# Workflow to Develop and Validate a Prediction model in Multiply Imputed data

#### Martijn W Heymans

#### 2021-09-23

# Introduction

This vignette shows how to develop, internally and externally validate a (logistic) regression prediction model with `mice` and `psfmi`.

# Steps

- Step 1 - Install the psfmi and mice package
- Step 2 - Impute the missing data with mice
- Step 3 - Develop the model
- Step 4 - Determine model performance
- Step 5 - Internally validate the model
- Step 6 - Shrink pooled coefficients and determine new intercept
- Step 7 - Externally validate the model

## Step 1 - Install the psfmi and mice package

You can install the released version of psfmi with:

    install.packages("psfmi")

And the development version from GitHub with:

    # install.packages("devtools")
    devtools::install_github("mwheymans/psfmi")

You can install the released version of mice with:

    install.packages("mice")

Back to Steps

## Step 2 - Impute the missing data with mice

I generated 5 imputed datasets to handle the missing values in the lbp_orig dataset that is included in the `psfmi` package. Depending on the amount of missing data, the number of imputed datasets may be increased (see on-line book: Applied Missing Data analysis).

      imp <- mice(lbp_orig, m=5, maxit=10, seed = 750)

    ##
    ##  iter imp variable
    ##   1   1  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   1   2  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   1   3  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   1   4  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   1   5  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   2   1  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   2   2  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   2   3  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   2   4  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   2   5  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   3   1  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   3   2  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   3   3  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   3   4  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   3   5  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   4   1  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   4   2  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   4   3  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   4   4  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   4   5  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   5   1  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   5   2  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   5   3  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   5   4  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   5   5  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   6   1  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   6   2  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   6   3  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   6   4  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   6   5  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   7   1  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   7   2  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   7   3  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   7   4  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   7   5  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   8   1  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   8   2  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   8   3  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   8   4  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   8   5  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   9   1  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   9   2  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   9   3  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   9   4  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   9   5  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   10   1  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   10   2  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   10   3  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   10   4  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport
    ##   10   5  Carrying  Pain  Tampascale  Function  Radiation  Age  Satisfaction  JobControl  JobDemands  SocialSupport

Use the `complete` function of the `mice` package to extract the completed datasets. By setting `action = "long"` and `include = FALSE`, the imputed datasets are stacked under each other to form one long dataset and the original dataset (that included missing values) is not included in that long dataset.

      data_comp <- complete(imp, action = "long", include = FALSE)

Back to Steps

## Step 3 - Develop the model

Use the `psfmi_lr` function in the `psfmi` package to perform backward selection over the 5 multiply imputed datasets with a p-value of 0.05 and as selection method D1.
Note that with this function it is possible to include cubic spline coefficients in case of non-linear relationships during backward selection and that predictors may be forced in the model during backward selection by including them at the setting `keep.predictors`. When the setting `direction` is changed to FW, forward selection is done.

      pool_lr <- psfmi_lr(data=data_comp, nimp=5, impvar=".imp",
                  formula=Chronic ~ Gender + Age + JobControl + Tampascale +
                  Pain + Radiation + JobDemands + SocialSupport + Smoking +
                  factor(Satisfaction) + factor(Carrying) + rcs(Function, 3),
                  keep.predictors = "Radiation",
                  p.crit = 0.157, method="D1", direction = "BW")

    ## Removed at Step 1 is - Gender

    ## Removed at Step 2 is - JobDemands

    ## Removed at Step 3 is - Smoking

    ## Removed at Step 4 is - Age

    ## Removed at Step 5 is - Tampascale

    ## Removed at Step 6 is - rcs(Function,3)

    ## Removed at Step 7 is - JobControl

    ## Removed at Step 8 is - SocialSupport

    ##
    ## Selection correctly terminated,
    ## No more variables removed from the model

      pool_lr$RR_model_final

    ## $`Step 9`
    ##                    term   estimate std.error  statistic        df      p.value
    ## 1           (Intercept) -5.6290593 1.1754061 -4.7890336  39.95530 2.318419e-05
    ## 2                  Pain  0.9354091 0.1851492  5.0521901  67.68397 3.532584e-06
    ## 3             Radiation  0.5186631 0.5151459  1.0068276  91.48901 3.166731e-01
    ## 4 factor(Satisfaction)2 -0.4282950 0.6531511 -0.6557364  44.49919 5.153699e-01
    ## 5 factor(Satisfaction)3 -2.6431297 0.9429219 -2.8031270  26.83928 9.283867e-03
    ## 6     factor(Carrying)2  1.2554087 0.6254666  2.0071556  78.40085 4.817958e-02
    ## 7     factor(Carrying)3  1.6871723 0.6473849  2.6061349 115.47451 1.036350e-02
    ##            OR    lower.EXP  upper.EXP
    ## 1 0.003591953 0.0003338808  0.0386429
    ## 2 2.548255652 1.7610642412  3.6873197
    ## 3 1.679780430 0.6037849102  4.6732905
    ## 4 0.651619185 0.1747816872  2.4293596
    ## 5 0.071138280 0.0102714430  0.4926917
    ## 6 3.509272384 1.0103539396 12.1887907
    ## 7 5.404177760 1.4991278521 19.4814186

