Logistic回归分析使用Logit模型研究二元因变量和一组独立（解释）变量之间的关联。然而，在匹配研究中，无条件的logistic regression是偏见的（高估了OR）。条件logistic回归是由Norman Breslow, Nicholas Day, Katherine Halvorsen, Ross L. Prentice和C. Sabai在1978年提出，是logistic回归的延伸，允许人们考虑到分层和匹配，通常用于具有特定条件或属性的病例受试者与没有该条件的n个对照受试者相匹配。一般来说，可能有1至m个病例与1至n个对照组相匹配。然而，最常见的设计是1：1匹配，其次是1：n匹配，其中n从1到5不等。 它的主要应用领域是观察性研究，特别是流行病学。比较是在每个层内，没有估计截距，因此没有预测的概率，所以没有ROC或Hosmer-Lemeshow测试。
CLR有以下特点：
1. CLR提供与至少在一个阶层内变化的自变量（通常称为协变量）相关的回归系数的估计值。同样，CLR也不提供与独立变量相关的任何回归系数的估计值，这些独立变量在各阶层内不发生变化。
2. 随着研究样本量的增加，阶层（群组）的数量也以同样的速度增加。
3. 模型中出现了分层指示变量，但没有显示逐层输出。
4. 当匹配组有不同数量的病例和对照组时，可以使用CLR。
在R中可以用‘Survival’包中的clogit()函数，及Epi包中的clogistic()函数实现：
Using survival:clogit()
library(survival)
resp <- levels(logan$occupation)
n <- nrow(logan)
indx <- rep(1:n, length(resp))
logan2 <- data.frame(logan[indx,],
id = indx,
tocc = factor(rep(resp, each=n)))
logan2$case <- (logan2$occupation == logan2$tocc)
logan2 <- logan2[order(logan2$id),]
## Show dataset for first three strata
logan2[logan2$id %in% c(1,2,3), ]
occupation focc education race id tocc case
1 sales professional 14 non-black 1 farm FALSE
1.1 sales professional 14 non-black 1 operatives FALSE
1.2 sales professional 14 non-black 1 craftsmen FALSE
1.3 sales professional 14 non-black 1 sales TRUE
1.4 sales professional 14 non-black 1 professional FALSE
2 craftsmen sales 13 non-black 2 farm FALSE
2.1 craftsmen sales 13 non-black 2 operatives FALSE
2.2 craftsmen sales 13 non-black 2 craftsmen TRUE
2.3 craftsmen sales 13 non-black 2 sales FALSE
2.4 craftsmen sales 13 non-black 2 professional FALSE
3 sales professional 16 non-black 3 farm FALSE
3.1 sales professional 16 non-black 3 operatives FALSE
3.2 sales professional 16 non-black 3 craftsmen FALSE
3.3 sales professional 16 non-black 3 sales TRUE
3.4 sales professional 16 non-black 3 professional FALSE
id为每个组别，匹配比例为1:3
## clogit实现
res.clogit <- clogit(case ~ tocc + tocc:education + strata(id), logan2)
summ.clogit <- summary(res.clogit)
summ.clogit
Call:
coxph(formula = Surv(rep(1, 4190L), case) ~ tocc + tocc:education +
strata(id), data = logan2, method = "exact")
n= 4190, number of events= 838
coef exp(coef) se(coef) z Pr(>|z|)
toccfarm -1.896463 0.150099 1.380782 -1.37 0.1696
toccoperatives 1.166750 3.211539 0.565646 2.06 0.0391 *
toccprofessional -8.100549 0.000303 0.698724 -11.59 < 2e-16 ***
toccsales -5.029230 0.006544 0.770086 -6.53 6.5e-11 ***
tocccraftsmen:education -0.332284 0.717283 0.056868 -5.84 5.1e-09 ***
toccfarm:education -0.370286 0.690537 0.116410 -3.18 0.0015 **
toccoperatives:education -0.422219 0.655591 0.058433 -7.23 5.0e-13 ***
toccprofessional:education 0.278247 1.320812 0.051021 5.45 4.9e-08 ***
toccsales:education NA NA 0.000000 NA NA
