Regression Models and Life-Tables
Journal of the Royal Statistical Society Series B: Statistical Methodology, 1972 · paper · Mathematics and statistics
Considers analysis of censored failure times when explanatory variables are recorded for each individual. The hazard is modelled as a function of the variables and regression coefficients multiplied by an arbitrary unknown function of time, and a conditional likelihood gives inference on the coefficients.
10 findings
Substance Section 10 (p. 198) states "the total numbers of failures 9 and 22 respectively" and uses "the F distribution with (18,44) degrees of freedom". But Table 1 lists 21 uncensored control (sample 1) times, so the counts would give 21 failures and (18,42) degrees of freedom.
- p. 13 Section 10, paragraph on exponential distributions
of failures 9 and 22 respectively
- p. 13 Section 10, same paragraph
F distribution with (18,44) degrees
- p. 11 Table 1, Sample 1 (control)
1, 1, 2, 2, 3, 4, 4, 5, 5, 8, 8, 8, 8, 11, 11, 12, 12, 15, 17, 22, 23
- p. 12 Text below Table 2
21β–28γ
Only the authors can say what is right here.
Judge's note: Checked the page images. On p. 196 (PDF 11), Table 1 Sample 1 lists 21 values and none has an asterisk. The text on p. 198 also says "For sample 1 this gives the ordinary sample survivor function, there being no censoring." Those 21 values sum to 182, which matches the 182 weeks in the text. Sample 0 has 9 uncensored values summing to 359, which also matches. On p. 197 (PDF 12), Table 2 gives 21 at risk in sample 1 at time 1, and the multiplicities m(i) sum to 30 = 9 + 21. The text there uses the first term 21β. So 22 failures, and 44 = 2×22 degrees of freedom, conflict with the paper's own data. I did not settle whether the printed limits 0·83 and 2·43 were computed with 22 or 21; reviewers' recomputations disagree. The authors would need to say whether the limits need recomputing.
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Substance "We can also attribute a standard error to the statistics U_k, given, asymptotically, by √[2/{n(p + 1)φ(p)}]. In the example this takes value 0·79." The formula is the square root of a diagonal element of I⁻¹ (standard error of an estimated β_k), not of U_k, whose variance by the printed information matrix is npφ(p) (standard error about 1·55).
- p. 20 Downton's contribution, after the test statistic value
We can also attribute a standard error to the statistics
- p. 20 Same paragraph
In the example this takes value 0·79.
- p. 19 Information matrix
so that the information matrix is given by
Only the authors can say what is right here.
Judge's note: Read pp. 204-205 images. Information matrix printed as I = nφ(p)[p on the diagonal, −1 off], inverse [n(p+1)φ(p)]⁻¹[2 on the diagonal, 1 off]. With p = 3, n = 5: φ(3) = 5/16 − (1/12)(1 + 1/2 + 1/3) = 0.1597. 2/(n(p+1)φ) = 0.626, square root 0.79, matching the printed value, so 0·79 is the diagonal of I⁻¹. The variance of U_k is the diagonal of I, npφ = 2.396, square root 1.55. I also reproduced U'I⁻¹U = 7.75 from the printed U_k, so the matrices are as printed. With a standard error of 0·79, U_1 = −35/12 would be 3.7 standard errors from zero, inconsistent with an overall statistic just below the 5 per cent point. The sentence may have intended the standardised estimate I⁻¹U, so only the author can say which quantity was meant.
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Reference or label Breslow's contribution (p. 216) refers to "the conditional likelihood of Section 2". Section 2 is 'The product-limit method', and the conditional likelihood is in Section 5.
- p. 31 Breslow's contribution, first sentence
puzzled by the conditional likelihood of Section 2
- p. 5 Section heading
5. A CONDITIONAL LIKELIHOOD
Judge's note: Section headings in the text layer: 2 is THE PRODUCT-LIMIT METHOD and 5 is A CONDITIONAL LIKELIHOOD. Section 2 (text pp. 188–189) contains no conditional likelihood. The page image of p. 216 shows 'Section 2'.
