# How to add estimability checking to your model’s `predict` method

#### estimability package, Version 2.0.0

The goal of this short vignette is to show how you can easily add estimability checking to your package’s `predict()` methods. Suppose that you have developed a model class that has elements `$coefficients`, `$formula`, etc. Suppose it also has an `$env` element, an environment that can hold miscellaneous information. This is not absolutely necessary, but handy if it exists. Your model class involves some kind of linear predictor.

We are concerned with models that:

- Allow rank deficiencies (where some predictors may be excluded)
- Allow predictions for new data

For any such model, it is important to add estimability checking to your predict method, because the regression coefficients are not unique – and hence that predictions may not be unique. It can be shown that predictions on new data are unique only for cases that fall within the row space of the model matrix. The estimability package is designed to check for this.

The recommended design for accommodating rank-deficient models is to follow the example of `stats::lm` objects, where any predictors that are excluded have a corresponding regression coefficient of `NA`. Please note that this `NA` code actually doesn’t actually means the coefficient is missing; it is a code that means that that coefficient has been constrained to be zero. In what follows, we assume that this convention is used.

First note that estimability checking is not needed unless you are predicting for new data. So that’s where you need to incorporate estimability checking. The `predict` method should be coded something like this:

    predict.mymod <- function(object, newdata, ...) {
        # ... some setup code ...
        if (!missing(newdata)) {
            X <-  # ... code to set up the model matrix for newdata ...

            b <- coef(object)
            if (any(is.na(b))) {  # we have rank deficiency so test estimability
                if (is.null (nbasis <- object$env$nbasis))
                    nbasis <- object$nbasis <-
                        estimability::nonest.basis(model.matrix(object))
                b[is.na(b)] <- 0
                pred <- X %*% b
                pred[!estimability::is.estble(X, nbasis)] <- NA
            }
            else
                pred <- X %*% coef(object)
        }
        # ... perhaps more code ...
        pred
    }

That’s it – and this is the fancy version, where we can save `nbasis` for use with possible future predictions. Any non-estimable cases are flagged as `NA` in the `pred` vector.

An alternative way to code this would be to exclude the columns of `X` and elements of `b` that correspond to `NA`s in `b`. But be careful, because you need all the columns in `X` in order to check estimability.

The only other thing you need to do is add `estimability` to the `Imports` list in your `Description file.

## A complete example

Below is a complete model-fitting function that illustrates how to navigate the various practical aspects of incorporating estimability checking in your code. This code emphasises readability rather than computational efficiency, and lacks many aspects of robustness or checking of inputs. The function returns information about the model terms and factor levels, the computed coefficients `coef` and their variance-covariance matrix `vcov`, and the basis `nbasis` of the null space of the model matrix. It utilizes the Cholesky decomposition with pivoting. Pivoting involves rearranging the order of predictors, managing rank-deficient cases by moving some linearly dependent predictors to the end of the line. We can the ignore or “discard” those dependent predictors, which in estimability parlance is obtaining a solution by constraining the coefficients of those predictors to be zero.

Much of what is shown here is how to deal with the `pivot` attribute, which comprises the indices of the reordered predictors. The `rank` attribute tells us how many nonzero coefficients there are. We compute `coef` using the first `rank` pivoted predictors, and by putting `NA`s for the coefficients constrained to zero, while `vcov` is only for the non-`NA` elements of `coef`. As is also true in the standard `lm` function, the `NA` elements are just used to signal which coefficients were constrained to zero. The `nonest.basis()` function sees the `pivot` attribute and re-orders the predictors accordingly, so we don’t un-pivot that result. The last few lines of code are necessary for passing needed information to the `predict` method.

    mylm <- function(formula, data) {
        y <- data[[all.vars(formula)[1]]]
        X <- model.matrix(formula[-2], data = data)
        ch <- chol(t(X) %*% X, pivot = TRUE) |> suppressWarnings()
        rank = attr(ch, "rank")
        pivot <- attr(ch, "pivot")
        XpXinv <- chol2inv(ch, size = rank)
        coef <- rep(NA, ncol(X))
        names(coef) <- colnames(X)
        coef[pivot[1:rank]] <- XpXinv %*% (t(X[, pivot[1:rank], drop = FALSE]) %*% y)
        nonNA <- which(!is.na(coef))
        fit <- X[ , nonNA, drop = FALSE] %*% coef[nonNA]
        mse <- sum((y - fit)^2) / (length(y) - rank)
        ord <- order(pivot[1:rank])
        vcov <- mse * XpXinv[ord, ord, drop = FALSE]
        dimnames(vcov) <- list(names(coef)[nonNA], names(coef)[nonNA])
        nbasis <- estimability::nonest.basis(ch)
        terms <- terms(formula) |> delete.response()
        xlev = list()
        for (v in all.vars(terms))
            if(!is.null(lv <- levels(data[[v]])))
                xlev[[v]] <- lv
        obj <- list(terms = terms, xlev = xlev, coef = coef, vcov = vcov, nbasis = nbasis)
        class(obj) <- "mylm"
        obj
    }

The following code is a corresponding S3 method to provide predictions and standard errors for new data, with estimability checking. The `model.frame` call ensures that we incorporate the original factor coding in the new data. We then construct the corresponding model matrix `X` for the new data, and test the rows for estimability. Once that is determined, we obtain the matrix `XX` with only the estimable rows and corresponding to non-`NA` elements of `coef`. The predictions and SEs of any non-estimable rows are set to `NA`.

    predict.mylm <- function(mod, newdata) {
        mf <- model.frame(mod$terms, data = newdata, xlev = mod$xlev)
        X <- model.matrix(mod$terms, data = mf)
        pred <- se <- rep(NA, nrow(X))
        estble <- estimability::is.estble(X, mod$nbasis)
        XX <- X[estble, !is.na(mod$coef), drop = FALSE]
        pred[estble] <- XX %*% mod$coef[!is.na(mod$coef)]
        se[estble] <- sqrt(diag(XX %*% mod$vcov %*% t(XX)))
        list(pred = pred, se = se)
    }

Finally, here is a test of these functions, using a subset of the `warpbreaks` data with two complete cells excluded along with a few more observations. Then we obtain predictions for all factor combinations. We find that the two excluded cells are non-estimable.

    # test code
    warp = warpbreaks[11:40, ]
    warp.mylm = mylm(breaks ~ wool*tension, warp)

    new = do.call(expand.grid, warp.mylm$xlev)
    cbind(new, predict(warp.mylm, newdata = new))

    ##   wool tension     pred       se
    ## 1    A       L       NA       NA
    ## 2    B       L 28.22222 3.324365
    ## 3    A       M 24.75000 3.526021
    ## 4    B       M 25.75000 4.986547
    ## 5    A       H 24.55556 3.324365
    ## 6    B       H       NA       NA