R: Transformations/Imputations using Canonical Variates

transcan {Hmisc} | R Documentation

## Transformations/Imputations using Canonical Variates

### Description

`transcan` is a nonlinear additive transformation and imputation function, and there are several functions for using and operating on its results. `transcan` automatically transforms continuous and categorical variables to have maximum correlation with the best linear combination of the other variables. There is also an option to use a substitute criterion - maximum correlation with the first principal component of the other variables. Continuous variables are expanded as restricted cubic splines and categorical variables are expanded as contrasts (e.g., dummy variables). By default, the first canonical variate is used to find optimum linear combinations of component columns. This function is similar to `ace` except that transformations for continuous variables are fitted using restricted cubic splines, monotonicity restrictions are not allowed, and `NA`s are allowed. When a variable has any `NA`s, transformed scores for that variable are imputed using least squares multiple regression incorporating optimum transformations, or `NA`s are optionally set to constants. Shrinkage can be used to safeguard against overfitting when imputing. Optionally, imputed values on the original scale are also computed and returned. For this purpose, recursive partitioning or multinomial logistic models can optionally be used to impute categorical variables, using what is predicted to be the most probable category.

By default, `transcan` imputes `NA`s with “best guess” expected values of transformed variables, back transformed to the original scale. Values thus imputed are most like conditional medians assuming the transformations make variables' distributions symmetric (imputed values are similar to conditionl modes for categorical variables). By instead specifying `n.impute`, `transcan` does approximate multiple imputation from the distribution of each variable conditional on all other variables. This is done by sampling `n.impute` residuals from the transformed variable, with replacement (a la bootstrapping), or by default, using Rubin's approximate Bayesian bootstrap, where a sample of size n with replacement is selected from the residuals on n non-missing values of the target variable, and then a sample of size m with replacement is chosen from this sample, where m is the number of missing values needing imputation for the current multiple imputation repetition. Neither of these bootstrap procedures assume normality or even symmetry of residuals. For sometimes-missing categorical variables, optimal scores are computed by adding the “best guess” predicted mean score to random residuals off this score. Then categories having scores closest to these predicted scores are taken as the random multiple imputations (`impcat = "rpart"` is not currently allowed with `n.impute`). The literature recommends using `n.impute = 5` or greater. `transcan` provides only an approximation to multiple imputation, especially since it “freezes” the imputation model before drawing the multiple imputations rather than using different estimates of regression coefficients for each imputation. For multiple imputation, the `aregImpute` function provides a much better approximation to the full Bayesian approach while still not requiring linearity assumptions.

When you specify `n.impute` to `transcan` you can use `fit.mult.impute` to re-fit any model `n.impute` times based on `n.impute` completed datasets (if there are any sometimes missing variables not specified to `transcan`, some observations will still be dropped from these fits). After fitting `n.impute` models, `fit.mult.impute` will return the fit object from the last imputation, with `coefficients` replaced by the average of the `n.impute` coefficient vectors and with a component `var` equal to the imputation-corrected variance-covariance matrix. `fit.mult.impute` can also use the object created by the `mice` function in the mice library to draw the multiple imputations, as well as objects created by `aregImpute`. The following components of fit objects are also replaced with averages over the `n.impute` model fits: `linear.predictors`, `fitted.values`, `stats`, `means`, `icoef`, `scale`, `center`, `y.imputed`.

The `summary` method for `transcan` prints the function call, R^2 achieved in transforming each variable, and for each variable the coefficients of all other transformed variables that are used to estimate the transformation of the initial variable. If `imputed=TRUE` was used in the call to transcan, also uses the `describe` function to print a summary of imputed values. If `long = TRUE`, also prints all imputed values with observation identifiers. There is also a simple function `print.transcan` which merely prints the transformation matrix and the function call. It has an optional argument `long`, which if set to `TRUE` causes detailed parameters to be printed. Instead of plotting while `transcan` is running, you can plot the final transformations after the fact using `plot.transcan` or `ggplot.transcan`, if the option `trantab = TRUE` was specified to `transcan`. If in addition the option `imputed = TRUE` was specified to `transcan`, `plot` and `ggplot` will show the location of imputed values (including multiples) along the axes. For `ggplot`, imputed values are shown as red plus signs.

`impute` method for `transcan` does imputations for a selected original data variable, on the original scale (if `imputed=TRUE` was given to `transcan`). If you do not specify a variable to `impute`, it will do imputations for all variables given to `transcan` which had at least one missing value. This assumes that the original variables are accessible (i.e., they have been attached) and that you want the imputed variables to have the same names are the original variables. If `n.impute` was specified to `transcan` you must tell `impute` which `imputation` to use. Results are stored in `.GlobalEnv` when `list.out` is not specified (it is recommended to use `list.out=TRUE`).

The `predict` method for `transcan` computes predicted variables and imputed values from a matrix of new data. This matrix should have the same column variables as the original matrix used with `transcan`, and in the same order (unless a formula was used with `transcan`).