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
toccfarm 0.150099 6.662 0.0100243 2.24750
toccoperatives 3.211539 0.311 1.0598246 9.73178
toccprofessional 0.000303 3296.278 0.0000771 0.00119
toccsales 0.006544 152.815 0.0014466 0.02960
tocccraftsmen:education 0.717283 1.394 0.6416297 0.80186
toccfarm:education 0.690537 1.448 0.5496656 0.86751
toccoperatives:education 0.655591 1.525 0.5846481 0.73514
toccprofessional:education 1.320812 0.757 1.1951206 1.45972
toccsales:education NA NA NA NA
Rsquare= 0.147 (max possible= 0.475 )
Likelihood ratio test= 666 on 8 df, p=0
Wald test = 414 on 8 df, p=0
Score (logrank) test = 682 on 8 df, p=0
tocc对结局影响具有统计学意义，tocc与education存在交互作用
##非条件性logistic回归 (不推荐)，矫正id
res.logit1 <- glm(case ~ tocc + tocc:education + factor(id), logan2, family = binomial(link = "logit"))
summary(res.logit1)$coef[c(1:5,843:846),]
Estimate Std. Error z value Pr(>|z|)
(Intercept) 4.1110 1.43414 2.867 4.150e-03
toccfarm -2.7176 1.43439 -1.895 5.815e-02
toccoperatives 1.7751 0.68699 2.584 9.768e-03
toccprofessional -11.5476 0.82091 -14.067 6.076e-45
toccsales -5.8349 0.82293 -7.090 1.337e-12
tocccraftsmen:education -0.3800 0.06062 -6.270 3.619e-10
toccfarm:education -0.3779 0.12012 -3.146 1.657e-03
toccoperatives:education -0.5139 0.06384 -8.050 8.298e-16
toccprofessional:education 0.4981 0.05956 8.363 6.108e-17
exp(coef(res.logit1)[c(1:5,843:846)]) #得到OR值
(Intercept) toccfarm toccoperatives toccprofessional
61.007195773 0.066033211 5.901084569 0.000009659
toccsales tocccraftsmen:education toccfarm:education toccoperatives:education
0.002923648 0.683836589 0.685320145 0.598166153
toccprofessional:education
1.645564209
## 非条件性logistic回归 ，忽略分层（不适合）
res.logit1b <- glm(case ~ tocc + tocc:education, logan2, family = binomial(link = "logit"))
summary(res.logit1b)$coef
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.85771 0.43803 4.241 2.225e-05
toccfarm -3.22405 1.13783 -2.834 4.604e-03
toccoperatives 1.66071 0.65561 2.533 1.131e-02
toccprofessional -10.00008 0.72279 -13.835 1.559e-43
toccsales -4.67442 0.69818 -6.695 2.155e-11
tocccraftsmen:education -0.22820 0.03357 -6.798 1.057e-11
toccfarm:education -0.18559 0.08352 -2.222 2.627e-02
toccoperatives:education -0.35063 0.03810 -9.204 3.448e-20
toccprofessional:education 0.53842 0.03999 13.464 2.535e-41
toccsales:education 0.06351 0.03830 1.658 9.728e-02
exp(coef(res.logit1b))
(Intercept) toccfarm toccoperatives toccprofessional
6.4090214 0.0397935 5.2630345 0.0000454
toccsales tocccraftsmen:education toccfarm:education toccoperatives:education
0.0093309 0.7959686 0.8306101 0.7042440
toccprofessional:education toccsales:education
1.7133004 1.0655733
比较从不同模型中得到的OR值
