BIOSTATISTICAL METHODS

Transcription

1 BIOSTATISTICAL METHOS FOR TRANSLATIONAL & CLINICAL RESEARCH ROC Crve: IAGNOSTIC MEICINE

2 iagnostic tests have been presented as alwas having dichotomos otcomes. In some cases, the reslt of the test ma be binar, bt in man cases it is based on the dichotomization of a continos separator or biomarker some factor correlated the absence or presence of the disease. To deal with a continos separator, we need a wellknown graph called the Receiver Operating Characteristic crve or ROC crve.

3 If the idea, in the developmental stage, was to classified people as diseased condition present or health condition absent based on certain continos measrement from blood or rinar components; then we need to dichotomize the measrement: for example, if the measrement is high then he s classified as diseased if it s low, the sbject is health. Bt the basic qestion is How high is high? or How low is low? - i.e. where shold be the ct-point?

4 A SIMPLE PLAUSIBLE MOEL T- ctpoint T+ Separator Y is normall distribted with the same variance, bt different means; no matter where o ct, both errors reslt! More important, specificit & sensitivit are fnctions of the ctpoint.

5 ASSUMPTION In the case of man diseases, the larger vales of the separator Y are associated with the diseased poplation also called poplation of the cases and smaller vales are associated with the control or non-diseased or health poplation e.g. blood glcose for diabetes, PSA for prostate cancer, antibodies for infections, For man others, the smaller vales of the separator Y are associated with the diseased poplation and larger vales are associated with the non-diseased poplation static admittance for Otitis Media, TSH for hperthroidism. We will assme, withot loss of generalit, that larger vales of Y are associated with the diseased poplation.

6 If, in fact, smaller vales of Y are associated with the diseased poplation, methods presented here cold be applied b simpl reversing the roles of cases sbjects with the disease and controls sbjects withot the disease.

7 SENSITIVITY With or assmption that larger vales of Y are associated with the diseased poplation, the sensitivit, PrT+ +, associated with a ctpoint Y is: S + PrY> + tre positive rate - PrY + - F + where F + PrY + is the cmlative distribtion fnction cdf of Y for the diseased poplation or poplation of cases.

8 SPECIFICITY With or assmption that larger vales of Y are associated with the diseased poplation, the specificit, PrT- -, associated with a ctpoint Y is: S - PrY - F -, or - S - - F - false positive rate where F - x is the cmlative distribtion fnction cdf of Y for the non-diseased or health poplation.

9 The sensitivit, S + - F +, and the -specificit, - S - - F - are srvival fnctions. And the vales of the sensitivit and of the specificit both are fnctions of the ct-point which cold be set arbitraril.

10 ROC FUNCTION & ROC CURVE A fnction R from [0,] to [0,] that maps false positive rate to tre positive rate, -F - to -F +, is called the ROC fnction : R[-F - ] -F + or R[-S - ] S + The graph of R. is called the ROC crve The ROC crve, the graph of sensitivit, S +, verss - specificit, -S -, is generated as the ctpoint moves throgh its range of possible vales.

11 The ROC fnction maps sensitivit against -specificit or tre positive rate against false positive rate. It maps a srvival fnction against another srvival fnction.

12 In statistical terms, an ROC crve maps statistical power against tpe I error : It is a tool to present how good a statistical test of significance is. Sa, one can draw an ROC crve for the two-sample t-test and another one for the Wilcoxon test; then compare.

13 , sensitivit -F + S + tre positive rate 0 -specificit, -F - -S - false positive rate ROC Crve

14 STATISTICAL EXPRESSION -cdf is called the Srvival Fnction, St; let ] ] [ [0, ]; [S S R H + t S t S R t F t S t F t S H H

15 Isse #: How to estimate the ROC crve given two independent samples, { 0i ; i,,n 0 } and { j, j,, n } from n 0 controls and n cases?

16 EMPIRICAL ESTIMATE The simplest wa to estimate R. is to replace cdfs F + and F - b their empirical estimates p + and p - ; p + is the proportion of the n observations j s of the cases which are less than or eqal to, and p - is defined similarl. This is a non-parametric estimate and {-p -, -p + } is an nbiased estimator of {-F -, -F + } If there are no ties in the combined sample of 0i s and j s, there n 0 * n points. One cold simpl connect the dots to form a formal graphical estimate of the ROC crve;

17 A RANOM WALK Steck 97 made an statistical attempt to connect the dots, trning them into a step fnction. He combined 2 samples & in the sal increasing order. He described the empirical estimator as a random walk from the bottom-left corner 0,0 to the top-right corner, whose next step is /n p or /n 0 to the right according to whether the next observation in the ordered combined sample is a case s measrement or a control s measrement 0.

18 EXAMPLE #A: < 0 < 2 < 02 < 03 2nd 4th 3rd 5th st 0 It s like an empirical cdf of size 2 with weights /2 at points 0 & /3

19 Index for IAGNOSTIC ACCURACY ROC crve is a graphical device to show all possible combinations of sensitivit and specificit bt, for simplicit, it is desirable to redce an entire crve to a single qantitative index of diagnostic accrac. Possibilities inclde the difference between means of Y for the two poplations, those with disease and those withot; and the ratio of variances. However, the most poplar one has been the area nder the ROC crve. The area nder the crve has a powerfl interpretation and it is related to other well-known statistics making it easier to learn its statistical properties.

