# Pooling and Selection of Logistic Regression Models

#### Martijn W Heymans

#### 2021-09-23

# Introduction

With the `psfmi` package you can pool logistic regression models by using
the following pooling methods: RR (Rubin’s Rules), D1, D2, D3 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

- [Logistic Regression]
  - Pooling without BS and method D1
  - Pooling with BS and method D1
  - Pooling with BS and method MPR
  - Pooling with BS including several 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_lr <- psfmi_lr(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Chronic ~ Gender + Smoking +
                          Function + JobControl + JobDemands + SocialSupport,
                          method="D1")

      pool_lr$RR_model
    #> $`Step 1 - no variables removed -`
    #>            term    estimate  std.error    statistic        df     p.value
    #> 1   (Intercept) -0.02145084 2.49485297 -0.008598036 104.09644 0.993156301
    #> 2        Gender -0.35445151 0.41807427 -0.847819477 141.28927 0.397972465
    #> 3       Smoking  0.07565036 0.34084592  0.221948835 147.74179 0.824660215
    #> 4      Function -0.14188458 0.04337897 -3.270815252 132.02927 0.001368147
    #> 5    JobControl  0.00690354 0.02053384  0.336203110  88.93815 0.737509628
    #> 6    JobDemands  0.00227508 0.03872846  0.058744401 103.72259 0.953268722
    #> 7 SocialSupport  0.04434046 0.05750883  0.771019941 126.70867 0.442130487
    #>          OR   lower.EXP   upper.EXP
    #> 1 0.9787776 0.006951596 137.8108760
    #> 2 0.7015581 0.306989710   1.6032584
    #> 3 1.0785854 0.549958398   2.1153353
    #> 4 0.8677214 0.796369271   0.9454664
    #> 5 1.0069274 0.966670925   1.0488604
    #> 6 1.0022777 0.928182101   1.0822882
    #> 7 1.0453382 0.932895897   1.1713332

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## Pooling with BS and method D1

Pooling Logistic 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_lr <- psfmi_lr(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Chronic ~ Gender + Smoking +
                          Function + JobControl + JobDemands + SocialSupport,
                          keep.predictors = "Smoking", method="D1", p.crit=0.05,
                          direction="BW")
    #> Removed at Step 1 is - JobDemands
    #> Removed at Step 2 is - JobControl
    #> Removed at Step 3 is - SocialSupport
    #> Removed at Step 4 is - Gender
    #>
    #> Selection correctly terminated,
    #> No more variables removed from the model
      pool_lr$RR_model_final
    #> $`Step 5`
    #>          term    estimate  std.error  statistic       df    p.value        OR
    #> 1 (Intercept)  1.20696975 0.48230894  2.5024827 138.9717 0.01349142 3.3433381
    #> 2     Smoking  0.06427314 0.33804675  0.1901309 151.8086 0.84946055 1.0663836
    #> 3    Function -0.14058914 0.04225212 -3.3273866 121.6357 0.00115993 0.8688462
    #>   lower.EXP upper.EXP
    #> 1 1.2883439 8.6761852
    #> 2 0.5468337 2.0795610
    #> 3 0.7991282 0.9446467
      pool_lr$multiparm_final
    #> $`Step 5`
    #>           p-values D1 F-statistic
    #> Smoking  0.8492075865  0.03614977
    #> Function 0.0009162863 11.07150131
      pool_lr$predictors_out
    #>         Gender Smoking Function JobControl JobDemands SocialSupport
    #> Step 1       0       0        0          0          1             0
    #> Step 2       0       0        0          1          0             0
    #> Step 3       0       0        0          0          0             1
    #> 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 Logistic 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_lr <- psfmi_lr(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Chronic ~ Gender + Smoking +
                          Function + JobControl + JobDemands + SocialSupport,
                          keep.predictors = "Smoking", method="MPR", p.crit=0.05,
                          direction="BW")
    #> Removed at Step 1 is - JobDemands
    #> Removed at Step 2 is - JobControl
    #> Removed at Step 3 is - SocialSupport
    #> Removed at Step 4 is - Gender
    #>
    #> Selection correctly terminated,
    #> No more variables removed from the model
      pool_lr$RR_model_final
    #> $`Step 5`
    #>          term    estimate  std.error  statistic       df    p.value        OR
    #> 1 (Intercept)  1.20696975 0.48230894  2.5024827 138.9717 0.01349142 3.3433381
    #> 2     Smoking  0.06427314 0.33804675  0.1901309 151.8086 0.84946055 1.0663836
    #> 3    Function -0.14058914 0.04225212 -3.3273866 121.6357 0.00115993 0.8688462
    #>   lower.EXP upper.EXP
    #> 1 1.2883439 8.6761852
    #> 2 0.5468337 2.0795610
    #> 3 0.7991282 0.9446467
      pool_lr$multiparm_final
    #> $`Step 5`
    #>           p-value MPR
    #> Smoking  0.8734388519
    #> Function 0.0003502948
      pool_lr$predictors_out
    #>         Gender Smoking Function JobControl JobDemands SocialSupport
    #> Step 1       0       0        0          0          1             0
    #> Step 2       0       0        0          1          0             0
    #> Step 3       0       0        0          0          0             1
    #> Step 4       1       0        0          0          0             0
    #> Removed      1       0        0          1          1             1

