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  • 1. Ben A. Dwamena, MD Department of Radiology, University of Michigan Medical School Nuclear Medicine Service, VA Ann Arbor Health Care System Ann Arbor, Michigan METAGRAPHITI
  • 2. Statistical Graphics For Interpretation, Exploration And Presentation Of Meta-analysis Data METAGRAPHITI
  • 3. VISUOGRAPHIC FRAMEWORK FOR  Exploring distributional assumptions  Testing and correcting for publication bias  Investigating heterogeneity  Summary of Data and Sensitivity Analyses METAGRAPHITI
  • 4.  Avoid potential misrepresentation by faulty distributional and other statistical assumptions.  Facilitates greater interaction between the researcher and the data by highlighting interesting and unusual aspects of the quantitative data. METAGRAPHITI
  • 5.  User-friendlier summaries of large, complicated quantitative data sets  Preliminary exploration before definite data synthesis  Effective emphasis of important features rather than details of data METAGRAPHITI
  • 6. CONTINGENCY TABLE FOR SINGLE STUDY
  • 7.  True Positives =Experimental Group With Outcome Present (a).  False Positives = Control Group With Outcome Present (b).  False Negatives=Experimental Group With Outcome Absent (c).  True Negatives= Control Group With Outcome Absent (d). DIAGNOSTIC VERSUS TREATMENT TRIAL
  • 8. Odds Ratio (OR) =(a x d)/(b x c). Relative risk in experimental group {[a/(a + c)]/[b/(b+ d)]} =Likelihood Ratio for a Positive Test. Relative Risk in Control Group = Likelihood Ratio for a Negative Test. DIAGNOSTIC VERSUS TREATMENT TRIAL
  • 9. Box plots Normal quantile plots Stem-and-Leaf plots DISTRIBUTION PLOTS
  • 10.  Displays important characteristics of the dataset based on the five-number summary of the data.  “Box” covers inter-quartile range.  “Beltline” of box represents the median value.  “Whiskers” include all but outlier observations. BOX AND WHISKER PLOT
  • 11. 123456 BOX AND WHISKER PLOT
  • 12. 1** | 47 1** | 81,93 2** | 20,48 2** | 51,86 3** | 04,22 3** | 59,81,85 4** | 10 4** | 59,67,67,68 5** | 24,34,48 5** | 57,58,67 6** | 06 rounded to nearest multiple of .01 plot in units of .01 STEM-AND-LEAF PLOT
  • 13.  Plot of standardized effect size, Ei/√Vi vs. normal distribution.  Deviations from linearity ⇒ deviations from normality.  Slope of regression line =standard deviation of data= 1 for effect size if the studies from a single population and have large samples.   The y-intercept of the regression =the mean. NORMAL QUANTILE PLOT
  • 14. NORMAL QUANTILE PLOT
  • 15. NORMAL QUANTILE PLOT
  • 16. NORMAL QUANTILE PLOT
  • 17. NORMAL QUANTILE PLOTS
  • 18. Selective publication of articles showing certain types of results over those of showing other types of results Commonly, tendency to publish only studies with statistical significant results PUBLICATION BIAS
  • 19. Published studies do not represent all studies on a specific topic.  Trend towards publishing statistically significant (p < 0.05) or clinically relevant results.  Publication bias assessed by examining asymmetry of funnel plots of estimates of odds ratios vs. precision. INVESTIGATING PUBLICATION BIAS
  • 20. Funnel plot Begg’s rank correlation plot Egger’s regression plot Harbord’s modified radial plot INVESTIGATING PUBLICATION BIAS