      pool_lr$multiparm_final

    ## $`Step 9`
    ##                       p-values D1 F-statistic
    ## Pain                 1.275830e-06   25.524625
    ## Radiation            3.148421e-01    1.013702
    ## factor(Satisfaction) 2.426042e-02    4.267156
    ## factor(Carrying)     3.261509e-02    3.483376

Back to Steps

## Step 4 - Determine model performance

The performance of the model selected at step 3 can be determined by using the `pool_performance` function.

      perf <- pool_performance(data=data_comp, nimp=5, impvar=".imp",
        formula = pool_lr$formula_final[[1]],
        cal.plot=TRUE, plot.method="mean", groups_cal = 10)

      perf

    ## $ROC_pooled
    ##                     95% Low C-statistic 95% Up
    ## C-statistic (logit)  0.8144      0.8978 0.9462
    ##
    ## $coef_pooled
    ##           (Intercept)                  Pain             Radiation
    ##            -5.6290593             0.9354091             0.5186631
    ## factor(Satisfaction)2 factor(Satisfaction)3     factor(Carrying)2
    ##            -0.4282950            -2.6431297             1.2554087
    ##     factor(Carrying)3
    ##             1.6871723
    ##
    ## $R2_pooled
    ## [1] 0.5842591
    ##
    ## $Brier_Scaled_pooled
    ## [1] 0.4882759
    ##
    ## $nimp
    ## [1] 5
    ##
    ## $HLtest_pooled
    ##         D2          p        df1        df2
    ##  0.7163405  0.6758435  8.0000000 42.8715173
    ##
    ## $model_type
    ## [1] "binomial"

## Step 5 - Internally validate the model

To internally validate the model we use the `psfmi_` function. With this function five different methods can be used to internally validate models in MI data see these Vignettes, Three methods use cross-validation and two bootstrapping in combination with MI.

We will use cross-validation and more specific the method cv_MI_RR. With this method it is possible to integrate backward selection into the cross-validation procedure. The last model that is selected by the psfmi_lr function is internally validated. So, if we want to apply backward selection during cross-validation from the full model we first have to apply the psfmi_lr function without variable selection. That is what we apply here, because we know that variable selection is one of the main reasons that prediction models are over-fitted.

      pool_val <- psfmi_lr(data=data_comp, formula = Chronic ~ Gender + Age + JobControl + Tampascale +
                          Pain + Radiation + JobDemands + SocialSupport + Smoking + factor(Satisfaction) +
                          factor(Carrying) + rcs(Function, 3), p.crit = 1, direction="BW",
                          nimp=5, impvar=".imp", method="D1")

      set.seed(200)
      res_cv <- psfmi_validate(pool_val, val_method = "cv_MI_RR", data_orig = lbp_orig, folds = 5,
                         p.crit=0.05, BW=TRUE, nimp_mice = 10, miceImp = miceImp, printFlag = FALSE)

