Both the BIC and the AIC criterion are a trade-off between fit (-2 log L) and model complexity (2p for the AIC and p log(n) for BIC).
However, the trade-off between fit and model complexity is different for the AIC and the BIC.
In order to favour a more complex model,
Because log(n) > 2 as soon as n >= 8, the BIC criterion will penalise more for the increase in model complexity and it will therefore favour smaller models than the AIC criterion, which punishes less for the increase in model complexity.
model
\[ Y = V1 + 2* V2 + 4*V3 +\epsilon \]
#> Start: AIC=10.31
#> y ~ V1 + V2 + V3 + V4 + V5 + V6 + V7 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V7 1 0.31 89.28 8.657
#> - V4 1 0.37 89.33 8.722
#> - V9 1 0.66 89.63 9.053
#> - V10 1 1.15 90.12 9.599
#> - V6 1 1.20 90.17 9.652
#> <none> 88.97 10.311
#> - V5 1 5.07 94.04 13.855
#> - V8 1 5.66 94.63 14.476
#> - V1 1 127.33 216.30 97.150
#> - V2 1 320.34 409.30 160.929
#> - V3 1 1518.09 1607.06 297.699
#>
#> Step: AIC=8.66
#> y ~ V1 + V2 + V3 + V4 + V5 + V6 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V4 1 0.56 89.83 7.278
#> - V9 1 0.63 89.91 7.359
#> - V10 1 1.05 90.32 7.824
#> - V6 1 1.15 90.43 7.942
#> <none> 89.28 8.657
#> - V5 1 5.01 94.28 12.113
#> - V8 1 5.59 94.87 12.731
#> - V1 1 127.24 216.52 95.251
#> - V2 1 325.09 414.37 160.158
#> - V3 1 1572.65 1661.92 299.056
#>
#> Step: AIC=7.28
#> y ~ V1 + V2 + V3 + V5 + V6 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V9 1 0.64 90.47 5.984
#> - V10 1 1.13 90.96 6.527
#> - V6 1 1.18 91.01 6.583
#> <none> 89.83 7.278
#> - V5 1 4.58 94.41 10.249
#> - V8 1 5.25 95.08 10.956
#> - V1 1 127.10 216.93 93.441
#> - V2 1 327.62 417.46 158.901
#> - V3 1 1584.28 1674.11 297.787
#>
#> Step: AIC=5.98
#> y ~ V1 + V2 + V3 + V5 + V6 + V8 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V10 1 1.49 91.96 5.620
#> - V6 1 1.52 91.99 5.654
#> <none> 90.47 5.984
#> - V5 1 4.44 94.91 8.776
#> - V8 1 5.83 96.30 10.228
#> - V1 1 130.29 220.76 93.190
#> - V2 1 327.02 417.49 156.909
#> - V3 1 1588.71 1679.18 296.089
#>
#> Step: AIC=5.62
#> y ~ V1 + V2 + V3 + V5 + V6 + V8
#>
#> Df Sum of Sq RSS AIC
#> - V6 1 1.33 93.29 5.058
#> <none> 91.96 5.620
#> - V5 1 4.59 96.55 8.491
#> - V8 1 4.88 96.84 8.788
#> - V1 1 128.80 220.76 91.190
#> - V2 1 336.71 428.67 157.553
#> - V3 1 1587.72 1679.68 294.119
#>
#> Step: AIC=5.06
#> y ~ V1 + V2 + V3 + V5 + V8
#>
#> Df Sum of Sq RSS AIC
#> <none> 93.29 5.058
#> - V8 1 3.94 97.23 7.193
#> - V5 1 4.11 97.41 7.371
#> - V1 1 127.53 220.83 89.221
#> - V2 1 348.90 442.19 158.658
#> - V3 1 1608.86 1702.16 293.448#>
#> Call:
#> lm(formula = y ~ V1 + V2 + V3 + V5 + V8, data = x)
#>
#> Coefficients:
#> (Intercept) V1 V2 V3 V5 V8
#> 9.9709 0.9868 1.9685 3.8627 -0.2056 0.2121realPredAIC <- sum(names(lmAIC$coefficients) %in% pred)
falsePredAIC <- length(lmAIC$coefficients) - realPredAIC - 1The AIC criterion selects the model with the lowest AIC.
This model correctly selects 3 out of 3 real predictors. However, it also selects 2 predictors that are not associated with the response!
We can do this in the step function by specifying k. By default k=2.
If we define k=log(n) than we use the BIC.