- clogit: conditional logistic regression (unbiased)
- logit1: unconditional logistic regression, 使用分层变量作为哑变量
- logit2: unconditional logistic regression, 忽略分层变量
clogit logit1 logit2
toccfarm 0.1501 0.0660 0.0398
toccoperatives 3.2115 5.9011 5.2630
toccprofessional 0.0003 0.0000 0.0000
toccsales 0.0065 0.0029 0.0093
tocccraftsmen:education 0.7173 0.6838 0.7960
toccfarm:education 0.6905 0.6853 0.8306
toccoperatives:education 0.6556 0.5982 0.7042
toccprofessional:education 1.3208 1.6456 1.7133
toccsales:education NA ? 1.0656
Using Epi::clogistic()
library(Epi)
data(bdendo)
## Show dataset for first three strata
bdendo[bdendo$set %in% c(1,2,3),]
set d gall hyp ob est dur non duration age cest agegrp age3
1 1 1 No No Yes Yes 4 Yes 96 74 3 70-74 65-74
2 1 0 No No <NA> No 0 No 0 75 0 70-74 65-74
3 1 0 No No <NA> No 0 No 0 74 0 70-74 65-74
4 1 0 No No <NA> No 0 No 0 74 0 70-74 65-74
5 1 0 No No Yes Yes 3 Yes 48 75 1 70-74 65-74
6 2 1 No No No Yes 4 Yes 96 67 3 65-69 65-74
7 2 0 No No No Yes 1 No 5 67 3 65-69 65-74
8 2 0 No Yes Yes No 0 Yes 0 67 0 65-69 65-74
9 2 0 No No No Yes 3 No 53 67 2 65-69 65-74
10 2 0 No No No Yes 2 Yes 45 68 2 65-69 65-74
11 3 1 No Yes Yes Yes 1 Yes 9 76 1 75-79 75+
12 3 0 No Yes Yes Yes 4 Yes 96 76 2 75-79 75+
13 3 0 No Yes No Yes 1 Yes 3 76 1 75-79 75+
14 3 0 No Yes Yes Yes 2 Yes 15 76 2 75-79 75+
15 3 0 No No No Yes 2 Yes 36 77 1 75-79 75+
## Analysis
res.clogistic <- clogistic(d ~ cest + dur, strata = set, data = bdendo)
res.clogistic
Call:
clogistic(formula = d ~ cest + dur, strata = set, data = bdendo)
coef exp(coef) se(coef) z p
cest.L 0.240 1.271 2.276 0.105 0.92
cest.Q 0.890 2.435 1.812 0.491 0.62
cest.C 0.113 1.120 0.891 0.127 0.90
dur.L 1.965 7.134 2.222 0.884 0.38
dur.Q -0.716 0.489 1.858 -0.385 0.70
dur.C 0.136 1.146 1.168 0.117 0.91
dur^4 NA NA 0.000 NA NA
Likelihood ratio test=35.3 on 6 df, p=0.0000038, n=254
## clogit
res.clogit2 <- clogit(d ~ cest + dur + strata(set), bdendo)
summary(res.clogit2)
Call:
coxph(formula = Surv(rep(1, 315L), d) ~ cest + dur + strata(set),
data = bdendo, method = "exact")
n= 295, number of events= 54
(20 observations deleted due to missingness)
coef exp(coef) se(coef) z Pr(>|z|)
cest.L 0.240 1.271 2.276 0.11 0.92
cest.Q 0.890 2.435 1.812 0.49 0.62
cest.C 0.113 1.120 0.891 0.13 0.90
dur.L 1.965 7.134 2.222 0.88 0.38
dur.Q -0.716 0.489 1.858 -0.39 0.70
dur.C 0.136 1.146 1.168 0.12 0.91
dur^4 NA NA 0.000 NA NA
exp(coef) exp(-coef) lower .95 upper .95
cest.L 1.271 0.787 0.0147 109.94
cest.Q 2.435 0.411 0.0698 84.93
cest.C 1.120 0.893 0.1953 6.42
dur.L 7.134 0.140 0.0916 555.94
dur.Q 0.489 2.046 0.0128 18.67
dur.C 1.146 0.873 0.1161 11.31
dur^4 NA NA NA NA
Rsquare= 0.113 (max possible= 0.439 )
