# Pooling and Selection of Linear Regression Models

#### Martijn W Heymans

#### 2021-09-23

# Introduction

With the `psfmi_lm` function you can pool Linear regression models by using
the following pooling methods: RR (Rubin’s Rules), D1, D2 and MPR (Median R Rule).

You can also use forward or backward selection from the pooled model.

This vignette show you examples of how to apply these procedures.

# Examples

- [Linear Regression]
  - Pooling without BS and method D1
  - Pooling with BS and method D1
  - Pooling with BS and method MPR
  - Pooling with BS including interaction terms and method D2
  - Pooling with BS and forcing interaction terms and method D1
  - Pooling with BS including spline coefficient and method D1

## Pooling without BS and method D1

      library(psfmi)
      pool_lm <- psfmi_lm(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Pain ~ Gender + Smoking +
                          Function + JobControl + JobDemands + SocialSupport,
                          method="D1")

      pool_lm$RR_model
    #> $`Step 1 - no variables removed -`
    #>            term     estimate  std.error   statistic        df     p.value
    #> 1   (Intercept)  7.626750501 2.37470136  3.21166721 103.21605 0.001760151
    #> 2        Gender -0.549897436 0.41763180 -1.31670395  97.10997 0.191036859
    #> 3       Smoking -0.184822738 0.35459284 -0.52122524  60.23783 0.604120893
    #> 4      Function -0.126983721 0.04264394 -2.97776686  46.48759 0.004600709
    #> 5    JobControl -0.018201443 0.01884372 -0.96591573 117.54453 0.336069460
    #> 6    JobDemands  0.015351105 0.03590006  0.42760673 121.85071 0.669692207
    #> 7 SocialSupport -0.003435975 0.05621115 -0.06112622  96.21255 0.951385488

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## Pooling with BS and method D1

Pooling linear regression models over 5 imputed datasets with backward selection using a p-value of 0.05 and as method D1 and forcing the predictor “Smoking” in the models during backward selection.

      library(psfmi)
      pool_lm <- psfmi_lm(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Pain ~ Gender + Smoking +
                          Function + JobControl + JobDemands + SocialSupport,
                          keep.predictors = "Smoking", method="D1", p.crit=0.05,
                          direction="BW")
    #> Removed at Step 1 is - SocialSupport
    #> Removed at Step 2 is - JobDemands
    #> Removed at Step 3 is - JobControl
    #> Removed at Step 4 is - Gender
    #>
    #> Selection correctly terminated,
    #> No more variables removed from the model
      pool_lm$RR_model_final
    #> $`Step 5`
    #>          term   estimate  std.error  statistic       df     p.value
    #> 1 (Intercept)  6.7504947 0.47607990 14.1793314 78.48419 0.000000000
    #> 2     Smoking -0.1998222 0.35556369 -0.5619871 57.20990 0.576320084
    #> 3    Function -0.1403048 0.04077998 -3.4405314 51.97198 0.001153144
      pool_lm$multiparm_final
    #> $`Step 5`
    #>           p-values D1 F-statistic
    #> Smoking  0.5753099186   0.3158295
    #> Function 0.0008794238  11.8372561
      pool_lm$predictors_out
    #>         Gender Smoking Function JobControl JobDemands SocialSupport
    #> Step 1       0       0        0          0          0             1
    #> Step 2       0       0        0          0          1             0
    #> Step 3       0       0        0          1          0             0
    #> Step 4       1       0        0          0          0             0
    #> Removed      1       0        0          1          1             1

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## Pooling with BS and method MPR

Pooling linear regression models over 5 imputed datasets with backward selection using a p-value of 0.05 and as method D1 and forcing the predictor “Smoking” in the models during backward selection.

      library(psfmi)
      pool_lm <- psfmi_lm(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Pain ~ Gender + Smoking +
                          Function + JobControl + JobDemands + SocialSupport,
                          keep.predictors = "Smoking", method="MPR", p.crit=0.05,
                          direction="BW")
    #> Removed at Step 1 is - SocialSupport
    #> Removed at Step 2 is - JobDemands
    #> Removed at Step 3 is - JobControl
    #> Removed at Step 4 is - Gender
    #>
    #> Selection correctly terminated,
    #> No more variables removed from the model
      pool_lm$RR_model_final
    #> $`Step 5`
    #>          term   estimate  std.error  statistic       df     p.value
    #> 1 (Intercept)  6.7504947 0.47607990 14.1793314 78.48419 0.000000000
    #> 2     Smoking -0.1998222 0.35556369 -0.5619871 57.20990 0.576320084
    #> 3    Function -0.1403048 0.04077998 -3.4405314 51.97198 0.001153144
      pool_lm$multiparm_final
    #> $`Step 5`
    #>           p-value MPR
    #> Smoking  0.6019832504
    #> Function 0.0001268997
      pool_lm$predictors_out
    #>         Gender Smoking Function JobControl JobDemands SocialSupport
    #> Step 1       0       0        0          0          0             1
    #> Step 2       0       0        0          0          1             0
    #> Step 3       0       0        0          1          0             0
    #> Step 4       1       0        0          0          0             0
    #> Removed      1       0        0          1          1             1