The `Function` function is a generic function generator. `Function.transcan` creates R functions to transform variables using transformations created by `transcan`. These functions are useful for getting predicted values with predictors set to values on the original scale.

The `vcov` methods are defined here so that imputation-corrected variance-covariance matrices are readily extracted from `fit.mult.impute` objects, and so that `fit.mult.impute` can easily compute traditional covariance matrices for individual completed datasets.

The subscript method for `transcan` preserves attributes.

The `invertTabulated` function does either inverse linear interpolation or uses sampling to sample qualifying x-values having y-values near the desired values. The latter is used to get inverse values having a reasonable distribution (e.g., no floor or ceiling effects) when the transformation has a flat or nearly flat segment, resulting in a many-to-one transformation in that region. Sampling weights are a combination of the frequency of occurrence of x-values that are within `tolInverse` times the range of `y` and the squared distance between the associated y-values and the target y-value (`aty`).

### Usage

    transcan(x, method=c("canonical","pc"),
             categorical=NULL, asis=NULL, nk, imputed=FALSE, n.impute,
             boot.method=c('approximate bayesian', 'simple'),
             trantab=FALSE, transformed=FALSE,
             impcat=c("score", "multinom", "rpart"),
             mincut=40,
             inverse=c('linearInterp','sample'), tolInverse=.05,
             pr=TRUE, pl=TRUE, allpl=FALSE, show.na=TRUE,
             imputed.actual=c('none','datadensity','hist','qq','ecdf'),
             iter.max=50, eps=.1, curtail=TRUE,
             imp.con=FALSE, shrink=FALSE, init.cat="mode",
             nres=if(boot.method=='simple')200 else 400,
             data, subset, na.action, treeinfo=FALSE,
             rhsImp=c('mean','random'), details.impcat='', ...)

    ## S3 method for class 'transcan'
    summary(object, long=FALSE, digits=6, ...)

    ## S3 method for class 'transcan'
    print(x, long=FALSE, ...)

    ## S3 method for class 'transcan'
    plot(x, ...)

    ## S3 method for class 'transcan'
    ggplot(data, mapping, scale=FALSE, ..., environment)

    ## S3 method for class 'transcan'
    impute(x, var, imputation, name, pos.in, data,
           list.out=FALSE, pr=TRUE, check=TRUE, ...)

    fit.mult.impute(formula, fitter, xtrans, data, n.impute, fit.reps=FALSE,
                    dtrans, derived, vcovOpts=NULL, pr=TRUE, subset, ...)

    ## S3 method for class 'transcan'
    predict(object, newdata, iter.max=50, eps=0.01, curtail=TRUE,
            type=c("transformed","original"),
            inverse, tolInverse, check=FALSE, ...)

    Function(object, ...)

    ## S3 method for class 'transcan'
    Function(object, prefix=".", suffix="", pos=-1, ...)

    invertTabulated(x, y, freq=rep(1,length(x)),
                    aty, name='value',
                    inverse=c('linearInterp','sample'),
                    tolInverse=0.05, rule=2)

