psfmi_lr Pooling and backward or forward selection of Logistic regression models in multiply imputed data using selection methods RR, D1, D2, D3 and MPR.

  formula = NULL,
  nimp = 5,
  impvar = NULL,
  Outcome = NULL,
  predictors = NULL,
  cat.predictors = NULL,
  spline.predictors = NULL,
  int.predictors = NULL,
  keep.predictors = NULL,
  nknots = NULL,
  p.crit = 1,
  method = "RR",
  direction = NULL



Data frame with stacked multiple imputed datasets. The original dataset that contains missing values must be excluded from the dataset. The imputed datasets must be distinguished by an imputation variable, specified under impvar, and starting by 1.


A formula object to specify the model as normally used by glm. See under "Details" and "Examples" how these can be specified. If a formula object is used set predictors, cat.predictors, spline.predictors or int.predictors at the default value of NULL.


A numerical scalar. Number of imputed datasets. Default is 5.


A character vector. Name of the variable that distinguishes the imputed datasets.


Character vector containing the name of the outcome variable.


Character vector with the names of the predictor variables. At least one predictor variable has to be defined. Give predictors unique names and do not use predictor name combinations with numbers as, age2, gender10, etc.


A single string or a vector of strings to define the categorical variables. Default is NULL categorical predictors.


A single string or a vector of strings to define the (restricted cubic) spline variables. Default is NULL spline predictors. See details.


A single string or a vector of strings with the names of the variables that form an interaction pair, separated by a “:” symbol.


A single string or a vector of strings including the variables that are forced in the model during predictor selection. All type of variables are allowed.


A numerical vector that defines the number of knots for each spline predictor separately.


A numerical scalar. P-value selection criterium. A value of 1 provides the pooled model without selection.


A character vector to indicate the pooling method for p-values to pool the total model or used during predictor selection. This can be "RR", D1", "D2", "D3" or "MPR". See details for more information. Default is "RR".


The direction of predictor selection, "BW" means backward selection and "FW" means forward selection.


An object of class pmods (multiply imputed models) from which the following objects can be extracted:

  • data imputed datasets

  • RR_model pooled model at each selection step

  • RR_model_final final selected pooled model

  • multiparm pooled p-values at each step according to pooling method

  • multiparm_final pooled p-values at final step according to pooling method

  • multiparm_out (only when direction = "FW") pooled p-values of removed predictors

  • formula_step formula object at each step

  • formula_final formula object at final step

  • formula_initial formula object at final step

  • predictors_in predictors included at each selection step

  • predictors_out predictors excluded at each step

  • impvar name of variable used to distinguish imputed datasets

  • nimp number of imputed datasets

  • Outcome name of the outcome variable

  • method selection method

  • p.crit p-value selection criterium

  • call function call

  • model_type type of regression model used

  • direction direction of predictor selection

  • predictors_final names of predictors in final selection step

  • predictors_initial names of predictors in start model

  • keep.predictors names of predictors that were forced in the model


The basic pooling procedure to derive pooled coefficients, standard errors, 95 confidence intervals and p-values is Rubin's Rules (RR). However, RR is only possible when the model included continuous or dichotomous variables. Specific procedures are available when the model also included categorical (> 2 categories) or restricted cubic spline variables. These pooling methods are: “D1” is pooling of the total covariance matrix, ”D2” is pooling of Chi-square values, “D3” is pooling Likelihood ratio statistics (method of Meng and Rubin) and “MPR” is pooling of median p-values (MPR rule). Spline regression coefficients are defined by using the rcs function for restricted cubic splines of the rms package. A minimum number of 3 knots as defined under knots is required.

A typical formula object has the form Outcome ~ terms. Categorical variables has to be defined as Outcome ~ factor(variable), restricted cubic spline variables as Outcome ~ rcs(variable, 3). Interaction terms can be defined as Outcome ~ variable1*variable2 or Outcome ~ variable1 + variable2 + variable1:variable2. All variables in the terms part have to be separated by a "+". If a formula object is used set predictors, cat.predictors, spline.predictors or int.predictors at the default value of NULL.



Eekhout I, van de Wiel MA, Heymans MW. Methods for significance testing of categorical covariates in logistic regression models after multiple imputation: power and applicability analysis. BMC Med Res Methodol. 2017;17(1):129.

Enders CK (2010). Applied missing data analysis. New York: The Guilford Press.

Meng X-L, Rubin DB. Performing likelihood ratio tests with multiply-imputed data sets. Biometrika.1992;79:103-11.

van de Wiel MA, Berkhof J, van Wieringen WN. Testing the prediction error difference between 2 predictors. Biostatistics. 2009;10:550-60.

Marshall A, Altman DG, Holder RL, Royston P. Combining estimates of interest in prognostic modelling studies after multiple imputation: current practice and guidelines. BMC Med Res Methodol. 2009;9:57.

Van Buuren S. (2018). Flexible Imputation of Missing Data. 2nd Edition. Chapman & Hall/CRC Interdisciplinary Statistics. Boca Raton.

EW. Steyerberg (2019). Clinical Prediction MOdels. A Practical Approach to Development, Validation, and Updating (2nd edition). Springer Nature Switzerland AG.


Martijn Heymans, 2020


pool_lr <- psfmi_lr(data=lbpmilr, formula = Chronic ~ Pain + factor(Satisfaction) + rcs(Tampascale,3) + Radiation + Radiation*factor(Satisfaction) + Age + Duration + BMI, p.crit = 0.05, direction="FW", nimp=5, impvar="Impnr", keep.predictors = c("Radiation*factor(Satisfaction)", "Age"), method="D1")
#> Entered at Step 1 is - Pain
#> #> Selection correctly terminated, #> No new variables entered the model
#> $`Final model` #> term estimate std.error statistic #> 1 (Intercept) -4.709498778 1.43056951 -3.292044710 #> 2 Pain 1.000998071 0.17550142 5.703646588 #> 3 Radiation 1.290636824 1.09101880 1.182964787 #> 4 Age -0.015400069 0.02511643 -0.613147208 #> 5 factor(Satisfaction)2 0.006745828 0.72630877 0.009287825 #> 6 factor(Satisfaction)3 -2.245109289 1.28811308 -1.742944248 #> 7 Radiation:factor(Satisfaction)2 -1.112425071 1.29072002 -0.861863964 #> 8 Radiation:factor(Satisfaction)3 -0.328027439 1.69820886 -0.193160834 #> df p.value OR lower.EXP upper.EXP #> 1 121.39346 1.302896e-03 0.009009292 0.0005305503 0.1529871 #> 2 86.27941 1.612718e-07 2.720996221 1.9196231945 3.8569134 #> 3 30.54412 2.459495e-01 3.635100740 0.3922493500 33.6876464 #> 4 102.60849 5.411357e-01 0.984717906 0.9368661635 1.0350137 #> 5 94.66955 9.926090e-01 1.006768633 0.2380619605 4.2576440 #> 6 23.29136 9.453270e-02 0.105915964 0.0073877578 1.5184839 #> 7 40.99328 3.937762e-01 0.328760726 0.0242557883 4.4559927 #> 8 19.52844 8.488278e-01 0.720343254 0.0207352364 25.0247643 #>