  • 21. A funnel diagram (a.k.a. funnel plot, funnel graph, bias plot) Special type of scatter plot with an estimate of sample size on one axis vs. effect-size estimate on the other axis FUNNEL PLOT
  • 22. Based on statistical principle that sampling error decreases as sample size increases Used to search for publication bias and to test whether all studies come from a single population FUNNEL PLOT
  • 23. FUNNEL PLOTS
  • 24. metafunnel ldor seldor, xlab(0(2)8) xtitle (Log odds ratio) ytitle(Standard error of log OR) saving(zfunnel, replace) metafunnel ldor seldor, xlab(0(2)8) xtitle(Log odds ratio) ytitle (Standard error of log OR) egger saving (eggerfunnel, replace) FUNNEL PLOT: STATA SYNTAX
  • 25. FUNNEL PLOT: STATA DIALOG
  • 26. 0.511.52 StandarderroroflogOR 0 2 4 6 8 Log odds ratio Funnel plot with pseudo 95% confidence limits FUNNEL PLOT: EXAMPLE
  • 27. 0.511.52 StandarderroroflogOR 0 2 4 6 8 Log odds ratio Funnel plot with pseudo 95% confidence limits FUNNEL PLOT: WITH REGRESSION LINE
  • 28. An adjusted rank correlation method to assess the correlation between effect estimates and their variances. Deviation of Spearman's rho from zero=estimate of funnel plot asymmetry. Positive values=a trend towards higher levels of effect sizes in studies with smaller sample sizes BEGG’S BIAS TEST
  • 29. metabias LogOR seLogOR, graph(b) saving(beggplot, replace) BEGG’S BIAS TEST: STATA SYNTAX
  • 30. BEGG’S BIAS TEST: STATA DIALOG
  • 31. Begg's funnel plot with pseudo 95% confidence limits logOR s.e. of: logOR 0 .5 1 1.5 2 0 5 10 BEGG’S BIAS PLOT
  • 32. Adjusted Kendall's Score (P-Q) = 26 Std. Dev. of Score = 40.32 Number of Studies = 24 z = 0.64 Pr > |z| = 0.519 z = 0.62 (continuity corrected) Pr >|z| = 0.53(continuity corrected) BEGG’S BIAS TEST: STATISTICS
  • 33.  Assesses potential association b/n effect size and precision.  Regression equation: SND = A + B x SE(d)-1. 1. SND=standard normal deviate (effect, d divided by its standard error SE(d)); 2. A =intercept 3. B=slope. . EGGER’S REGRESSION TEST
  • 34. The intercept value (A) = estimate of asymmetry of funnel plot Positive values (A > 0) indicate higher levels of effect size in studies with smaller sample sizes. EGGER’S REGRESSION METHOD
  • 35. EGGER’S BIAS PLOT
  • 36. metabias logOR selogOR, graph(e) saving(eggerplot, replace) EGGER’S BIAS TEST: STATA SYNTAX
  • 37. EGGER’S BIAS TEST: STATA DIALOG
  • 38. Egger's publication bias plot standardizedeffect precision 0 1 2 3 4 0 2 4 6 8 EGGER’S BIAS PLOT: EXAMPLE
  • 39.   ------------------------------------------------------------- Std_Eff | Coef. P>|t| [95% CI] -------------+----------------------------------------------- slope | 1.737492 0.001 .8528166 2.622168 bias | 1.796411 0.002 .7487423 2.84408 ------------------------------------------------------------- EGGER’S BIAS TEST: STATISTICS
  • 40. Test for funnel-plot asymmetry Regresses Z/sqrt(V) vs. sqrt (V), where Z is the efficient score and V is Fisher's information (the variance of Z under the null hypothesis). Modified Galbraith plot of Z/sqrt(V) vs. sqrt(V) with the fitted regression line and a confidence interval around the intercept. MODIFIED BIAS TEST(HARBORD)
  • 41. metamodbias tp fn fp tn, graph z(Z) v(V) mlabel(index) saving(HarbordPlot, replace) MODIFIED BIAS TEST: STATA SYNTAX