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## Pooling with BS including several interaction terms and method D2

Pooling Logistic 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_lr <- psfmi_lr(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Chronic ~ 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 - JobControl
    #> Removed at Step 2 is - factor(Satisfaction)
    #> Removed at Step 3 is - Gender*Smoking
    #> Removed at Step 4 is - Gender
    #> Removed at Step 5 is - Smoking*factor(Carrying)
    #> Removed at Step 6 is - Smoking
    #> Removed at Step 7 is - Function
    #>
    #> Selection correctly terminated,
    #> No more variables removed from the model
      pool_lr$RR_model_final
    #> $`Step 8`
    #>                term  estimate std.error statistic       df      p.value
    #> 1       (Intercept) -1.582393 0.3773100 -4.193880 151.2668 4.652717e-05
    #> 2 factor(Carrying)2  1.391554 0.4709708  2.954650 144.1330 3.657304e-03
    #> 3 factor(Carrying)3  2.248897 0.4750324  4.734198 151.4269 5.010441e-06
    #>          OR  lower.EXP  upper.EXP
    #> 1 0.2054828 0.09750312  0.4330445
    #> 2 4.0210931 1.58510710 10.2006922
    #> 3 9.4772792 3.70747175 24.2264342
      pool_lr$multiparm_final
    #> $`Step 8`
    #>                   p-values D2 F-statistic
    #> factor(Carrying) 1.251841e-05    11.29016
      pool_lr$predictors_out
    #>         Gender Smoking Function JobControl factor(Carrying)
    #> Step 1       0       0        0          1                0
    #> Step 2       0       0        0          0                0
    #> Step 3       0       0        0          0                0
    #> Step 4       1       0        0          0                0
    #> Step 5       0       0        0          0                0
    #> Step 6       0       1        0          0                0
    #> Step 7       0       0        1          0                0
    #> Removed      1       1        1          1                0
    #>         factor(Satisfaction) Smoking*factor(Carrying) Gender*Smoking
    #> Step 1                     0                        0              0
    #> Step 2                     1                        0              0
    #> Step 3                     0                        0              1
    #> Step 4                     0                        0              0
    #> Step 5                     0                        1              0
    #> Step 6                     0                        0              0
    #> Step 7                     0                        0              0
    #> Removed                    1                        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_lr <- psfmi_lr(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Chronic ~ 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 - Gender*Smoking
    #> Removed at Step 2 is - Gender
    #> Removed at Step 3 is - factor(Satisfaction)
    #> Removed at Step 4 is - Function
      pool_lr$RR_model_final
    #> $`Step 5`
    #>                        term     estimate std.error  statistic        df
    #> 1               (Intercept) -0.810522255 1.3650185 -0.5937812  55.33263
    #> 2                   Smoking -1.796541680 1.1699026 -1.5356336  65.48650
    #> 3                JobControl -0.004625312 0.0216596 -0.2135455  58.16427
    #> 4         factor(Carrying)2  0.723452199 0.6214600  1.1641171 107.17663
    #> 5         factor(Carrying)3  1.534813529 0.5908820  2.5974958 107.09227
    #> 6 Smoking:factor(Carrying)2  2.093737680 1.3149790  1.5922214  66.01171
    #> 7 Smoking:factor(Carrying)3  2.370029492 1.3934064  1.7008889  51.83267
    #>      p.value         OR  lower.EXP  upper.EXP
    #> 1 0.55507827  0.4446258 0.02884811   6.852861
    #> 2 0.12944599  0.1658715 0.01604000   1.715297
    #> 3 0.83164826  0.9953854 0.95315370   1.039488
    #> 4 0.24696141  2.0615378 0.60139911   7.066751
    #> 5 0.01071096  4.6404601 1.43831227  14.971624
    #> 6 0.11611271  8.1151905 0.58760877 112.075110
    #> 7 0.09495675 10.6977078 0.65294658 175.268475
      pool_lr$multiparm_final
    #> $`Step 5`
    #>                          p-values D1 F-statistic
    #> Smoking                  0.313723002   1.1877289
    #> JobControl               0.831291534   0.0456017
    #> factor(Carrying)         0.000903194   4.7060389
    #> Smoking*factor(Carrying) 0.188926654   1.6841757
      pool_lr$predictors_out
    #>         Gender Smoking Function JobControl factor(Carrying)
    #> Step 1       0       0        0          0                0
    #> Step 2       1       0        0          0                0
    #> Step 3       0       0        0          0                0
    #> Step 4       0       0        1          0                0
    #> Removed      1       0        1          0                0
    #>         factor(Satisfaction) Smoking*factor(Carrying) Gender*Smoking
    #> Step 1                     0                        0              1
    #> Step 2                     0                        0              0
    #> Step 3                     1                        0              0
    #> Step 4                     0                        0              0
    #> Removed                    1                        0              1