    ## Default S3 method:
    vcov(object, regcoef.only=FALSE, ...)

    ## S3 method for class 'fit.mult.impute'
    vcov(object, regcoef.only=TRUE,
                    intercepts='mid', ...)

### Arguments

`x` | a matrix containing continuous variable values and codes for categorical variables. The matrix must have column names (`dimnames`). If row names are present, they are used in forming the `names` attribute of imputed values if `imputed = TRUE`. `x` may also be a formula, in which case the model matrix is created automatically, using data in the calling frame. Advantages of using a formula are that `categorical` variables can be determined automatically by a variable being a `factor` variable, and variables with two unique levels are modeled `asis`. Variables with 3 unique values are considered to be `categorical` if a formula is specified. For a formula you may also specify that a variable is to remain untransformed by enclosing its name with the identify function, e.g. `I(x3)`. The user may add other variable names to the `asis` and `categorical` vectors. For `invertTabulated`, `x` is a vector or a list with three components: the x vector, the corresponding vector of transformed values, and the corresponding vector of frequencies of the pair of original and transformed variables. For `print`, `plot`, `ggplot`, `impute`, and `predict`, `x` is an object created by `transcan`.
`formula` | any R model formula
`fitter` | any R, `rms`, modeling function (not in quotes) that computes a vector of `coefficients` and for which `vcov` will return a variance-covariance matrix. E.g., `fitter = lm`, `glm`, `ols`. At present models involving non-regression parameters (e.g., scale parameters in parametric survival models) are not handled fully.
`xtrans` | an object created by `transcan`, `aregImpute`, or `mice`
`method` | use `method="canonical"` or any abbreviation thereof, to use canonical variates (the default). `method="pc"` transforms a variable instead so as to maximize the correlation with the first principal component of the other variables.
`categorical` | a character vector of names of variables in `x` which are categorical, for which the ordering of re-scored values is not necessarily preserved. If `categorical` is omitted, it is assumed that all variables are continuous (or binary). Set `categorical="*"` to treat all variables as categorical.
`asis` | a character vector of names of variables that are not to be transformed. For these variables, the guts of `lm.fit` `method="qr"` is used to impute missing values. You may want to treat binary variables `asis` (this is automatic if using a formula). If `imputed = TRUE`, you may want to use "categorical" for binary variables if you want to force imputed values to be one of the original data values. Set `asis="*"` to treat all variables `asis`.
`nk` | number of knots to use in expanding each continuous variable (not listed in `asis`) in a restricted cubic spline function. Default is 3 (yielding 2 parameters for a variable) if \var{n} < 30, 4 if 30 \eq \var{n} < 100, and 5 if \var{n} >= 100 (4 parameters).
`imputed` | Set to `TRUE` to return a list containing imputed values on the original scale. If the transformation for a variable is non-monotonic, imputed values are not unique. `transcan` uses the `approx` function, which returns the highest value of the variable with the transformed score equalling the imputed score. `imputed=TRUE` also causes original-scale imputed values to be shown as tick marks on the top margin of each graph when `show.na=TRUE` (for the final iteration only). For categorical predictors, these imputed values are passed through the `jitter` function so that their frequencies can be visualized. When `n.impute` is used, each `NA` will have `n.impute` tick marks.
`n.impute` | number of multiple imputations. If omitted, single predicted expected value imputation is used. `n.impute=5` is frequently recommended.
`boot.method` | default is to use the approximate Bayesian bootstrap (sample with replacement from sample with replacement of the vector of residuals). You can also specify `boot.method="simple"` to use the usual bootstrap one-stage sampling with replacement.
`trantab` | Set to `TRUE` to add an attribute `trantab` to the returned matrix. This contains a vector of lists each with components `x` and `y` containing the unique values and corresponding transformed values for the columns of `x`. This is set up to be used easily with the `approx` function. You must specify `trantab=TRUE` if you want to later use the `predict.transcan` function with `type = "original"`.
`transformed` | set to `TRUE` to cause `transcan` to return an object `transformed` containing the matrix of transformed variables
`impcat` | This argument tells how to impute categorical variables on the original scale. The default is `impcat="score"` to impute the category whose canonical variate score is closest to the predicted score. Use `impcat="rpart"` to impute categorical variables using the values of all other transformed predictors in conjunction with the `rpart` function. A better but somewhat slower approach is to use `impcat="multinom"` to fit a multinomial logistic model to the categorical variable, at the last iteraction of the `transcan` algorithm. This uses the `multinom` function in the nnet library of the MASS package (which is assumed to have been installed by the user) to fit a polytomous logistic model to the current working transformations of all the other variables (using conditional mean imputation for missing predictors). Multiple imputations are made by drawing multinomial values from the vector of predicted probabilities of category membership for the missing categorical values.
`mincut` | If `imputed=TRUE`, there are categorical variables, and `impcat = "rpart"`, `mincut` specifies the lowest node size that will be allowed to be split. The default is 40.
`inverse` | By default, imputed values are back-solved on the original scale using inverse linear interpolation on the fitted tabulated transformed values. This will cause distorted distributions of imputed values (e.g., floor and ceiling effects) when the estimated transformation has a flat or nearly flat section. To instead use the `invertTabulated` function (see above) with the `"sample"` option, specify `inverse="sample"`.
`tolInverse` | the multiplyer of the range of transformed values, weighted by `freq` and by the distance measure, for determining the set of x values having y values within a tolerance of the value of `aty` in `invertTabulated`. For `predict.transcan`, `inverse` and `tolInverse` are obtained from options that were specified to `transcan` by default. Otherwise, if not specified by the user, these default to the defaults used to `invertTabulated`.
`pr` | For `transcan`, set to `FALSE` to suppress printing R^2 and shrinkage factors. Set `impute.transcan=FALSE` to suppress messages concerning the number of `NA` values imputed. Set `fit.mult.impute=FALSE` to suppress printing variance inflation factors accounting for imputation, rate of missing information, and degrees of freedom.
`pl` | Set to `FALSE` to suppress plotting the final transformations with distribution of scores for imputed values (if `show.na=TRUE`).
`allpl` | Set to `TRUE` to plot transformations for intermediate iterations.
`show.na` | Set to `FALSE` to suppress the distribution of scores assigned to missing values (as tick marks on the right margin of each graph). See also `imputed`.
`imputed.actual` | The default is "none" to suppress plotting of actual vs. imputed values for all variables having any `NA` values. Other choices are "datadensity" to use `datadensity` to make a single plot, "hist" to make a series of back-to-back histograms, "qq" to make a series of q-q plots, or "ecdf" to make a series of empirical cdfs. For `imputed.actual="datadensity"` for example you get a rug plot of the non-missing values for the variable with beneath it a rug plot of the imputed values. When `imputed.actual` is not "none", `imputed` is automatically set to `TRUE`.
`iter.max` | maximum number of iterations to perform for `transcan` or `predict`. For `predict`, only one iteration is used if there are no `NA` values in the data or if `imp.con` was used.
`eps` | convergence criterion for `transcan` and `predict`. `eps` is the maximum change in transformed values from one iteration to the next. If for a given iteration all new transformations of variables differ by less than `eps` (with or without negating the transformation to allow for “flipping”) from the transformations in the previous iteration, one more iteration is done for `transcan`. During this last iteration, individual transformations are not updated but coefficients of transformations are. This improves stability of coefficients of canonical variates on the right-hand-side. `eps` is ignored when `rhsImp="random"`.
`curtail` | for `transcan`, causes imputed values on the transformed scale to be truncated so that their ranges are within the ranges of non-imputed transformed values. For `predict`, `curtail` defaults to `TRUE` to truncate predicted transformed values to their ranges in the original fit (`xt`).
