3. general considerations for the analysis of case-control ... - IARC

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LOGISTIC REGRESSION FOR MATCHED SETS 255 Table 7.3 Results of fitting the conditional logistic regression model to matched pairs consisting of the case and first matched control: Los Angeles study of endometrial cancer No. of Goodness of fit Score testa Regression coefficients f standard error for each variable in equation parameters (G) A. Any oestrogens EST 2.269 f 0.606 B. Gall-bladder disease and age GALL GALL x AGEGP 3.56 0.956 f 0.526 1.68 1.946 f 1.069 -1.540 k 1.249 0.3Sb 1.052 k 0.566 GALL x (AGE-70) C. HypertensionIGall-bladder disease GALL HYP 0.81 0.325 + 0.364 3.61 0.970 f 0.531 0.348 + 0.364 2.01 1.51 7 k 0.699 0.627 f 0.435 D. Gall-bladder disease1Hypertension GALL HYP 1 83.65 3.56 0.956 f.0.526 2 82.79 0.86 0.970 k 0.531 0.348 f 0.377 3 80.84 2.01 1.517 f 0.699 0.627 f 0.435 GALL x HYP GALL x HYP -1.548f 1.125 E. Dose levels of conjugated oestrogen F. Coded dose of conjugated oestrogen DOS DOS x AGE 1 65.50 14.71 0.690 k 0.202 2 65.50 0.00 0.693 f 0.282 -0.001 f 0.403 "Score statistic comparing each model with the preceding model in each set, unless otherwise indicated. The first model in each set is compared with the model in which all p's are 0. After fitting one parameter model with GALL only reflects .the fact that the conditional likelihood (7.2) is identical (up to a constant of proportionality) to that used earlier as a basis of inference (5.3), so that the two analyses are entirely equivalent. Likewise, the score statistic for the test of the null hypothesis, Ho: 11, = 1, is identical with the uncorrected (for continuity) value of the x2 defined in (5.4), namely
256 BRESLOW & DAY This illustrates the point that many of the elementary tests are in fact score tests based on the model (Day & Byar, 1979). The corrected chi-square value is of course the more accurate and preferred one, but it has not been incorporated in the computer programme written for the general regression analysis, since it is not applicable in other situations. Two other statistics are available for testing the null hypothesis. These are the differences in goodness-of-fit measures, 87.34-62.89 = 24.45, and the square of the standardized regression coefficient, (2.269/0.606)2 = 13.99, each of which also has a nominal X; distribution under the null hypothesis. Although the three values are somewhat disparate with these data, they all indicate a highly significant effect. The test based on the corrected score statistic is preferred when available, as this comes closest to the corresponding exact test. Asymptotic 95% confidence limits for 11, are calculated as exp(2.269 + 1.96 x 0.606) = (2.9, 31.7), the upper limit being noticeably smaller than that based on the exact conditional (binomial) distribution (VU = 49.6) or the normal approximation to it (vu = 39.7) which were calculated in 5 5.2. Part B of Table 7.3 presents the relative risk estimate for gall-bladder disease and its relationship to age. Just as for EST, the estimate of relative risk associated with GALL, exp(0.956) = 2.6 = 13/5, and the (uncorrected) score statistic, 3.56 = (13-5)2/18, must agree with the values found earlier. There is better concordance between the three available tests of the null hypothesis in this (less extreme) case: 87.34-83.65 = 3.69 for the test based on G, and (0.956/0.526)2 = 3.30 for that based on the standardized coefficient, are the other two values besides the score test. For the second model in Part B the coefficient of GALL represents the log relative risk for those under 70 years of age, exp(1.946) = 7.0 = 7/1, while the sum of the coefficients for GALL and GALLx AGEGP gives the log relative risk for those 70 and over, exp(1.946-1.540) = 1.50 = 6/4. These are the same results as found before. Similarly, the score statistic for the additional parameter GALL x AGEGP, which tests the equality of the relative risk estimates in the two age groups, is identical to the uncorrected chi-square test for equality of the proportions 7/8 and 6/10, namely In 5 5.2 we reported the corrected value of this chi-square as x2 = 0.59. The third line of Part B of the table introduces an interaction term with the continuous matching variable AGE. Here the coefficient of GALL gives the estimated relative risk for someone aged 70, exp(1.052) = 2.86, while the relative risk for other ages is determined from exp(1.052-0.066(AGE-70)). In other words, the RR is estimated to decline by a factor exp(4.066) = 0.936 for each year of age above 70 and increase by a factor exp(0.066) = 1.068 for each year below. However this tendency has no statistical significance; all three of the available tests for homogeneity give a chi-square of about 0.35 (p = 0.56). Such continuous variable modelling is of course not available with the elementary techniques. Part C of Table 7.3 illustrates the increased analytical power which is available using regression methods. In order to estimate and test the relative risk of gall-bladder disease, while controlling for hypertension, we start with an equation containing the
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WORLD HEALTH ORGANIZATION INTERNATI
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N.E. Breslow & N. E. Day (1980) Sta
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CONTENTS Foreword .................