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## Pooling with BS including spline coefficient and method D1

Pooling Logistic 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_lr <- psfmi_lr(data=lbpmilr, nimp=5, impvar="Impnr",
                          formula = Chronic ~ 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 - JobControl
    #> Removed at Step 2 is - Gender
    #> Removed at Step 3 is - rcs(Function,3)
    #> Removed at Step 4 is - factor(Satisfaction)
    #> Removed at Step 5 is - Smoking*factor(Carrying)
    #> Removed at Step 6 is - Smoking
    #>
    #> Selection correctly terminated,
    #> No more variables removed from the model
      pool_lr$RR_model_final
    #> $`Step 7`
    #>                term  estimate std.error statistic       df      p.value
    #> 1       (Intercept) -1.582393 0.3773100 -4.193880 151.2668 4.652717e-05
    #> 2 factor(Carrying)2  1.391554 0.4709708  2.954650 144.1330 3.657304e-03
    #> 3 factor(Carrying)3  2.248897 0.4750324  4.734198 151.4269 5.010441e-06
    #>          OR  lower.EXP  upper.EXP
    #> 1 0.2054828 0.09750312  0.4330445
    #> 2 4.0210931 1.58510710 10.2006922
    #> 3 9.4772792 3.70747175 24.2264342
      pool_lr$multiparm_final
    #> $`Step 7`
    #>                   p-values D1 F-statistic
    #> factor(Carrying) 2.257078e-05     10.8085
      pool_lr$predictors_out
    #>         Gender Smoking JobControl factor(Carrying) factor(Satisfaction)
    #> Step 1       0       0          1                0                    0
    #> Step 2       1       0          0                0                    0
    #> Step 3       0       0          0                0                    0
    #> Step 4       0       0          0                0                    1
    #> Step 5       0       0          0                0                    0
    #> Step 6       0       1          0                0                    0
    #> Removed      1       1          1                0                    1
    #>         rcs(Function,3) Smoking*factor(Carrying)
    #> Step 1                0                        0
    #> Step 2                0                        0
    #> Step 3                1                        0
    #> Step 4                0                        0
    #> Step 5                0                        1
    #> Step 6                0                        0
    #> Removed               1                        1

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