Statistical Methods in Medical Research 4ed

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Table 19.5 Combination of relative risks from 10 retrospective surveys on smoking and lung cancer (Cornfield, 1956; Gart 1962). Study number Lung cancer patients Control patients Smokers ai Nonsmokers bi Smokers ci Nonsmokers di ^c i ln ^c i Woolf 's method Mantel±Haenszel (1) 1=ai ‡ 1=bi ‡ 1=ci ‡ 1=di wi ˆ 1 …1† wi ln ^c i ni aidi=ni bici=ni E…ai† var…ai† 1 83 3 72 14 5 38 1 683 0 4307 2 3 3 87 172 6 756 1 256 77 50 3 85 2 90 3 227 43 5 68 1 737 0 3721 2 7 4 69 363 10 661 1 876 81 21 7 68 3 129 7 81 19 4 32 1 463 0 2156 4 6 6 73 236 10 386 2 403 121 02 5 67 4 412 32 299 131 5 64 1 730 0 0447 22 4 38 75 874 61 753 10 947 361 19 33 18 5 1350 7 1296 61 9 08 2 206 0 1608 6 2 13 68 2714 30 343 3 343 1323 00 16 58 6 60 3 106 27 5 09 1 627 0 3965 2 5 4 07 196 8 265 1 622 53 36 5 57 7 459 18 534 81 3 87 1 353 0 0720 13 9 18 81 1092 34 047 8 802 433 76 22 17 8 499 19 462 56 3 18 1 157 0 0747 13 4 15 50 1036 26 973 8 473 480 50 17 41 9 451 39 1729 636 4 25 1 447 0 0300 33 3 48 19 2855 100 468 23 619 374 15 73 30 10 260 5 259 28 5 62 1 726 0 2434 4 1 7 08 522 13 188 2 346 249 16 7 76 Total 3793 136 5065 1096 105 4 161 37 302 840 64 687 3554 85 193 17 19.5 Relative risk 679
680 Statistical methods in epidemiology 1 543 1 96 0 0901 ˆ 1 366 and 1 720: The corresponding limits for c are the exponentials, 3 92 and 5 58. The point estimate and the confidence limits are very similar indeed to those derived using Woolf 's method. Details of the calculation are not given here but (19.24) gives SE (ln RMH) ˆ 0 0977 to give 95% confidence limits of 3 86 and 5 67, similar to both sets of limits derived earlier. Another frequently occurring situation is when the factor, instead of having just two levels, presence or absence, has three or more levels that have a definite order. For example, instead of classifying people as non-smokers or smokers, the smokers could be divided further according to the amount smoked. Such a situation was discussed in §15.7. Mantel (1963) extended the Mantel±Haenszel procedure to provide a test for trend and to combine a number of subsets of the data (15.21). An example of the method is given in Example 15.8. The methods discussed above are powerful in the commonly occurring situation where it is required to analyse the association between a disease and a single exposure factor, making allowance for one other factor, or at most a few, and where the effect of the other factors may be adequately represented by subdivision of the data into strata. The methods are inconvenient where several exposure factors are of interest and for each factor allowance must be made for the others, as was the case in Example 14.1. Also, where several factors are to be adjusted for, stratification would involve a large number of strata; in the limit each stratum would contain only one subject, so that comparisons within strata would be impossible. In these cases more general methods must be used and logistic regression (§14.2) is appropriate. This is because of the combination of two features. First, as observed above, the odds ratio can be estimated from a case±control study and, secondly, the logarithm of the odds ratio is a difference of logits. For a case-control study all the coefficients of (14.6) may be estimated except for the constant term b 0, which is distorted by the investigator's choice of the size of the case and control groups. The estimate of a regression coefficient measures the logarithm of the odds ratio for a change of unity in the corresponding variable, and so the exponentials of the regression coefficients give estimated odds ratios or approximate relative risks. It was observed above that, for a case±control study where a matched control is chosen for each case, the Mantel±Haenszel method may be applied, and leads to McNemar's test. When the number of cases available is limited, the precision of a study may be increased, to some extent, by choosing more than one matched control per case. The Mantel±Haenszel method can be applied, treating each set of a case and its controls as a subset. The analysis may be simplified by grouping together sets providing identical information; for example, with three controls per case those sets where the case and just two of the three controls are positive for the exposure factor all provide identical information. This is an extension of
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# 1971, 1987, 1994, 2002 by Blackwe
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vi Contents 9.3 Factorial designs,
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Preface to the fourth edition In th
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Preface to the fourth edition xi Ho
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2 The scope of statistics other pat
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4 The scope of statistics groups is
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6 The scope of statistics pressure
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2 Describing data 2.1 Diagrams One
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10 Describing data 1- 4 weeks Under
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12 Describing data Table 2.1 Outcom
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14 Describing data Your height: ...