Likelihood ratio test= 35.3 on 6 df, p=0.0000038
Wald test = 27.7 on 6 df, p=0.000105
Score (logrank) test = 36.8 on 6 df, p=0.0000019
## 非条件性logistic回归 (not appropriate)
res.logit2 <- glm(d ~ cest + dur + factor(set), bdendo, family = binomial(link = "logit"))
..glm.ratio(res.logit2)[1:10,]
OR 2.5 % 97.5 % P
(Intercept) 0.21 0.01 3.04 0.296
cest.L 1.29 0.01 260.47 0.925
cest.Q 4.17 0.06 317.18 0.507
cest.C 1.15 0.14 9.34 0.892
dur.L 17.44 0.10 3614.14 0.280
dur.Q 0.36 0.00 27.69 0.643
dur.C 1.22 0.08 19.62 0.887
dur^4 NA NA NA 0.809
factor(set)2 0.63 0.01 33.31 0.815
factor(set)3 1.56 0.03 81.06 0.996
## 非条件性logistic回归, 忽略分层 (not appropriate)
res.logit2b <- glm(d ~ cest + dur, bdendo, family = binomial(link = "logit"))
..glm.ratio(res.logit2b)
OR 2.5 % 97.5 % P
(Intercept) 0.33 0.21 0.49 0.000
cest.L 1.07 0.01 81.75 0.976
cest.Q 2.85 0.10 96.14 0.549
cest.C 1.02 0.19 5.41 0.982
dur.L 7.20 0.11 569.14 0.364
dur.Q 0.38 0.01 13.12 0.593
dur.C 1.28 0.14 11.96 0.827
dur^4 NA NA NA 0.000
OR obtained from different methods
- clogit: conditional logistic regression (unbiased)
- logit1: unconditional logistic regression, 使用分层变量为哑变量
- logit2: unconditional logistic regression, 忽略分层
clogit logit1 logit2
cest.L 1.27 1.29 1.07
cest.Q 2.43 4.17 2.85
cest.C 1.12 1.15 1.02
dur.L 7.13 17.44 7.20
dur.Q 0.49 0.36 0.38
dur.C 1.15 1.22 1.28
Kleinbaum MI 例子（clogit函数）
数据特征：
MI患者和非MI患者的1:2匹配。
所关注的暴露是吸烟状况。
在年龄、种族、性别和医院方面进行匹配。
在SBP和ECG上不匹配
## Load data
library(foreign)
midat <- read.dta("http://www.sph.emory.edu/~dkleinb/datasets/mi.dta")
## show first three strata
midat[midat$match %in% c(1,2,3),]
match person mi smk sbp ecg survtime
1 1 1 1 0 160 1 1
2 1 2 0 0 140 0 2
3 1 3 0 0 120 0 2
4 2 4 1 0 160 1 1
5 2 5 0 0 140 0 2
6 2 6 0 0 120 0 2
7 3 7 1 0 160 0 1
8 3 8 0 0 140 0 2
9 3 9 0 0 120 0 2
## Conditional logistic regression
res.clogit3.int <- clogit(mi ~ smk + sbp + ecg + smk:sbp + smk:ecg + strata(match), midat)
res.clogit3 <- clogit(mi ~ smk + sbp + ecg + strata(match), midat)
## No interaction (smaller) model is adequate
anova(res.clogit3.int, res.clogit3)
Analysis of Deviance Table
Cox model: response is Surv(rep(1, 117L), mi)
Model 1: ~ smk + sbp + ecg + smk:sbp + smk:ecg + strata(match)
Model 2: ~ smk + sbp + ecg + strata(match)
loglik Chisq Df P(>|Chi|)
1 -31.5
2 -31.7 0.55 2 0.76
## Show result
summary(res.clogit3)
Call:
coxph(formula = Surv(rep(1, 117L), mi) ~ smk + sbp + ecg + strata(match),
data = midat, method = "exact")
n= 117, number of events= 39
coef exp(coef) se(coef) z Pr(>|z|)
smk 0.7291 2.0731 0.5613 1.30 0.1940
sbp 0.0456 1.0467 0.0152 2.99 0.0028 **
ecg 1.5993 4.9494 0.8534 1.87 0.0609 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