  • 42. 8 18 213 13 7 15 6 4 9 22 11 10 12 24 17 1 2 14 16 19 5 23 20 051015 Z/sqrt(V) 0 1 2 3 4 sqrt(V) Study regression line 90% CI for intercept MODIFIED BIAS PLOT
  • 43. ----------------------------------------------------------------------------- ZoversqrtV | Coef. Std. Err. P>|t| [90% Conf. Interval] --+-------------------------------------------------------------------------- sqrtV| 2.406756 .3464027 0.000 1.811933 3.00158 bias| .9965934 .6383554 0.133 -.0995549 2.092742 ----------------------------------------------------------------------------- MODIFIED BIAS TEST: STATISTICS
  • 44.  A rank-based data augmentation technique  used to estimate the number of missing studies and to produce an adjusted estimate of test accuracy by imputing suspected missing studies.  Both random and fixed effect models may be used to assess the impact of model choice on publication bias. TRIM-AND-FILL METHOD
  • 45. • metatrim LogOR seLogOR, eform funnel print graph id(author)saving(tweedieplot, replace) TRIM-AND-FILL TEST: STATA SYNTAX
  • 46. TRIM AND FILL: STATA DIALOG
  • 47. TRIM-AND-FILL BIAS PLOT Filled funnel plot with pseudo 95% confidence limits theta,filled s.e. of: theta, filled 0 .5 1 1.5 2 -2 0 2 4 6
  • 48.  When effect sizes differences are attributable to only sampling error, studies are homogeneous.  Heterogeneity means that there is between-study variation and variability in effect sizes exceeds that expected from sampling error. HETEROGENEITY
  • 49. Potential sources of heterogeneity: 1. Characteristics of study population 2. Variation in study design 3. Statistical methods 4. Covariates adjusted for (if relevant) HETEROGENEITY
  • 50. DEALING WITH HETEROGENEITY  Use analysis of variance with the log odds ratio as dependent variable and categorical variables for subgroups as factors to look for differences among subgroups
  • 51. DEALING WITH HETEROGENEITY  Repeat analysis after excluding outliers  Conduct analysis with predefined subgroups  Construct multivariate models to search for the independent effect of study characteristics
  • 52. Standardized effect vs. reciprocal of the standard error. Small studies/less precise results appear on the left side and the largest trials on the right end . GALBRAITH’S PLOT
  • 53. A regression line , through the origin, represents the overall log-odds ratio. Lines +/- 2 above regression line =95 per cent boundaries of the overall log-odds ratio.  The majority of points within area of +/- 2 in the absence of heterogeneity. GALBRAITH’S PLOT
  • 54. galbr LogOR seLogOR, id(index) yline(0) saving(gallplot, replace) GALBRAITH’S PLOT: STATA SYNTAX
  • 55. GALBRAITH PLOT: STATA DIALOG
  • 56. b/se(b) 1/se(b) b/se(b) Fitted values 0 3.80729 -2-2 0 2 13.1045 . 7 8 1315 11 18 321 1222 1419 17 6 9 2 24 1 4 10 23 16 5 20 GALBRAITH PLOT: EXAMPLE
  • 57. This plots the event rate in the experimental (intervention) group against the event rate in the control group  Visual aid to exploring the heterogeneity of effect estimates within a meta-analysis. L’ABBE PLOT
  • 58. labbe tp fn fp tn, s(O) xlab(0,0.25,0.50,0.75,1) ylab(0,0.25,0.50,0.75,1) l1("TPR) b2("FPR") saving(flabbeplot, replace) L’ABBE PLOT: STATA SYNTAX
  • 59. L’ABBE PLOT: STATA DIALOG