`imp.con` | for `transcan`, set to `TRUE` to impute `NA` values on the original scales with constants (medians or most frequent category codes). Set to a vector of constants to instead always use these constants for imputation. These imputed values are ignored when fitting the current working transformation for asingle variable.
`shrink` | default is `FALSE` to use ordinary least squares or canonical variate estimates. For the purposes of imputing `NA`s, you may want to set `shrink=TRUE` to avoid overfitting when developing a prediction equation to predict each variables from all the others (see details below).
`init.cat` | method for initializing scorings of categorical variables. Default is "mode" to use a dummy variable set to 1 if the value is the most frequent value (this is the default). Use "random" to use a random 0-1 variable. Set to "asis" to use the original integer codes asstarting scores.
`nres` | number of residuals to store if `n.impute` is specified. If the dataset has fewer than `nres` observations, all residuals are saved. Otherwise a random sample of the residuals of length `nres` without replacement is saved. The default for `nres` is higher if `boot.method="approximate bayesian"`.
`data` | Data frame used to fill the formula. For `ggplot` is the result of `transcan` with `trantab=TRUE`.
`subset` | an integer or logical vector specifying the subset of observations to fit
`na.action` | These may be used if `x` is a formula. The default `na.action` is `na.retain` (defined by `transcan`) which keeps all observations with any `NA` values. For `impute.transcan`, `data` is a data frame to use as the source of variables to be imputed, rather than using `pos.in`. For `fit.mult.impute`, `data` is mandatory and is a data frame containing the data to be used in fitting the model but before imputations are applied. Variables omitted from `data` are assumed to be available from frame1 and do not need to be imputed.
`treeinfo` | Set to `TRUE` to get additional information printed when `impcat="rpart"`, such as the predicted probabilities of category membership.
`rhsImp` | Set to "random" to use random draw imputation when a sometimes missing variable is moved to be a predictor of other sometimes missing variables. Default is `rhsImp="mean"`, which uses conditional mean imputation on the transformed scale. Residuals used are residuals from the transformed scale. When "random" is used, `transcan` runs 5 iterations and ignores `eps`.
`details.impcat` | set to a character scalar that is the name of a category variable to include in the resulting `transcan` object an element `details.impcat` containing details of how the categorical variable was multiply imputed.
`...` | arguments passed to `scat1d` or to the `fitter` function (for `fit.mult.impute`). For `ggplot.transcan`, these arguments are passed to `facet_wrap`, e.g. `ncol=2`.
`long` | for `summary`, set to `TRUE` to print all imputed values. For `print`, set to `TRUE` to print details of transformations/imputations.
`digits` | number of significant digits for printing values by `summary`
`scale` | for `ggplot.transcan` set `scale=TRUE` to scale transformed values to [0,1] before plotting.
`mapping,environment` | not used; needed because of rules about generics
`var` | For `impute`, is a variable that was originally a column in `x`, for which imputated values are to be filled in. `imputed=TRUE` must have been used in `transcan`. Omit `var` to impute all variables, creating new variables in position `pos` (see `assign`).
`imputation` | specifies which of the multiple imputations to use for filling in `NA` values
`name` | name of variable to impute, for `impute` function. Default is character string version of the second argument (`var`) in the call to `impute`. For `invertTabulated`, is the name of variable being transformed (used only for warning messages).
`pos.in` | location as defined by `assign` to find variables that need to be imputed, when all variables are to be imputed automatically by `impute.transcan` (i.e., when no input variable name is specified). Default is position that contains the first variable to be imputed.
`list.out` | If `var` is not specified, you can set `list.out=TRUE` to have `impute.transcan` return a list containing variables with needed values imputed. This list will contain a single imputation. Variables not needing imputation are copied to the list as-is. You can use this list for analysis just like a data frame.
`check` | set to `FALSE` to suppress certain warning messages
`newdata` | a new data matrix for which to compute transformed variables. Categorical variables must use the same integer codes as were used in the call to `transcan`. If a formula was originally specified to `transcan` (instead of a data matrix), `newdata` is optional and if given must be a data frame; a model frame is generated automatically from the previous formula. The `na.action` is handled automatically, and the levels for factor variables must be the same and in the same order as were used in the original variables specified in the formula given to `transcan`.
`fit.reps` | set to `TRUE` to save all fit objects from the fit for each imputation in `fit.mult.impute`. Then the object returned will have a component `fits` which is a list whose ith element is the ith fit object.
`dtrans` | provides an approach to creating derived variables from a single filled-in dataset. The function specified as `dtrans` can even reshape the imputed dataset. An example of such usage is fitting time-dependent covariates in a Cox model that are created by “start,stop” intervals. Imputations may be done on a one record per subject data frame that is converted by `dtrans` to multiple records per subject. The imputation can enforce consistency of certain variables across records so that for example a missing value of sex will not be imputed as male for one of the subject's records and female as another. An example of how `dtrans` might be specified is `dtrans=function(w) {w$age <- w$years + w$months/12; w}` where `months` might havebeen imputed but `years` was never missing. An outline for using 'dtrans' to impute missing baseline variables in a longitudinal analysis appears in Details below.
`derived` | an expression containing R expressions for computing derived variables that are used in the model formula. This is useful when multiple imputations are done for component variables but the actual model uses combinations of these (e.g., ratios or other derivations). For a single derived variable you can specified for example `derived=expression(ratio <- weight/height)`. For multiple derived variables use the form `derived=expression({ratio <- weight/height; product <- weight*height})` or put the expression on separate input lines. To monitor the multiply-imputed derived variables you can add to the `expression` a command such as `print(describe(ratio))`. See the example below. Note that `derived` is not yet implemented.
`vcovOpts` | a list of named additional arguments to pass to the `vcov` method for `fitter`. Useful for `orm` models for retaining all intercepts (`vcovOpts=list(intercepts='all')`) instead of just the middle one.
`type` | By default, the matrix of transformed variables is returned, with imputed values on the transformed scale. If you had specified `trantab=TRUE` to `transcan`, specifying `type="original"` does the table look-ups with linear interpolation to return the input matrix `x` but with imputed values on the original scale inserted for `NA` values. For categorical variables, the method used here is to select the category code having a corresponding scaled value closest to the predicted transformed value. This corresponds to the default `impcat`. Note: imputed values thus returned when `type="original"` are single expected value imputations even in `n.impute` is given.
`object` | an object created by `transcan`, or an object to be converted to R function code, typically a model fit object of some sort
`prefix, suffix` | When creating separate R functions for each variable in `x`, the name of the new function will be `prefix` placed in front of the variable name, and `suffix` placed in back of the name. The default is to use names of the form .varname, where varname is the variable name.
`pos` | position as in `assign` at which to store new functions (for `Function`). Default is `pos=-1`.
`y` | a vector corresponding to `x` for `invertTabulated`, if its first argument `x` is not a list
`freq` | a vector of frequencies corresponding to cross-classified `x` and `y` if `x` is not a list. Default is a vector of ones.
`aty` | vector of transformed values at which inverses are desired
`rule` | see `approx`. `transcan` assumes `rule` is always 2.
`regcoef.only` | set to `TRUE` to make `vcov.default` delete positions in the covariance matrix for any non-regression coefficients (e.g., log scale parameter from `psm` or `survreg`)
`intercepts` | this is primarily for `orm` objects. Set to `"none"` to discard all intercepts from the covariance matrix, or to `"all"` or `"mid"` to keep all elements generated by `orm` (`orm` only outputs the covariance matrix for the intercept corresponding to the median). You can also set `intercepts` to a vector of subscripts for selecting particular intercepts in a multi-intercept model.

