# Compare Models with AIC

The AIC (Akaike Information Criterion) ranks models by balancing goodness-of-fit against complexity. In a GAM, complexity is measured by the total effective degrees of freedom (EDF) rather than the raw number of parameters, so AIC automatically accounts for how wiggly each smooth is.

Use AIC when you have several candidate models and want a principled way to decide which predictors to include.


# Fit the candidate models

Load the `abalone` dataset and fit three nested models, starting with a single smooth and adding predictors one at a time.


``` python
import whittaker as wk

# Load data and fit nested candidate models
data = wk.load_dataset("abalone")

m1 = wk.GAM("rings ~ s(length)").fit(data)
m2 = wk.GAM("rings ~ s(length) + s(diameter)").fit(data)
m3 = wk.GAM("rings ~ s(length) + s(diameter) + s(height) + s(shucked_weight)").fit(data)
```


# Compare AIC values

Extract the `.aic` attribute from each model and display them together.


``` python
# Compare AIC values across models
m1.aic, m2.aic, m3.aic
```


    (2052.895482172216, 2054.8931824650294, 1847.3973234458235)


Lower AIC is better. A difference (ΔAIC) of less than 2 means the models are essentially equivalent in predictive quality. A ΔAIC greater than 10 means the simpler model is decisively worse and the extra predictors are earning their place.


# Check total EDF

AIC penalises via total EDF, so it is worth inspecting how complex each model actually is.


``` python
# Compare total EDF across models
m1.edf_total, m2.edf_total, m3.edf_total
```


    (4.329619009488265, 5.339006719376878, 19.625034023054337)


The full model carries more EDF. If its AIC is not substantially lower, the extra terms are not paying off after the penalty is applied.


# Inspect the winning model

Look at the summary of the winning model. In this dataset `m3` has the lowest AIC.


``` python
m3.summary()
```


    GAM fit summary
    ============================================================
    Formula:    rings ~ s(length) + s(diameter) + s(height) + s(shucked_weight)
    Family:     Gaussian(link='identity')
    Observations: 500
    Coefficients: 37

    Parametric coefficients:
      Term                       Estimate    Std.Err    t value    p-value
      ------------------------ ---------- ---------- ---------- ----------
      (Intercept)                  6.8680     0.0673    102.038    < 1e-16

    Approximate significance of smooth terms:
      Term                        EDF Ref.df     Chi.sq    p-value
      ------------------------ ------ ------ ---------- ----------
      s(length)                  6.27      7     42.371  4.411e-07
      s(diameter)                1.00      2      0.002     0.9991
      s(height)                  5.99      6     25.650  0.0002587
      s(shucked_weight)          5.36      6    245.380    < 1e-16

    Total EDF:  19.63
    Deviance:   1088.1502
    Null dev:   6745.2880
    Dev. expl:  83.9%
    GCV score:  2.357752
    Scale est:  2.265210
    AIC:        1847.40
    BIC:        1930.11


Check [deviance_explained](../reference/GAM.md#whittaker.GAM.deviance_explained) and the per-smooth EDF values. Any smooth with EDF near 1.0 is nearly linear (it contributes little beyond a simple slope).


# AIC rule of thumb

| ΔAIC from best model | Interpretation                                     |
|----------------------|----------------------------------------------------|
| 0-2                  | Essentially equivalent; prefer simpler             |
| 2-10                 | Some evidence for the more complex model           |
| \> 10                | Strong evidence; simpler model is decisively worse |

AIC selects the model that will generalise best on average, not the model that fits the training data best. When predictive accuracy on held-out data is the primary goal, pair AIC comparison with cross-validation (see recipe 26).