  • 60. TPR FPR 0 .25 .5 .75 1 0 .25 .5 .75 1 L’ABBE PLOT: EXAMPLE
  • 61. twoway (rcap dorlo dorhi Study, horizontal blpattern(dash))(scatter Study dor, ms(O)msize(medium) mcolor(black))(scatter DOR_with_CIs eb_dor, yaxis(2) msymbol(i) msize(large) mcolor(black))(scatteri 26 83, msymbol(diamond) msize(large)), ylabel(1(1)25 26 "OVERALL", valuelabels angle(horizontal)) xlabel(0 10 100 1000 10000) xscale(log) ylabel(1(1)25 26 "Pooled Estimate", valuelabels angle(horizontal) axis(2)) legend(off) xtitle(Odds Ratio) xline(83, lstyle(foreground)) saving(OddsForest, replace) DATA SUMMARY: STATA SYNTAX
  • 62. metan tp fn fp tn, or random nowt sortby(year) label(namevar=author, yearvar=year) t1(Summary DOR, Random Effects) b2(Diagnostic Odds Ratio) saving(SDORRE, replace) force xlabel(0,1,10,100,1000) DATA SYNTHESIS: RANDOM EFFECTS
  • 63. • metan tp fn fp tn, or fixed nowt sortby(year) label(namevar=author, yearvar=year) t1(Summary DOR, Fixed Effects) b2(Diagnostic Odds Ratio) saving(SDORFE, replace) force xlabel(0,1,10,100,1000) DATA SYNTHESIS: FIXED EFFECTS
  • 64. 106.33(6.40 - 1765.75 107.23(40.31 - 285.28 107.67(6.57 - 1763.99 12.33(2.04 - 74.41) 17.50(2.65 - 115.66) 175.00(28.22 - 1085.2 189.00(11.36 - 3143.3 191.67(12.02 - 3055.9 21.00(1.43 - 308.21) 240.88(68.75 - 843.96 25.00(1.72 - 364.11) 262.66(24.39 - 2829.1 266.00(33.46 - 2114.7 32.82(5.43 - 198.37) 33.00(8.83 - 123.26) 4.33(1.05 - 17.90) 429.00(14.65 - 12562. 45.00(3.75 - 539.38) 46.93(17.48 - 125.96) 56.08(4.62 - 680.33) 6.10(3.95 - 9.42) 6.86(1.81 - 26.01) 7.09(1.06 - 47.42) 9.00(0.58 - 140.71) 98.80(15.24 - 640.50) Pooled Estimate DOR_with_CIs Adler 1997 Avril 1996 Bassa 1996 Danforth 2002 Greco 2001 Guller 2002 Hubner 2000 Inoue 2004 Lin 2002 Nakamoto(a) 2002 Nakamoto(b) 2002 Noh 1998 Ohta 2000 Palmedo 1997 Rostom 1999 Scheidhauer 1996 Schirrmeister 2001 Smith 1998 Tse 1992 Utech 1996 Wahl 2004 Yang 2001 Yutani 2001 Zornoza 2004 van Hoeven 2002 OVERALL 0 10 100 1000 10000 Odds Ratio FOREST PLOT: STATA GRAPHICS
  • 65. Assumes homogeneity of effects across the studies being combined. There is a common true effect size for all studies. In the summary estimate, only the variance of each study is taken into account. FIXED EFFECTS META-ANALYSIS
  • 66. Summary DOR, Fixed Effects Odds ratio .1 1 100 1000 10000 Study Odds ratio (95% CI) 36.27 (4.27,308.02)Adler (1997) 98.80 (10.66,916.11)Avril (1996) 21.00 (0.86,515.50)Bassa (1996) 4.33 (0.80,23.49)Danforth (2002) 107.23 (33.42,344.07)Greco (2001) 12.00 (1.23,117.41)Guller (2002) 429.00 (7.67,23982.81)Hubner (2000) 189.00 (6.63,5384.60)Lin (2002) 17.50 (1.84,166.04)Nakamoto (2002) 6.86 (1.40,33.57)Nakamoto (2002) 208.33 (7.72,5621.57)Noh (1998) 60.23 (3.09,1174.51)Ohta (2000) 106.33 (3.74,3023.90)Palmedo (1997) 289.00 (15.83,5276.04)Rostom (1999) 107.67 (3.85,3013.13)Scheidhauer (1996) 46.93 (14.47,152.17)Schirrmeister (2001) 266.00 (22.50,3145.19)Smith (1998) 9.00 (0.34,238.21)Tse (1992) 262.66 (15.47,4459.70)Utech (1996) 6.10 (3.64,10.24)Wahl (2004) 25.00 (1.03,608.09)Yang (2001) 45.00 (2.33,867.81)Yutani (2001) 240.88 (54.08,1072.94)Zornoza (2004) 12.33 (1.45,104.97)van Hoeven (2002) 19.85 (14.16,27.82)Overall (95% CI) FIXED EFECTS META-ANALYSIS