### Details

The starting approximation to the transformation for each variable is taken to be the original coding of the variable. The initial approximation for each missing value is taken to be the median of the non-missing values for the variable (for continuous ones) or the most frequent category (for categorical ones). Instead, if `imp.con` is a vector, its values are used for imputing `NA` values. When using each variable as a dependent variable, `NA` values on that variable cause all observations to be temporarily deleted. Once a new working transformation is found for the variable, along with a model to predict that transformation from all the other variables, that latter model is used to impute `NA` values in the selected dependent variable if `imp.con` is not specified.

When that variable is used to predict a new dependent variable, the current working imputed values are inserted. Transformations are updated after each variable becomes a dependent variable, so the order of variables on `x` could conceivably make a difference in the final estimates. For obtaining out-of-sample predictions/transformations, `predict` uses the same iterative procedure as `transcan` for imputation, with the same starting values for fill-ins as were used by `transcan`. It also (by default) uses a conservative approach of curtailing transformed variables to be within the range of the original ones. Even when `method = "pc"` is specified, canonical variables are used for imputing missing values.

Note that fitted transformations, when evaluated at imputed variable values (on the original scale), will not precisely match the transformed imputed values returned in `xt`. This is because `transcan` uses an approximate method based on linear interpolation to back-solve for imputed values on the original scale.