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Reference or label "Instead the log likelihood (15) was computed" and "Note that in (15) the first term is 21β–28γ": (15) defines the weighted average A_(ξi)(β); the conditional log likelihood is (13) (with ties, the unnumbered display after (22)).
- p. 12 Section 10, above Table 2
Instead the log likelihood (15) was computed
- p. 12 Section 10, below Table 2
Note that in (15) the first term is
- p. 6 Equation (15)
the sum being over
- p. 6 Section 5, equation (15)
(15)
Judge's note: PDF p. 6: (13) is L(β) = Σ z(i)β − Σ log[...], whose first term is linear in the parameters as the text on PDF p. 12 describes; (15) is A_(ξi)(β) = Σ z exp(zβ)/Σ exp(zβ), a ratio with no such first term. Both citations of (15) on PDF p. 12 are wrong. Since the example has ties and uses s(i) sums, the display after (22) on PDF p. 7 is the exact form used.
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Reference or label "A test of the global null hypothesis β = γ = 0 could be done via the test statistic (20)": (20) is the information matrix element 𝓘ξη(0) = Σ C(ξηi); the test statistic is (18).
- p. 12 Section 10, above Table 2
could be done via the test statistic
- p. 7 Equation (20)
The form of weighted sampling associated with general
- p. 12 Section 10, text above Table 2
could be done via the test statistic (20) but is not very relevant here
- p. 6 Section 5, equation (18)
{U(0)}T{𝓘(0)}−1{U(0)} (18)
Judge's note: PDF p. 6 shows (18) is the quadratic form {U(0)}^T{𝓘(0)}^−1{U(0)}, stated on PDF p. 7 to be asymptotically chi-squared with p degrees of freedom under the null hypothesis. (20) on PDF p. 7 is only the null information matrix. Same offset of two as the adjacent citation of (15) for (13).
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Reference or label "Indeed the same is true for any of the results in Section 4 provided that the z's are not functions of time." Section 4 contains no results; the rank-invariant conditional-likelihood results are in Section 5.
- p. 8 Section 7, last paragraph
for any of the results in Section 4 provided that the z's are not functions of time
- p. 8 Section 7, last paragraph
Indeed the same is true for any of the results in Section 4 provided that the z's are not functions of time.
- p. 8 end of Section 7
for any of the results in Section 4 provided
Only the authors can say what is right here.
Judge's note: Read Section 4 (PDF p. 5): a discussion of approaches (exponential, Weibull, step function, arbitrary λ0) with no formulae or results, and the parametric methods it mentions are not invariant under monotone time transformations. The results that depend only on ranks, (12)-(20), are in Section 5 (PDF pp. 6-7). The pointer is very probably meant to be Section 5 (perhaps with Section 6), but only the author can say which.
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Reference or label Equation (32), second line: "= n1 − Σ_{l=1}^{n} e_{n c_l}" has upper summation limit n; it should be n1, as in the first line of (32) and in (33).
- p. 9 Equation (32), second line
= n1 −
Judge's note: Viewed a 3x enlargement of the equation on PDF p. 9: the upper limit on the second line of (32) is a bare "n" with no subscript, whereas the line above and both sums in (33) show n1. c_l is defined only for l = 1, ..., n1.
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Reference or label Glassborow's contribution (p. 209) says "In Section 8 Professor Cox analyses two causes of failure". Two types of failure time are treated in Section 9, 'Bivariate life tables'.
- p. 24 Glassborow's contribution
In Section 8
- p. 33 Author's reply
the discussion of Section 9 is concerned with the possibly
- p. 24 Discussion, Mr Glassborow
In Section 8 Professor Cox analyses two causes of failure
- p. 10 Section headings
9. BIVARIATE LIFE TABLES
Judge's note: Section 8 is 'Estimation of distribution of failure-time'; Section 9, 'Bivariate life tables' (p. 195), covers two types of failure time and their independence. The author's reply to Glassborow (p. 218) cites 'the discussion of Section 9'. The discussant may have worked from a preprint, but the pointer is wrong as printed.
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Reference or label Downton's paired-comparison log likelihood is printed "L(β) = rβ − n log (1 + e^β)" although r is defined as the number of pairs in which the A specimen failed first and z = 0 for an A failure, which gives (n − r)β − n log(1 + e^β).