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16 Describing data be transferred f
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18 Describing data extracted from t
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20 Describing data many variables a
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22 Describing data cannot be less t
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24 Describing data Bufotenin (nmol/
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26 Describing data Frequency 20 18
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28 Describing data The number of as
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30 Describing data Frequency Measur
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32 Describing data may be abbreviat
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34 Describing data variables follow
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36 Describing data 107 01. The geom
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38 Describing data If the number of
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40 Describing data Therefore the me
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42 Describing data xi x will then n
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44 Describing data taking the antil
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46 Describing data against their ef
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48 Probability example, is sometime
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50 Probability It will appear in du
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52 Probability In general, pairs of
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54 Probability the five types be cl
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56 Probability converted to a ratio
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58 Probability Correct Equivocal In
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60 Probability Probability 1 . 0 0
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62 Probability Probability density
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64 Probability As a second example,
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66 Probability coefficient. In the
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68 Probability n r pr …1 p† n r
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70 Probability π = 0 Probability .
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72 Probability 0 } — T n Fig. 3.8
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74 Probability Probability 0 . 4 0
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76 Probability Table 3.4 Binomial a
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78 Probability where exp(z) is a co
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80 Probability approaches the shape
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82 Probability Probabililty density
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84 Analysing means and proportions
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148 Analysing variances Probability
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150 Analysing variances headings 0
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152 Analysing variances Method AB N
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154 Analysing variances Example 5.2
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156 Analysing variances Comparison
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158 Analysing variances With contin
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160 Analysing variances Let y ˆ x1
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162 Analysing variances p … log x
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164 Analysing variances as expected
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166 Bayesian methods This argument
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168 Bayesian methods nothing of the
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170 Bayesian methods in this way. F
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172 Bayesian methods In some situat
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174 Bayesian methods difference bet
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176 Bayesian methods the distributi
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178 Bayesian methods In the unpaire
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180 Bayesian methods The examples d
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182 Bayesian methods difference bet
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184 Bayesian methods variation may
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186 Bayesian methods The resulting
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188 Regression and correlation Infa
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190 Regression and correlation y x
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192 Regression and correlation on s
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194 Regression and correlation y ,
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196 Regression and correlation y (a
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198 Regression and correlation incr
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200 Regression and correlation From
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202 Regression and correlation y ,
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204 Regression and correlation Valu
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206 Regression and correlation disc
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8 Comparison of several groups 8.1
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210 Comparison of several groups P
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212 Comparison of several groups s
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214 Comparison of several groups to
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216 Comparison of several groups sh
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218 Comparison of several groups s
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220 Comparison of several groups su
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222 Comparison of several groups No
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224 Comparison of several groups yc
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226 Comparison of several groups Q
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228 Comparison of several groups te
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230 Comparison of several groups Ta
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232 Comparison of several groups Ta
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234 Comparison of several groups th
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9 Experimental design 9.1 General r
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238 Experimental design the estimat
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240 Experimental design Table 9.1 N
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242 Experimental design Similarly,
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244 Experimental design The sum of
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246 Experimental design If, in a tw
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248 Experimental design Table 9.5 S
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250 Experimental design interaction
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252 Experimental design difference
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254 Experimental design Table 9.7 C
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256 Experimental design interaction
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258 Experimental design litters and
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260 Experimental design randomizati
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262 Experimental design rows, colum
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264 Experimental design type of des
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266 Experimental design Main unit S
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268 Experimental design categories
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270 Experimental design Example 9.6
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10 Analysing non-normal data 10.1 D
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274 Analysing non-normal data The s
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276 Analysing non-normal data Estim
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278 Analysing non-normal data betwe
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Table 10.1 Some properties of three
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282 Analysing non-normal data This
