1.70  glm_startd
    #> 33 stageStage III     1.77      0.385   1.01       2.52  glm_startd
    #> 34 receptorLow        0.444     0.197   0.0578     0.829 glm_startd