- p. 18 Downton's contribution, paired comparison
L(β) = rβ – n log (1 + e^β)
- p. 18 Downton's contribution, preceding sentence
where z = 0 or 1, according as the failure
- p. 18 Discussion, Professor Downton, paired comparison
where z = 0 or 1, according as the failure was of the specimen given treatment A or B, respectively
Only the authors can say what is right here.
Judge's note: Read p. 203 image. Text: 'r specimens given treatment A "failed" before their paired specimens'; hazards λ0(t) and λ0(t)e^β for A and B; probability e^{zβ}/(1+e^β) with 'z = 0 or 1, according as the failure was of the specimen given treatment A or B, respectively'. Sum of z over pairs is therefore n − r, not r, so L(β) and U(β) = r − ne^β/(1+e^β) carry the wrong label (r and n − r swapped, or z coded the other way round). The test statistic 4(r − n/2)²/n is symmetric in r and n − r, so the result is unaffected; hence a labelling error rather than a wrong result. Either the definition of r or of z could be the one to change, so author query.
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Wording "At the ith largest observed failure time, individuals with ranks n, n−1, ..., i are at risk": this risk set (n − i + 1 individuals) is that of the ith smallest failure time.
- p. 9 Section 7, before equation (30)
At the ith largest observed failure time, individuals with ranks
- p. 9 Section 7, before equation (30)
At the ith largest observed failure time, individuals with ranks n, n−1, ..., i are at risk
Judge's note: PDF p. 9: ranks are defined by c1 < c2 < ... in the ranking of the full data, and (30) uses the divisor n − i + 1 with H(c_l − i), i.e. individuals with rank ≥ i are at risk. That is the ith failure time in increasing order (ith smallest). At the ith largest time only i individuals would be at risk. The formulae are right; the wording is wrong. No statement elsewhere defines ranks in decreasing order.
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19 minor (typos and the like; most checked by the judge only)
Reference or label "the estimation of the distribution associated with the hazard (10) either for z = 0, or for some other given value of z": (10) is the two-sample time-dependent hazard ψe^{β2 t}λ0(t) of Example 2; the general model with z is (9).
- p. 9 Section 8, first sentence
the estimation of the distribution associated with the hazard (10) either for
- p. 9 Section 8, first sentence
the estimation of the distribution associated with the hazard (10) either for z = 0, or
Only the authors can say what is right here.
Judge's note: PDF p. 4 shows (9) is λ(t; z) = exp(zβ)λ0(t) and (10) is the special hazard for the second sample in Example 2, which has no general z. Section 8 (PDF p. 9) is written for general z, so (9) fits the sentence. (10) is a special case of (9), so the sentence is not meaningless; probable stale equation number.
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Reference or label "Dependence on further variables z can be indicated in the same way as for (11)." Equation (11) is the reparametrized factor ρ exp{β2(t − t*)} and contains no z; the multiplicative dependence on z is model (9).
- p. 11 Section 9, after equation (42)
can be indicated in the same way as for (11)
- p. 11 Section 9, after eq. (42)
Dependence on further variables z can be indicated in the same way as for (11).
Only the authors can say what is right here.
Judge's note: PDF p. 4: (11) is the reparametrization of (10) for Example 2; (9) is exp(zβ)λ0(t). The next sentence on PDF p. 11 ("the same function of z multiplying all four hazard functions") describes the structure of (9). The reference is probably meant to be (9), but the author's intention cannot be settled from the files.
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Reference or label Table 2 column head "Multiplicity" is printed directly above A(i) (the proportions 0·1429, 0·2222, ...), while the multiplicity column m(i) has no word heading.
- p. 12 Table 2, column headers
Multiplicity
- p. 11 Section 10 text
their multiplicities m(i) are listed; A(i) is the proportion of the relevant risk population in sample 1
Judge's note: Image of PDF p. 12: "Multiplicity" sits over the A(i) column. The text on PDF p. 11 says m(i) are the multiplicities and A(i) is the proportion of the risk population in sample 1. The symbols under the heading are correct, so a reader is not misled about the values; the word heading is misplaced.
