Statistical Methods in Medical Research 4ed

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11.10 More on data transformation 377 that there is no a priori reason why a single transformation should achieve all three aims, and in §10.8 an example is given of a transformation that leads to a linear relationship but not to normality. In such cases classical multiple regression, as described in this chapter, cannot be applied, but generalized linear models which allow a general form for the distribution of errors about the regression are appropriate.
12 Further regression models for a continuous response 12.1 Polynomial regression Reference was made in §11.6 to the possibility of creating new predictor variables defined as the squares of existing variables, to cope with non-linear or curvilinear relationships. This is an important idea, and is most easily studied in situations in which there is originally only one predictor variable x. Instead of the linear regression equation introduced in §7.2, we consider the polynomial model E…y† ˆa ‡ bx …12:1† E…y† ˆa ‡ b 1x ‡ b 2x 2 ‡ ...‡ b px p : …12:2† The highest power of x, denoted here by p, is called the degree of the polynomial. Some typical shapes of low-degree polynomial curves are shown in Fig. 12.1. The curve for p ˆ 2, when the term in x 2 is added, is called quadratic; that for p ˆ 3 cubic, and that for p ˆ 4 quartic. Clearly, a wide variety of curves can be represented by polynomials. The quadratic curve has one peak or trough; the cubic has at most two peaks or troughs; and so on. A particular set of data may be fitted well by a portion of a low-degree polynomial even though no peaks or troughs are present. In particular, data showing a moderate amount of curvature can often be fitted adequately by a quadratic curve. The general principle of polynomial regression analysis is to regard the successive powers of x as separate predictor variables. Thus, to fit the p-degree polynomial (12.2), we could define x1 ˆ x, x2 ˆ x 2 , ...xp ˆ x p and apply the standard methods of §11.6. It will often be uncertain which degree of polynomial is required. Considerations of simplicity suggest that as low an order as possible should be used; for example, we should normally use linear regression unless there is any particular reason to use a higher-degree polynomial. The usual approach is to use a slightly higher degree than one supposes to be necessary. The highest-degree terms can then be dropped successively so long as they contribute, separately or together, increments to the sum of squares (SSq) which are non-significant when compared with the Residual SSq. Some problems arising from this approach are illustrated in Example 12.1, p. 379. 378
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To J.O. Irwin Mentor and friend
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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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Page 391 and 392: 380 Further regression models Table
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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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628 Clinical trials on different oc
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630 Clinical trials Table 18.5 Numb
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632 Clinical trials Treatment perio
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634 Clinical trials in cases where
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636 Clinical trials independent, an
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638 Clinical trials Sample size The
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640 Clinical trials and the restric
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642 Clinical trials original data.
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644 Clinical trials if the odds rat
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646 Clinical trials 3 2 1 0 -1 -2 -
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19 Statistical methods in epidemiol
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650 Statistical methods in epidemio
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Table 19.1 Death rate for two popul
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718 Laboratory assays such cases, t
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720 Laboratory assays and YT ˆ yT
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722 Laboratory assays significant a
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724 Laboratory assays The general p
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726 Laboratory assays This relation
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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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