  • 67. Heterogeneity is incorporated into the pooled estimate by including a between study component of variance.  Assumes sample of studies included in the analysis is drawn from a population of studies.  Each sample of studies has a true effect size. RANDOM EFFECTS META-ANALYSIS
  • 68. Summary DOR, Random Effects Odds ratio .1 1 100 1000 10000 Study Odds ratio (95% CI) 36.27 (4.27,308.02)Adler (1997) 98.80 (10.66,916.11)Avril (1996) 21.00 (0.86,515.50)Bassa (1996) 4.33 (0.80,23.49)Danforth (2002) 107.23 (33.42,344.07)Greco (2001) 12.00 (1.23,117.41)Guller (2002) 429.00 (7.67,23982.81)Hubner (2000) 189.00 (6.63,5384.60)Lin (2002) 17.50 (1.84,166.04)Nakamoto (2002) 6.86 (1.40,33.57)Nakamoto (2002) 208.33 (7.72,5621.57)Noh (1998) 60.23 (3.09,1174.51)Ohta (2000) 106.33 (3.74,3023.90)Palmedo (1997) 289.00 (15.83,5276.04)Rostom (1999) 107.67 (3.85,3013.13)Scheidhauer (1996) 46.93 (14.47,152.17)Schirrmeister (2001) 266.00 (22.50,3145.19)Smith (1998) 9.00 (0.34,238.21)Tse (1992) 262.66 (15.47,4459.70)Utech (1996) 6.10 (3.64,10.24)Wahl (2004) 25.00 (1.03,608.09)Yang (2001) 45.00 (2.33,867.81)Yutani (2001) 240.88 (54.08,1072.94)Zornoza (2004) 12.33 (1.45,104.97)van Hoeven (2002) 42.54 (20.88,86.68)Overall (95% CI) RANDOM EFFECTS META-ANALYSIS
  • 69.  Process of “prospectively” performing a new or updated analysis every time another trial is published  Provides answers regarding effectiveness of an intervention at the earliest possible date in time CUMULATIVE META-ANALYSIS
  • 70.  Studies are sequentially pooled by adding each time one new study according to an ordered variable.  For instance, the year of publication; then, a pooling analysis will be done every time a new article appears. CUMULATIVE META-ANALYSIS
  • 71.  In theory, the effect of any continuous or ordinal study-related variable can be assessed  Ex: sample size, study quality score, baseline risk etc CUMULATIVE META-ANALYSIS
  • 72. metacum LogOR seLogOR, eform id(author) effect(f) graph cline saving(year_fcummplot, replace) CUMULATIVE META-ANALYSIS: SYNTAX
  • 73. CUMULATIVE META-ANALYSIS: DIALOG 1
  • 74. CUMULATIVE META-ANALYSIS: DIALOG 2
  • 75. LogOR 7.67387 23982.8 Wahl 2004 Yang 2001 Yutani 2001 Zornoza 2004 Inoue 2004 Nakamoto(b) 2002 Nakamoto(a) 2002 van Hoeven 2002 Adler 1997 Rostom 1999 Avril 1996 Guller 2002 Schirrmeister 2001 Smith 1998 Utech 1996 Greco 2001 Ohta 2000 Hubner 2000 Bassa 1996 Tse 1992 Noh 1998 Scheidhauer 1996 Palmedo 1997 Lin 2002 Danforth 2002 CUMULATIVE META-ANALYSIS: PLOT
  • 76.  Studies are pooled according to influence of a trial on overall effect defined as the difference between the effect estimated with and without the trial INFLUENCE ANALYSIS
  • 77. metaninf tp fn fp tn, id(author) saving(influplot, replace) save(infcoeff, replace) INFLUENCE ANALYSIS: STATA SYNTAX
  • 78. INFLUENCE ANALYSIS: STATA DIALOG
  • 79. INFLUENCE ANALYSIS: STATA DIALOG
  • 80. INFLUENCE ANALYSIS: PLOT 4.75 6.535.41 7.88 10.30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Lower CI Limit Estimate Upper CI Limit Meta-analysis estimates, given named study is omitted
  • 81. INFLUENCE ANALYSIS: STATA DIALOG
  • 82. A scatter plot of true positive fraction (sensitivity) vs. false positive fraction (1- specificity) Aids in visualization of range of results from primary studies ROC PLANE