#> Start: AIC=38.97
#> y ~ V1 + V2 + V3 + V4 + V5 + V6 + V7 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V7 1 0.31 89.28 34.71
#> - V4 1 0.37 89.33 34.77
#> - V9 1 0.66 89.63 35.10
#> - V10 1 1.15 90.12 35.65
#> - V6 1 1.20 90.17 35.70
#> <none> 88.97 38.97
#> - V5 1 5.07 94.04 39.91
#> - V8 1 5.66 94.63 40.53
#> - V1 1 127.33 216.30 123.20
#> - V2 1 320.34 409.30 186.98
#> - V3 1 1518.09 1607.06 323.75
#>
#> Step: AIC=34.71
#> y ~ V1 + V2 + V3 + V4 + V5 + V6 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V4 1 0.56 89.83 30.72
#> - V9 1 0.63 89.91 30.81
#> - V10 1 1.05 90.32 31.27
#> - V6 1 1.15 90.43 31.39
#> <none> 89.28 34.71
#> - V5 1 5.01 94.28 35.56
#> - V8 1 5.59 94.87 36.18
#> - V1 1 127.24 216.52 118.70
#> - V2 1 325.09 414.37 183.60
#> - V3 1 1572.65 1661.92 322.50
#>
#> Step: AIC=30.72
#> y ~ V1 + V2 + V3 + V5 + V6 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V9 1 0.64 90.47 26.83
#> - V10 1 1.13 90.96 27.37
#> - V6 1 1.18 91.01 27.42
#> <none> 89.83 30.72
#> - V5 1 4.58 94.41 31.09
#> - V8 1 5.25 95.08 31.80
#> - V1 1 127.10 216.93 114.28
#> - V2 1 327.62 417.46 179.74
#> - V3 1 1584.28 1674.11 318.63
#>
#> Step: AIC=26.83
#> y ~ V1 + V2 + V3 + V5 + V6 + V8 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V10 1 1.49 91.96 23.856
#> - V6 1 1.52 91.99 23.891
#> <none> 90.47 26.825
#> - V5 1 4.44 94.91 27.013
#> - V8 1 5.83 96.30 28.465
#> - V1 1 130.29 220.76 111.426
#> - V2 1 327.02 417.49 175.145
#> - V3 1 1588.71 1679.18 314.325
#>
#> Step: AIC=23.86
#> y ~ V1 + V2 + V3 + V5 + V6 + V8
#>
#> Df Sum of Sq RSS AIC
#> - V6 1 1.33 93.29 20.689
#> <none> 91.96 23.856
#> - V5 1 4.59 96.55 24.122
#> - V8 1 4.88 96.84 24.419
#> - V1 1 128.80 220.76 106.821
#> - V2 1 336.71 428.67 173.184
#> - V3 1 1587.72 1679.68 309.750
#>
#> Step: AIC=20.69
#> y ~ V1 + V2 + V3 + V5 + V8
#>
#> Df Sum of Sq RSS AIC
#> - V8 1 3.94 97.23 20.219
#> - V5 1 4.11 97.41 20.397
#> <none> 93.29 20.689
#> - V1 1 127.53 220.83 102.247
#> - V2 1 348.90 442.19 171.683
#> - V3 1 1608.86 1702.16 306.474
#>
#> Step: AIC=20.22
#> y ~ V1 + V2 + V3 + V5
#>
#> Df Sum of Sq RSS AIC
#> - V5 1 3.54 100.77 19.187
#> <none> 97.23 20.219
#> - V1 1 124.15 221.38 97.891
#> - V2 1 389.45 486.68 176.664
#> - V3 1 1620.25 1717.48 302.765
#>
#> Step: AIC=19.19
#> y ~ V1 + V2 + V3
#>
#> Df Sum of Sq RSS AIC
#> <none> 100.77 19.187
#> - V1 1 123.13 223.90 94.419
#> - V2 1 393.16 493.93 173.539
#> - V3 1 1618.02 1718.79 298.236#>
#> Call:
#> lm(formula = y ~ V1 + V2 + V3, data = x)
#>
#> Coefficients:
#> (Intercept) V1 V2 V3
#> 10.0389 0.9633 2.0267 3.8315realPredBIC <- sum(names(lmBIC$coefficients) %in% pred)
falsePredBIC <- length(lmBIC$coefficients) - realPredBIC - 1The BIC criterion selects the model with the lowest BIC.
This model correctly selects 3 out of 3 real predictors. However, it selects 0 predictors that are not associated with the response!
If the noise increases it will be harder to select the correct model and we still can expect the AIC to result in more complex models than the BIC.
We will use the same seed so that the difference in the response is not induced by the random generator but only by the difference in variance.