Shrinkage uses the method of Van Houwelingen and Le Cessie (1990) (similar to Copas, 1983). The shrinkage factor is

[1 - (1 - \var{R2})(\var{n} - 1)/(\var{n} - \var{k} - 1)]/\var{R2}

where R2 is the apparent R-squared for predicting the variable, n is the number of non-missing values, and k is the effective number of degrees of freedom (aside from intercepts). A heuristic estimate is used for k: `A - 1 + sum(max(0,Bi - 1))/m + m`, where A is the number of d.f. required to represent the variable being predicted, the Bi are the number of columns required to represent all the other variables, and m is the number of all other variables. Division by m is done because the transformations for the other variables are fixed at their current transformations the last time they were being predicted. The + \var{m} term comes from the number of coefficients estimated on the right hand side, whether by least squares or canonical variates. If a shrinkage factor is negative, it is set to 0. The shrinkage factor is the ratio of the adjusted R-squared to the ordinary R-squared. The adjusted R-squared is

1 - (1 - R2)(n - 1)/(n - k - 1)

which is also set to zero if it is negative. If `shrink=FALSE` and the adjusted R-squares are much smaller than the ordinary R-squares, you may want to run `transcan` with `shrink=TRUE`.

Canonical variates are scaled to have variance of 1.0, by multiplying canonical coefficients from `cancor` by sqrt(\var{n} - 1).

When specifying a non-rms library fitting function to `fit.mult.impute` (e.g., `lm`, `glm`), running the result of `fit.mult.impute` through that fit's `summary` method will not use the imputation-adjusted variances. You may obtain the new variances using `fit$var` or `vcov(fit)`.

When you specify a rms function to `fit.mult.impute` (e.g. `lrm`, `ols`, `cph`, `psm`, `bj`, `Rq`, `Gls`, `Glm`), automatically computed transformation parameters (e.g., knot locations for `rcs`) that are estimated for the first imputation are used for all other imputations. This ensures that knot locations will not vary, which would change the meaning of the regression coefficients.

Warning: even though `fit.mult.impute` takes imputation into account when estimating variances of regression coefficient, it does not take into account the variation that results from estimation of the shapes and regression coefficients of the customized imputation equations. Specifying `shrink=TRUE` solves a small part of this problem. To fully account for all sources of variation you should consider putting the `transcan` invocation inside a bootstrap or loop, if execution time allows. Better still, use `aregImpute` or a package such as as mice that uses real Bayesian posterior realizations to multiply impute missing values correctly.

It is strongly recommended that you use the Hmisc `naclus` function to determine is there is a good basis for imputation. `naclus` will tell you, for example, if systolic blood pressure is missing whenever diastolic blood pressure is missing. If the only variable that is well correlated with diastolic bp is systolic bp, there is no basis for imputing diastolic bp in this case.

At present, `predict` does not work with multiple imputation.

When calling `fit.mult.impute` with `glm` as the `fitter` argument, if you need to pass a `family` argument to `glm` do it by quoting the family, e.g., `family="binomial"`.

`fit.mult.impute` will not work with proportional odds models when regression imputation was used (as opposed to predictive mean matching). That's because regression imputation will create values of the response variable that did not exist in the dataset, altering the intercept terms in the model.

You should be able to use a variable in the formula given to `fit.mult.impute` as a numeric variable in the regression model even though it was a factor variable in the invocation of `transcan`. Use for example `fit.mult.impute(y ~ codes(x), lrm, trans)` (thanks to Trevor Thompson trevor@hp5.eushc.org).

Here is an outline of the steps necessary to impute baseline variables using the `dtrans` argument, when the analysis to be repeated by `fit.mult.impute` is a longitudinal analysis (using e.g. `Gls`).

1. Create a one row per subject data frame containing baseline variables plus follow-up variables that are assigned to windows. For example, you may have dozens of repeated measurements over years but you capture the measurements at the times measured closest to 1, 2, and 3 years after study entry

2. Make sure the dataset contains the subject ID

3. This dataset becomes the one passed to `aregImpute` as `data=`. You will be imputing missing baseline variables from follow-up measurements defined at fixed times.

4. Have another dataset with all the non-missing follow-up values on it, one record per measurement time per subject. This dataset should not have the baseline variables on it, and the follow-up measurements should not be named the same as the baseline variable(s); the subject ID must also appear

5. Add the dtrans argument to `fit.mult.impute` to define a function with one argument representing the one record per subject dataset with missing values filled it from the current imputation. This function merges the above 2 datasets; the returned value of this function is the merged data frame.

6. This merged-on-the-fly dataset is the one handed by `fit.mult.impute` to your fitting function, so variable names in the formula given to `fit.mult.impute` must matched the names created by the merge

### Value

For `transcan`, a list of class transcan with elements

`call` | (with the function call)
`iter` | (number of iterations done)
`rsq, rsq.adj` | containing the R-squares and adjusted R-squares achieved in predicting each variable from all the others
`categorical` | the values supplied for `categorical`
`asis` | the values supplied for `asis`
`coef` | the within-variable coefficients used to compute the first canonical variate
`xcoef` | the (possibly shrunk) across-variables coefficients of the first canonical variate that predicts each variable in-turn.
`parms` | the parameters of the transformation (knots for splines, contrast matrix for categorical variables)
`fillin` | the initial estimates for missing values (`NA` if variable never missing)
`ranges` | the matrix of ranges of the transformed variables (min and max in first and secondrow)
`scale` | a vector of scales used to determine convergence for a transformation.
`formula` | the formula (if `x` was a formula)