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## Pooling with BS including interaction terms and method D2

Pooling linear regression models over 5 imputed datasets with BS using a p-value of 0.05 and as method D2. Several interaction terms, including a categorical predictor, are part of the selection procedure.

      library(psfmi)
      pool_lm <- psfmi_lm(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Pain ~ Gender + Smoking +
                            Function + JobControl + factor(Carrying) +
                            factor(Satisfaction) +
                            factor(Carrying):Smoking + Gender:Smoking,
                          method="D2", p.crit=0.05,
                          direction="BW")
    #> Removed at Step 1 is - Function
    #> Removed at Step 2 is - Gender*Smoking
    #> Removed at Step 3 is - Smoking*factor(Carrying)
    #> Removed at Step 4 is - Smoking
    #> Removed at Step 5 is - JobControl
    #> Removed at Step 6 is - Gender
    #>
    #> Selection correctly terminated,
    #> No more variables removed from the model
      pool_lm$RR_model_final
    #> $`Step 7`
    #>                    term  estimate std.error  statistic        df      p.value
    #> 1           (Intercept) 3.8156476 0.3621860 10.5350490 129.49014 0.000000e+00
    #> 2     factor(Carrying)2 0.8759161 0.3761904  2.3283850 113.24274 2.166656e-02
    #> 3     factor(Carrying)3 1.8001704 0.3746799  4.8045553 145.87249 3.811778e-06
    #> 4 factor(Satisfaction)2 0.1385358 0.3729809  0.3714288 108.41273 7.110431e-01
    #> 5 factor(Satisfaction)3 1.4420012 0.4685986  3.0772635  74.55846 2.921715e-03
      pool_lm$multiparm_final
    #> $`Step 7`
    #>                       p-values D2 F-statistic
    #> factor(Carrying)     1.653888e-05   11.150999
    #> factor(Satisfaction) 7.789587e-03    5.372204
      pool_lm$predictors_out
    #>         Gender Smoking Function JobControl factor(Carrying)
    #> Step 1       0       0        1          0                0
    #> Step 2       0       0        0          0                0
    #> Step 3       0       0        0          0                0
    #> Step 4       0       1        0          0                0
    #> Step 5       0       0        0          1                0
    #> Step 6       1       0        0          0                0
    #> Removed      1       1        1          1                0
    #>         factor(Satisfaction) Smoking*factor(Carrying) Gender*Smoking
    #> Step 1                     0                        0              0
    #> Step 2                     0                        0              1
    #> Step 3                     0                        1              0
    #> Step 4                     0                        0              0
    #> Step 5                     0                        0              0
    #> Step 6                     0                        0              0
    #> Removed                    0                        1              1

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## Pooling with BS and forcing interaction terms and method D1

Same as above but now forcing several predictors, including interaction terms, in the model during BS.