#> Start: AIC=470.83
#> y ~ V1 + V2 + V3 + V4 + V5 + V6 + V7 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V7 1 30.79 8927.6 469.17
#> - V4 1 36.64 8933.5 469.24
#> - V9 1 66.24 8963.1 469.57
#> - V10 1 115.29 9012.1 470.12
#> - V6 1 120.08 9016.9 470.17
#> - V1 1 127.61 9024.5 470.25
#> - V2 1 144.13 9041.0 470.44
#> <none> 8896.8 470.83
#> - V5 1 507.13 9404.0 474.37
#> - V8 1 565.69 9462.5 474.99
#> - V3 1 834.14 9731.0 477.79
#>
#> Step: AIC=469.17
#> y ~ V1 + V2 + V3 + V4 + V5 + V6 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V4 1 55.60 8983.2 467.79
#> - V9 1 62.93 8990.6 467.88
#> - V10 1 104.79 9032.4 468.34
#> - V6 1 115.47 9043.1 468.46
#> - V1 1 126.64 9054.3 468.58
#> - V2 1 158.35 9086.0 468.93
#> <none> 8927.6 469.17
#> - V5 1 500.69 9428.3 472.63
#> - V8 1 559.12 9486.8 473.25
#> - V3 1 803.42 9731.1 475.79
#>
#> Step: AIC=467.79
#> y ~ V1 + V2 + V3 + V5 + V6 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V9 1 63.68 9046.9 466.50
#> - V10 1 112.93 9096.2 467.04
#> - V6 1 118.00 9101.2 467.10
#> - V1 1 125.12 9108.4 467.18
#> - V2 1 169.56 9152.8 467.66
#> <none> 8983.2 467.79
#> - V5 1 457.91 9441.1 470.77
#> - V8 1 524.92 9508.2 471.47
#> - V3 1 836.85 9820.1 474.70
#>
#> Step: AIC=466.5
#> y ~ V1 + V2 + V3 + V5 + V6 + V8 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V1 1 145.47 9192.4 466.10
#> - V10 1 149.20 9196.1 466.14
#> - V6 1 152.40 9199.3 466.17
#> - V2 1 162.80 9209.7 466.28
#> <none> 9046.9 466.50
#> - V5 1 444.13 9491.0 469.29
#> - V8 1 582.94 9629.9 470.75
#> - V3 1 854.90 9901.8 473.53
#>
#> Step: AIC=466.1
#> y ~ V2 + V3 + V5 + V6 + V8 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V10 1 120.54 9312.9 465.40
#> - V6 1 128.99 9321.4 465.49
#> - V2 1 161.82 9354.2 465.84
#> <none> 9192.4 466.10
#> - V5 1 424.91 9617.3 468.61
#> - V8 1 512.28 9704.7 469.52
#> - V3 1 812.18 10004.6 472.56
#>
#> Step: AIC=465.4
#> y ~ V2 + V3 + V5 + V6 + V8
#>
#> Df Sum of Sq RSS AIC
#> - V6 1 115.07 9428.0 464.63
#> <none> 9312.9 465.40
#> - V2 1 198.94 9511.9 465.51
#> - V8 1 436.36 9749.3 467.98
#> - V5 1 440.21 9753.1 468.02
#> - V3 1 787.70 10100.6 471.52
#>
#> Step: AIC=464.63
#> y ~ V2 + V3 + V5 + V8
#>
#> Df Sum of Sq RSS AIC
#> <none> 9428.0 464.63
#> - V2 1 247.31 9675.3 465.22
#> - V8 1 355.83 9783.8 466.33
#> - V5 1 397.89 9825.9 466.76
#> - V3 1 718.83 10146.8 469.97#>
#> Call:
#> lm(formula = y ~ V2 + V3 + V5 + V8, data = x)
#>
#> Coefficients:
#> (Intercept) V2 V3 V5 V8
#> 9.715 1.657 2.579 -2.022 2.004realPredAIC10 <- sum(names(lmAIC10$coefficients) %in% pred)
falsePredAIC10 <- length(lmAIC10$coefficients) - realPredAIC10 - 1The AIC criterion selects the model with the lowest AIC.
This model correctly selects 2 out of 3 real predictors. However, it also selects 2 predictors that are not associated with the response!
We can do this in the step function by specifying k. By default k=2.
If we define k=log(n) than we use the BIC.
