In OxWearables/epicoda: Supports epidemiological analyses using compositional exposure variables

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Only 2-sided 95% confidence intervals are currently implemented.

Presenting Compositional Data Analyses

The results of Compositional Data Analyses can be presented graphically, using plots of model predictions for pairwise balances of parts (plot_transfers) or model predictions at particular compositions (forest_plot_comp). In this package, these both use predict_fit_and_ci internally. This function can also be used in a standalone fashion to explore model predictions and confidence intervals.

Wherever results are presented, there are two options. The default is to set terms = TRUE, and present the predicted difference in the outcome at the different levels of the compositional variables. Implicitly, in this case, all other covariates are fixed.

The alternative is to set terms = FALSE. In this case, the fixed_values argument is used (if fixed_values is NULL, the default, the generate_fixed_values function is used to set this argument internally, using mean, modal and compositional mean values as appropriate). The plot shows model predictions for the specified composition and for additional (non-compositional) covariate values as given in fixed_values.

Confidence intervals when terms = TRUE

In all cases, using predict with type = "terms" gives fitted values of each specified term on the linear predictor scale i.e. for $x_1$ it returns $\beta_1 x_1$. If the (transformed) compositional variables are $z_1, z_2, ... z_n$ and these are the specified terms, then predict returns estimates of $\beta_1 z_1, \beta_2 z_2, ... \beta_n z_n$ which can be used to calculate $$\overline{\Delta y} = \beta_1 z_1 + \beta_2 z_2 + ... + \beta_n z_n.$$ We now need to calculate the uncertainty on this quantity.

This can be derived in the same way as the confidence interval on a model prediction is derived in general; the difference here is the use of a partial sum of terms, rather than the full sum of terms. For a linear regression model, the standard error of $\overline{\Delta y}$ is given by: $$ SE(\overline{\Delta y}) = \sigma \sqrt{diag\Big((x - \overline{x})(ZZ^{T})^{-1}(x-\overline{x})^T\Big)} $$ where $Z$ is the design matrix of the model, $\sigma$ is the residual variance, $x$ is the new values of the explanatory variables and $\overline{x}$ is the mean value of the explanatory variables in the data used to calculate the model.

As $$ \sigma ^2 (ZZ^{T})^{-1} $$ is the variance-covariance matrix of the coefficients of the fitted model (henceforth written $V$), this can be calculated as: $$ SE(\overline{\Delta y}) = \sqrt{diag\Big((x - \overline{x})V(x-\overline{x})^T\Big)}. $$ This generalises immediately to the sum of linear predictors in logistic regression or Cox model (using the respective variance-covariance metrics for the coefficients), and is the version implemented in the package.

Linear models

This is the simplest case. Using the above, we directly estimate $\overline{\Delta y}$ and the confidence interval on this, $(\overline{\Delta y_{min}}, \overline{\Delta y_{max}})$.

Logistic models

Here, the terms given by the predict function are on the scale of the linear predictors i.e. we estimate $\overline{\Delta y}$ on the scale of the linear predictors and the confidence interval on this, $(\overline{\Delta y_{min}}, \overline{\Delta y_{max}})$. Then, the odds ratio and uncertainty on it is given by exponentiating the estimate ($\exp(\overline{\Delta y})$) and confidence interval ($(\exp(\overline{\Delta y_{min}}), \exp(\overline{\Delta y_{max}}))$.

Cox models

Here, the terms given by the predict function are on the scale of the linear predictors i.e. we estimate $\overline{\Delta y}$ on the scale of the linear predictors and the confidence interval on this, $(\overline{\Delta y_{min}}, \overline{\Delta y_{max}})$. Then, the hazard ratio and uncertainty on it is given by exponentiating the estimate ($\exp(\overline{\Delta y})$) and confidence interval ($(\exp(\overline{\Delta y_{min}}), \exp(\overline{\Delta y_{max}}))$.

Confidence intervals when terms = FALSE

In this case, for linear, logistic and Cox models, the confidence intervals are the standard confidence intervals on the prediction, as calculated by the predict method. Note the intervals are not the prediction intervals (the intervals within which 95% of new observations $\overline{\Delta y}$ might be expected to fall given their $x$ values).

OxWearables/epicoda documentation built on Dec. 7, 2022, 9:07 p.m.