1. 1 The Receiver Operating Characteristic (ROC) Curve EPP 245 Statistical Analysis of Laboratory Data

2. November 30, 2006EPP 245 Statistical Analysis of Laboratory Data 2 Binary Classification Suppose we have two groups for which each case is a member of one or the other, and that we know the correct classification (“truth”). Suppose we have a prediction method that produces a single numerical value, and that small values of that number suggest membership in group 1 and large values suggest membership in group 2

3. November 30, 2006EPP 245 Statistical Analysis of Laboratory Data 3 If we pick a cutpoint t, we can assign any case with a predicted value ≤ t to group 1 and the others to group 2. For that value of t, we can compute the number correctly assigned to group 2 and the number incorrectly assigned to group 2 (true positives and false positives). For t small enough, all will be assigned to group 2 and for t large enough all will be assigned to group 1. The ROC curve is a plot of true positives vs. false positives

4. November 30, 2006EPP 245 Statistical Analysis of Laboratory Data 4 Juul's IGF data Description: The 'juul' data frame has 1339 rows and 6 columns. It contains a reference sample of the distribution of insulin-like growth factor (IGF-I), one observation per subject in various ages with the bulk of the data collected in connection with school physical examinations. Variables: age a numeric vector (years). menarche a numeric vector. Has menarche occurred (code 1: no, 2: yes)? sex a numeric vector (1: boy, 2: girl). igf1 a numeric vector. Insulin-like growth factor ($mu$g/l). tanner a numeric vector. Codes 1-5: Stages of puberty a.m. Tanner. testvol a numeric vector. Testicular volume (ml).

5. November 30, 2006EPP 245 Statistical Analysis of Laboratory Data 5 Predicting Menarche Subset Juul data to only females between 8 and 20 years old Predict menarch from age as a quantitative variable and Tanner score as a qualitative variable using dummy variables Menarch re-coded to be 0/1

6. November 30, 2006EPP 245 Statistical Analysis of Laboratory Data 6. logistic men1 age tan2 tan3 tan4 tan5 Logistic regression Number of obs = 519 LR chi2(5) = 568.74 Prob > chi2 = 0.0000 Log likelihood = -75.327218 Pseudo R2 = 0.7906 ------------------------------------------------------------------------------ men1 | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | 3.944062.7162327 7.56 0.000 2.762915 5.630151 tan2 |.0444044.0486937 -2.84 0.005.0051761.3809341 tan3 |.1369598.095596 -2.85 0.004.0348712.5379227 tan4 |.6969611.3898228 -0.65 0.519.2328715 2.085935 tan5 | 9.169558 7.638664 2.66 0.008 1.791671 46.9287 ------------------------------------------------------------------------------. predict pmen (option p assumed; Pr(men1)). predict pmen1, xb

7. November 30, 2006EPP 245 Statistical Analysis of Laboratory Data 7. histogram pmen. graph export pmenhist.wmf. histogram pmen if men1==0, title("Pre-Menarch"). graph export pmenhist0.wmf. histogram pmen if men1==1, title("Post-Menarch"). graph export pmenhist1.wmf. histogram pmen1. graph export pmen1hist.wmf. hist pmen1 if men1==0, title("Pre-Menarche"). graph export pmen1hist0.wmf. hist pmen1 if men1==1, title("Post-Menarche"). graph export pmen1hist1.wmf. lroc Logistic model for men1 number of observations = 519 area under ROC curve = 0.9867. graph export pmenroc.wmf

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