, and optionally a vector of shrinkage factors used for predicting each variable from the others. For `asis` variables, the scale is the average absolute difference about the median. For other variables it is unity, since canonical variables are standardized. For `xcoef`, row i has the coefficients to predict transformed variable i, with the column for the coefficient of variable i set to `NA`. If `imputed=TRUE` was given, an optional element `imputed` also appears. This is a list with the vector of imputed values (on the original scale) for each variable containing `NA`s. Matrices rather than vectors are returned if `n.impute` is given. If `trantab=TRUE`, the `trantab` element also appears, as described above. If `n.impute > 0`, `transcan` also returns a list `residuals` that can be used for future multiple imputation.

`impute` returns a vector (the same length as `var`) of class impute with `NA` values imputed.

`predict` returns a matrix with the same number of columns or variables as were in `x`.

`fit.mult.impute` returns a fit object that is a modification of the fit object created by fitting the completed dataset for the final imputation. The `var` matrix in the fit object has the imputation-corrected variance-covariance matrix. `coefficients` is the average (over imputations) of the coefficient vectors, `variance.inflation.impute` is a vector containing the ratios of the diagonals of the between-imputation variance matrix to the diagonals of the average apparent (within-imputation) variance matrix. `missingInfo` is Rubin's rate of missing information and `dfmi` is Rubin's degrees of freedom for a t-statistic for testing a single parameter. The last two objects are vectors corresponding to the diagonal of the variance matrix. The class `"fit.mult.impute"` is prepended to the other classes produced by the fitting function.

`fit.mult.impute` stores `intercepts` attributes in the coefficient matrix and in `var` for `orm` fits.

### Side Effects

prints, plots, and `impute.transcan` creates new variables.

### Author(s)

Frank Harrell
Department of Biostatistics
Vanderbilt University
f.harrell@vanderbilt.edu

### References

Kuhfeld, Warren F: The PRINQUAL Procedure. SAS/STAT User's Guide, Fourth Edition, Volume 2, pp. 1265–1323, 1990.

Van Houwelingen JC, Le Cessie S: Predictive value of statistical models. Statistics in Medicine 8:1303–1325, 1990.

Copas JB: Regression, prediction and shrinkage. JRSS B 45:311–354, 1983.

He X, Shen L: Linear regression after spline transformation. Biometrika 84:474–481, 1997.

Little RJA, Rubin DB: Statistical Analysis with Missing Data. New York: Wiley, 1987.

Rubin DJ, Schenker N: Multiple imputation in health-care databases: An overview and some applications. Stat in Med 10:585–598, 1991.

Faris PD, Ghali WA, et al:Multiple imputation versus data enhancement for dealing with missing data in observational health care outcome analyses. J Clin Epidem 55:184–191, 2002.

### See Also

`aregImpute`, `impute`, `naclus`, `naplot`, `ace`, `avas`, `cancor`, `prcomp`, `rcspline.eval`, `lsfit`, `approx`, `datadensity`, `mice`, `ggplot`

### Examples

    ## Not run:
    x <- cbind(age, disease, blood.pressure, pH)
    #cbind will convert factor object `disease' to integer
    par(mfrow=c(2,2))
    x.trans <- transcan(x, categorical="disease", asis="pH",
                        transformed=TRUE, imputed=TRUE)
    summary(x.trans)  #Summary distribution of imputed values, and R-squares
    f <- lm(y ~ x.trans$transformed)   #use transformed values in a regression
    #Now replace NAs in original variables with imputed values, if not
    #using transformations
    age            <- impute(x.trans, age)
    disease        <- impute(x.trans, disease)
    blood.pressure <- impute(x.trans, blood.pressure)
    pH             <- impute(x.trans, pH)
    #Do impute(x.trans) to impute all variables, storing new variables under
    #the old names
    summary(pH)       #uses summary.impute to tell about imputations
                      #and summary.default to tell about pH overall
    # Get transformed and imputed values on some new data frame xnew
    newx.trans     <- predict(x.trans, xnew)
    w              <- predict(x.trans, xnew, type="original")
    age            <- w[,"age"]            #inserts imputed values
    blood.pressure <- w[,"blood.pressure"]
    Function(x.trans)  #creates .age, .disease, .blood.pressure, .pH()
    #Repeat first fit using a formula
    x.trans <- transcan(~ age + disease + blood.pressure + I(pH),
                        imputed=TRUE)
    age <- impute(x.trans, age)
    predict(x.trans, expand.grid(age=50, disease="pneumonia",
            blood.pressure=60:260, pH=7.4))
    z <- transcan(~ age + factor(disease.code),  # disease.code categorical
                  transformed=TRUE, trantab=TRUE, imputed=TRUE, pl=FALSE)
    ggplot(z, scale=TRUE)
    plot(z$transformed)