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Typo Surname printed "Haenzel" throughout ("Mantel and Haenzel (1959)", "Mantel–Haenzel procedure", "MANTEL, N. and HAENZEL, W. (1959)"); the usual spelling is Haenszel.
- p. 2 Section 1
and Haenzel (1959)
- p. 8 Section 7
Mantel and Haenzel, 1959
- p. 17 References, Mantel (1963)
Haenzel procedure
- p. 17 References
MANTEL, N. and HAENZEL, W. (1959)
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that the co-author of the 1959 J. Nat. Cancer Inst. paper spells the name Haenszel. Page images confirm the paper prints "Haenzel" on PDF pp. 2, 8, 17 (twice, including the title of Mantel 1963) and 35. The spelling is internally consistent, so nothing in the files contradicts it.
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Typo Breslow's contribution (p. 217) cites "Granander (1956)"; both reference lists and Section 2 spell the name Grenander.
- p. 32 Breslow's contribution, first line of page
Granander (1956)
- p. 35 References in the discussion
GRENANDER, U. (1956)
- p. 32
starting point from which Granander (1956) derives ML estimates in the
- p. 32 Discussion, Professor Breslow, first line of page
This is also the starting point from which Granander (1956) derives ML estimates
Judge's note: The page image of p. 217, first line, reads 'Granander'. The discussion reference list on p. 220 reads 'GRENANDER, U. (1956)', and the main text and main reference list also use Grenander.
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Typo "to be rigorous extension of the usual conditions for maximum-likelihood formulae and theory would be involved to cover censoring": a comma (or article) is missing after "rigorous".
- p. 5 Section 4, paragraph 1
to be rigorous extension of the usual conditions for maximum-
Judge's note: Image of PDF p. 5 shows the text as quoted. The sentence parses as "to be rigorous, extension of the usual conditions ... would be involved", so no words are missing, only punctuation at most. Cosmetic.
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Typo Stray comma: "analysis of response times in bioassay, see, Sampford (1954)."
- p. 5 Section 4, paragraph 1, last sentence
analysis of response times in bioassay, see, Sampford (1954)
Judge's note: Image of PDF p. 5 shows "see, Sampford (1954)" with a comma after "see", while the same sentence has "see Sampford and Taylor (1959)" without one. Cosmetic.
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Typo "we are really interested in some standard stress, say s = 1, and which to use other values of s to get quick laboratory results": "which" should be "wish".
- p. 15 Section 11, first paragraph
and which to use other
- p. 15
and which to use other values of s to get quick
- p. 15 Section 11, paragraph 1
and which to use other values of s
Judge's note: Image of PDF p. 15 shows "and which to use other values of s"; not an OCR artefact. The sentence only makes sense with "wish".
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Typo Misspelling: "failure corresponds to the excedence of the activation energy" (exceedance).
- p. 16 Section 11, paragraph 2
excedence of the activation energy
Judge's note: Image of PDF p. 16 shows "excedence". Not a recognised variant spelling.
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Typo Reference list gives the Sampford series title as "The estimation of response-time distributions, II" (1952) but "The estimation of response-time distribution, III" (1954).
- p. 17 References, Sampford (1954)
The estimation of response-time distribution, III
- p. 17 References, Sampford (1952)
The estimation of response-time distributions, II
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: whether the original Part III title has the plural 'distributions'. The page image of p. 202 shows plural for part II and singular for part III of the same series, so the inconsistency is in print; which form each original article carries cannot be checked from the files.
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Typo Gehan (1965) is listed as "A generalized Wilcoxon test for comparing arbitrarily single-censored samples"; the reviewer says the article title reads 'singly-censored'.
- p. 17 References, Gehan (1965)
arbitrarily single-censored
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the true title of Gehan's 1965 Biometrika article ('singly-censored'). The page image confirms the reference list prints 'single-censored'; nothing in the files can confirm the original title.
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Typo Peto's contribution (p. 206) misspells regression as "a few regresssion coefficients", with three s's.
- p. 21 Peto's contribution, last paragraph on page
regresssion
- p. 21
have a few regresssion coefficients and significance levels to look
- p. 21 Discussion, Mr Peto, last paragraph of page
than if we just have a few regresssion
- p. 21 Last paragraph, last line
a few regresssion coefficients
Judge's note: The page image of p. 206 (PDF 21), last line, clearly reads 'regresssion'.