#> Start: AIC=499.49
#> y ~ V1 + V2 + V3 + V4 + V5 + V6 + V7 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V7 1 30.79 8927.6 495.23
#> - V4 1 36.64 8933.5 495.29
#> - V9 1 66.24 8963.1 495.62
#> - V10 1 115.29 9012.1 496.17
#> - V6 1 120.08 9016.9 496.22
#> - V1 1 127.61 9024.5 496.30
#> - V2 1 144.13 9041.0 496.49
#> <none> 8896.8 499.49
#> - V5 1 507.13 9404.0 500.42
#> - V8 1 565.69 9462.5 501.04
#> - V3 1 834.14 9731.0 503.84
#>
#> Step: AIC=495.23
#> y ~ V1 + V2 + V3 + V4 + V5 + V6 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V4 1 55.60 8983.2 491.24
#> - V9 1 62.93 8990.6 491.32
#> - V10 1 104.79 9032.4 491.79
#> - V6 1 115.47 9043.1 491.91
#> - V1 1 126.64 9054.3 492.03
#> - V2 1 158.35 9086.0 492.38
#> <none> 8927.6 495.23
#> - V5 1 500.69 9428.3 496.08
#> - V8 1 559.12 9486.8 496.69
#> - V3 1 803.42 9731.1 499.24
#>
#> Step: AIC=491.24
#> y ~ V1 + V2 + V3 + V5 + V6 + V8 + V9 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V9 1 63.68 9046.9 487.34
#> - V10 1 112.93 9096.2 487.89
#> - V6 1 118.00 9101.2 487.94
#> - V1 1 125.12 9108.4 488.02
#> - V2 1 169.56 9152.8 488.51
#> <none> 8983.2 491.24
#> - V5 1 457.91 9441.1 491.61
#> - V8 1 524.92 9508.2 492.31
#> - V3 1 836.85 9820.1 495.54
#>
#> Step: AIC=487.34
#> y ~ V1 + V2 + V3 + V5 + V6 + V8 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V1 1 145.47 9192.4 484.33
#> - V10 1 149.20 9196.1 484.37
#> - V6 1 152.40 9199.3 484.41
#> - V2 1 162.80 9209.7 484.52
#> <none> 9046.9 487.34
#> - V5 1 444.13 9491.0 487.53
#> - V8 1 582.94 9629.9 488.98
#> - V3 1 854.90 9901.8 491.77
#>
#> Step: AIC=484.33
#> y ~ V2 + V3 + V5 + V6 + V8 + V10
#>
#> Df Sum of Sq RSS AIC
#> - V10 1 120.54 9312.9 481.03
#> - V6 1 128.99 9321.4 481.12
#> - V2 1 161.82 9354.2 481.47
#> - V5 1 424.91 9617.3 484.25
#> <none> 9192.4 484.33
#> - V8 1 512.28 9704.7 485.15
#> - V3 1 812.18 10004.6 488.19
#>
#> Step: AIC=481.03
#> y ~ V2 + V3 + V5 + V6 + V8
#>
#> Df Sum of Sq RSS AIC
#> - V6 1 115.07 9428.0 477.65
#> - V2 1 198.94 9511.9 478.54
#> - V8 1 436.36 9749.3 481.00
#> <none> 9312.9 481.03
#> - V5 1 440.21 9753.1 481.04
#> - V3 1 787.70 10100.6 484.54
#>
#> Step: AIC=477.65
#> y ~ V2 + V3 + V5 + V8
#>
#> Df Sum of Sq RSS AIC
#> - V2 1 247.31 9675.3 475.64
#> - V8 1 355.83 9783.8 476.75
#> - V5 1 397.89 9825.9 477.18
#> <none> 9428.0 477.65
#> - V3 1 718.83 10146.8 480.40
#>
#> Step: AIC=475.64
#> y ~ V3 + V5 + V8
#>
#> Df Sum of Sq RSS AIC
#> - V5 1 437.44 10112.7 475.45
#> <none> 9675.3 475.64
#> - V8 1 548.90 10224.2 476.55
#> - V3 1 723.85 10399.1 478.25
#>
#> Step: AIC=475.45
#> y ~ V3 + V8
#>
#> Df Sum of Sq RSS AIC
#> <none> 10113 475.45
#> - V8 1 492.56 10605 475.60
#> - V3 1 693.98 10807 477.49#>
#> Call:
#> lm(formula = y ~ V3 + V8, data = x)
#>
#> Coefficients:
#> (Intercept) V3 V8
#> 10.020 2.533 2.281realPredBIC10 <- sum(names(lmBIC10$coefficients) %in% pred)
falsePredBIC10 <- length(lmBIC10$coefficients) - realPredBIC10 - 1The BIC criterion selects the model with the lowest BIC.
This model correctly selects 1 out of 3 real predictors. However, it selects 1 predictor that is not associated with the response!
Note, that AIC and BIC are a good estimate of the insample error, however, when building prediction models we are interested in using the model for predictor patterns that are not observed in the training set. So it is better to build a model based on an estimate of the outsample error.
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