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284 Analysing non-normal data High
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286 Analysing non-normal data which
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288 Analysing non-normal data Table
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290 Analysing non-normal data 1 It
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292 Analysing non-normal data It is
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294 Analysing non-normal data proba
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296 Analysing non-normal data Fract
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298 Analysing non-normal data h=…
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300 Analysing non-normal data This
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302 Analysing non-normal data Boots
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304 Analysing non-normal data more
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306 Analysing non-normal data Sampl
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308 Analysing non-normal data 3 squ
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310 Analysing non-normal data mG ˆ
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11 Modelling continuous data 11.1 A
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314 Modelling continuous data Examp
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316 Modelling continuous data n k i
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318 Modelling continuous data consi
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320 Modelling continuous data basis
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322 Modelling continuous data 11.4
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324 Modelling continuous data var
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326 Modelling continuous data Table
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328 Modelling continuous data Again
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330 Modelling continuous data SSq D
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332 Modelling continuous data Two g
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334 Modelling continuous data which
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336 Modelling continuous data the i
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338 Modelling continuous data predi
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340 Modelling continuous data equat
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344 Modelling continuous data Somet
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346 Modelling continuous data multi
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348 Modelling continuous data (i) D
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350 Modelling continuous data The r
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352 Modelling continuous data In th
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354 Modelling continuous data The c
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356 Modelling continuous data If th
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358 Modelling continuous data at th
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360 Modelling continuous data about
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362 Modelling continuous data y −
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364 Modelling continuous data y −
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366 Modelling continuous data outli
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370 Modelling continuous data Patie
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372 Modelling continuous data Norma
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374 Modelling continuous data Cumul
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12 Further regression models for a
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380 Further regression models Table
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382 Further regression models repre
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384 Further regression models where
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386 Further regression models Popul
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388 Further regression models and S
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390 Further regression models s(x)
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392 Further regression models SS ˆ
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394 Further regression models A…a
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396 Further regression models and t
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398 Further regression models that
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400 Further regression models Regre
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402 Further regression models M…T
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404 Further regression models a0
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406 Further regression models Maxim
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408 Further regression models using
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410 Further regression models suppo
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412 Further regression models Dose
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414 Further regression models Condu
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416 Further regression models the p
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418 Further regression models To be
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420 Further regression models 9.5.
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422 Further regression models in Ex
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424 Further regression models param
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426 Further regression models namel
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428 Further regression models All t
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430 Further regression models A not
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432 Further regression models Appro
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434 Further regression models form
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436 Further regression models times
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438 Further regression models block
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440 Further regression models This
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442 Further regression models Here
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444 Further regression models follo
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446 Further regression models missi
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448 Further regression models probl
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450 Further regression models perio
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452 Further regression models betwe
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454 Further regression models Spell
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456 Multivariate methods 13.2 Princ
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458 Multivariate methods High loadi
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Table 13.1 Correlation matrix of 20
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462 Multivariate methods eigenvalue
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464 Multivariate methods The place
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466 Multivariate methods allocation
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468 Multivariate methods would pred
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470 Multivariate methods and We fol
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472 Multivariate methods In the dis
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474 Multivariate methods Paired dat
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476 Multivariate methods Mean SE (m
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478 Multivariate methods x (3) x (5
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480 Multivariate methods Allocation
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482 Multivariate methods diagram in
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484 Multivariate methods explanator
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486 Modelling categorical data In t
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488 Modelling categorical data calc
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490 Modelling categorical data as t
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492 Modelling categorical data DF.