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PREFACE Twenty years have elapsed s
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ACKNOWLEDGEMENTS Since the initial
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LIST OF PARTICIPANTS AT IARC WORKSH
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1.1 The case-control study in cance
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INTRODUCTION 15 a specific disease
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History INTRODUCTION 17 In 1926 Lan
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INTRODUCTION 19 day human affairs c
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INTRODUCTION 21 cording to age, sex
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INTRODUCTION 1.5 Planning The case
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INTRODUCTION 25 determinants of sur
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INTRODUCTION 27 like exposed subjec
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INTRODUCTION 29 warrants careful co
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INTRODUCTION 31 strata, and formati
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INTRODUCTION 33 in which at least 1
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INTRODUCTION 35 analytical techniqu
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INTRODUCTION 37 ity; is it understo
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INTRODUCTION 39 Hutchison, G. B. (1
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2. FUNDAMENTAL MEASURES OF DISEASE
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MEASURES OF DISEASE 43 operate prio
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MEASURES OF DISEASE 4 5 change is r
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MEASURES OF DISEASE 4 7 veillance s
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MEASURES OF DISEASE 49 Fig. 2.3 Age
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MEASURES OF DISEASE t A (t) = 2 l(n
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MEASURES OF DISEASE Table 2.4 Cumul
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MEASURES OF DISEASE 55 exposed vers
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MEASURES OF DISEASE 57 Example: Fig
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MEASURES OF DISEASE 59 explore thes
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31 5 42.5 47.5 52.5 57.5 62.5 67.5
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MEASURES OF DISEASE 63 Fig. 2.8 Rat
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MEASURES OF DISEASE 65 Fig. 2.9 Age
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MEASURES OF DISEASE 67 Table 2.8 Jo
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2.7 Logical properties of the relat
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MEASURES OF DISEASE 71 Example: Sup
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MEASURES OF DISEASE 73 or similar b
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MEASURES OF DISEASE 75 categories i
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MEASURES OF DISEASE 77 RAR = AR2-AR
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MEASURES OF DISEASE 79 Court Brown,
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MEASURES OF DISEASE 8 1 Plk P2 k R
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CHAPTER I11 GENERAL CONSIDERATIONS
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86 BRESLOW & DAY 3.2 Criteria for a
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88 BRESLOW & DAY Fig. 3.1 Relative
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90 BRESLOW & DAY Considerations ext
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92 BRESLOW & DAY of this type can b
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94 BRESLOW & DAY decreases breast c
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96 BRESLOW & DAY E and those not ex
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98 BRESLOW & DAY The confounding ef
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BRESLOW & DAY Comparison of w* with
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Factor C+ Exposure E + - BRESLOW &
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104 BRESLOW & DAY there were limita
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106 BRESLOW & DAY to have a strong
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1 08 BRESLOW & DAY 3. If a factor i
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110 BRESLOW & DAY Table 3.6 Risk fo
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112 BRESLOW & DAY risk for some fac
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114 BRESLOW & DAY an additional cat
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116 BRESLOW & DAY Boice, J.D. & Mon
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118 BRESLOW & DAY Report of the Sur
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4. CLASSICAL METHODS OF ANALYSIS OF
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ANALYSIS OF GROUPED DATA Table 4.1
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ANALYSIS OF GROUPED DATA 125 such a
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Estimation of q ANALYSIS OF GROUPED
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ANALYSIS OF GROUPED DATA 129 for th
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ANALYSIS OF GROUPED DATA 131 analog
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ANALYSIS OF GROUPED DATA 133 Referr
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ANALYSIS OF GROUPED DATA 135 Halper
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ANALYSIS OF GROUPED DATA 137 are pa
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ANALYSIS OF GROUPED DATA 1.39 signi
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ANALYSIS OF GROUPED DATA 141 strata
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ANALYSIS OF GROLIPED DATA 143 appro
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ANALYSIS OF GROLIPED DATA 145 Table
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ANALYSIS OF GROUPED DATA Exposure l
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ANALYSIS OF GROUPED DATA 149 Nor wi
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ANALYSIS OF GROUPED DATA 151 Alcoho
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ANALYSIS OF GROUPED DATA 153 relati
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ANALYSIS OF GROUPED DATA 155 obtain
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ANALYSIS OF GROUPED DATA 157 Gart,
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ANALYSIS OF GROUPED DATA 159 Cov(x,
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CHAPTER V CLASSICAL METHODS OF ANAL
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164 BRESLOW & DAY 5.2 Matched pairs
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166 BRESLOW & DAY Approximate confi
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168 BRESLOW & DAY where 2.42 = F.,,
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170 BRESLOW & DAY The first and las
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BRESLOW & DAY This is a special cas
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ANALYSIS OF MATCHED DATA Case Expos
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176 BRESLOW & DAY or limits of 3.8-
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BRESLOW & DAY Table 5.3 Data layout
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180 BRESLOW & DAY Case : control ra
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182 BRESLOW & DAY = 11.82, from whi
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184 BRESLOW & DAY observed and expe
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BRESLOW & DAY Their solution, obtai
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188 BRESLOW & DAY Miettinen, 0. S.