    ## End(Not run)

    # Multiple imputation and estimation of variances and covariances of
    # regression coefficient estimates accounting for imputation
    set.seed(1)
    x1 <- factor(sample(c('a','b','c'),100,TRUE))
    x2 <- (x1=='b') + 3*(x1=='c') + rnorm(100)
    y  <- x2 + 1*(x1=='c') + rnorm(100)
    x1[1:20] <- NA
    x2[18:23] <- NA
    d <- data.frame(x1,x2,y)
    n <- naclus(d)
    plot(n); naplot(n)  # Show patterns of NAs
    f  <- transcan(~y + x1 + x2, n.impute=10, shrink=FALSE, data=d)
    options(digits=3)
    summary(f)

    f  <- transcan(~y + x1 + x2, n.impute=10, shrink=TRUE, data=d)
    summary(f)

    h <- fit.mult.impute(y ~ x1 + x2, lm, f, data=d)
    # Add ,fit.reps=TRUE to save all fit objects in h, then do something like:
    # for(i in 1:length(h$fits)) print(summary(h$fits[[i]]))

    diag(vcov(h))

    h.complete <- lm(y ~ x1 + x2, na.action=na.omit)
    h.complete
    diag(vcov(h.complete))

    # Note: had the rms ols function been used in place of lm, any
    # function run on h (anova, summary, etc.) would have automatically
    # used imputation-corrected variances and covariances

    # Example demonstrating how using the multinomial logistic model
    # to impute a categorical variable results in a frequency
    # distribution of imputed values that matches the distribution
    # of non-missing values of the categorical variable

    ## Not run:
    set.seed(11)
    x1 <- factor(sample(letters[1:4], 1000,TRUE))
    x1[1:200] <- NA
    table(x1)/sum(table(x1))
    x2 <- runif(1000)
    z  <- transcan(~ x1 + I(x2), n.impute=20, impcat='multinom')
    table(z$imputed$x1)/sum(table(z$imputed$x1))

    # Here is how to create a completed dataset
    d <- data.frame(x1, x2)
    z <- transcan(~x1 + I(x2), n.impute=5, data=d)
    imputed <- impute(z, imputation=1, data=d,
                      list.out=TRUE, pr=FALSE, check=FALSE)
    sapply(imputed, function(x)sum(is.imputed(x)))
    sapply(imputed, function(x)sum(is.na(x)))

    ## End(Not run)

    # Example where multiple imputations are for basic variables and
    # modeling is done on variables derived from these

    set.seed(137)
    n <- 400
    x1 <- runif(n)
    x2 <- runif(n)
    y  <- x1*x2 + x1/(1+x2) + rnorm(n)/3
    x1[1:5] <- NA
    d <- data.frame(x1,x2,y)
    w <- transcan(~ x1 + x2 + y, n.impute=5, data=d)
    # Add ,show.imputed.actual for graphical diagnostics
    ## Not run:
    g <- fit.mult.impute(y ~ product + ratio, ols, w,
                         data=data.frame(x1,x2,y),
                         derived=expression({
                           product <- x1*x2
                           ratio   <- x1/(1+x2)
                           print(cbind(x1,x2,x1*x2,product)[1:6,])}))

    ## End(Not run)

    # Here's a method for creating a permanent data frame containing
    # one set of imputed values for each variable specified to transcan
    # that had at least one NA, and also containing all the variables
    # in an original data frame.  The following is based on the fact
    # that the default output location for impute.transcan is
    # given by the global environment

    ## Not run:
    xt <- transcan(~. , data=mine,
                   imputed=TRUE, shrink=TRUE, n.impute=10, trantab=TRUE)
    attach(mine, use.names=FALSE)
    impute(xt, imputation=1) # use first imputation
    # omit imputation= if using single imputation
    detach(1, 'mine2')

    ## End(Not run)

    # Example of using invertTabulated outside transcan
    x    <- c(1,2,3,4,5,6,7,8,9,10)
    y    <- c(1,2,3,4,5,5,5,5,9,10)
    freq <- c(1,1,1,1,1,2,3,4,1,1)
    # x=5,6,7,8 with prob. .1 .2 .3 .4 when y=5
    # Within a tolerance of .05*(10-1) all y's match exactly
    # so the distance measure does not play a role
    set.seed(1)      # so can reproduce
    for(inverse in c('linearInterp','sample'))
     print(table(invertTabulated(x, y, freq, rep(5,1000), inverse=inverse)))

    # Test inverse='sample' when the estimated transformation is
    # flat on the right.  First show default imputations
    set.seed(3)
    x <- rnorm(1000)
    y <- pmin(x, 0)
    x[1:500] <- NA
    for(inverse in c('linearInterp','sample')) {
    par(mfrow=c(2,2))
      w <- transcan(~ x + y, imputed.actual='hist',
                    inverse=inverse, curtail=FALSE,
                    data=data.frame(x,y))
      if(inverse=='sample') next
    # cat('Click mouse on graph to proceed\n')
    # locator(1)
    }

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