<jupyter_start><jupyter_text>**Why Normative Modeling?**  **Context** - Currently, there are no biological tests to diagnose psychiatric disorders. They are diagnosed using clinical symptoms, which leads heterogeneous groups on the biological level.- The overwhelming majority of analysis in psychiatry & neuroimaging is focussed on group averages, which ignore individual differences.- This leads to losing information about individual subjects and missing patterns that aren't consistent across the entire group.From Marquand et al. 2016:![image1](image1.jpg)>The classical case-control approach assumes that cases and controls each form a well-defined group. This may often be a reasonable assumption, but in practice many other scenarios are possible. >- A Cases and controls each form a well-defined group>- B The clinical population may be composed of multiple groups, each having distinct pathology >- C Disease-related variation may be nested within healthy variation>- D The clinical group may be diffuse and heterogeneous as a result of misdiagnosis, comorbidities, or an aggregation of different pathologies.  **Example where case-control fails**We'll construct an example in which:- We know there is a difference between probands and controls (as we've created it)- Case-control analysis misses it  Generate DataFirst we generate the data:- We model score = 2*log(age) + gaussian noise- Probands have a random offset to their score- Probands are not distributed evenly throughout the dataset<jupyter_code># Define Functions
def sample_x(low=1,high=100,n_subs=1000,sampling='full'):
    if sampling =='full':
        x = np.random.uniform(low=low,high=high,size=n_subs)
    else:
        x = np.concatenate([np.random.normal(20,10,size=int(n_subs/2)),np.random.normal(80,10,size=int(n_subs/2))])
        x = x[(x<high) & (x > low)]
    return x
    
def f(x):
    return 2*np.log(x) + np.random.randn()

def dataset(seed=10,sampling='full'):
    # Set seed
    np.random.seed(seed)

    # Age param
    x = np.sort(sample_x(sampling=sampling))

    # PROB = 1/CTR = 0
    status1 =  np.random.binomial(1,0.2,size=(int(0.75*x.shape[0])))
    status2 =  np.random.binomial(1,0.07,size=(x.shape[0] - status1.shape[0]))
    status = np.concatenate([status1, status2])

    # Score param
    score = np.zeros(x.shape[0])
    for i in range(x.shape[0]):
        s = f(x[i])

        # Random offset for probands
        s = s + status[i]*np.random.normal(0.7,1)

        score[i] = s

    return pd.DataFrame([x,status,score],index=['age','status','score']).transpose()
# Create a dataset
df = dataset()
df.head()<jupyter_output><empty_output><jupyter_text>Visualize data- CTR = 0- PROB = 1We can see below that extreme scores for a given age are more frequently probands.<jupyter_code>plt.figure(figsize=(15,7))
sns.scatterplot(x='age',y='score',hue='status',data=df)<jupyter_output><empty_output><jupyter_text>Case-control- Here we perform a simple case-control analysis using the status column of the dataset (PROB/CTR).- The result is not significant (p = 0.99)<jupyter_code>def case_control(df, group='status',score='score'):
    dmat = pat.dmatrix('C({})'.format(group), df, return_type='dataframe',NA_action='raise')
    results = sm.OLS(df[score],dmat).fit()
    return results.pvalues[1]

print('p-value =', case_control(df))<jupyter_output>p-value = 0.8760719899903657
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