      library(psfmi)
      pool_lm <- psfmi_lm(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Pain ~ Gender + Smoking +
                          Function + JobControl + factor(Carrying) + factor(Satisfaction) +
                            factor(Carrying):Smoking + Gender:Smoking,
                          keep.predictors = c("Smoking*Carrying", "JobControl"), method="D1",
                          p.crit=0.05, direction="BW")
    #> Removed at Step 1 is - Function
    #> Removed at Step 2 is - Gender*Smoking
    #> Removed at Step 3 is - Gender
    #>
    #> Selection correctly terminated,
    #> No more variables removed from the model
      pool_lm$RR_model_final
    #> $`Step 4`
    #>                        term    estimate  std.error  statistic        df
    #> 1               (Intercept)  5.05673749 1.11537162  4.5336796  87.35469
    #> 2                   Smoking -0.75879295 0.59455328 -1.2762405  50.60796
    #> 3                JobControl -0.01558801 0.01737846 -0.8969733  87.93658
    #> 4         factor(Carrying)2  0.51735642 0.51915658  0.9965325 132.99359
    #> 5         factor(Carrying)3  1.31863192 0.50113424  2.6312948 126.77358
    #> 6     factor(Satisfaction)2  0.11077123 0.37320587  0.2968100 117.98206
    #> 7     factor(Satisfaction)3  1.44590689 0.48154484  3.0026423  64.65768
    #> 8 Smoking:factor(Carrying)2  0.81312389 0.77812973  1.0449721  87.32029
    #> 9 Smoking:factor(Carrying)3  1.13073244 0.79050622  1.4303903 104.46161
    #>        p.value
    #> 1 1.832877e-05
    #> 2 2.076965e-01
    #> 3 3.721823e-01
    #> 4 3.208012e-01
    #> 5 9.561386e-03
    #> 6 7.671335e-01
    #> 7 3.802284e-03
    #> 8 2.989200e-01
    #> 9 1.555895e-01
      pool_lm$multiparm_final
    #> $`Step 4`
    #>                           p-values D1 F-statistic
    #> Smoking                  0.5399398352   0.7214117
    #> JobControl               0.3705279273   0.8045610
    #> factor(Carrying)         0.0001017566   5.9155402
    #> factor(Satisfaction)     0.0025183119   6.2392581
    #> Smoking*factor(Carrying) 0.3318368885   1.1068101
      pool_lm$predictors_out
    #>         Gender Smoking Function JobControl factor(Carrying)
    #> Step 1       0       0        1          0                0
    #> Step 2       0       0        0          0                0
    #> Step 3       1       0        0          0                0
    #> Removed      1       0        1          0                0
    #>         factor(Satisfaction) Smoking*factor(Carrying) Gender*Smoking
    #> Step 1                     0                        0              0
    #> Step 2                     0                        0              1
    #> Step 3                     0                        0              0
    #> Removed                    0                        0              1

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## Pooling with BS including spline coefficient and method D1

Pooling linear regression models over 5 imputed datasets with BS using a p-value of 0.05 and as method D1. A spline predictor and interaction term are part of the selection procedure.

      library(psfmi)
      pool_lm <- psfmi_lm(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Pain ~ Gender + Smoking +
                          JobControl + factor(Carrying) + factor(Satisfaction) +
                          factor(Carrying):Smoking + rcs(Function, 3),
                          method="D1",
                          p.crit=0.05, direction="BW")
    #> Removed at Step 1 is - rcs(Function,3)
    #> Removed at Step 2 is - Smoking*factor(Carrying)
    #> Removed at Step 3 is - Smoking
    #> Removed at Step 4 is - JobControl
    #> Removed at Step 5 is - Gender
    #>
    #> Selection correctly terminated,
    #> No more variables removed from the model
      pool_lm$RR_model_final
    #> $`Step 6`
    #>                    term  estimate std.error  statistic        df      p.value
    #> 1           (Intercept) 3.8156476 0.3621860 10.5350490 129.49014 0.000000e+00
    #> 2     factor(Carrying)2 0.8759161 0.3761904  2.3283850 113.24274 2.166656e-02
    #> 3     factor(Carrying)3 1.8001704 0.3746799  4.8045553 145.87249 3.811778e-06
    #> 4 factor(Satisfaction)2 0.1385358 0.3729809  0.3714288 108.41273 7.110431e-01
    #> 5 factor(Satisfaction)3 1.4420012 0.4685986  3.0772635  74.55846 2.921715e-03
      pool_lm$multiparm_final
    #> $`Step 6`
    #>                       p-values D1 F-statistic
    #> factor(Carrying)     1.752967e-05   11.125118
    #> factor(Satisfaction) 2.477744e-03    6.275617
      pool_lm$predictors_out
    #>         Gender Smoking JobControl factor(Carrying) factor(Satisfaction)
    #> Step 1       0       0          0                0                    0
    #> Step 2       0       0          0                0                    0
    #> Step 3       0       1          0                0                    0
    #> Step 4       0       0          1                0                    0
    #> Step 5       1       0          0                0                    0
    #> Removed      1       1          1                0                    0
    #>         rcs(Function,3) Smoking*factor(Carrying)
    #> Step 1                1                        0
    #> Step 2                0                        1
    #> Step 3                0                        0
    #> Step 4                0                        0
    #> Step 5                0                        0
    #> Removed               1                        1

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