  • 83. twoway (scatter TPF FPF, sort ) (lfit uTPR FPF, sort range(0 1) clcolor(black) clpat(dash) clwidth(vthin) connect(direct)) (lfit sTPR FPF, sort range(0 1) clcolor(black) clpat(dot) clwidth(vthin) connect(direct)), ytitle(Sensitivity) ylabel(0(.1)1, grid) xtitle(1-Specificity) xlabel(0(.1)1, grid) title(ROC Plot of SENSITIVITY vs. 1-SPECIFICITY, size(medium)) legend(pos(3) col(1) lab(1 "Observed Data") lab(2 "Uninformative Test") lab(3 "Symmetry Line")) saving(ROCplot, replace) plotregion(margin(zero)) ROC PLANE: STATA SYNTAX
  • 84. 0.1.2.3.4.5.6.7.8.91 Sensitivity 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 1-Specificity Observed Data Uninformative Test Symmetry Line ROC Plot of SENSITIVITY vs. 1-SPECIFICITY ROC PLANE: PLOT
  • 85.  ORDINARY LEAST SQUARES METHOD: Studies are weighted equally  WEIGHTED LEAST SQUARES METHOD: Weighted by the inverse variance weights of the odds ratio, or simply the sample size  ROBUST-RESISTANT METHOD: Minimizes the influence of outliers SROC: LINEAR REGRESSION MODELS
  • 86.  Logit transformations of the TP rate (sensitivity) and FP rate (1 - specificity). D=ln(DOR) =logit(TPR) – logit(FPR)  Differences in logit transformations, D, regressed on sums of logit transformations, S. S=logit(TPR)+logit(FPR)  Logit(TPR)=natural log odds of a TP result and logit(FPR) =natural log of the odds of a FP test result. SROC: LOGIT TRANSFORMATION
  • 87. twoway (scatter D S, sort msymbol(circle)) (lfit tfitted S, clcolor(black) clpat(solid) clwidth(thin) connect(direct))(lfit wfitted S, clcolor(black) clpat(dash) clwidth(thin) connect(direct)), ytitle(Discriminatory Power/D) xtitle(Diagnostic Threshold/S) title(REGRESSION PLOT) legend(lab(1 "Observed Data")lab(2 "EWLSR")lab(3 "VWLSR"))saving(regplot, replace) xline(0) yscale(noline) ACCURACY/THRESHOLD PLOT
  • 88. 123456 DiscriminatoryPower/D -4 -2 0 2 4 Diagnostic Threshold/S Observed Data EWLSR VWLSR REGRESSION PLOT ACCURACY/THRESHOLD PLOT
  • 89.  Back transformation of logistic regression to conventional axes of sensitivity [TPR] vs. (1 – specificity) [FPR]) with the equation  TPR = 1/{1 + exp[- a/(1 - b )]} [(1 - FPR)/(FPR)](1+b)/(1-b) .  Slope b and intercept a are obtained from the linear regression analyses SUMMARY ROC CURVE
  • 90. twoway (scatter TPF FPF, sort msymbol(circle) msize(medium) mcolor(black))(fpfit tTPR FPF, clpat(dash)clwidth(medium) connect(direct ))(fpfit wTPR FPF, clpat(solid)clwidth(medium) connect(direct ))(lfit uTPR FPF, sort range(0 1) clcolor(black) clpat(dash) clwidth(thin) connect(direct)) (lfit sTPR FPF, sort range(0 1) clcolor(black) clpat(dot) clwidth(medium) connect(direct)), ytitle(Sensitivity/TPF) yscale(range(0 1)) ylabel( 0(.2)1,grid ) xtitle(1- Specificity/FPF) xscale(range(0 1)) xlabel(0(.2)1, grid) legend(lab(1 "Observed Data")lab(2 "EWLSR")lab(3 "VWLSR")lab(4 "RRLSR")lab(5 "Uninformative Test") lab(6 "Symmetry Line") pos(3) col(1)) title(SUMMARY ROC CURVES) graphregion(margin(zero)) saving(aSROCplot, replace) SUMMARY ROC CURVE: STATA SYNTAX
  • 91. 0.2.4.6.81 Sensitivity/TPF 0 .2 .4 .6 .8 1 1-Specificity/FPF Observed Data EWLSR VWLSR RRLSR Uninformative Test SUMMARY ROC CURVES SUMMARY ROC CURVE: EXAMPLE
  • 92. SUMMARY ROC : SUBGROUP ANALYSIS
  • 93. STATA 8.2 (Stata Corp, College Station, Texas, USA) SOFTWARE

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