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494 Modelling categorical data in a
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496 Modelling categorical data if p
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498 Modelling categorical data When
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500 Modelling categorical data Tabl
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502 Modelling categorical data The
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504 Empirical methods for categoric
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16 Further Bayesian methods 16.1 Ba
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530 Further Bayesian methods Analyt
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532 Further Bayesian methods compli
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534 Further Bayesian methods One ap
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536 Further Bayesian methods result
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538 Further Bayesian methods (i) th
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540 Further Bayesian methods From (
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542 Further Bayesian methods A stra
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544 Further Bayesian methods the st
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546 Further Bayesian methods but ma
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548 Further Bayesian methods σ 30
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550 Further Bayesian methods practi
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552 Further Bayesian methods which
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554 Further Bayesian methods Densit
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556 Further Bayesian methods recurr
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558 Further Bayesian methods a ˆ m
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560 Further Bayesian methods values
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562 Further Bayesian methods distri
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564 Further Bayesian methods the se
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566 Further Bayesian methods probab
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17 Survival analysis 17.1 Introduct
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570 Survival analysis Table 17.1 Cu
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572 Survival analysis Table 17.2 Li
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574 Survival analysis otherwise the
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576 Survival analysis qtj ˆ dj=n 0
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578 Survival analysis quantities in
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580 Survival analysis Survival % 10
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582 Survival analysis logrank test
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584 Survival analysis the mean surv
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586 Survival analysis the death rat
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588 Survival analysis McGilchrist a
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590 Survival analysis The martingal
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592 Clinical trials trials. In drug
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594 Clinical trials first type of e
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596 Clinical trials . Proposed numb
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598 Clinical trials If the response
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600 Clinical trials methods of Baye
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602 Clinical trials represented by
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604 Clinical trials physicians find
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606 Clinical trials very minor proc
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608 Clinical trials each patient's
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610 Clinical trials Table 18.1 Resu
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612 Clinical trials solution is ava
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614 Clinical trials Administrative
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616 Clinical trials non-sequential
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618 Clinical trials Table 18.3 Maxi
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620 Clinical trials Standardized de
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622 Clinical trials A somewhat diff
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624 Clinical trials A trial showing
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626 Clinical trials Publication bia
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Page 647 and 648: 636 Clinical trials independent, an
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Page 659 and 660: 19 Statistical methods in epidemiol
Page 661 and 662: 650 Statistical methods in epidemio
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Page 729 and 730: 718 Laboratory assays such cases, t
Page 731 and 732: 720 Laboratory assays and YT ˆ yT
Page 733 and 734: 722 Laboratory assays significant a
Page 735 and 736: 724 Laboratory assays The general p
Page 737 and 738: 726 Laboratory assays This relation
Page 739 and 740: 728 Laboratory assays Proportion po
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730 Laboratory assays P contribute
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732 Laboratory assays transformatio
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734 Laboratory assays Table 20.1 Es
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736 Laboratory assays P m iˆ1 ri l
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738 Laboratory assays synthesize hi
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740 Laboratory assays estimate of a
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Appendix tables 743
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2 0 0 02275 0 02222 0 02169 0 02118
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16 6 91 93 1 11 91 153 4 19 3 7 23
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16 0 128 0 258 0 392 0 535 0 690 0
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5 0 05 6 61 5 79 5 41 5 19 5 05 4 9
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3 0 0 05 4 17 3 32 2 92 2 69 2 53 2
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14 0.05 3 03 3 70 4 11 4 41 4 64 4
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Table A7 Percentage points for the
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Table A9 Sample size for detecting
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Bailey N.T.J. (1975) The Mathematic
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Buyse M.E., Staquet M.J. and Sylves
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eceptor interactions. J. Ster. Bioc
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Etzioni R.D. and Weiss N.S. (1998)
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Goetghebeur E. and Lapp K. (1997) T
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sample data in randomized block and
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Lehmann E.L. (1975) Nonparametrics:
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Mehta C.R. (1994) The exact analysi
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Peto R., Pike M., Day N. et al. (19
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Scott J.E.S., Hunter E.W., Lee R.E.
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Thompson S.G. and Barber J.A. (2000
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Zhang H., Crowley J., Sox H.C. and
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786 Author Index Brooks, S.P. 549,
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788 Author Index Gart, J.J. 127, 67
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790 Author Index Laurence, D.R. 519
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792 Author Index RamoÂn, J.M. 536
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794 Author Index Westley-Wise, V.J.
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796 Subject Index Assays (cont.) ra
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798 Subject Index Chronic obstructi
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800 Subject Index Continuous variab
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802 Subject Index Dot diagram 23, 2
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804 Subject Index Growth curves 409
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806 Subject Index McNemar's test 12
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808 Subject Index Normal (cont.) st
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810 Subject Index Proportions Bayes
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812 Subject Index Roughness penalty
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814 Subject Index Standard (cont.)
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816 Subject Index Tumour (cont.) in
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