    ##
    ## fold 1

    ## Removed at Step 1 is - Age

    ## Removed at Step 2 is - Smoking

    ## Removed at Step 3 is - Gender

    ## Removed at Step 4 is - Tampascale

    ## Removed at Step 5 is - JobDemands

    ## Removed at Step 6 is - rcs(Function,3)

    ## Removed at Step 7 is - SocialSupport

    ## Removed at Step 8 is - JobControl

    ## Removed at Step 9 is - Radiation

    ## Removed at Step 10 is - factor(Carrying)

    ##
    ## Selection correctly terminated,
    ## No more variables removed from the model

    ##
    ## fold 2

    ## Removed at Step 1 is - rcs(Function,3)

    ## Removed at Step 2 is - Radiation

    ## Removed at Step 3 is - Gender

    ## Removed at Step 4 is - JobControl

    ## Removed at Step 5 is - Age

    ## Removed at Step 6 is - SocialSupport

    ## Removed at Step 7 is - Smoking

    ## Removed at Step 8 is - JobDemands

    ## Removed at Step 9 is - Tampascale

    ## Removed at Step 10 is - factor(Carrying)

    ##
    ## Selection correctly terminated,
    ## No more variables removed from the model

    ##
    ## fold 3

    ## Removed at Step 1 is - Gender

    ## Removed at Step 2 is - JobDemands

    ## Removed at Step 3 is - Tampascale

    ## Removed at Step 4 is - Smoking

    ## Removed at Step 5 is - JobControl

    ## Removed at Step 6 is - rcs(Function,3)

    ## Removed at Step 7 is - Age

    ## Removed at Step 8 is - SocialSupport

    ## Removed at Step 9 is - Radiation

    ##
    ## Selection correctly terminated,
    ## No more variables removed from the model

    ##
    ## fold 4

    ## Removed at Step 1 is - Gender

    ## Removed at Step 2 is - Smoking

    ## Removed at Step 3 is - JobDemands

    ## Removed at Step 4 is - JobControl

    ## Removed at Step 5 is - Radiation

    ## Removed at Step 6 is - factor(Carrying)

    ## Removed at Step 7 is - SocialSupport

    ## Removed at Step 8 is - Age

    ##
    ## Selection correctly terminated,
    ## No more variables removed from the model

    ##
    ## fold 5

    ## Removed at Step 1 is - JobDemands

    ## Removed at Step 2 is - Gender

    ## Removed at Step 3 is - Smoking

    ## Removed at Step 4 is - Tampascale

    ## Removed at Step 5 is - rcs(Function,3)

    ## Removed at Step 6 is - Age

    ## Removed at Step 7 is - Radiation

    ## Removed at Step 8 is - SocialSupport

    ## Removed at Step 9 is - JobControl

    ## Removed at Step 10 is - factor(Carrying)

    ##
    ## Selection correctly terminated,
    ## No more variables removed from the model

      res_cv

    ## $stats
    ##                  Train      Test
    ## AUC          0.8877807 0.8467882
    ## Brier scaled 0.4657961 0.3374208
    ## Rsq          0.5622000 0.5117315
    ##
    ## $slope
    ##  Intercept      Slope
    ## -0.1494438  0.8449148

Back to Steps

## Step 6 - Shrink pooled coefficients and determine new intercept

We can use the slope value of 0.8449148 that was estimated at the previous step as a shrinkage factor to prevent our model from being overfitted in new data. We do this by multiplying the pooled coefficients with the shrinkage factor and also to determine a new intercept value that is aligned with the shrunken coefficients.

We use the `pool_intadj` function for this that will provide the adjusted coefficients and new intercept value.

      pool_select <- psfmi_lr(data=data_comp, nimp=5, impvar=".imp",
      formula = pool_lr$formula_final[[1]],
      p.crit = 1, method="D1")

      res <- pool_intadj(pool_select, shrinkage_factor = 0.8449148)

      res$int_adj

    ## [1] -4.795145

      res$coef_shrink_pooled

    ##                  Pain             Radiation factor(Satisfaction)2
    ##             0.7903410             0.4382261            -0.3618728
    ## factor(Satisfaction)3     factor(Carrying)2     factor(Carrying)3
    ##            -2.2332194             1.0607134             1.4255169

The last step is to externally validate this adjusted model.

Back to Steps

## Step 7 - Externally validate the model

We validate the model in an external dataset that is imputed five times due to missing data.

      res_extval <- mivalext_lr(data.val = lbpmi_extval, nimp = 5, impvar = "Impnr",
        formula = pool_lr$formula_final[[1]], lp.orig = c(res$int_adj,
        res$coef_shrink_pooled), cal.plot = TRUE, plot.method = "mean")

    ##
    ## Pooled performance measures over m = 5 imputed external validation datasets
    ##             correctly estimated

      res_extval$ROC

    ##                     95% Low C-statistic 95% Up
    ## C-statistic (logit)  0.7382      0.8482 0.9172

      res_extval$R2

    ## [1] 0.45587

      res_extval$HLtest

    ##          D2           p         df1         df2
    ##   0.7147875   0.6781956   8.0000000 149.5561959

      res_extval$LP_pooled_ext

    ## intercept     slope
    ##  -0.01812   0.88741

In the external dataset the AUC is 0.8482 and the R-squared 0.45587. The Hosmer and Lemeshow test has a p-value of 0.67820, which means that the fit is satisfactory. This is also confirmed by the calibration plot where the calibration curves averaged across multiply imputed datasets is near the optimal (dashed) line. The slope (regression coefficients) has a value of 0.88741 and slightly deviates from the value of 1, which means that the coefficient values differ between the development and external dataset.

Back to Steps