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Typo Lindley: "so that ψt̄/s̄ is F on (2n′, 2m′) d.f.: here s = S/m′ and t = T/n′"; the means are used with bars but defined without them.
- p. 24 Lindley's contribution, after (*)
here s = S/m′ and t = T/n′
Judge's note: Checked a 250 dpi render of the line on p. 209: bars are clearly present over t and s in 'ψt̄/s̄' and absent in 's = S/m′ and t = T/n′'. Unbarred s and t are already in use for the observation times s_i, t_j. Meaning is recoverable; cosmetic.
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Typo Benjamin's estimates and figure label the samples "Sample I" and "Sample O" (letters), whereas the paper calls them Sample 0 and Sample 1.
- p. 27 Benjamin's estimates and Fig. I labels
Sample I 0·649 (0·0155) 0·358 (0·0569)
Judge's note: p. 212 image: the text lines and the curve labels read 'Sample I' and 'Sample O' with a capital letter O, distinct from the figure's own numerals (0·10, 0). The paper's Section 10 uses 'Sample 0 (drug 6-MP)' and 'Sample 1 (control)'. A label-style inconsistency only; identity of the samples is clear. The 'Fig. I' part of the finding is not a defect: discussion figures are numbered in Roman numerals (Fig. I, Fig. II) to distinguish them from the paper's Fig. 1.
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Typo Author spelt "de Haas (1964)" in Meshalkin and Kagan's contribution but "DE HASS, J. H. (1964). Changing Mortality Patterns and Cardiovascular Diseases" in the discussion reference list.
- p. 28 Meshalkin and Kagan's contribution
de Haas (1964)
- p. 35 References in the discussion
DE HASS, J. H. (1964)
- p. 35
DE HASS, J. H. (1964). Changing Mortality Patterns
- p. 28 Discussion, Drs Meshalkin and Kagan, point 2
as used, for example in de Haas (1964).
Judge's note: Both spellings are visible in the page images (p. 213 and p. 220). The two forms cannot both be right; that 'de Haas' is the correct one is the natural reading but is not verifiable from the files.
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Typo The author's reply (p. 217) misspells censoring: "extra assumption that consoring operates equally on all groups".
- p. 32 Author's reply, paragraph on Mr Peto
consoring operates equally on all groups
- p. 32
extra assumption that consoring operates equally on all groups, is important.
Judge's note: The page image of p. 217, in the paragraph on Mr Peto, shows 'consoring'.
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Typo The author's reply (p. 218) misspells explanatory: "at least when the expanatory variables are independent of time".
- p. 33 Author's reply, paragraph on Lindley and Zelen
at least when the expanatory variables are independent
- p. 33
when the expanatory variables are independent of time
- p. 33 Reply, (c)(iii) paragraph beginning 'Both Professors Lindley and Zelen'
the expanatory variables are independent
- p. 33
which, at least when the expanatory variables are independent
Judge's note: The page image of p. 218, in the paragraph on Lindley and Zelen, shows 'expanatory'.
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Typo The author's reply (p. 219) misspells proceeding: "is only one way of proceding".
- p. 34 Author's reply, final paragraph
is only one way of proceding
- p. 34
is only one way of proceding and the possibility
- p. 34 Last paragraph of the reply
only one way of proceding
- p. 34
only one way of proceding and the possibility
Judge's note: The page image of p. 219, in the final paragraph of the reply, shows 'proceding'.
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Typo Running header on PDF page 35 (printed p. 220) reads "[No 2,." where the other even pages print "[No. 2,": the full stop is misplaced after the comma instead of after "No".
- p. 35 Running header, right
[No 2,.
Judge's note: Viewed the page image for PDF page 35: the right-hand running header clearly prints "[No 2,." (no stop after No, stray stop after the comma). Compared with PDF page 33 (printed p. 218), whose header prints "[No. 2,"; the OCR text shows the same "[No. 2," form on eight other even pages. Visible in the page image, so not an OCR artefact. Purely cosmetic typesetting slip in a running head.
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