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6. UNCONDITIONAL LOGISTIC REGRESSIO
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LOGISTIC REGRESSION FOR LARGE STRAT
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Page 202 and 203: LOGISPIC REGRESSION FOR LARGE STRAT
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Page 222 and 223: 6.8 Regression adjustment for confo
Page 224 and 225: 6.9 Analysis of continuous data LOG
Page 244 and 245: 7. CONDITIONAL LOGISTIC REGRESSION
Page 246 and 247: LOGISTIC REGRESSION FOR MATCHED SET
Page 256 and 257: LOGIS-I-IC REGRESSION FOR MATCHED S
Page 260 and 261: Table 7.7 Matched multivariate anal
Page 264 and 265: Table 7.9 (contd) C. Dose, duration
Page 270 and 271: LOGIS'TIC REGRESSION FOR MATCHED SE
Page 272 and 273: Table 7.12 Coefficients (f standard
Page 278 and 279: APPENDIX I AGE (YRS) ALCOHOL (GM/DA
Page 280 and 281: APPENDIX 11: GROUPED DATA FROM THE
Page 282 and 283: APPENDIX II YEAR OF DEATH AGE AT DE
Page 284 and 285: APPENDIX II YEAR OF DEATH AGE AT DE
Page 286 and 287: APPENDIX 111: MATCHED DATA FROM THE
Page 288 and 289: 292 APPENDIX Ill CASE OR CONTROL AG
Page 290 and 291: 294 APPENDIX Ill CASE OR CONTROL AG
Page 292 and 293: 296 APPENDIX Ill CASE OR CONTROL CA
Page 294 and 295: APPENDIX IV MAIN PROGRAM .C MASTER
Page 296 and 297: 300 APPENDIX IV 103 FORMAT(1H ,'NUM
Page 298 and 299: 302 APPENDIX IV C UPON CONVERGENCE
Page 300 and 301: APPENDIX IV MODEL WITH 2 VARIABLES:
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APPENDIX IV MODEL WITH 4 VARIABLES:
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APPENDIX V MAIN PROGRAM DIMENSION N
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31 0 APPENDIX V C REFERENCES : C N.
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APPENDIX V CALL SYMINV~COVI,NP,COV,
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APPENDIX V SUBROUTINE SYMINVCA, N,
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APPENDIX V SUBROUTINE TWIDL(X, Y, Z
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APPENDIX V MODEL WITH SINGLE VARIAB
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APPENDIX V MODEL WITH 3 VARIABLES:
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APPENDIX VI LISTING OF PROGRAM LOGO
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C C C C C C APPENDIX VI SUBPROGRAM
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APPENDIX VI SUBROUTINE CALC(NTAB,P,
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328 APPENDIX VI SUBROUTINE MYSTCT,
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APPENDIX VI DOUBLE PRECISION FUNCTI
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332 APPENDIX VI C C C C C C REDUCE
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MODEL WITH LINEAR EFFECT OF YEAR RE
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TABLE A 0 E 31 12 10.64 32 4 5.12 3
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0 03 TABLE A B 0 E 0 E 9 1 6 7.06 6
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340 SUBJECT INDEX Biological monito
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342 SUBJECT INDEX Confounding risk
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344 SUBJECT INDEX Grouped data anal
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346 SUBJECT INDEX Matched case-cont
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348 SUB.IECT INDEX Poisson variabil
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350 SUB-IECT INDEX Tobacco consumpt
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