smk 2.07 0.482 0.690 6.23
sbp 1.05 0.955 1.016 1.08
ecg 4.95 0.202 0.929 26.36
Rsquare= 0.173 (max possible= 0.519 )
Likelihood ratio test= 22.2 on 3 df, p=0.0000592
Wald test = 13.7 on 3 df, p=0.00338
Score (logrank) test = 19.7 on 3 df, p=0.000198
## 非条件性的logistic回归，使用匹配作为分类变量（不适合）
res.logit3 <- glm(mi ~ smk + sbp + ecg + factor(match), midat, family = binomial(link = "logit"))
..glm.ratio(res.logit3)
OR 2.5 % 97.5 % P
(Intercept) 0.00 0.00 0.00 0.001
smk 3.38 0.85 14.55 0.090
sbp 1.08 1.04 1.12 0.000
ecg 16.18 2.06 203.43 0.015
factor(match)2 1.00 0.00 346.51 1.000
factor(match)3 3.76 0.03 745.27 0.615
factor(match)4 3.76 0.03 745.27 0.615
factor(match)5 3.76 0.03 745.27 0.615
factor(match)6 3.76 0.03 745.27 0.615
factor(match)7 3.76 0.03 745.27 0.615
factor(match)8 3.76 0.03 745.27 0.615
factor(match)9 3.76 0.03 745.27 0.615
factor(match)10 3.76 0.03 745.27 0.615
factor(match)11 6.61 0.04 1391.13 0.483
factor(match)12 20.54 0.19 4213.57 0.246
factor(match)13 20.54 0.19 4213.57 0.246
factor(match)14 4.74 0.05 804.49 0.542
factor(match)15 0.70 0.01 82.52 0.886
factor(match)16 0.23 0.00 30.38 0.580
factor(match)17 0.55 0.00 311.53 0.866
factor(match)18 0.55 0.00 311.53 0.866
factor(match)19 0.60 0.00 94.07 0.843
factor(match)20 0.15 0.00 30.80 0.509
factor(match)21 2.13 0.01 523.61 0.785
factor(match)22 13.05 0.11 2769.44 0.330
factor(match)23 3.01 0.03 533.36 0.670
factor(match)24 2.92 0.03 532.64 0.680
factor(match)25 2.92 0.03 532.64 0.680
factor(match)26 2.23 0.02 435.65 0.759
factor(match)27 13.05 0.11 2769.44 0.330
factor(match)28 0.82 0.00 238.63 0.945
factor(match)29 2.92 0.03 532.64 0.680
factor(match)30 2.13 0.01 523.61 0.785
factor(match)31 3.01 0.03 533.36 0.670
factor(match)32 0.32 0.00 137.94 0.722
factor(match)33 0.08 0.00 16.05 0.374
factor(match)34 0.53 0.00 78.14 0.813
factor(match)35 1.70 0.01 384.23 0.845
factor(match)36 0.12 0.00 14.70 0.412
factor(match)37 1.26 0.01 298.55 0.933
factor(match)38 0.32 0.00 61.02 0.670
factor(match)39 6.08 0.05 1387.95 0.501
## 非条件性logistic回归, 忽略分层变量(not appropriate)
res.logit3b <- glm(mi ~ smk + sbp + ecg, midat, family = binomial(link = "logit"))
..glm.ratio(res.logit3b)
OR 2.5 % 97.5 % P
(Intercept) 0.00 0.00 0.01 0.000
smk 1.79 0.70 4.59 0.222
sbp 1.05 1.02 1.08 0.000
ecg 2.12 0.76 5.96 0.149
OR obtained from different methods
- clogit: conditional logistic regression (unbiased)
- logit1: unconditional logistic regression, 使用分层变量为哑变量
- logit2: unconditional logistic regression,忽略分层
clogit logit1 logit2
smk 2.07 3.38 1.79
sbp 1.05 1.08 1.05
ecg 4.95 16.18 2.12可以看到在OR的估计上有所区别，匹配研究中使用非条件性logistic回归可能会错误估计变量对结局的影响程度。
ref:Conditional Logistic Regression (netdna-ssl.com)