20 Sppose that an observation is randoml sampled from the diseased poplation and another random observation Y 0 is independentl sampled from the non-diseased poplation; and let PrY >Y 0 denote the probabilit of the event that the Y observation is larger than the Y 0 observation; we have: A Pr Y > Y 0 A F + d F A A 0 R d Area nder ROC crve

21 There are man different was to obtain standard error and/or confidence intervals bt nmerical reslts are ver similar. Pick or choice & learn

22 AN ALTERNATIVE INEX ctpoint T- T+ Separator Y is normall distribted with the same variance, bt different means; no matter where o ct, both errors reslt! The sizes of these errors depend on the standardized distance d μ μ H /σ

23 Are the two indices, A and d different? Yes, different nmerical vales bt statisticall eqivalent. If we let Φ -. denote inverse of the standard normal cmlative distribtion fnction, for example Φ , then Simpson and Fitter 973 showed that: d 2 Φ A So, what is special abot index d? It has a ver powerfl interpretation in terms of disease development!

24 LOGISTIC REGRESSION The probabilit of disease development and the vale Y of the separator Y are related b the Logistic Regression Model : ln π π π β0 + β e Pr + Y β0 + + e β 0 + β β, or

25 USE OF BAYES RULE π π Pr Pr + Y Y π π Pr Y Pr Y + Pr Pr + / / Pr Y Pr Y π π Pr Y Pr Y + Pr Pr + ln π π Constant + ln[ PrY PrY + ]

26 RESULT Sppose Y is normall distribted with the same variance, different means for PrY + and PrY -, we have: σ β d Y Y H H Constant ln ] } / exp{ } / exp{ ln[ Constant ln ] Pr Pr ln[ Constant ln σ µ µ π π σ µ σ µ π π π π

27 INTERPRETATION OF d Under logistic model and Sppose Y is normall distribted with the same variance bt different means for PrY + and PrY -, then: d β σ The vale of Index d is eqal to the logodds Ratio de to a change of one S in the vale of the marker Y

28 The Optimization Problem

29 We all know that, for example, high PSA likel indicates prostate cancer; bt how high it is to classif a man as having prostate cancer? If we set the ct-point too high, we wold miss cases that is low sensitivit ; if we set the ct-point too low, we wold have man false positives that is low specificit!

30 For a continos marker/predictor sch as PSA ; the basic qestion is How high is high? or How low is low?. In practice, ctpoints are formed arbitraril becase we fail to form and jstif a criterion or criteria. We need an optimal ctpoint ; bt what do we mean b optimal? Good, bt what it is good for? Ma be more than one soltion becase there are different criteria.

31 Basic Strateg/Criterion: To determine an optimal ctpoint for a continos marker b maximizing the Yoden s Index of the dichotomized test. Using this strateg, when sing the reslting dichotomized test in a prevalence srve we wold obtain an estimate with minimal error. There are other gains too.

32 SOLUTION #: EMPIRICAL Pool the two samples and arrange in increasing order At each midwa between two data points, calclate the Sensitivit S + and Specificit S - ; then the Yoden s Index J S + +S - - Locate the ctpoint corresponding to the maximm vale of J. Ver simple!

33 SOLUTION #2: NON-PARAMETRIC The ROC fnction R. maps U -S - on the horizontal axis to V S + on the vertical axis: V RU The Yoden s Index J S + +S - - RU - U is maximized when: 0 R U -, or R U. Process: i Smooth empirical estimate b an smoothing techniqe eg. Lowess, ii Locate the point with slope to obtain specificit, then iii Go to control sample to get ct-point.

34 , sensitivit -F + S + 0 -specificit, -F - -S - If the ROC crve is smmetric between 0,0 and,, the point on the crve with slope is closest to corner 0,.

35 SOLUTION #3: SEMI-PARAMETRIC Still looking for the point on the crve with Slope bt, first fitting empirical data with a smooth crve Y RU θ becase it wold take less data to do a better job than nonparametric smoothing we need a model bt can check for goodness-of-fit. The two components needed are: i Choosing a meaningfl parameter θ, ii Choosing a fnctional form for R.; one possibilit is the Proportional Hazard model

36 EXAMPLE: PROSTATE CANCER There were 53 patients with prostate cancer; 20 of them with nodal involvement and 33 withot. We examined level of acid phosphatase in blood serm x00. ata are reprodced from Miller et al 980 and are as follows: Patients withot Nodal Involvement: 40, 40, 46, 47, 48, 48, 49, 49, 50, 50, 50, 50, 50, 52, 52, 55, 55, 56, 59, 62, 62, 63, 65, 66, 7, 75, 76, 78, 83, 95, 98, 02, 87. Patients with Nodal Involvement: 486, 499, 56, 562.5, 6730, 6730, 6730, 7032,5, , 7235, , 7840, 84, , , 8445, 8946, 9949, 265, 3652; nmbers in parentheses are the ranks in the combined sample, mid-ranks are sed for tied observations.

37 SOLUTION #4: PARAMETRIC 0 ; [0,] ]; [ ] [ ' ' ' R R J S R J S S R t S t S R S t F t S S t F t S H H H + +

38 LOG-LOGISTIC ISTRIBUTION If lnx is distribted as logistic, X is distribted as log-logistic; the loglogistic distribtion is similar to log-normal distribtion bt with thicker tails so fits better real non-negative measrements. S t ρ ν + e σ µ,, ρ t ν ; where μ is Mean where σ St eviation

39 BOTH ARE LOG-LOGISTIC ISTRIBUTIONS β σ µ µ σ σ σ ρ ν exp : R Then t t S H H + + +

40 β 0 < µ exp < for β σ µ µ > H µ H

41 exp exp : : ] [ ' 2 ' d where S S Optimal R R R H σ µ µ β β β β β β β β β

42 SCREENING VALUE OF BIOMARKERS d S-S+ 62% 2 73% 3 82% 4 88%